Cadieu et al. (2014) Deep neural networks rival the representation of primate IT cortex for core visual object recognition

Cadieu et al. (2014) Deep neural networks rival the representation of primate IT cortex for core visual object recognition
Deep Neural Networks Rival the Representation of
Primate IT Cortex for Core Visual Object Recognition
Charles F. Cadieu1*, Ha Hong1,2, Daniel L. K. Yamins1, Nicolas Pinto1, Diego Ardila1, Ethan A. Solomon1,
Najib J. Majaj1, James J. DiCarlo1
1 Department of Brain and Cognitive Sciences and McGovern Institute for Brain Research, Massachusetts Institute of Technology, Cambridge, Massachusetts, United States
of America, 2 Harvard–MIT Division of Health Sciences and Technology, Institute for Medical Engineering and Science, Massachusetts Institute of Technology, Cambridge,
Massachusetts, United States of America
The primate visual system achieves remarkable visual object recognition performance even in brief presentations, and
under changes to object exemplar, geometric transformations, and background variation (a.k.a. core visual object
recognition). This remarkable performance is mediated by the representation formed in inferior temporal (IT) cortex. In
parallel, recent advances in machine learning have led to ever higher performing models of object recognition using
artificial deep neural networks (DNNs). It remains unclear, however, whether the representational performance of DNNs
rivals that of the brain. To accurately produce such a comparison, a major difficulty has been a unifying metric that accounts
for experimental limitations, such as the amount of noise, the number of neural recording sites, and the number of trials,
and computational limitations, such as the complexity of the decoding classifier and the number of classifier training
examples. In this work, we perform a direct comparison that corrects for these experimental limitations and computational
considerations. As part of our methodology, we propose an extension of ‘‘kernel analysis’’ that measures the generalization
accuracy as a function of representational complexity. Our evaluations show that, unlike previous bio-inspired models, the
latest DNNs rival the representational performance of IT cortex on this visual object recognition task. Furthermore, we show
that models that perform well on measures of representational performance also perform well on measures of
representational similarity to IT, and on measures of predicting individual IT multi-unit responses. Whether these DNNs rely
on computational mechanisms similar to the primate visual system is yet to be determined, but, unlike all previous bioinspired models, that possibility cannot be ruled out merely on representational performance grounds.
Citation: Cadieu CF, Hong H, Yamins DLK, Pinto N, Ardila D, et al. (2014) Deep Neural Networks Rival the Representation of Primate IT Cortex for Core Visual
Object Recognition. PLoS Comput Biol 10(12): e1003963. doi:10.1371/journal.pcbi.1003963
Editor: Matthias Bethge, University of Tübingen and Max Planck Institute for Biologial Cybernetics, Germany
Received June 23, 2014; Accepted October 3, 2014; Published December 18, 2014
Copyright: ß 2014 Cadieu et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The authors confirm that all data underlying the findings are fully available without restriction. All relevant data are available from http://
Funding: This work was supported by the U.S. National Eye Institute (NIH NEI: 5R01EY014970-09), the National Science Foundation (NSF: 0964269), and the
Defense Advanced Research Projects Agency (DARPA: HR0011-10-C-0032). CFC was supported by the U.S. National Eye Institute (NIH: F32 EY022845-01). The
funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* Email: [email protected]
areas extending from primary visual area V1, through visual areas
V2 and V4, and culminating in inferior temporal (IT) cortex. At
the end of the ventral stream, IT cortex creates a representation of
visual stimuli that is selective for object identity and tolerant to
nuisance parameters such as object position, scale, pose, and
background [7–10]. The responses of IT neurons are remarkable
because they indicate that the ventral stream has transformed the
complicated, non-linear object recognition problem at the retinae
into a new neural representation that separates objects based on
their category [6,11]. Results using linear classifiers have shown
that the IT neural representation creates a simpler object
recognition problem that can often be solved with a linear
function predictive of object category [9,10]. It is thought that this
transformation is achieved through the ventral stream by a series
of recapitulated modules that each produce a non-linear
transformation of their input that becomes selective for objects
and tolerant to nuisance variables unrelated to object identity [6].
A number of bio-inspired models have sought to replicate the
phenomenology observed in the primate ventral stream (see e.g.
Primate vision achieves a remarkable proficiency in object
recognition, even in brief visual presentations and under changes to
object exemplar, geometric transformations, and background
variation. Humans [1] and macaques [2] are known to solve this
task with high accuracy at low latency for presentation times shorter
than 100 ms [3,4]. This ability is likely related to the presence and
rate of saccadic eye movements, which for natural viewing typically
occur at a rate of one saccade every 200–250 ms [5]. Therefore,
when engaged in natural viewing the primate visual system is
proficient at recognizing and making rapid and accurate judgements about the objects present within a single saccadic fixation.
While not encompassing all of primate visual abilities, this ability is
an important subproblem that we operationally define and refer to
as ‘‘core visual object recognition’’ [6].
A key to this primate visual object recognition ability is the
representation that the cortical ventral stream creates from visual
signals from the eye. The ventral stream is a series of cortical visual
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DNNs Rival the Representation of IT Cortex for Core Object Recognition
recorded neural sites, or the number of recorded stimulus
presentations (see e.g. [9,10,27]). Our methodology makes explicit
these limitations by either correcting for, or modifying model
representations to arrive at a fair comparison to neural representation. We find that these corrections have a dramatic effect on our
results and shed light on previous comparisons that we believe may
have been misleading.
Second, previous attempts have utilized fixed complexity
classifiers and have not addressed the relationship between
classifier complexity and decision boundary accuracy (see e.g.
[9,10,27]). In our methodology we utilize a novel extension of
‘‘kernel analysis,’’ formulated in the works of [28–30], to measure
the accuracy of a representation as a function of the complexity of
the task decision boundary. This allows us to identify representations that achieve high accuracy for a given complexity and
avoids a measurement confound that arises when using crossvalidated accuracy: the decision boundary’s complexity and/or
constraints are dependent on the size and choice of the training
dataset, factors that can strongly affect accuracy scores.
Third, previous attempts have not measured the variations in
the neural or model spaces that are relevant to class-level object
classification [31]. For example the work in [31] examined the
variation present in neural populations to visual stimuli presentations and compared this variation to the variation produced in
model feature spaces to the same stimuli. This methodology does
not address representational performance and does not provide an
accuracy-complexity analysis (however, see [32] and [33], for
discussion of methodologies to account for dissimilarity matrices
by class-distance matrices). Our methodology of analyzing
absolute representational performance using kernel analysis
provides a novel and complementary finding to the results in
[27,32,34]. Because of this complementarity, in this paper we also
directly measure the amount of IT neural variance captured by
deep neural networks as IT encoding models and by measuring
representational similarity.
Finally, our approach utilizes a dataset that is an order of
magnitude larger than previous datasets, and captures a degree of
stimulus complexity that is critical for assessing IT representational
performance. For example, the analysis in [10] utilized 150 images
and the comparison in [31] utilized 96 images, while in this work
we utilize an image set of 1960 images. The larger number of
images allows our dataset to span and sample a relatively high
degree of stimulus variation, which includes variation due to object
exemplar, geometric transformations (position, scale, and rotation/pose) and background. Importantly this variation is critical to
distinguish between models based on object classification performance: only in the presence of high variation are models
distinguishable from each other [35,36] and from IT [27].
In this work, we propose an object categorization task and
establish measurements of human performance for brief visual
presentations. We then present our novel extension of kernel
analysis and show that the latest deep neural networks achieve
higher representational performance on this visual task compared
to previous generation bio-inspired models. We next compare
model representational performance to the IT cortex neural
representation on the same task and images by matching the
number of model features to the number of IT recordings and to
the amount of observed experimental noise for both multi-unit
recordings and single-unit recordings. We find that the latest
DNNs match IT performance whereas previous models significantly lag the IT neural representation. In addition, we replicate
the findings using a linear classifier approach. Finally, we show
that the latest DNNs also provide compelling models of the actual
IT neural response by measuring encoding model predictions and
Author Summary
Primates are remarkable at determining the category of a
visually presented object even in brief presentations, and
under changes to object exemplar, position, pose, scale,
and background. To date, this behavior has been
unmatched by artificial computational systems. However,
the field of machine learning has made great strides in
producing artificial deep neural network systems that
perform highly on object recognition benchmarks. In this
study, we measured the responses of neural populations in
inferior temporal (IT) cortex across thousands of images
and compared the performance of neural features to
features derived from the latest deep neural networks.
Remarkably, we found that the latest artificial deep neural
networks achieve performance equal to the performance
of IT cortex. Both deep neural networks and IT cortex
create representational spaces in which images with
objects of the same category are close, and images with
objects of different categories are far apart, even in the
presence of large variations in object exemplar, position,
pose, scale, and background. Furthermore, we show that
the top-level features in these models exceed previous
models in predicting the IT neural responses themselves.
This result indicates that the latest deep neural networks
may provide insight into understanding primate visual
[12–16]) and recent, related models in the machine learning
community, generally referred to as ‘‘deep neural networks’’ share
many properties with these bio-inspired models. The computational concepts utilized in these models date back to early models
of the primate visual system in the work of Hubel and Wiesel
[17,18], who hypothesized that within primary visual cortex more
complex functional responses (‘‘complex’’ cells) were constructed
from more simplistic responses (‘‘simple’’ cells). Models of
biological vision have extended this hypothesis by suggesting that
higher visual areas recapitulate this mechanism and form a
hierarchy [12,13,19–22]. In the last few years, a series of visual
object recognition systems have been produced that utilize deep
neural networks and have achieved state-of-the-art performance
on computer vision benchmarks (see e.g. [23–26]). These deep
neural networks implement architectures containing successive
layers of operations that resemble the simple and complex cell
hierarchy first described by Hubel and Wiesel. However, unlike
previous bio-inspired models, these latest deep neural networks
contain many layers of computation (typically 7–9 layers, while
previous models contained 3–4) and adapt the parameters of the
layers using supervised learning on millions of object-labeled
images (the parameters of previous models were either handtuned, adapted through unsupervised learning, or trained on just
thousands of labeled images). Given the increased complexity of
these deep neural networks and the dramatic increases in
performance over previous models, it is relevant to ask, ‘‘how
close are these models to achieving object recognition representational performance that is similar to that observed in IT cortex?’’
In this work we seek to address this question.
Our methodology directly compares the representational
performance of IT cortex to deep neural networks and overcomes
the shortcoming of previous comparisons. There are four areas
where our approach has advantages over previous attempts.
Although previous attempts have addressed one or two of these
shortcomings, none has addressed all four. First, previous attempts
have not corrected for a number of experimental limitations
including the amount of experimental noise, the number of
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DNNs Rival the Representation of IT Cortex for Core Object Recognition
because it determines specific choices in the experimental setup.
The behavioral context that we seek to address is a sub-problem of
general visual behavior: vision in a natural duration fixation, or
visual object recognition within one fixation without contextual
influence, eye movements, or shifts in attention (also called ‘‘core
visual object recognition’’ [6]). In our neural experiments we have
chosen a presentation time of 100 milliseconds (ms) so as to be
relevant for this behavior (see Discussion for further justification
and Supporting Information (SI) for behavioral measurements on
this task).
As a first step to evaluate the neural representation, we recorded
multi-unit and single-unit neural activity from awake behaving
rhesus macaques during passive fixation. We recorded activity
using large scale multi-electrode arrays placed in either IT cortex
or visual area V4. To create a neural feature vector, which we use
to assess object representational performance, we presented each
image (1960 images in total) for 100 ms and measured the
normalized, background subtracted firing-rate in a window from
70 ms to 170 ms post image onset, averaged over 47 repetitions
(see Methods). Over two macaques we measured 168 multi-unit
sites in IT cortex, and 128 multi-unit sites in V4. From these
recordings we also isolated single-units from IT and V4 cortex.
Using conservative criteria (see Methods), we isolated 40 singleunits from IT and 40 single-units from V4 with 6 repetitions per
image for each single-unit.
To evaluate the performance of neural or model representations
we utilize a novel extension of kernel analysis. Kernel analysis
evaluates the efficacy of the representation by measuring how the
precision of the category regression problem changes as we allow
the complexity of the regression function to increase [30].
Intuitively, more effective representations will achieve higher
precision at the same level of complexity because they have
removed irrelevant variability from the original representational
space (here irrelevant variability in the original space is due to
object exemplar, geometric transformation, and background). To
measure precision vs. complexity of the regression function, we
perform kernel ridge regression using a Gaussian kernel (see
Methods for details). We define complexity as the inverse of the
regularization parameter (1=l) and precision as 1 minus the
normalized mean-squared leave-one-example-out generalization
error, such that a precision value of 0 is chance performance and 1
is perfect performance. The regularization parameter restricts the
complexity of the resulting regression function. By choosing a
Gaussian kernel we can move between regression functions that
are effectively linear, to functions that interpolate between the data
points (a ‘‘complex’’ regression function) [40]. Note that complex
regression functions may not generalize if there are not enough
training examples (known as ‘‘sample complexity’’), which will
result in saturation or reduction in accuracy as complexity
performing a representational similarity analysis. We conclude
with a discussion of the limitations of the current approach and
future directions for studying models of visual recognition and
primate object recognition.
To evaluate the question of representational performance we
must first make a choice about the task to be analyzed. The task
we examine here is visual object category recognition in a natural
duration fixation. This task is a well studied subproblem in visual
perception and tests a core problem of visual perception: context
independent basic-level object recognition within brief visual
presentation. The task is to determine the category of an object
instance that is presented under the effect of image variations due
to object exemplar, geometric transformations (position, scale, and
rotation/pose), and background. This task is well supported by
behavioral measurements: humans [1] and macaques [2] are
known to solve this task with high proficiency. It is well supported
by neural measurements: evidence from IT cortex indicates that
the neural representation supports and performs highly on this task
[37]. Furthermore, this task provides a computationally challenging problem on which previous computational models have been
shown to severely underperform [35,36]. Therefore, this task is
difficult computationally and is performed at high proficiency by
primates, with evidence that the primate ventral visual stream
produces an effective representation in IT cortex.
Methodologically, the task is defined through an image
generation process. An image is constructed by first choosing
one of seven categories, then one of seven 3D object exemplars
from that category, then a randomly chosen background image
(each background image is used only once), and finally the
variation parameters are drawn from a distribution to span two full
octaves of scale variation, the full width of the image for translation
variation, and the full sphere for pose variation. For each object
exemplar we generated 40 unique images using this process,
resulting in 1960 images in total. See Fig. 1 for example images
organized by object category and Methods for further description
of the image generation process. The resulting image set has
several advantages and disadvantages. Advantageously, this
procedure eliminates dependencies between objects and backgrounds that may be found in real-world images [38], and
introduces a controlled amount of variability or difficulty in the
task, which we have used to produce image datasets that are
known to be difficult for algorithms [35,36,39]. Though arguably
not fully ‘‘natural’’, the resulting images are highly complex (see
Discussion for further advantages and disadvantages).
In evaluating the neural representational performance we must
also define the behavioral context within which the neural
representation supports behavior. This definition is important
Fig. 1. Example images used to measure object category recognition performance. Two of the 1960 tested images are shown from the
categories Cars, Fruits, and Animals (we also tested the categories Planes, Chairs, Tables, and Faces). Variability within each category consisted of
changes to object exemplar (e.g. 7 different types of Animals), geometric transformations due to position, scale, and rotation/pose, and changes to
background (each background image is unique).
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We compared the neural representation to three convolutional
DNNs and three other biologically relevant representations. Note
that the development of these representations did not utilize the
1960 images we use here for testing in any way. The three recent
convolutional DNNs we examine are described in Krizhevsky et
al. 2012 [24], Zeiler & Fergus 2013 [25], and Yamins et al. 2014
[27,34]. The Krizhevsky et al. 2012 and Zeiler & Fergus 2013
DNNs are of note because they have each successively surpassed
the state-of-the-art performance on the ImageNet Large Scale
Visual Recognition Challenge (ILSVRC) datasets. Note that
results have continued to improve on this challenge since we ran
our analysis. See the latest results.
The DNN presented in Yamins et al. 2014 [27] is created using a
supervised optimization procedure called hierarchical modular
optimization (we refer to this model by the abbreviation HMO).
The HMO DNN has been shown to match closely representational dissimilarity matrices of the ventral stream and to be
predictive of IT and V4 neural responses [27]. We also evaluated
an instantiation of the HMAX model of invariant object
recognition that uses sparse localized features [41] and has
previously been shown to be a relatively high performing model
among artificial systems [16]. Finally, we also evaluated a V2-like
model and a V1-like model that each attempt to capture a firstorder account of secondary (V2) [42] and primary visual cortex
(V1) [35], respectively.
Each of the three convolutional DNNs was developed,
implemented, and trained by their respective researchers and for
those developed outside of our group we obtained features from
each DNN computed on our test images. The convolutional DNN
described in Krizhevsky et al. 2012 [24] was trained by supervised
learning on the ImageNet 2011 Fall release (,15 M images, 22K
categories) with additional training on the LSVRC-2012 dataset
(1000 categories). The authors computed the features in the
penultimate layer of their model (4096 features) on the 1960
images we used to measure the neural representation. The similar
8-layer deep neural network of Zeiler & Fergus 2013 [25] was
trained using supervised learning on the LSVRC-2012 dataset
augmented with random crops and left-right flips. This model took
advantage of hyper-parameter tuning informed by visualizations of
the intermediate network layers. The 4096 dimensional feature
representation was produced by taking the penultimate layer
features and averaging them over 10 image crops (the 4 corners,
center, and horizontal flips for each). The model of Yamins et al.
2014 [27] is an extension of the high-throughput optimization
strategy described in [16] that produces a heterogeneous
combination of hierarchical convolutional models optimized on
a supervised object recognition task through hyperparameter
optimization using boosting and error-based reweighing (see [27]
for details). The total output feature space per image for the HMO
model is 1250 dimensional.
Before comparing the representational performance of the
neural and model representations, we first evaluate the absolute
representational performance of these models on the task to verify
that the task we have chosen is computationally difficult. As
described in our previous work [35], we determined that a task is
computationally difficult if ‘‘simple’’ computational models fail on
the task. For the models tested here, the V1-like and V2-like
models represent these computationally simple models. Using
kernel analysis we evaluated both the DNNs and the bio-inspired
models on the task and plot the precision vs. the complexity curves
for each model representation in Fig. 2. This analysis indicates
that both the V1-like and V2-like models perform near chance on
this task over the entire range of complexity. Furthermore, the
HMAX model performs only slightly better on this task. If we
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reduce the difficulty of the task by reducing the magnitude range
of the variations we introduce (not shown here, but see [35,43] for
such an analysis), these models are known to perform well on this
task; therefore, it is object recognition under variation that makes
this problem difficult, and the magnitude range we have chosen for
this task is quite difficult in that the HMAX model performs
poorly. In contrast, the three DNNs perform at much higher
precision levels over the complexity range. A clear ranking is
observed with the Zeiler & Fergus 2013 [25] model followed by
the Krizhevsky et al. 2012 [24] model and the HMO model [27].
These results indicate that these models outperform models of
early visual areas, and we next ask which model, if any, can match
the performance of high-level visual area IT.
In order to directly compare the representational performance
of the IT neural representation to the model representations we
take a number of steps to produce a fair comparison. The
experimental procedure that we used to measure the neural
representation is limited by the number of neural samples (sites or
number of neurons) that we can measure and by noise induced by
uncontrolled experimental variability and/or intrinsic neural
noise. To equalize the sampling between the neural representation
and the model representations we fix the number of neural
samples (80 for the multi-unit analysis and 40 for the single-unit
analysis) and model features (we will vary this number in later
experiments). To correct for the observed experimental noise, we
add noise to the model representations. To add noise to the models
we estimate an experimental neural noise model. Following the
observation that spike counts of neurons are approximately
Poisson [44,45] and similar analyses of our own recordings, we
model response variability as being proportional to the mean
response. Precisely, the estimated noise model is additive to the
mean response and is a zero-mean Gaussian random variable with
variance being a linear function of the mean response. We
estimate the parameters of the noise model from the empirical
distribution of multi-unit responses and single-unit responses. Note
that our empirical estimate of these quantities is influenced by both
uncontrolled experimental variability (e.g. variability across
recording sessions) as well as intrinsic neural noise. See Methods
for further description and S1 Fig. for a verification that the noise
model reduces performance more greatly than empirical noise,
thus demonstrating that the noise model is conservative and overpenalizes models. To produce noise-matched model representations, we sample the model response dependent noise and measure
the representational performance of the resulting representation
using kernel analysis. We repeat this procedure 10 times to
measure the variability produced by the additive noise model.
We compare the sample and noise corrected model representations to the multi-unit neural representations in Fig. 3A. The
kernel analysis curves are plotted for neural and model representations sampled at 80 neural samples or 80 model features,
respectively. The model representations have been corrected for
the neural noise observed in the multi-unit IT neural measurement. Note that we do not attempt to correct the V4 sample to the
noise level observed in IT because we observed similar noise
between the V4 and IT neural measurements and each sample is
averaged over the same number of trials (47 trials). Compared to
the model representational performance in Fig. 2, model performance is reduced because of the subsampling and because of the
added noise correction (without added noise and subsampling
maximum precision is above 0.5 and with noise and subsampling
does not pass 0.35). Consistent with previous work [10,36], we
observed that the sampled IT neural representation significantly
exceeds the similarly-sampled V4 neural representation. Unsurprisingly, HMAX, V2-like, and V1-like representations perform
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DNNs Rival the Representation of IT Cortex for Core Object Recognition
Fig. 2. Kernel analysis curves of model representations. Precision, one minus loss (1{looe(l)), is plotted against complexity, the inverse of the
regularization parameter (1=l). Shaded regions indicate the standard deviation of the measurement over image set randomizations, which are often
smaller than the line thickness. The Zeiler & Fergus 2013, Krizhevsky et al. 2012 and HMO models are all hierarchical deep neural networks. HMAX [41]
is a model of the ventral visual stream and the V1-like [35] and V2-like [42] models attempt to replicate response properties of visual areas V1 and V2,
respectively. These analyses indicate that the task we are measuring proves difficult for V1-like and V2-like models, with these models barely moving
from 0.0 precision for all levels of complexity. Furthermore, the HMAX model, which has previously been shown to perform relatively well on object
recognition tasks, performs only marginally better. Each of the remaining deep neural network models performs drastically better, with the Zeiler &
Fergus 2013 model performing best for all levels of complexity. These results indicate that the visual object recognition task we evaluate is
computationally challenging for all but the latest deep neural networks.
representation, and is outperformed by the Zeiler & Fergus 2013
[25] representation. Furthermore, a comparison of the relative
performance of the multi-unit sample and the single-unit sample
indicates that the multi-unit sample outperforms the single-unit
sample. See Discussion for elaboration of this finding and S4 Fig.
for trial corrected performance comparison between single- and
In Figs. 4A and 4B we analyze the representational performance as a function of neural sites or model features for multi-unit
and single-unit neural measurements. To achieve a summary
number from the kernel analysis curves we compute the areaunder-the-curve and we omit the HMAX, V2-like, and V1-like
models because they are near zero performance in this regime. In
Fig. 4A we vary the number of multi-unit recording samples and
the number of features. Just as in Fig. 3A, we correct for neural
noise by adding a matched neural noise level to the model
representations. Fig. 4A indicates that the representational
performance relationship we observed at 80 samples is robust
near chance. All three recent DNNs perform better than the V4
representation. The IT representation performs quite well,
especially considering the sampling and noise limitations of our
recordings and would be quite competitive if directly compared to
the model results in Fig. 2. After correcting for sampling and
noise, the IT representation is only matched by the top performing
DNN of Zeiler & Fergus 2013. Interestingly, this relationship holds
for the entire complexity range.
We present the equivalent representational comparison between
models and neural representations for the single-unit neural
recordings in Fig. 3B. Because of the increased noise and fewer
trials collected for the single-unit measurements compared to our
multi-unit measurements, the single-unit noise and sample
corrected model representations achieve lower precision vs.
complexity curves than under the multi-unit noise and sample
correction (compare to Fig. 3A). This analysis shows that the
single-unit IT representation performs better than the HMO
representation, slightly worse than the Krizhevsky et al. 2012
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Fig. 3. Kernel analysis curves of sample and noise matched neural and model representations. Plotting conventions are the same as in
Fig. 2. Multi-unit analysis is presented in panel A and single-unit analysis in B. Note that the model representations have been modified such that they
are both subsampled and noisy versions of those analyzed in Fig. 2 and this modification is indicated by the { symbol for noise matched to the multiunit IT cortex sample and by the { symbol for noise matched to the single-unit IT cortex sample. To correct for sampling bias, the multi-unit analysis
uses 80 samples, either 80 neural multi-units from V4 or IT cortex, or 80 features from the model representations, and the single-unit analysis uses 40
samples. To correct for experimental and intrinsic neural noise, we added noise to the subsampled model representation (no additional noise is
added to the neural representations) that is commensurate to the observed noise from the IT measurements. Note that we observed similar noise
between the V4 and IT Cortex samples and we do not attempt to correct the V4 cortex sample of the noise observed in the IT cortex sample. We
observed substantially higher noise levels in IT single-unit recordings than multi-unit recordings due to both higher trial-to-trial variability and more
trials for the multi-unit recordings. All model representations suffer decreases in accuracy after correcting for sampling and adding noise (compare
absolute precision values to Fig. 2). All three deep neural networks perform significantly better than the V4 cortex sample. For the multi-unit analysis
(A), IT cortex sample achieves high precision and is only matched in performance by the Zeiler & Fergus 2013 representation. For the single-unit
analysis (B), both the Krizhevsky et al. 2012 and the Zeiler & Fergus 2013 representations surpass the IT representational performance.
Fig. 4. Effect of sampling the neural and noise-corrected model representations. We measure the area-under-the-curve of the kernel
analysis measurement as we change the number of neural sites (for neural representations), or the number of features (for model representations).
Measured samples are indicated by filled symbols and measured standard deviations indicated by error bars. Multi-unit analysis is shown in panel A
and single-unit analysis in B. The model representations are noise corrected by adding noise that is matched to the IT multi-unit measurements (A, as
indicated by the { symbol) or single-unit measurements (B, as indicated by the { symbol). For the multi-unit analysis, the Zeiler & Fergus 2013
representation rivals the IT cortex representation over our measured sample. For the single-unit analysis, the Krizhevsky et al. 2012 representation
rivals the IT cortex representation for low number of features and slightly surpasses it for higher number of features. The Zeiler & Fergus 2013
representation surpasses the IT cortex representation over our measured sample.
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DNNs Rival the Representation of IT Cortex for Core Object Recognition
single-unit analyses and the analysis of performance as a function
of the number of neural sites or features (see S4 Fig.).
While the goal of our analysis has been to measure representational performance of neural and machine representations it is
also informative to measure neural encoding metrics and measures
of representational similarity. Such analyses are complementary
because representational performance relates to the task goals (in
this case category labels) and encoding models and representational similarity metrics are informative about a model’s ability to
capture image-dependent neural variability, even if this variability
is unrelated to task goals. We measured the performance of the
model representations as encoding models of the IT multi-unit
responses by estimating linear regression models from the model
representations to the IT multi-unit responses. We estimated
models on 80% of the images and tested on 20%, repeating the
procedure 10 times (see Methods). The median predictions
averaged over the 10 splits are presented in Fig. 6A. For
comparison, we also estimated regression models using the V4
multi-unit responses to predict IT multi-unit responses. The results
show that the Krizhevsky et al. 2012 and the Zeiler & Fergus 2013
DNNs achieve higher prediction accuracies than the HMO model,
which was previously shown to achieve high predictions on a
similar test [27]. These predictions are similar in explained
variance to the predictions achieved by V4 multi-units. However,
no model is able to fully account for the explainable variance in
the IT multi-unit responses. In Fig. 6B we show the mean
explained variance of each IT multi-unit site as predicted by the
between 10 samples and 160 samples. Fig. 4B indicates that the
performance of the IT single-unit representation is comparatively
worse than the multi-unit, with the single-unit representation
falling below the performance of the Krizhevsky et al. 2012
representation for much of the range of our analysis.
These results indicate that after correcting for noise and
sampling effects, the Zeiler & Fergus 2013 DNN rivals the
performance of the IT multi-unit representation and that both the
Krizhevsky et al. 2012 and Zeiler & Fergus 2013 DNNs surpasses
the performance of the IT single-unit representation. The
performance of these two DNNs in the low-complexity regime is
especially interesting because it indicates that they perform
comparably to the IT representation in the low-sample regime
(i.e. low number of training examples), where restricted representational complexity is essential for generalization (e.g. [46]).
To verify the results of the kernel analysis procedure we
measured linear-SVM generalization performance on the same
task for each neural and model representation (Fig. 5). We used a
cross-validated procedure to train the linear-SVM on 80% of the
images and test on 20% (regularization parameters were estimated
from the training set). We repeated the procedure for 10
randomizations of the training-testing split. The linear-SVM
results reveal a similar relationship to the results produced using
kernel analysis (Fig. 3A). This indicates that the Zeiler & Fergus
2013 representation achieves generalization comparable to the IT
multi-unit neural sample for a simple linear decision boundary.
We also found near identical results to kernel analysis for the
Fig. 5. Linear-SVM generalization performance of neural and model representations. Testing set classification accuracy averaged over 10
randomly-sampled test sets is plotted and error bars indicate standard deviation over the 10 random samples. Chance performance is ,14.3%. V4
and IT Cortex Multi-Unit Sample are the values measured directly from the neural samples. Following the analysis in Fig. 3A, the model
representations have been modified such that they are both subsampled and have noise added that is matched to the observed IT multi-unit noise.
We indicate this modification by the { symbol. Both model and neural representations are subsampled to 80 multi-unit samples or 80 features.
Mirroring the results using kernel analysis, the IT cortex multi-unit sample achieves high generalization accuracy and is only matched in performance
by the Zeiler & Fergus 2013 representation.
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DNNs Rival the Representation of IT Cortex for Core Object Recognition
Fig. 6. Neural and model representation predictions of IT multi-unit responses. A) The median predictions of IT multi-unit responses
averaged over 10 train/test splits is plotted for model representations and V4 multi-units. Error bars indicate standard deviation over the 10 train/test
splits. Predictions are normalized to correct for trial-to-trial variability of the IT multi-unit recording and calculated as percentage of explained,
explainable variance. The HMO, Krizhevsky et al. 2012, and Zeiler & Fergus 2013 representations achieve IT multi-unit predictions that are comparable
to the predictions produced by the V4 multi-unit representation. B) The mean predictions over the 10 train/test splits for the V4 cortex multi-unit
sample and the Zeiler & Fergus 2013 DNN are plotted against each other for each IT multi-unit site.
V4 cortex multi-unit sample and the Zeiler & Fergus 2013 DNN.
There is a relatively weak relationship between the encoding
performance of the neural V4 and DNN representations (r = 0.48
between V4 and Zeiler & Fergus 2013, compared to r = 0.96 and
r = 0.74 for correlations between Krizhevsky et al. 2012 and Zeiler
& Fergus 2013, and HMO and Zeiler & Fergus 2013,
respectively), indicating that V4 and DNN representations may
account for different sources of variability in IT (see Discussion).
Finally, we measured representational similarity using the
analysis methodology proposed in [32]. This analysis methodology
measures how similar two representations are and is robust to
global scalings and rotations of the representational spaces. To
compute the representational similarity between the IT multi-unit
and model representations, we computed object-level representational dissimilarity matrices (RDMs) for model and neural
representations (matrices are 49x49 dimensional as there are 49
total objects). We then measured the Spearman rank correlations
between the model derived RDM and the IT multi-unit RDM (see
Methods). In Fig. 7A we show the results of the representational
similarity measurements for the model representations and in
Fig. 7B we show depictions of the RDMs for select representations. For comparison we present the result between the V4 multiunit representation and the IT multi-unit representation. To
determine the variability due to the IT neural sample, we also
present the similarity measurement between one-half of the IT
multi-units and the other half (IT Cortex Split-Half). In addition,
we provide results following the methodology in [27], which first
predicts the IT multi-unit site responses from the model
representation and then uses these predictions to form a new
representation. We refer to these representations with an
appended ‘‘+IT-fit’’. Our measurements of the HMO + IT-fit
representation are in general agreement with the results in [27] but
vary slightly because of differences in the image set used to
produce these measurements and details of the methodology used
to produce the IT predictions. Interestingly, by fitting a linear
transform at the image-level to IT multi-units, the Krizhevsky et
al. 2012 and Zeiler & Fergus 2013 DNNs fall within the noise limit
PLOS Computational Biology |
of the IT split-half object-level RDM measurement. However, the
HMO, Krizhevsky et al. 2012, and Zeiler & Fergus 2013
representations, without the added linear mapping, have deviations from the IT representation that are unexplained by noise
variation. While it is informative that a linear mapping can
produce RDMs in correspondence with the IT RDM, we
conclude that there remains a gap between DNN models and
IT representation when measured with object-level representational similarity.
In summary, our measurements indicate that the latest DNNs
rival the representational performance of IT cortex on a rapid
object category recognition task. We evaluated representational
performance using a novel kernel analysis methodology, which
measures precision as a function of classifier complexity. Kernel
analysis allows us to measure a desirable property of a
representation: a good representation is highly performant with a
simple classification function and can thus accurately predict class
labels from few examples, while a poor representation is only
performant with complex classification functions and thus requires
a large number of training examples to accurately predict (see
Methods for elaboration on this point). Importantly, we made
comparisons between models and neural measurements by
correcting the models for experimental limitations due to
sampling, noise, and trials. In this analysis we found that the
Zeiler & Fergus 2013 DNN achieved comparable representational
performance to the IT cortex multi-unit representation and both
the Krizhevsky et al. 2012 and Zeiler & Fergus 2013 representations surpassed the performance of the IT cortex single-unit
representation. These results reflect substantial progress of
computational object recognition systems since our previous
evaluations of model representations using a similar object
recognition task [35,36]. These results extend our understanding
over recent, complimentary studies, which have examined
representational similarity [27], by evaluating directly absolute
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DNNs Rival the Representation of IT Cortex for Core Object Recognition
Fig. 7. Object-level representational similarity analysis comparing model and neural representations to the IT multi-unit
representation. A) Following the proposed analysis in [32], the object-level dissimilarity matrix for the IT multi-unit representation is compared to
the matrices computed from the model representations and from the V4 multi-unit representation. Each bar indicates the similarity between the
corresponding representation and the IT multi-unit representation as measured by the Spearman correlation between dissimilarity matrices. Error
bars indicate standard deviation over 10 splits. The IT Cortex Split-Half bar indicates the deviation measured by comparing half of the multi-unit sites
to the other half, measured over 50 repetitions. The V1-like, V2-like, and HMAX representations are highly dissimilar to IT cortex. The HMO
representation produces comparable deviations from IT as the V4 multi-unit representation while the Krizhevsky et al. 2012 and Zeiler & Fergus 2013
representations fall in-between the V4 representation and the IT cortex split-half measurement. The representations with an appended ‘‘+ IT-fit’’
follow the methodology in [27], which first predicts IT multi-unit responses from the model representation and then uses these predictions to form a
new representation (see text). B) Depictions of the object-level RDMs for select representations. Each matrix is ordered by object category (animals,
cars, chairs, etc.) and scaled independently (see color bar). For the ‘‘+ IT-fit’’ representations, the feature for each image was averaged across testing
set predictions before computing the RDM (see Methods).
representational performance for this task. In contrast to the
representational performance results, all models that we have
tested failed to capture the full explainable variation in IT
responses (Figs. 6 and 7). Nonetheless, our results, in conjunction
with the results in Yamins et al. 2014 [27], indicate that the latest
DNNs provide compelling models of primate object recognition
representations that predict neural responses in IT cortex [27] and
rival the representational performance of IT cortex.
To address the behavioral context of core visual object
recognition our neural recordings were made using 100 ms
presentation times. We chose only a single presentation time (as
opposed to rerunning the experiment at different presentation
times) to maximize the number of images and repetitions per
image given time and cost constraints in neurophysiological
recordings. This choice is justified by previous results that indicate
human subjects are performant on similar tasks with just 13 ms
presentation times [4], that human performance on similar tasks
rapidly increases from 14 ms to 56 ms and has diminishing returns
between 56 ms and 111 ms [3], that decoding from IT at 111 ms
presentation times achieves nearly the same performance at
222 ms presentation times [3], that for 100 ms presentation times
the first spikes after stimulus onset in IT are informative and peak
decoding performance is at 125 ms [9], and that maximal
information rates in high-level visual cortex are achieved at a
rate of 56 ms/stimulus [47]. Furthermore, we have measured
human performance on our task and observed that the mean
response accuracy at 100 ms presentation times is within 92% of
the accuracy at 2000 ms presentation times (see S2 Fig.). While
reducing presentation time below 50 ms likely would lead to
reduced representational performance measurements in IT (see
[3]), the presentation time of 100 ms we used for our evaluation is
applicable for the core recognition behavior, has previously been
shown to be performant behaviorally and physiologically, and in
our own measurements on this task captures the large majority of
long-presentation-time (2 second) human performance.
The images we have used to define the computational task allow
us to precisely control variations to object exemplar, geometric
transformations, and background; however, they have a number of
disadvantages that can be improved upon in further studies. For
example, this image set does not expose contextual effects that are
present in the real world and may be used by both neural and
machine systems, and it does not include other relevant variations,
e.g. lighting, texture, natural deformations, or occlusion. We view
these current disadvantages as opportunities for future datasets
and neural measurements, as the approach taken here can
naturally be expanded to encompass these issues.
There are a number of issues related to our measurement of
macaque visual cortex, including viewing time, behavioral
paradigm, and mapping the neural recording to a neural feature,
that will be necessary to address in determining the ultimate
representational measurement of macaque visual cortex. The
PLOS Computational Biology |
presentation time of the images shown to the animals was
intentionally brief (100 ms), but is close to typical single-fixation
durations during natural viewing (,200 ms), and human behavioral testing (S2 Fig.) shows that the visual system achieves high
performance at this viewing time. It will be interesting to measure
how the neural representational space changes with increased
viewing time and multiple fixations. Another aspect to consider is
that during the experimental procedure, animals were engaged in
passive viewing and human subjects were necessarily performing
an active task. Does actively performing a task influence the neural
representation? While several studies report that such effects are
present, but weak at the single-unit level [48–51], no study has yet
examined the quantitative impact of these effects at the population
level for the type of object recognition task we examined. Active
task performance may be related to what are commonly referred
to as attentional phenomena [e.g. biased competition]. In addition,
the mapping from multi-unit and single-unit recordings to the
neural feature vector we have used for our analysis is only one
possible mapping, but it is a parsimonious first choice. Finally,
visual experience or learning may impact the representations
observed in IT cortex. Interestingly, the macaques involved in
these studies have had little or no real-world experience with a
number of the object categories used in our evaluation, though
they do benefit from millions of years of evolution and years of
postnatal experience. However, significant learning effects in adult
IT cortex have been observed [52–54], even during passive
viewing [55]. We have examined the performance of computational algorithms in terms of their absolute representational
performance. It is also interesting to examine the necessary
processing time and energy efficiency of these algorithms in
comparison to the primate visual system. While a more in depth
analysis of this issue is warranted, from a ‘‘back-of-the-envelope’’
calculation (see SI) we conclude that model processing times are
currently competitive with primate behavioral reaction times but
model energy requirements are 2 to 3 orders of magnitude higher
than the primate visual system.
How do our measurements of representational performance
relate to overall system performance for this task? Measuring
representational performance fundamentally relies on a measure
of the representation, which we have assumed is a neural measure
such as single-unit response or multi-unit response. This poses
difficulties for obtaining an accurate measure of human representational performance. Using only behavioral measurements the
representation must be inferred, which may be possible through an
investigation of the psychological space of visually presented
objects. However, more direct methods may be fruitful using fMRI
(see [31]), or a process that first equates macaque and human
performance and uses the macaque neural representation as a
proxy for the human neural representation. One approach to
directly measure the overall system performance is to replicate the
cross-validated procedure used to measure models in humans.
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DNNs Rival the Representation of IT Cortex for Core Object Recognition
Such a procedure should control the human exposure to the
training set and provide the correct labels on the training set. The
procedure for measuring human performance presented in the SI
does not follow this procedure. However, a comparison between
human performance at 100 ms presentation times (see S2 Fig.) and
overall DNN model performance on the test-set (see S3 Fig.)
indicates that there is likely a gap between human performance
(85% mean accuracy) and DNN performance (77% mean
accuracy) on this task because allowing the human subjects
exposure to 80% of the images with the correct labels is only likely
to increase the human performance number. Furthermore, there is
individual variability in the human performance with some
individuals performing well above the mean. Therefore, while
we have not attempted to make a direct comparison between
human performance and DNN performance, we infer that human
performance exceeds current DNN performance.
Our methodology and approach relates to the encoding and
decoding approaches in systems neuroscience, which, in our view,
provide complementary insights into neural visual processing. The
kernel analysis methodology we use here is a neural decoding
approach because it measures the relationship between the neural
(or model) representation and unobserved characteristics of the
stimuli (class labels). The linear-SVM methodology is also a
decoding approach because it tests the generalization performance
of predicting the unobserved class label from the neural (or model)
representation. The approaches of predicting IT multi-unit
response (Fig. 6) and measuring representational similarity to IT
representation (Fig. 7) are encoding approaches because they
measure the relationship between functions or measurements
derived from the stimuli (pixels in the images) and the neural
variation present in IT. The complementary nature of these
approaches is demonstrated in our results. For example, while the
Zeiler & Fergus 2013 DNN rivals the decoding performance of IT
cortex, it fails to capture over 40% of the explainable variance in
the IT neural sample and therefore does not produce a complete
neural encoding model. Conversely, the V4 multi-unit representation severely underperforms the DNNs and IT cortex when
measured with decoding approaches but produces comparable
results to these representations when predicting IT multi-unit
responses with an encoding approach. It is currently unclear what
variation in the IT cortex multi-unit representation is not captured
by DNNs or the V4 multi-unit representation. Furthermore, the
IT variation that is captured by DNNs and V4 is, relative to
correlations between DNN models, weakly correlated (Fig. 6B).
Overall, the remaining unexplained IT variation may be exposed
through a decoding approach (by, for example, exploring
additional task labels), through an encoding approach (by
exploring additional stimulus transformations), or through approaches that take into account intrinsic neural dynamics (e.g.
[56]). The comparably high performance of the V4 multi-unit
representation at predicting IT multi-unit responses may be due to
its ability to capture intrinsic neural dynamics present in IT that
are unrelated to stimulus derived variables.
We found that multi-units outperform single-units in our
evaluation, as evidenced in the relative performance increase of
DNNs over single-units in Figs. 3 and 4 and in a trial corrected
comparison between multi- and single-units in S4 Fig. It may be
surprising that multi-units outperform single-units on this task. A
priori, we might assume that the averaging process introduced by
the multi-unit recording, which aggregates the spikes from
multiple spatially-localized single-units, would ‘‘wash-out’’ or
average away neural selectivity for visual objects. However, when
considering generalization performance, it is possible that averaged single-units could outperform the original single-units by
PLOS Computational Biology |
removing noise or non-class specific signal variation. In this way,
multi-units may provide a form of regularization that is
appropriate for this task. This regularization may be due to
averaging out single-unit noise, and/or reducing variation in the
representational space that is irrelevant, and therefore spurious, for
the task. Alternatively, the single-unit variation may be appropriate for different tasks that we have not measured, such as fine
distinctions between objects or 3-dimensional object surface
representation. The spatial arrangement of functional cortical
responses (topographic maps, and functional clustering) also
indicates a current limitation of DNNs as models of high-level
visual cortex. There is no notion of tissue or cortical space in the
DNN layers that we utilize as features: the features correspond to
fully connected layers which do not have convolutional topology,
which may be trivially mapped to physical space, nor do they have
localized operations, such as local divisive normalization. For this
reason, it is non-trivial to include a biologically realistic averaging
process to the DNN representations.
Could the regularizing properties of multi-unit responses be
indicative of broader regularization mechanisms related to spatial
organization in cortex (topographic maps, and functional clustering)? Just as our multi-unit recordings average together a number
of single-units, neurons ‘‘reading-out’’ signals from IT cortex could
average over cortical topography and thus regularize the
classification decision boundary. This suggests the future goal of
finding an appropriate mapping of biological phenomenology and
physical mechanism to the computational concepts of kernels,
regularizers, and representational spaces. The overall performance
of learning algorithms strongly depends on the interconnections
between the choice of kernel, regularizer, and representation. In
our current work (and the predominant mode in the field), we have
made specific choices on the kernel and regularizer and examined
aspects of the representational space. However, a full account of
biological learning and decision making must determine accurate
descriptions for all three of these computational components in
biological systems.
Interestingly, many of the computational concepts utilized in the
high performing DNNs that we have measured extend back to
early models of the primate visual system. All three DNNs we
examined [24,25,27] implement architectures containing successive layers of operations that resemble the simple and complex cell
hierarchy first described by Hubel and Wiesel [17,18]. In
particular, max-pooling was proposed in [13] and is a prominent
feature of all three DNNs. Additional computational concepts are
convolution or weight sharing, which was introduced in [57] and
utilized in [24,25], and backpropagation [58], which is utilized in
The success of the Krizhevsky et al. 2012 and the Zeiler &
Fergus 2013 DNNs raises a number of interesting questions. The
categories we used for testing (the 7 classes used in the kernel
analysis measurements) are a small fraction of the 1000 classes that
these models were trained on, and it is not clear if there is a direct
correspondence between the classes in the two image sets. At this
point it is not clear how the non-relevant classes in the set used to
train the models affects our performance estimate. As more
detailed analyses are conducted it will be interesting to determine
which categories are necessary to replicate ventral stream
performance and similarity. For example, there may be biases in
the necessary category distribution toward ecologically relevant
categories, such as faces. Of biological relevance, it is not clear if
natural primate development is comparable to the 15M labeled
images used to train these DNNs and it seems likely that more
innate knowledge of the visual world (acquired during evolution)
and/or more unsupervised training (during development) are
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center of the screen, indicated by a small (0.25u) red dot.
Presentations with larger eye movements were discarded. In each
experimental block, we recorded responses to all images. Within
one block each image was repeated once. Over all recording
sessions, this resulted in the collection of 47 image repetitions,
collected over multiple days. All surgical and experimental
procedures were done in accordance with the National Institute
of Health guidelines and the Massachusetts Institute of Technology Committee on Animal Care.
To arrive at the multi-unit neural representation, we converted
the raw multi-unit neural responses to a neural representation
through the following normalization process. For each image in a
block, we compute the vector of raw firing rates across
measurement sites by counting the number of spikes between
70 ms and 170 ms after the onset of the image for each site. We
then subtracted the background firing rate, which is the average
firing rate during presentation of a gray background (‘‘blank’’
image), from the evoked response. In order to minimize the effect
of variable external noise, we normalize each site by the standard
deviation of each site’s response to a block of images. Finally, the
neural representation is calculated by taking the mean across the
repetitions for each image and for each site, producing a scalar
valued matrix of neural sites by images. This post-processing
procedure is only our current best-guess at a neural code, which
has been shown to quantitatively account for human performance
[60]. Therefore, it may be possible to develop a more effective
neural decoding for example influenced by intrinsic cortical
variability [56], or dynamics [61,62].
To arrive at the single-unit neural representation, we followed a
similar normalization process as the multi-unit representation, but
first conducted spike-sorting on the multi-unit recordings. We
sorted single-units from the multi-unit IT and V4 data by using
affinity propagation [63] together with the method described in
[64]. Using a conservative criteria we isolated 160 single-units
from the IT recordings and 95 single-units from the V4 recordings
with 6 repetitions per image for each single-unit. Given these
single-unit responses for each image we followed a processing
procedure identical to the multi-unit procedure, which included
counting the number of spikes between 70 ms and 170 ms after
the onset of the image, subtracting the background firing rate, and
normalizing by the standard deviation of the site’s response to a
block of images. Finally, we selected from these single-units the top
40 based on response consistency over trials on a separate image
set (units with high SNR). Specifically, we measured the
response to 280 images not included in our evaluations of
performance but drawn from a similar stimulus distribution.
These images contained 7 unique objects not contained in the
original set with 1 object from each of the 7 categories. For each
object there are 40 images, each with a unique background and
with the object position, scale, and pose randomly sampled. For
each single-unit, we separated the responses over trials into two
groups, averaged across trials, and measured the correlation of
these response vectors. We then sorted the single-units based on
this correlation and selected the 40 with highest correlation and
therefore most consistent single-units. We repeated this procedure separately for V4 and IT. The resulting consistency
measurement for the IT single-units was comparable to other
measurements using single-unit electrophysiology [10]. The
consistency of the 40 IT single-units was higher than the 15th
percentile of single-units measured in [10]. In other words, the
least consistent IT single-unit in the group of 40 was more
consistent than the bottom 15% of single-units analyzed in [10].
Unless otherwise noted, all analyses of single-units use these 40
high-SNR single-units.
utilized in biological systems. Finally, given their similar architectures, it is unclear why some DNNs perform well and others do
not. However, our analyses provide cursory evidence that models
with more layers perform better and models that effectively reduce
the dimensionality of the original problem perform better. More
work is necessary to determine best practices using these
architectures and to determine the importance of hierarchical
representations and representations that reduce dimensionality.
The principled approach we have provided here allows for
practical evaluations between models and neurons, and may
provide a tool in assessing progress in the development of DNNs.
Going forward, we would ideally like a better theoretical
understanding of these architectures that would lead to more
consistent implementations and would produce a detailed,
mechanistic hypothesis for ventral stream processing (see [59] for
an example of such an effort).
Ethics statement
This study was performed in strict accordance with the
recommendations in the Guide for the Care and Use of
Laboratory Animals of the National Institutes of Health. All
surgical and experimental procedures were approved by the
Massachusetts Institute of Technology Committee on Animal
Care (Animal protocol: 0111-003-014). All human behavioral
measurements were approved by the Massachusetts Institute of
Technology’s Committee on the Use of Humans as Experimental
Subjects (number: 0812003043).
Image dataset generation
Synthetic images of objects were generated using POV-Ray, a
free, high-quality ray tracing software package (http://www. 3-d models (purchased from Dosch Design and
TurboSquid) were converted to the POV-Ray format. This
general approach allowed us to generate image sets with arbitrary
numbers of different objects, undergoing controlled ranges of
identity preserving object variation/transformation. The 2-d
projection of the 3-d model was then combined with a randomly
chosen background. In our image set, no two images had the same
background, in some cases the background was, by chance,
correlated with the object (plane on a sky background, car on a
street) but more often they were uncorrelated, giving no
information about the identity of the object. A circular aperture
with radial fall-off was applied to create each final image.
Neural data collection
We collected neural data from V4 and IT across two adult male
rhesus monkeys (Macaca mulatta, 7 and 9 kg) by using a multielectrode array recording system (BlackRock Microsystems,
Cerebus System). We chronically implanted three arrays per
animal and recorded the 128 most visually driven neural
measurement sites (determined by separate pilot images) in one
animal (58 IT, 70 V4) and 168 in another (110 IT, 58 V4). During
image presentation we recorded multi-unit neural responses to our
images from the V4 and IT sites. Images were presented on an
LCD screen (Samsung, SyncMaster 2233RZ at 120 Hz) one at a
time. Each image was presented for 100 ms with a diameter of 8u
(visual angle) at the center of the screen on top of the half-gray
background and was followed by a 100 ms half-gray ‘‘blank’’
period. The animal’s eye position was monitored by a video eye
tracking system (SR Research, EyeLink II), and the animal was
rewarded upon the successful completion of 6–8 image presentations while maintaining good eye fixation (within 62u) at the
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a set of n data points f(x1 ,y1 ),:::,(xn ,yn )g drawn independently
from p(x,y) we evaluate a representation defined as a mapping
x.w(x). For our case, the inputs x are images, the y are
normalized category labels, and the w denotes a feature extraction
As suggested by [30], we utilize the Gaussian kernel because this
kernel implies a smoothness of the task of interest in the input
space [67] and does not bias against representations that may be
more adapted to non-linear regression functions. We compute the
kernel matrix Ks associated to the data set as
Kernel analysis methodology
We would also like to measure accuracy as we change the
complexity of the prediction function. To accomplish this, we use
an extension of the work presented in [30], which is based on
theory presented in [28], and [29], and we refer to as kernel
analysis. We provide a brief description of this measure and refer
the reader to those references for additional details and
justification on measuring precision against complexity. This
procedure is a derivative of regularized least squares [65], which
arrises as a Tikhonov minimization problem [46], and can be
viewed as a form of Gaussian process regression [66]. We do not
elaborate on the relationships between these views, but each is a
valid interpretation on the procedure.
The measurement procedure, which we refer to here as kernel
analysis, utilizes regularized kernel ridge regression to determine
how well a task in question can be solved by a regularized kernel.
A highly regularized kernel will allow only a simple prediction
function in the feature space, and a weakly regularized kernel will
allow a complex prediction function in the feature space. A good
representation will have high variability in relation to the task in
question and can effectively perform the task even under high
regularization. Therefore, if the eigenvectors of the kernel with the
largest eigenvectors are effective at predicting the categories, a
highly regularized kernel will still be effective for that task. In
contrast, an ineffective representation will have very little variation
relevant for the task in question and variation relevant for the task
is only contained in the eigenvectors corresponding to the smallest
eigenvalues of the kernel and only a weakly regularized kernel will
be capable of performing the task efficiently. Changing the
amount of regularization changes the complexity of the resulting
decision function: highly regularized kernels allow for only simple
decision functions in the feature space and weakly regularized
kernels allow for complex decision functions. Intuitively, a good
representation is one that learns a simple boundary (highly
regularized) from a small number of randomly-chosen examples,
while a poor representation makes a more complicated boundary
(weakly regularized), requiring many examples to do so.
In our formulation, kernel analysis consists of solving a
regularized least squares or equivalently a kernel regression
problem over a range of regularization [65]. The regularization
parameter l controls the complexity of the function and the
precision or performance is measured as the leave-one-out
generalization error (looe). We refer to the inverse of the
regularization parameter (1=l) as the complexity and 1{looe(l)
as the precision. Thus, the curve 1{looe(l) provides us with a
measurement of the precision as a function of the model
complexity for the given representational space. The curves
produced by different representational spaces will inform us about
the simplicity of the task in that representational space, with higher
curves indicating that the problem is simpler for the representation.
One of the advantages of kernel analysis is that the kernel PCA
method converges favorably from a limited number of samples.
[29] shows that the kernel PCA projections obtained with a finite
and typically small number of samples n (images in our context)
are close with multiplicative errors to those that would be obtained
in the asymptotic case where n.?. This result is especially
important in our setting as the number of images we can
reasonably obtain from the neural measurements is comparatively
low. Therefore, kernel analysis provides us with a methodology for
assessing representational effectiveness that has favorable properties in the low image sample regime, here thousands of images.
We next present the specific computational procedure for
computing kernel analysis. Given the learning problem p(x,y) and
PLOS Computational Biology |
ks (w(x1 ),w(x1 )) :::
Ks ~B
@ ks (w(xn ),w(x1 )) :::
ks (w(x1 ),w(xn ))
ks (w(xn ),w(xn )) A
where the standard Gaussian kernel is defined
ks (x,x’)~ exp ({DDx{x’DD2 =2s2 ) with scale parameter s.
The regularized kernel regression problem is
EY {Ks HE22 z Ht Ks H,
2 i~1
where Y are the normalized vector of category labels, H is the
vector of regression parameters and l is the regularization scalar.
The solution to the regularized regression problem for a fixed s
and l is denoted as Hs (l) and is given as
Hs (l)~(Ks zlI){1 Y ,
where I is the identity matrix.
The leave-one-out error can be calculated as
LOOEs (l)~
Hs (l)
diag((Ks zlI){1 )
where diag(M) denotes the column vector satisfying
diag(M)i ~Mii and the division is elementwise. Note that
LOOEs (l) is a vector of errors for each example and we
compute the mean squared error over the examples as
(LOOEs (l)i )2 , which is considered a good
looes (l)~
empirical proxy for the error on future examples (generalization
error). We note that the leave-one-out error can be computed
efficiently given an eigendecomposition of the kernel matrix. We
refer to [65] for derivation and details on this computation.
To remove the dependence of the kernel on s we find the value
that minimizes looes (l) at that value of l: looe(l)~mins looes (l).
Finally, for convenience we plot precision (1{looe(l)) against
complexity (1=l). Note that because we optimize the value of s for
each value of complexity, s (the width of the Gaussian kernel) will
regularize the feature space when l does not provide sufficient
regularization. Because of this effect, the precision-complexity
curves in Fig. 2 plateau at high values of complexity because the
optimal value of s increases in the high complexity regime (low
values of l). We would otherwise expect that at high complexity,
the regression would over fit and produce poor generalization (low
1{looe(l) or high error). At low values of complexity (l) we
observe that there is an optimal value for s, which is dependent on
the representation, and is robust over image splits. Therefore, it is
not possible to achieve high precision at low complexity simply by
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DNNs Rival the Representation of IT Cortex for Core Object Recognition
reducing s; the variation in the representational space must be
aligned with the task in order to achieve high precision at low
Note that we have chosen to use a squared error loss function for
our multi-way classification problem. While it might be more
appropriate to evaluate a multi-way logistic loss function, we have
chosen to use the least-squares loss because it provides a stronger
requirement on the representational space to reduce variance within
category and to increase variance between categories, and it allows
us to distinguish representations that may be identical in terms of
separability for a certain complexity but still have differences in their
feature mappings. The kernel analysis of deep Boltzmann machines
in [68] also uses a mean squared loss function in the classification
problem setting and is widely used in machine learning.
In the discussion above, Y ~(y1 , . . . ,yn ) represents the vector of
task labels for the images (x1 , . . . ,xn ). In our specific case, the yi
are normalized category identity values (normalized such that
predicting 0 will result in a precision equal to 0 and perfect
prediction will result in a precision equal to 1). To generalize to the
case of multiway categorization, we use a version of the common
one-versus-all strategy. Assuming k distinct categories, for each
category j we compute the per-class leave-one-out error looes (l)j
by replacing Y in equations 2 and 3 with Yj . The overall leaveone-out error is then the average over categories of the percategory leave-one-out error. Minimization over s then proceeds
as in the single category case.
To evaluate both neural representations and machine representations using kernel analysis we measure the 1{looe(l) curve.
The image dataset consists of 1960 images containing seven object
categories with seven instances per object category. The categories
are Animals, Cars, Chairs, Faces, Fruits, Planes and Tables. To
measure statistical variation due to subsampling of image variation
parameters we compute the 1{looe(l) curve ten times, each time
sampling 80% of the images with replacement. The ten samples
are fixed for all representations and within each subset we equalize
the number of images from each category. For each representation, we maximize over the values of the Gaussian kernel s
parameter as follows. We evaluate the Gaussian kernel scale
parameter s for each representation at a range of values centered
on the median distance over the distribution of distances for that
representation. With sm denoting the value equal to the median
distance, we evaluated kernels with s~asm for 32 values of a in
the range of 0:1 to 10 with logarithmic spacing. We found that in
practice the values of s that minimized the leave-one-out error
were close to sm and well within this range. To span a large range
of complexity we evaluate looe(l) at 56 values of l from 10{4 to
103 with logarithmic spacing. For each representation, this
procedure produces a curve for each of the subsets, where the
mean and spread (range of the values over the 10 subsets for each
value of l) are shown in Fig. 2. As a summary value, we also
compute the area under the curve (KA-AUC) for each of the
image set randomizations and report the mean and standard
deviation. We use these mean KA-AUC values for the measurements in Figs. 4A and 4B.
Note that results have continued to improve on this challenge since
we ran our analysis. See the latest
Given the similarity of these DNNs to models of biological
vision and our particular interest in the primate visual system, we
also tested two representations that attempt to capture ventral
stream processing. The ‘‘V1-like’’ model attempts to replicate
functional responses in the primary visual area, the ‘‘V2-like’’
model similarly replicates response properties of the secondary
visual area, and the HMAX instantiation is a model of ventral
visual processing that implements the simple and complex cell
hierarchy proposed in [13].
V1-like [35]. We evaluate the V1-like representation from
Pinto et al.’s V1S+ [35]. This model attempts to capture a firstorder account of primary visual cortex (V1). It computes a
collection of locally-normalized, thresholded Gabor wavelet
functions spanning orientation and frequency. This model is a
simple, baseline biologically-plausible representation, against
which more sophisticated representations can be compared. This
model has 86400 dimensions.
V2-like [42]. We evaluated a recent proposal for the
functional role of visual area V2 [42]. This model constructs a
representation from conjunctions of Gabor outputs, which is
similar to the V1-like model. The Gabor outputs are combined
non-linearly and averaged within receptive field windows. The
representation formed by this model has been shown to
correspond to visual area V2 and explains visual crowding [42].
The output representation for the V2-like model has 24316
HMAX instantiation [41]. We evaluate the model in [41],
which is a biologically inspired hierarchical model utilizing sparse
localized features. This model has been shown to perform
relatively well on previous measures of invariant object recognition
[16], and to explain some aspects of ventral stream responses
[13,22]. This representation has 4096 dimensions. Counting the
simple-complex module as a single layer, this model has two layers.
HMO, Yamins et al. 2014 [27]. The model uses hierarchical modular optimization (HMO) to develop a large, deep network
that is a combination of convolutional neural networks and is
described in [27]. The HMO algorithm is an adaptive boosting
procedure that interleaves hyper-parameter optimization (see [27]
and references therein). The specific model we examine here is
identical to the one in Yamins et al. 2014 [27]. This model is
developed on a screening task that contains images of objects
placed on randomly selected backgrounds. This task is similar in its
construction to the task we use here to evaluate representational
performance, however, it contains entirely different objects in
totally non overlapping semantic categories, with none of the same
backgrounds and widely divergent lighting conditions and noise
levels. The specific model we examine was produced by the HMO
optimization procedure on this screening task and is a convolutional neural network with 1250 top-level outputs. Therefore, the
total dimensionality of the HMO representation is 1250 and the
model is composed of four layers. The model is identical to the one
presented in Yamins et al. 2014 [27]. However, the analysis
methodology in [27] has a number of important differences to the
analyses presented in this paper. The following differences are
noteworthy: in this paper we examine only the high variation task
and a subset of the images in the high variation task (7 categories
with 7 objects per category in this paper vs. 8 categories with 8
objects per category in [27]), and in this paper we compute
representational similarity on the raw HMO features, while in [27]
representational similarity is calculated on the HMO-based IT
model population.
Machine representations
We evaluate a number of model representations from the
literature, including several recent best of breed representational
learning algorithms and visual representation models. In particular
we examine three recent convolutional DNNs [24,25] + Yamins.
The DNNs in [24,25] are of note because they have each
successively surpassed the state-of-the-art performance on the
ImageNet Large Scale Visual Recognition Challenge (ILSVRC).
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DNNs Rival the Representation of IT Cortex for Core Object Recognition
with coefficients ai and bi . We then average these coefficients over
the sites to produce a single relationship between mean rate and
variance. For the multi-unit sites we estimate a~0:14 and b~0:92
and for the single-unit sites we estimate a~0:76 and b~0:71.
The noise model assumes that the empirically observed response
variation is due to an underlying and unobserved signal
contribution and a noise contribution. We can estimate the total
variance of the empirical response (s2signalznoise ) as,
Krizhevsky et al. 2012 [24] (SuperVision). We evaluate
the deep convolutional neural network model ‘‘SuperVision’’‘‘
described in [24], which is trained by supervised learning on the
ImageNet 2011 Fall release (,15M images, 22K categories)
with additional training on the LSVRC-2012 dataset (1000
categories). The authors computed the features of the penultimate layer of their model (4096 features) on the testing images
by cropping out the center 224 by 224 pixels (this is the input
size to their model). This mimics the procedure described in
[24], in which this feature is fed into logistic regression to
predict category labels. This model has seven layers as tested
(counting a layer for each linear-convolution or fully connected
Zeiler and Fergus 2013 [25]. The model is a very large
convolutional network with 8 layers [25] trained using supervised
learning on the LSVRC-2012 dataset to predict 1000 output
categories. The training data is augmented by taking random
crops and flips out of the 256 by 256 pixel images as in [24] to
prevent overfitting. Additionally, visualization experiments exploring what deep models learn lead to hyperparameter choices which
yield better performance. The feature representation is the 4096
features input to the softmax classifier averaged over crops from
the 4 corners, center, and horizontal flips (producing ten 4096
dimensional vectors that are averaged to produce the 4096
dimensional representation used here). This model also has seven
1 X 1X
rijk {m ,
NM ij
T k
~s2signal zs2noise :
Note that we cannot observe s2signal and we use the subscript
‘‘noise’’ to denote the standard error of the mean of the signal
estimate and not ordinary trial to trial variability. Therefore, the
estimate of the standard error of the signal is given as,
s2noise ~
1 X
(amij zb) =T:
NM ij
Note that we estimate the standard error of the mean using the
noise model, which has the benefit of being jointly estimated from
the population of neural responses. In other words, if different
neural sites share a similar noise model our estimate of that noise
model is improved by estimating its parameters from all of the sites
jointly. However, we have verified that using the empirical
estimate of s2noise gives nearly identical results.
To add noise to the model representations we first scale the total
variance of the representation such that the variance after adding
noise will be approximately equal to the variance observed in the
neural sample. To do this we scale the model representation such
Experimental noise matched model
In our evaluation of representational performance we are
limited by the observed noise in the neural representation. To
produce a fair comparison we alter the model representational
measurements by adding a level of noise that is matched to that
observed in the neural representation. Note that we are unable to
fully remove noise from the neural representation, and therefore
we add noise to the model representations. The sources of the
observed neural noise are various and we do not make an attempt
to distinguish the sources of noise. Broadly, these noise sources are
experimental in nature (e.g. physical electrode movement over
time) or are intrinsic to the system. Examples of noise variation
that are intrinsic to the system include the arousal state of the
animal, trial-to-trial variability of neural responses, and correlated
neural activity that is intrinsic to the system and not related to the
stimulus condition.
We estimate a rate-dependent additive noise model from either
the multi-unit or single-unit neural responses. The neural
responses entering this noise model estimation follow the same
averaging and background subtraction step as described previously. The rate-dependent additive noise model follows the common
observation that spike counts of neurons are approximately
Poisson [44,45]. We first normalize the variance of the neural
response to 1,
1 X
(rijk {m)2 ~1,
NMT ijk
1 X
(slj {n)2 ~s2signalznoise {s2noise ,
LM lj
where slj indicates the model representation value with index l
over L feature dimensions and n is the mean over image and
features for the model representation. Finally we add signal
dependent noise to the model representation:
^slj ~slj zN (0,(aslj zb)2 =T),
where N (m,s2 ) indicates a sample from the normal distribution
with mean m and variance s2 and ^s indicates the noise corrupted
model representation. We verified empirically that the resulting
variance of the noise corrupted model representation was close to
the empirical neural signal variance. We repeat this procedure for
the multi-units and single-units to arrive at model representations
that are corrected for the observers noise in the respective neural
See S1 Fig. for a validation of the experimental noise matched
where rijk indicates the neural response with indices i over N
neural sites, j over M images, and k over T trials, and m is the
mean over sites, images and trials. For each neural site we estimate
a linear fit of the relationship between the mean response over
trials (mij ) and the variance over trials (sij ) as,
Linear-SVM methodology
ai mij zbi ~sij
PLOS Computational Biology |
See Supporting Information.
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DNNs Rival the Representation of IT Cortex for Core Object Recognition
evaluate the IT multi-unit predictions using an object-level
representational similarity analysis.
Predicting IT multi-unit sites from model representations
We performed generalized linear model analysis using ridge
regression to predict the IT multi-unit responses from the model
representations [69]. We utilized the same cross-validation
procedure as the linear-SVM analysis, estimating encoding models
on 80% of the data and evaluating the performance on the
remaining 20% and repeating this for 10 randomizations. For
each IT multi-unit we estimated an encoding model, predicted the
testing responses, and determined the explained explainable
variance for that multi-unit. To measure explainable variance
we measured for each multi-unit site and each training set, the
Spearman-Brown corrected split-half self-consistency over image
presentations (trials). To arrive at a summary statistics we took the
median explained explainable variance over the multi-units for
each model representation. We also used the V4 multi-unit
responses to produce encoding models of the IT multi-unit
response. The results of this analysis are presented in Fig. 6. Note
that the predictions produced by the V2-like representation
proved to be unstable and produced high variance compared to
the other representations.
Supporting Information
Effects on kernel analysis performance of
empirical noise vs. induced noise model. In the top left
panel we show the performance measurements, as measured by
kernel analysis area-under-the-curve, of the IT cortex multi-unit
sample and of the IT cortex multi-unit sample with trial dependent
added noise as we vary the number of experimental trials
(repetitions per image) or the trials in the noise model (T in Eq.
10). In all plots error bars indicate standard deviations of the
measure over 10 repetitions of the analysis. Results are replotted
and divided by the maximum performance (Relative Performance)
in the lower left panel. The same analysis is performed for the IT
cortex single-unit sample in the right panels. These results indicate
that the noise model reduces our performance measurement over
the empirically observed noise and is therefore a conservative
model for inducing noise in model representations. In other words,
these results indicate that the model representations with neural
matched noise are likely overly penalized.
S1 Fig
Representational similarity analysis
To measure representational similarity we followed the analysis
methodology in [32]. We first computed a feature vector per
object for each representation by averaging the representational
vectors over image variations. The representational dissimilarity
matrix (RDM) is defined as
cov(w(xi ),w(xj ))
RDM(w(x))ij ~1{ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ,
var(w(xi )):var(w(xj ))
S2 Fig Human performance on the visual recognition
task as a function of presentation time. We plot the mean
block-accuracy for different stimulates presentation durations from
responses measured using Amazon Mechanical Turk. The mean
accuracy is plotted as diamond markers and the error bars indicate
the 95% confidence interval of the standard error of the mean over
block-accuracies. Chance performance is ,14% for this task. The
accuracy quickly increases such that at 100 ms stimulus duration it
is within 92% of the performance at 2 seconds. This indicates that
on this task, human subjects are able to perform relatively highly
even during brief presentations of 100 ms. We refer to this ability
as ‘‘core visual object recognition’’ [6] and we seek to measure the
neural representational performance that subserves this ability.
where w(x) indicates the representation averaged for each object, i
and j index the objects, cov indicates the covariance between the
vectors and var the variance of the vector. Because we have 49
unique objects in our task the resulting RDM is a 49x49 matrix.
To measure the relationship between two RDMs we measured the
Spearman rank correlation between the upper-triangular, nondiagonal elements of the RDMs. We computed the RDM on 20%
of the images and repeated the analysis 10 times. To compute
noise due to the neural sample, we computed the split-half
consistency between one half of the IT multi-units and the other
half. We repeated this measurement over 50 random groupings
and over the 10 image splits. Following the methodology in [27],
we also predicted IT multi-unit responses to form a new
representation, which we measured using representational similarity analysis. To produce IT multi-unit predictions for each
model representation, we followed the same methodology as
described previously (Predicting IT multi-unit sites from model
representations). For each image split, we estimated encoding
models on 80% of the images for each of the 168 IT multi-units
and produced predictions for each multi-unit on the remaining
20% of the images. We then used these predictions as a new
representation and computed the object-level RDM for the 20%
of held-out images. We repeated the procedure 10 times. Note that
the 20% of images used for each split was identical for all RDM
calculations and that the images used to estimate multi-unit
encoding models did not overlap with the images used to calculate
the RDM. The analysis of the representations with the additional
IT multi-unit fit can be seen as a different evaluation metric to the
results of predicting IT multi-units. In other words, in Fig. 6 we
evaluate the IT multi-unit predictions using explained variance at
the image-level, and in Fig. 7 for the ‘‘+ IT-fit’’ representations we
PLOS Computational Biology |
Linear-SVM performance of model representations without sample or noise correction. Testing set
classification accuracy averaged over 10 randomly-sampled test
sets is plotted and error bars indicate standard deviation over the
10 random samples. Chance performance is ,14.3%. Unlike in
Fig. 5, the model representations in this figure has not been
modified to correct for sampling or noise.
S3 Fig
S4 Fig Effect of sampling the neural and noise-correct-
ed model representations for the linear-SVM analysis.
We measure the mean testing-set linear-SVM generalization
performance as we change the number of neural sites (for neural
representations), or the number of features (for model representations). Measured samples are indicated by filled symbols and
measured standard deviations indicated by error bars. Multi-unit
analysis is shown in panel A and single-unit analysis in B. The
model representations are noise corrected by adding noise that is
matched to the IT multi-unit measurements (A, as indicated by the
{ symbol) or single-unit measurements (B, as indicated by the {
symbol). This analysis reveals a similar relationship to that found
using the kernel analysis methodology (compare to Fig. 5).
S5 Fig Comparison of IT multi-unit and single-unit
representations. In the left panel we plot the kernel analysis
AUC as a function of the number of single- or multi-unit sites. We
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DNNs Rival the Representation of IT Cortex for Core Object Recognition
plot results for two single-unit samples. ‘‘IT Cortex Single-Unit
All-SNR Sample’’ uses all 160 isolated single-units and ‘‘IT Cortex
Single-Unit High-SNR Sample’’ uses the 40 most consistent (least
noisy) single-units. In the right panel we show the same data, but
correct the number of multi-units to reflect an estimate of the
number of single-units contributing to each multi-unit recording,
thus plotting against the number of estimated neurons. Unlike
previous figures, these estimates have a fixed number of trials (6)
for both single- and multi-unit samples. Surprisingly, multi-unit
recordings surpass single-unit recordings in performance (left) and
are five times better per unit in performance to the all SNR singleunit sample (right).
presentation time; Linear-SVM methodology; Comparing IT
multi-unit and single-unit representations; Processing time and
energy consumption of computational models.
We thank Alex Krizhevsky, Matt Zeiler, Quoc Le, and Adam Coates for
their help in evaluating their models and comments on the paper. We also
thank Tomaso Poggio, Elias Issa, Christopher K.I. Williams, Lorenzo
Rosasco, and Rob Fergus for their helpful feedback and comments.
Author Contributions
Conceived and designed the experiments: CFC HH NP NJM JJD.
Performed the experiments: CFC HH DLKY DA EAS NJM. Analyzed the
data: CFC HH DLKY DA EAS. Contributed reagents/materials/analysis
tools: CFC HH DLKY NP DA EAS. Wrote the paper: CFC HH JJD.
Supporting Information for Methods and
Discussion. Validation of the experimental noise matched
model; Human performance on our task as a function of
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