On Sketching Matrix Norms and the Top Singular Vector Yi Li University of Michigan, Ann Arbor leeyi@umich.edu Huy L. Nguyễn Princeton University hlnguyen@princeton.edu David P. Woodruff IBM Research, Almaden dpwoodru@us.ibm.com Abstract 1 Sketching is a prominent algorithmic tool for processing large data. In this paper, we study the problem of sketching matrix norms. We consider two sketching models. The first is bilinear sketching, in which there is a distribution over pairs of r × n matrices S and n × s matrices T such that for any fixed n × n matrix A, from S · A · T one can approximate kAkp up to an approximation factor α ≥ 1 with constant probability, where kAkp is a matrix norm. The second is general linear sketching, in which there is a distribution over 2 linear maps L : Rn → Rk , such that for any fixed n × n 2 matrix A, interpreting it as a vector in Rn , from L(A) one can approximate kAkp up to a factor α. We study some of the most frequently occurring matrix norms, which correspond to Schatten p-norms for p ∈ {0, 1, 2, ∞}. The p-th Schatten Pr norm of a rank-r matrix A is defined to be kAkp = ( i=1 σip )1/p , where σ1 , . . . , σr are the singular values of A. When p = 0, kAk0 is defined to be the rank of A. The cases p = 1, 2, and ∞ correspond to the trace, Frobenius, and operator norms, respectively. For bilinear sketches we show: Sketching is an algorithmic tool for handling big data. A linear skech is a distribution over k × n matrices S, k n, so that for any fixed vector v, one can approximate a function f (v) from Sv with high probability. The goal is to minimize the dimension k. The sketch Sv thus provides a compression of v, and is useful for compressed sensing and for achieving lowcommunication protocols in distributed models. One can perform complex procedures on the sketch which would be too expensive to perform on v itself, and this has led to the fastest known approximation algorithms for fundamental problems in numerical linear algebra and nearest neighbor search. Sketching is also the only technique known for processing data in the turnstile model of data streams [22, 38], in which there is an underlying vector v initialized to 0n which undergoes a long sequence of additive positive and negative updates of the form vi ← vi + ∆ to its coordinates vi . Given an update of the form vi ← vi + ∆, resulting in a new vector v 0 = v + ∆ · ei , we can add S(∆ · ei ) to Sv to obtain Lv 0 (here ei is the i-th standard unit vector). A well-studied problem in the sketching literature is approximating the frequency moments Fp (v), which Pn is equivalent to estimating the p-norms kvkp = ( i=1 |vi |p )1/p , for p ∈ [0, ∞]. This problem was introduced by Alon, Matias, and Szegedy [1], and many insights in information complexity [6, 9] and sketching [1, 3, 7, 24] were made on the path to obtaining optimal bounds. In addition to being of theoretical interest, the problems have several applications. The value kvk0 , by continuity, is equal to the support size of x, also known as the number of distinct elements [16, 17]. The norm kvk1 is the Manhattan norm, which is a robust measure of distance and is proportional to the variation distance between distributions [19, 21, 28]. The Euclidean distance kvk2 is important in linear algebra problems [44], and corresponds to the self-join size in databases [1]. 1. For p = ∞ any sketch must have r · s = Ω(n2 /α4 ) dimensions. This matches an upper bound of Andoni and Nguyen (SODA, 2013), and implies one cannot approximate the top right singular vector v of A by a vector v 0 with kv 0 − vk2 ≤ 21 with r · s = õ(n2 ). 2. For p ∈ {0, 1} and constant α, any sketch must have r · s ≥ n1− dimensions, for arbitrarily small constant > 0. 3. For even integers p ≥ 2, we give a sketch with r · s = O(n2−4/p −2 ) dimensions for obtaining a (1 + )approximation. This is optimal up to logarithmic factors, and is the first general subquadratic upper bound for sketching the Schatten norms. For general linear sketches our results, though not optimal, are qualitatively similar, showing that for p = ∞, k = √ Ω(n3/2 /α4 ) and for p ∈ {0, 1}, k = Ω( n). These give separations in the sketching complexity of Schatten-p norms with the corresponding vector p-norms, and rule out a table lookup nearest-neighbor search for p = 1, making progress on a question of Andoni. Introduction Often one wishes to find or approximate the largest coordinates of x, known as the heavy hitters [10, 13], and kvk∞ is defined, by continuity, to equal maxi |vi |. In this paper, we are interested in the analogous problem of sketching matrix norms. We study some of the most frequently occurring matrix norms, which correspond to Schatten p-norms for p ∈ {0, 1, 2, ∞}. The p-th Schatten norm P of an n × n rank-r matrix A r is defined to be kAkp = ( i=1 σip )1/p , where σ1 , . . . , σr are the singular values of A. When p = 0, kAk0 is defined to be the rank of A. The cases p = 1, 2, and ∞ correspond to the trace norm, the Frobenius norm, and the operator norm, respectively. These problems have found applications in several areas; we refer the reader to [11] for graph applications for p = 0, to differential privacy [20, 33] and non-convex optimization [8, 15] for p = 1, and to the survey on numerical linear algebra for p ∈ {2, ∞} [35]. In nearest neighbor search (NNS), one technique often used is to first replace each of the input objects (points, images, matrices, etc.) with a small sketch, then build a lookup table for all possible sketches to support fast query time. In his talks at the Barriers in Computational Complexity II workshop and Mathematical Foundations of Computer Science conference, Alexandr Andoni states that a goal would be to design a NNS data structure for the Schatten norms, e.g., the trace or Schatten 1-norm (slide 31 of [2]). If a sketch for a norm has small size, then building a table lookup is feasible. Sketching Model. We give the first formal study of the sketching complexity of the Schatten-p norms. A first question is what does it mean to have a linear sketch of a matrix, instead of a vector as is typically studied. We consider two sketching models. The first model we consider is bilinear sketching, in which there is a distribution over pairs of r × n matrices S and n × s matrices T such that for any fixed n × n matrix A, from S · A · T one can approximate kAkp up to an approximation factor α ≥ 1 with constant probability, where kAkp is a matrix norm. The goal is to minimize r · s. This model has been used in several streaming papers for sketching matrices [4, 14, 23], and as far as we are aware, all known sketches in numerical linear algebra applications have this form. It also has the advantage that SAT can be computed quickly if S and T have fast matrix multiplication algorithms. The second model is more general, which we dub general linear sketching, and interprets the n × n matrix 2 A as a vector in Rn . The goal is then to design a 2 distribution over linear maps L : Rn → Rk , such that for any fixed n × n matrix A, interpreting it as a vector 2 in Rn , from L(A) one can approximate kAkp up to a factor α with constant probability. The goal is to minimize k. Previous Results. Somewhat surprisingly the only known o(n2 ) upper bound for either model is for p = 2, in which case one can achieve a bilinear sketch with r · s = O(1) [23]. Moreover, the only lower bounds known were those for estimating the pnorm of a vector v, obtained for p > 2 by setting A = diag(v) and are of the form k = Ω(n1−2/p log n) [5, 34, 40]. We note that the bit complexity lower bounds of [6, 9, 43] do not apply, since a single linear sketch (1, 1/M, 1/M 2 , 1/M 3 , . . . , 1/M n−1 )T v is enough to recover v if its entries are bounded by M . The sketching model thus gives a meaningful measure of complexity in the real RAM model. Thus, it was not even known if a sketching dimension of r · s = O(1) was sufficient for bilinear sketches to obtain a constant-factor approximation to the rank or Schatten 1-norm, or if k = Ω(n2 ) was required for general linear sketches. Our Results. We summarize our results for the two sketching models in Table 1. We note that, prior to our work, for all p ∈ / {2, ∞}, all upper bounds in the table were a trivial O(n2 ) while all lower bounds for p ≤ 2 were a trivial Ω(1), while for p > 2 they were a weaker Ω(n1−2/p log n). For the bilinear sketching model, we have the following results. For the spectral norm, p = ∞, we prove an Ω(n2 /α4 ) bound for achieving a factor αapproximation with constant probability, matching an upper bound achievable by an algorithm of [4]. This generalizes to Schatten-p norms for p > 2, for which we prove an Ω(n2−4/p ) lower bound, and give a matching O(n2−4/p ) upper bound for even integers p. For odd integers p we are only able to achieve this upper bound if we additionally assume that A is positive semi-definite (PSD). For the rank, p = 0, we prove an Ω(n2 ) lower bound, showing that no non-trivial sketching is possible. Finally for p = 1, we prove an n1−ε lower bound for arbitrarily small constant ε > 0. Note that our bounds are optimal in several cases, e.g., for p = ∞, for even integers p > 2, and for p = 0. For the general sketching model, we show the following. For the spectral norm, our bound is Ω(n3/2 /α3 ), which although is a bit weaker than the bilinear case, is super-linear in n. It implies, for example, that any general sketching algorithm for approximating the top right (or left) singular vector v of A by a vector v 0 with kv − v 0 k2 ≤ 21 requires k = Ω(n3/2 ). Indeed, otherwise, with a sketch of O(n) dimensions one could store gA, for a random Gaussian vector g, and together with v 0 obtain a constant approximation to kAk∞ , which our lower bound rules out. This bound naturally gen- Schatten p-norm p=∞ p>2 p=0 p=1 Bilinear sketches Lower Bound Upper Bound Ω(n2 /α4 ) O(n2 /α4 ) [4] 2−4/p Ω(n ) O(n2−4/p ) even p 2 Ω(n ) O(n2 ) 1−ε n for any ε > 0 O(n2 ) General Sketches Lower Bound Upper Bound Ω(n3/2 /α3 ) O(n2 /α4 ) [4] (3/2)(1−2/p) Ω(n √ ) O(n2−4/p ) even p Ω(√n) O(n2 ) Ω( n) O(n2 ) Table 1: Our results for approximating the Schatten-p norm up to a constant factor (except for the p = ∞ case) with constant probability in both the bilinear sketching and general sketching models. For the bilinear case, we look at the minimal r · s value, while for general linear sketches we look at the minimal value of k. For p = ∞, α ≥ 1 is the desired approximation factor. eralizes to an Ω(n(3/2)(1−2/p) ) bound for √ p > 2. For p ∈ {0, 1} our bound is now a weaker Ω( n). However, it is the first super-constant lower bound for rank and Schatten-1 norm, which in particular rules out a naïve table lookup solution to the NNS problem, addressing a question of Andoni. Our Techniques for Bilinear Sketches. A standard technique in proving lower bounds is Yao’s minimax principle which implies if there exists a distribution on sketches that succeeds on all n × n inputs matrices A with large probability, then for any distribution L on inputs A, there is a fixed pair S and T of r × n and n × s matrices, respectively, which succeeds with large probability over A ∼ L. Moreover, we can assume the rows of S are orthonormal, as well as the columns of T . This is because, given SAT , we can compute U SAT V , where U and V are arbitrary invertible r × r and s × s matrices, respectively. Thus, it suffices to give two distributions L1 and L2 on A for which the kAkp values differ by a factor α w.h.p. in the two distributions, but for any matrix S with orthonormal rows and T with orthonormal columns, the induced distributions L01 and L02 on SAT , when A ∼ L1 and A ∼ L2 , respectively, have low total variation distance dT V (L01 , L02 ). Since S has orthonormal rows and T has orthonormal columns, then if L1 and L2 are rotationally invariant distributions, then SAT is equal in distribution to an r × s submatrix√of A. This observation already suffices to get an Ω( n) bound on r · s for all Schatten p-norms for a fixed constant p 6= 2, √using a √ result of Jiang [26] which shows that square o( n) × o( n) submatrices of an n × n matrix of i.i.d. Gaussians and an n × n orthonormal matrix have o(1) variation distance. Note that both distributions are rotationally invariant and by the Marčenko-Pastur Law have constant factor difference in Schatten-p norm w.h.p. We slightly generalize Jiang’s proof to show the variation distance is o(1) for any r × s submatrix of these distributions provided r · s < n1−ε for arbitrarily small ε > 0. For our Ω(n2 ) bound for p = 0 and Ω(n2−4/p ) bound for p > 2, we propose the following rotationallyinvariant distributions with constant factor gap in Schatten norm: • For p = 0, L1 = U V T for n × n/2 i.i.d. Gaussian U and V , while L2 = U V T + γG for the same U and V and G an n × n i.i.d. Gaussian matrix with variance γ ≤ 1/poly(n). • For p > 2, L1 = G for an n × n i.i.d. Gaussian 1 uv T for the same matrix, while L2 = G + n1/2−1/p G and random n-dimensional Gaussian vectors u and v. The major difficulty is bounding dT V (L01 , L02 ). For p > 2 this amounts to distinguishing g from g + h, where g is an r × s matrix of Gaussians and h is a random r × s matrix with (i, j)-th entry equal to ui vj . The fact that h is random makes the probability density function of g + h intractable. Moreover, for each fixed h, the variation distance of g and g+h is much larger than for a random h, and the best lower bound we could obtain by fixing h is Ω(n1−2/p ) which would just match the vector lower bound (up to a log n factor). Instead of bounding dT V (L01 , L02 ) directly, we bound the χ2 -divergence of L01 and L02 , which if o(1) implies dT V (L01 , L02 ) = o(1). This idea was previously used in the context of sketching pnorms of vectors [5], improving the previous Ω(n1−2/p ) bound to Ω(n1−2/p log n). Surprisingly, for the Schatten p-norm, it improves a simple Ω(n1−2/p ) bound by a quadratic factor to Ω(n2−4/p ), which can similarly be used to show an Ω(n2 /α4 ) bound for α-approximating the spectral norm. One caveat is that if we were to directly compute the χ2 -divergence between L01 and L02 , it would be infinite once r · s ≥ n1−2/p . We fix this by conditioning L01 and L02 on a constant probability event, resulting in distributions Le01 and Le02 for which the χ2 divergence is small. For p = 0, the problem amounts to distinguishing an r × c submatrix Q of U V T from an r × s submatrix of U V T + γG. Working directly with the density function of U V T is intractable. We instead provide an algorithmic proof to bound the variation distance. See Theorem 3.5 for details. The proof also works for arbitrary Q of size O(n2 ), implying a lower bound of Ω(n2 ) to decide if an n × n matrix is of rank at most n/2 or ε-far from rank n/2 (for constant ε), showing an algorithm of Krauthgamer and Sasson is optimal [29]. Our Algorithm. Due to these negative results, a natural question is whether non-trivial sketching is possible for any Schatten p-norm, other than the Frobenius norm. To show this is possible, given an n × n matrix A, we left multiply by an n × n matrix G of i.i.d. Gaussians and right multiply by an n × n matrix H of i.i.d. Gaussians, resulting in a matrix A0 of the form G0 ΣH 0 , where G0 , H 0 are i.i.d. Gaussian and Σ is diagonal with the singular values of A on the diagonal. We then look at cyclesPin a submatrix n 0 of G0 ΣH 0 . The (i, j)-th entry of A0 is `=1 σ` G0i,` H`,j . Interestingly, for even p, for any distinct i1 , . . . , ip/2 and distinct j1 , . . . , jp/2 , E[(A0i1 ,j1 A0i2 ,j1 ) · (A0i2 ,j2 A0i3 ,j2 ) · · · (A0ip/2 ,jp/2 A0i1 ,jp/2 )] = kAkpp . The row indices of A0 read from left to right form a cycle (i1 , i2 , i2 , i3 , i3 , , ..., ip/2 , ip/2 , i1 ), which since also each column index occurs twice, results in an unbiased estimator. We need to average over many cycles to reduce the variance, and one way to obtain these is to store a submatrix of A0 and average over all cycles in it. While some of the cycles are dependent, their covariance is small, and we show that storing an n1−2/p × n1−2/p submatrix of A0 suffices. Our Techniques for General Linear Sketches: We follow the same framework for bilinear sketches. The crucial difference is that since the input A is now viewed as an n2 -dimensional vector, we are not able to design two rotationally invariant distributions L1 and 2 L2 , since unlike rotating A in Rn , rotating A in Rn does not preserve its Schatten p-norm. Fortunately, for both of our lower bounds (p > 2) and (p ∈ {0, 1}) we can choose L1 and L2 so that L01 is the distribution of a k-dimensional vector g of i.i.d. Gaussians. For p > 2, L02 is the distribution of g + h, where g is as 1 (uT L1 v, . . . , uT Lk v) for random before but h = n1/2−1/p n-dimensional Gaussian u and v and where Li is the i-th row of the the sketching matrix L, viewed as an n × n matrix. We again use the χ2 -divergence to bound dT V (L01 , L02 ) with appropriate conditioning. In this case the problem reduces to finding tail bounds for Gaussian chaoses of degree 4, namely, sums of the P form a,b,c,d Aa,b,c,d ua u0b vc vd0 for a 4th-order array P of n4 coefficients and independent n-dimensional Gaussian vectors u, u0 , v, v 0 . We use a tail bound of Latała [30], which generalizes the more familiar Hanson-Wright inequality for second order arrays P . For p ∈ {0, 1} we look at distinguishing an n × n Gaussian matrix G from a matrix (G0 , G0 M ), where G0 is an n × n/2 Gaussian random matrix and M is a random n/2 × n/2 orthogonal matrix. For all constants p 6= 2, the Schatten p-norms differ by a constant factor in the two cases. Applying our sketching matrix L, we have L01 distributed as N (0, Ik ), but L02 is the distribution of (Z1 , . . . , Zk ), where Zi = hAi , G0 i + hB i , G0 M i and each Li is written as the adjoined matrix (Ai , B i ) for n × n/2 dimensional matrices Ai and B i . For each fixed O, we can view Z as a k-dimensional Gaussian vector formed from linear combinations of entries of G0 . Thus the problem amounts to bounding the variation distance between two zero-mean k-dimensional Gaussian vectors with different covariance matrices. For L01 the covariance matrix is the identity Ik , while for L02 it is Ik +P for some perturbation matrix P . We show that with constant probability over M , the Frobenius norm √ kP kF is small enough to give us an k = Ω( n) bound, and so it suffices to fix M with this property. One may worry that fixing M reduces the variation distance— √ in this case one can show that with k = O( n), distributions L01 and L02 already have constant variation distance. We believe our work raises a number of intriguing open questions. Open Question 1: Is it possible that for every odd integer p < ∞, the Schatten-p norm requires k = Ω(n2 )? Interestingly, odd and even p behave very differently since for even p, we have kAkp = kA2 kp/2 , where A2 is PSD. Note that estimating Schatten norms of PSD matrices A can be much easier: in the extreme case of p = 1 the Schatten norm kAk1 is equal to the trace of A, which √ can be computed with k = 1, while we show k = Ω( n) for estimating kAk1 for non-PSD A. Open Question 2: For general linear sketches our lower bound for the operator norm is Ω(n3/2 /α3 ) for α-approximation. Can this be improved to Ω(n2 /α4 ), which would match our lower bound for bilinear sketches and the upper bound of [4]? Using the tightness of Latała’s bounds for Gaussian chaoses, this would either require a new conditioning of distributions L01 and L02 , or bounding the variation distance without using the χ2 -divergence. 2 Preliminaries Notation. Let Rn×d be the set of n × d real matrices and On the orthogonal group of degree n (i.e., the set of n × n orthogonal matrices). Let N (µ, Σ) denote the (multi-variate) normal distribution of mean µ and covariance matrix Σ. We write X ∼ D for a random variable X subject to a probability distribution D. We also use On to denote the uniform distribution over the orthogonal group of order n (i.e., endowed with the normalized Haar measure). We denote by G(m, n) the ensemble of random matrices with entries i.i.d. N (0, 1). For two n × n matrices P X and Y , we define hX, Y i as hX, Y i = tr(X T Y ) = i,j Xij Yij , i.e., the entrywise inner product of X and Y . Singular values and Schatten norms. Consider a matrix A ∈ Rn×d . Then AT A is√a positive semidefinite matrix. The eigenvalues of AT A are called the singular values of A, denoted by σ1 (A) ≥ σ2 (A) ≥ · · · ≥ σd (A) in decreasing order. Let r = rank(A). It is clear that σr+1 (A) = · · · = σd (A) = 0. The matrix A also has the following singular value decomposition (SVD) A = U ΣV T , where U ∈ On , V ∈ Od and Σ is an n × d diagonal matrix with diagonal entries σ1 (A), . . . , σmin{n,d} (A). Define dT V (µ, ν) ≤ p χ2 (µ||ν). Proposition 2.2. ([25, p97]) It holds that 0 χ2 (N (0, In ) ∗ µ||N (0, In )) ≤ Eehx,x i − 1, where x, x0 ∼ µ are independent. In the case of n = 1, if F (x) and G(x) are the cumulative distribution functions of µ and ν, respectively, the Kolmogorov distance is defined as dK (µ, ν) = sup |F (x) − G(x)|. x It follows easily that for continuous and bounded f , Z Z (2.1) f dµ − f dν ≤ kf k∞ · dK (µ, ν). If both µ and ν are compactly supported, it suffices to have f continuous and bounded on the union of the supports of µ and ν. 1 ! Hanson-Wright Inequality. Suppose that µ is a p r X distribution over R. We say µ is subgaussian if there p , p>0 kAkp = (σi (A)) exists a constant c > 0 such that Prx∼µ {|x| > t} ≤ i=1 1−ct2 e for all t ≥ 0. then kAkp is a norm over Rn×d , called the p-th Schatten The following form of the Hanson-Wright inequality norm, over Rn×d for p ≥ 1. When p = 1, it is also called on the tail bound of a quadratic form is contained in the trace norm or Ky-Fan norm. When p = 2, it is the proof of [41, Theorem 1.1] due to Rudelson and exactly the Frobenius norm kAkF , recalling that σi (A)2 Vershynin. are the eigenvalues of AT A and thus kAk2F = tr(AT A). Let kAk denote the operator norm of A when treating Theorem 2.1. (Hanson-Wright inequality) Let A as a linear operator from `d2 to `n2 . Besides, it holds u = (u1 , . . . , un ), v = (v1 , . . . , vn ) ∈ Rn be random that limp→∞ kAkp = σ1 (A) = kAk and limp→0+ kAkp = vectors such that u1 , . . . , un , v1 , . . . , vn are i.i.d. symrank(A). We define kAk∞ and kAk0 accordingly in this metric subgaussian variables. Let A ∈ Rn×n be a fixed limit sense. matrix, then Finally note that A and AT have the same non-zero t t2 T T singular values, so kAkp = kAT kp for all p. , Pr{|u Av−Eu Av| > t} ≤ 2 exp −c min kAk kAk2F Distance between probability measures. Suppose µ and ν are two probability measures over some for some constant c > 0, which depends only on the Borel algebra B on Rn such that µ is absolutely contin- constant of the subgaussian distribution. uous with respect to ν. For a convex function φ : R → R such that φ(1) = 0, we define the φ-divergence Latała’s Tail Bound. Suppose that gi1 , . . . , gid Z are i.i.d. N (0, 1) random variables. The following redµ Dφ (µ||ν) = φ dν. sult, duePto Latała [30], bounds the tails of Gaussian dν chaoses ai1 · · · aid gi1 · · · gid . The Hanson-Wright inIn general Dφ (µ||ν) is not a distance because it is not equality above, when restricted to Gaussian random variables, is a special case (d = 2) of this tail bound. symmetric. The total variation distance between µ and ν, The proof of Latała’s tail bound was later simplified by denoted by dT V (µ, ν), is defined as Dφ (µ||ν) for φ(x) = Lehec [32]. Suppose that A = (ai )1≤i1 ,...,id ≤n is a finite multi|x − 1|. It can be verified that this is indeed a distance. 2 indexed matrix of order d. For i ∈ [n]d and I ⊆ [d], The χ -divergence between µ and ν, denoted by 2 2 define i For disjoint nonempty subsets χ (µ||ν), is defined as Dφ (µ||ν) for φ(x) = (x − 1) or I = (ij )j∈I . 2 I , . . . , I ⊆ [d] define φ(x) = x − 1. It can be verified that these two choices 1 k of φ give exactly the same value of Dφ (µ||ν). ( X (1) (k) kAkI1 ,...,Ik = sup ai xiI · · · xiI : 1 k Proposition 2.1. ([45, p90]) It holds that i X iI1 (1) xiI 1 2 ≤ 1, . . . , X iIk (1) xiI k 2 ≤1 . Also denote by S(k, d) the set of all partitions of {1, . . . , d} into k nonempty disjoint sets I1 , . . . , Ik . It is easy to see that if a partition {I1 , . . . , Ik } is finer than another partition {J1 , . . . , J` }, then kAkI1 ,...,Ik ≤ kAkJ1 ,...,J` . Theorem 2.2. For any t > 0 and d ≥ 2, d X Y (j) ai Pr gij ≥ t j=1 i ( k2 ) t , ≤ Cd exp −cd min min 1≤k≤d (I1 ,...,Ik )∈S(k,d) kAkI1 ,...,Ik where Cd , cd > 0 are constants depending only on d. Distribution of Singular Values We shall need the following two lemmata. Theorem 3.1. Suppose that p > 2, ζ ∈ (0, 1) and rn , sn ≥ 4. Whenever rn sn ≤ cζ 2 n2(1−2/p) , it holds that dT V (L1 , L2 ) ≤ ζ + 0.009, where c > 0 is an absolute constant. Proof. For simplicity we write rn and sn as r and s, respectively. The distribution L1 is identical to N (0, Irs ), and the distribution L2 is a Gaussian mixture N (z, Irs ) with shifted mean z ∈ Rrs , where zij = 5n−1/2+1/p ui vj 1 ≤ i ≤ r, 1 ≤ j ≤ s. For a given t > 0, define the truncated Gaussian distribution, denoted by Nt (0, In ), as the marginal distribution of N (0, In ) conditioned on the event that 2 2 kxk ≤ t, where x ∼ N (0, In ). Since √ kxk ∼ χ (n), the probability pn := Pr{kxk > 2 n} < 0.004 by evaluating an integral of the p.d.f. of χ2 (n) distribution. Consider an auxiliary random vector z̃ ∈ Rrs defined as z̃ij = 5n−1/2+1/p ũi ṽj 1 ≤ i ≤ r, 1 ≤ j ≤ s, where (ũ1 , . . . , ũr ) is drawn from N2√r (0, Ir ) and (ṽ1 , . . . , ṽs ) is drawn from N2√s (0, Is ). Define an auxiliary distribution Le2 as N (z̃, Irs ). It is not difficult to Lemma 2.1. (Marčenko-Pastur Law [36, 18]) Suppose that X is a p × m matrix with i.i.d N (0, 1/p) see that entries. Consider the probability distribution FX (x) 1 e dT V (L2 , L2 ) ≤ max − 1, pr + ps < 0.009. associated with the spectrum of X T X as 1 − pr − ps 1 i : λi (X T X) ≤ x . FX (x) = So we need only bound dT V (L1 , Le2 ). It follows from m Proposition 2.1 and Proposition 2.2 that For γ ∈ (0, 1], define a distribution Gγ (x) with density p dT V (L1 , Le2 ) ≤ Eehz̃1 ,z̃2 i − 1, function pγ (x) as p where the expectation is taken over independent z̃1 and (b − x)(x − a) pγ (x) = , x ∈ [a, b], z̃2 , which are both identically distributed as z̃. Next we 2πγx compute Eehz̃1 ,z̃2 i . where √ √ Ez̃1 ,z̃2 exp(hz̃1 , z̃2 i) a = (1 − γ)2 , b = (1 + γ)2 . X −1+2/p 0 0 0 ,ṽ 0 exp = E 5n ũ ṽ ũ ṽ ũ,ṽ,ũ i j i j Then when m → ∞, p → ∞ and p/m → γ ∈ (0, 1) it i,j X X holds that the expected Kolmogorov distance (3.2) = Eũ,ṽ,ũ0 ,ṽ0 exp 5n−1+2/p ũi ũ0i ṽj ṽj0 . i j E sup |FX (x) − Gγ (x)| = O(n−1/2 ). P P x Now let us bound E| i ũũ0 |2k . Note that | i ũũ0 | ≤ 0 k2 ≤ 4r. By our assumption that r ≥ 4 it holds Lemma 2.2. ([46]) Suppose that X ∼ G(p, m). Then kũk2 kũ2k−1 that r ≥ 4, so t2 /r ≤ t for t ≤ 4r. Applying the −t2 /2 with probability at least 1−e , it holds that σ1 (X) ≤ √ √ Hanson-Wright inequality (Theorem 2.1) to the identity p + m + t. matrix Ir , we have that 2k Z 2k 3 Lower Bounds for Bilinear Sketches X X 4r 3.1 The case of p > 2 Fix rn ≤ n and sn ≤ n. E ũũ0 ≤ Pr ũi ũ0i > t dt 0 Let L1 = G(rn , sn ) and L2 denote the distribution of i i −1/2+1/p T k Z 4r the upper-left rn × sn block of G + 5n uv , 1/k r where G ∼ G(n, n), u, v ∼ N (0, In ) and G, u, v are k! ≤2 e−c1 t /r dt ≤ 2 c1 0 independent. for some absolute constant c1 > 0, where the integral is evaluated by variable substitution. We continue bounding (3.2) using the Taylor series as below: Ez̃1 ,z̃2 exp(hz̃1 , z̃2 i) P P 2 ∞ X (5n−1+ p )2k E| i ũũ0 |2k E| i ṽṽ 0 |2k ≤ (2k)! k=0 !k 2 ∞ X (k!)2 5n2(−1+ p ) rs ≤1+2 · 2 c1 (2k)! k=1 ∞ k X ζ2 ≤ 1 + ζ 2, ≤1+2 3 k=1 provided that 5rs/c21 n2(1−2/p) ≤ ζ 2 /3. Therefore kG + Xkp ≥ kXkp − kGkp ≥ (4.9 − 2.2)n1/p+1/2 ≥ 1.2 · 2.2n1/p+1/2 ≥ 1.2kGkp with high probability. 3.2 General p > 0 The following theorem is a generalization of a result of Jiang [26]. Following his notation, we let Zn denote the distribution of the upperleft rn × sn block of an orthonormal matrix chosen uniformly at random from On and Gn ∼ √1n G(rn , sn ). Theorem 3.3. If rn sn = o(n) and sn ≤ rn ≤ nd/(d+2) as n → ∞ for some integer d ≥ 1, then limn→∞ dT V (Zn , Gn ) = 0. dT V (L1 , L2 ) ≤ (ln Eehz̃1 ,z̃2 i )1/2 +dT V (L2 , Le2 ) ≤ ζ+0.009, Jiang’s original result restricts rn and sn to rn = o(√n) √ as claimed. The absolute constant c in the statement and sn = o( n). We follow the general notation can be taken as c = c21 /15. and outline in his paper, making a few modifications to remove this restriction. We postpone the proof to It is not difficult to see that kGkp and kG + Appendix A. 5n1/p−1/2 uv T kp differ by a constant factor with high The lower bound on bilinear sketches follows from probability. The lower bound on the dimension of the Theorem 3.3, with the hard pair of rotationally invariant bilinear sketch is immediate, using L1 and L2 as the distributions being a Gaussian random matrix versus a hard pair of distributions and the observations that (i) random orthonormal matrix. The proof follows from both distributions are rotationally invariant, and (ii) by Theorem 3.3 and is postponed to Appendix B. increasing the dimension by at most a constant factor, Theorem 3.4. Let A ∈ Rn×n and p > 0. Suppose that one can assume that rn ≥ 4 and sn ≥ 4. an algorithm takes a bilinear sketch SAT (S ∈ Rr×n Theorem 3.2. Let A ∈ Rn×n and p > 2. Suppose that and T ∈ Rn×s ) and computes Y with (1 − c )kAkp ≤ p p an algorithm takes a bilinear sketch SAT (S ∈ Rr×n Y ≤ (1 + c )kAkp for any A ∈ Rn×n with probability p p and T ∈ Rn×s ) and computes Y with (1 − cp )kAkpp ≤ |Ip −1| 3 1−η ) for Y ≤ (1 + cp )kAkpp for any A ∈ Rn×n with probability at least 4 , where cp = 2(Ip +1) . Then rs = Ω(n any constant η > 0. at least 3/4, where cp = (1.2p − 1)/(1.2p + 1). Then rs = Ω(n2(1−2/p) ). 3.3 Rank (p = 0) Let S ⊂ [n]×[n] be a set of indices Proof. It suffices to show that kGkp and kG + of an n × n matrix. For a distribution L over Rn×n , the 5n1/p−1/2 uv T kp differ by a constant factor with high entries of S induce a marginal distribution L(S) on R|S| probability. Let X = 5n1/p−1/2 uv T . Since X is of rank as one, the only non-zero singular value σ1 (X) = kXkF ≥ (Xp1 ,q1 , Xp2 ,q2 , . . . , Xp|S| ,q|S| ), X ∼ L. 4.9 · n1/p+1/2 with high probability, since kuv T k2F ∼ (χ2 (n))2 , which is tightly concentrated around n2 . Theorem 3.5. Let U, V ∼ G(n, d), G ∼ γG(n, n) for On the other hand, combining Lemma 2.1 and γ = n−14 . Consider two distributions L and L over 1 2 Lemma 2.2 as well as (2.1) with f (x) = xp on [0, 4], Rn×n defined by U V T and U V T + G respectively. Let we can see that with probability 1 − o(1) it holds for S ⊂ [n] × [n]. When |S| ≤ d2 , it holds that X ∼ √1n G(n, n) (note the normalization!) that (3.5) dT V (L1 (S), L2 (S)) ≤ C|S| n−2 + dcd , p (3.3) kXkp = (Ip + o(1)) n, where C > 0 and 0 < c < 1 are absolute constants. where p Z 4 Proof. (sketch, see Appendix D for the full proof.) We p (4 − x)x (3.4) Ip = x2 · dx ≤ 2p . give an algorithm which gives a bijection f : R|S| → 2πx 0 R|S| with the property that for all but a subset of |S| 1/p Hence kGkp ≤ 1.1 · Ip n1/2+1/p ≤ 1.1 · 2 · n1/2+1/p R of measure o(1) under both L1 (S) and L2 (S), the probability density functions of the two distributions are with high probability. By the triangle inequality, equal up to a multiplicative factor of (1 ± 1/poly(n)). The idea is to start with the row vectors U1 , . . . , Un of U and V1 , . . . , Vn of V , and to iteratively perturb them by adding γGi,j to U V T for each (i, j) ∈ S. We find new vectors U10 , . . . , Un0 and V10 , . . . , Vn0 of n × d matrices U 0 and V 0 so that (U 0 )(V 0 )T and U V T + γG are equal on S. We do this in a way for which kUi k2 = (1 ± 1/poly(n))kUi0 k2 and kVi k2 = (1 ± 1/poly(n))kVi0 k2 for all i, and so the marginal density function evaluated on Ui (or Vj ) is close to that evaluated on Ui0 (or Vj0 ), by definition. Moreover, our mapping is bijective, so the joint distribution of (U10 , . . . , Un0 , V10 , . . . , Vn0 ) is the same as that evaluated of (U1 , . . . , Un , V1 , . . . , Vn ) up to a (1 ± 1/poly(n))-factor. The bijection we create depends on properties of S, e.g., if the entry (U V T )i,j = hUi , Vj i is perturbed, and more than d entries of the i-th row of A appear in S, this places more than d constraints on Ui , but Ui is only d-dimensional. Thus, we must also change some of the vectors Vj . We change those Vj for which (i, j) ∈ Q and there are fewer than d rows i0 6= i for which (i0 , j) ∈ S; in this way there are fewer than d constraints on Vj so it is not yet fixed. We can find enough Vj with this property by the assumption that |S| ≤ d2 . Algorithm 1 The sketching algorithm for even p ≥ 4. Input: n, > 0, even integer p ≥ 4 and A ∈ Rn×n . 1: N ← Ω(−2 ) 2: Let {Gi } and {Hi } be independent n1−2/p × n matrices with i.i.d. N (0, 1) entries 3: Maintain each Gi AHiT , i = 1, . . . , N 4: Compute Z as defined in (4.6) 5: return Z As an aside, given that w.h.p. over A ∼ L2 in Theorem 3.5, A requires modifying Θ(n2 ) of its entries to reduce its rank to at most d if d ≤ n/2, this implies that we obtain an Ω(d2 ) bound on the non-adaptive query complexity of deciding if an n×n matrix is of rank at most d or ε-far from rank d (for constant ε), showing an algorithm of Krauthgamer and Sasson is optimal [29]. See Appendix E for the proof. A similar algorithm works for odd p and PSD matrices A. See Appendix F for details. sketches, which can thus be implemented in the most general turnstile data stream model (arbitrary number of positive and negative additive updates to entries given in an arbitrary order). The algorithm works for arbitrary A when p ≥ 4 is an even integer. Suppose that p = 2q. We define a cycle σ to be an ordered pair of a sequence of length q: σ = ((i1 , . . . , iq ), (j1 , . . . , jq )) such that ir , jr ∈ [k] for all r, ir 6= is and jr 6= js for r 6= s. Now we associate with σ Aσ = q Y Ai` ,j` Ai`+1 ,j` `=1 where we adopt the convention that ik+1 = i1 . Let C In the theorem above, choose d = n/2 and so denote the set of cycles. We define rank(U V T ) ≤ n/2 while rank(G) = n with probability N 1 X 1 X 1. Note that both distributions are rotationally invari(Gi AHiT )σ (4.6) Z= ant, and so the lower bound on bilinear sketches follows N i=1 |C| σ∈C immediately. for even p. Theorem 3.6. Let A ∈ Rn×n . Suppose that an alr×n gorithm takes a bilinear sketch SAT (S ∈ R and Theorem 4.1. With probability ≥ 3/4, the output Z T ∈ Rn×s ) and computes Y with (1 − cp ) rank(A) ≤ returned by Algorithm 1 satisfies (1 − )kAkp ≤ Z ≤ p Y ≤ (1 + cp ) rank(A) for any A ∈ Rn×n with probability (1 + )kAkp when p is even. The algorithm is a bilinear p at least 3/4, where cp ∈ (0, 1/3) is a constant. Then sketch with r · s = Op (−2 n2−4/p ). rs = Ω(n2 ). 4 Bilinear Sketch Algorithms By the Johnson-Lindenstrauss Transform, or in fact, any subspace embedding with near-optimal dimension (which can lead to better time complexity [44, 12, 37, 39]), we can reduce the problem of general matrices to square matrices (see Appendix C for details), and henceforth we shall assume the input matrices are square. We present a sketching algorithm to compute a (1+)-approximation of kAkpp for A ∈ Rn×n using linear 5 Lower Bounds for General Linear Sketches 5.1 Lower bound for p > 2 Let G ∼ G(n, n), u, v ∼ N (0, In ) be independent. Define two distributions: L1 = G(n, n), while L2 is the distribution induced by 1 1 G + 5n− 2 + p uv T . A unit linear sketch can be described using an n × n matrix L. Applying this linear sketch to an n × n matrix Q results in hL, Qi. More generally, consider k orthonormal linear sketches (which as was argued earlier, is equivalent to any k linear forms, since, given a sketch, one can left-multiply it by an arbitrary matrix to change its row space) corresponding to {Li }ki=1 with tr(LTi Lj ) = δij . Let X ∼ L2 and Zi = hLi , Xi. We define 1/2 k X X = (Lia,b )2 (Lic,d )2 a distribution Dn,k on Rk to be the distribution of (Z1 , . . . , Zk ). √ Theorem 5.1. Suppose that p > 2. For all suffi3 2 ciently large n, whenever k ≤ cn 2 (1− p ) , it holds that dT V (N (0, Ik ), Dn,k ) ≤ 0.24, where c > 0 is an absolute constant. Proof. It is clear that Dn,k is a Gaussian mixture with shifted mean = a,b,c,d i=1 k Partition of size 2 and 3. The norms √ are automatically upper-bounded by kAk{1,2,3,4} = k. Partition of size 4. The only partition is {1}, {2}, {3}, {4}. We have kAk{1},{2},{3},{4} = k X sup u,v,u0 ,v 0 ∈Sn−1 Xu,v = 5n−1/2+1/p (uT L1 v, uT L2 v, . . . , uT Lk v)T =: 5n−1/2+1/p Yu,v . ≤ sup u,v,u0 ,v 0 1 2 k X uT Li vu0T Li v 0 i=1 ! T i 2 0 0T i 2 huv , L i + hu v , L i ≤1 i=1 Without loss of generality we may assume √ that k ≥ 16. follows the fact that uv T is a unit Consider the event Eu,v = {kYu,v k2 ≤ 4 k}. Since The last inequality 2 n2 i vector in R and L ’s are orthonormal vectors in Rn . EkYu,v k22 = k, it follows from Markov’s inequality Latała’s inequality (Theorem 2.2) states that for en,k be the marginal √ that Pru,v {Eu,v } ≥ 15/16. Let D t ∈ [ k, k 2 ], distribution of Dn,k conditioned on Eu,v . Then o nX 0 0 A u u v v > t Pr a,b,c,d a c e b d dT V (Dn,k , Dn,k ) ≤ 1/8 a,b,c,d ( )! 2 t t2 t 3 √ en,k ). Resortand it suffices to bound dT V (N (0, In ), D ≤ C1 exp −c min √ , , 1 , t ing to χ2 divergence by invoking Proposition 2.1 and k k k3 ! Proposition 2.2, we have that 2 t3 p ≤ C1 exp −c · 1 en,k ) ≤ EehXu,v ,Xu0 ,v0 i − 1, dT V (N (0, In ), D k3 where u, v, u0 , v 0 ∼ N (0, In ) conditioned on Eu,v and with no conditions imposed on u, v, u0 , v 0 . It follows that Eu0 ,v0 . We first calculate that Pr |hYu,v , Yu0 ,v0 i| > tEu,v Eu0 ,v0 n X X 2 Pr {|hYu,v , Yu0 ,v0 | > t} (Li )ab (Li )cd ua u0b vc vd0 hXu,v , Xu0 ,v0 i = c2p n−1+ p ≤ Pr{Eu0 ,v0 } Pr{Eu,v } a,b,c,d=1 i ! 2 X √ 3 t 0 0 =: D Aa,b,c,d ua ub vc vd , ≤ C2 exp −c · 1 , t ∈ [ k, k 2 ]. k3 a,b,c,d 2 where D = c2p n−1+ p and Aa,b,c,d is an array of order 4 Note that conditioned on Eu,v and Eu0 ,v0 , such that |hYu,v , Yu0 ,v0 i| ≤ kYu,v k2 kYu0 ,v0 k2 ≤ 16k. k X Let = 1/8, then for t ∈ [k 1/2+ , 16k], it holds that Aa,b,c,d = Liab Licd . i=1 ct2/3 ct2/3 tD − 1/3 ≤ − 1/3 , We shall compute the partition norms of Aa,b,c,d as k 2k needed in Latała’s tail bound. 3 2 Partition of size 1. The only possible partition is provided that k ≤ c0 n 2 (1− p ) . Integrating the tail bound {1, 2, 3, 4}. We have gives that 1/2 !2 EeXu,v ,Xu0 ,v0 k X X Z 16k kAk{1,2,3,4} = Lia,b Lic,d = 1 + etD Pr{|hYu,v , Yu0 ,v0 i| > t}dt a,b,c,d i=1 0 1/2 Z k1/2+ k X X j j =1+D etD Pr{|hYu,v , Yu0 ,v0 i| > t}dt = Lia,b Lic,d La,b Lc,d a,b,c,d i,j=1 0 Z 16k where N (0, Ik ) is the standard k-dimensional Gaussian distribution. etD Pr{|hYu,v , Yu0 ,v0 i| > t}dt +D k1/2+ Z ≤1+D k1/2+ Z etD dt + C2 D ≤e 2/3 etD−ct /k1/3 dt k1/2+ 0 Dk1/2+ 16k Z 16k + C2 D 2/3 e − ct 1/3 dt ck 2/3 1/2+ ≤ exp(Dk ) + 16C2 kD · exp − 2 ! 2 cp ≤ exp 2 ( 14 − 32 )(1− p ) n ! 2 2 cc0 n(1− p ) ) 0 2 12 (1− p + 16C2 c cp n exp − 2 2k k1/2+ ≤ 1.01 when n is large enough. It follows immediately that en,k ) ≤ 1/10 and thus dT V (N (0, In ), D Proof. The sketch can be written as a matrix Φg, where 2 Φ ∈ Rk×n /2 is a random matrix that depends on A(i,σ) , (i,σ) B and M , and g ∼ G(n2 /2, 1). Assume that Φ has full row rank (we shall justify this assumption below). Fix Φ (by fixing M ). Then Φg ∼ N (0, ΦΦT ). It is known that ([27, Lemma 22]) q dT V (N (0, ΦΦT ), N (0, Ik )) ≤ tr(ΦΦT ) − k − ln |ΦΦT |, where λ1 , . . . , λk are the eigenvalues values of ΦΦT . Write ΦΦT = I + Define an event E = n o P. 12 k2 2 M : kP kF ≤ ζ · n . When E happens, the eigenvalq √k ues of P are bounded by 12 ζ · n ≤ 2/3. Let µ1 , . . . , µk be the eigenvalues of P , then λi = 1+µi and |µi | ≤ 2/3. Hence v u k uX T The lower bound on the number of linear sketches dT V (N (0, ΦΦ ), N (0, Ik )) ≤ t (µi − ln(1 + µi )) follows immediately as a corollary. i=1 v u k r q uX 2 12 k Theorem 5.2. Let X ∈ Rn×n and p > 2. Suppose t 2 2 ≤ · √ ≤ ζ, µi = kP kF ≤ that an algorithm takes k linear sketches of X and ζ 3 n i=1 computes Y with (1 − cp )kXkpp ≤ Y ≤ (1 + cp )kXkpp for any X ∈ Rn×n with probability at least 3/4, where where we use that x − ln(1 + x) ≤ x2 for x ≥ −2/3. cp = (1.2p − 1)/(1.2p + 1). Then k = Ω(n(3/2)(1−2/p) ). Therefore, when E happens, Φ is of full rank and we can apply the total variation bound above. We claim that 5.2 General p ≥ 0 Consider a random matrix EPij2 ≤ 4/n for all i, j and thus EkP k2F ≤ 4k 2 /n, it then (G, GM ), where G ∼ G(n, n/2) and M ∼ On/2 . follows that Pr(E) ≥ 1−ζ/3 by Markov’s inequality and A unit linear sketch can be described using an n × n 2 2 1 matrix L = (A, B), where A, B ∈ Rn×(n/2) such that dT V (Dn,k , N (0, Ik )) ≤ ζ + Pr(E c ) ≤ ζ + ζ = ζ 3 3 3 kAk2F + kBk2F = 1. Applying this linear sketch to an n×(n/2) as advertised. n × n matrix Q = (Q1 , Q2 ) (where Q1 , Q2 ∈ R ) Now we show that EPij2 ≤ 4/n for all i, j. Suppose results in hA, Q1 i + hB, Q2 i. More generally, consider k orthonormal linear that M = (mij ). Notice that the r-th row of Φ is X (r) sketches (which as was argued earlier, is equivalent to n (r) A + Bij m`j , i = 1, . . . , n, ` = 1, . . . , . i` any k linear forms, since, given a sketch, one can left2 j multiply it by an arbitrary matrix to change its row space) corresponding to {Li }ki=1 with tr(LTi Lj ) = δij . Hence by a straightforward calculation, the inner prodNow we define a probability distribution Dn,k on uct of r-th and s-th row is X (r) (s) X (s) (r) Rk . For each i write Li as Li = (A(i) , B (i) ). Then hΦ , Φ i = δ + A B m + Ai` Bij m`j (i) (j) (i) (j) r· s· rs `j ij i` by orthonormality, hA , A i + hB , B i = δi,j . (i) (i) i,j,` i,j,` Define Zi = hA , Gi + hB , GM i and Dn,k to be the X (r) (s) (s) (r) distribution of (Z1 , . . . , Zk ). = δrs + hA` , Bj i + hA` , Bj i m`j dT V (N (0, In ), Dn,k ) ≤ 1/10 + 1/8 < 0.24. Theorem 5.3. Let Dn,k be √ defined as above and ζ ∈ (0, 1). Then for k ≤ (ζ/3)3/2 n it holds that dT V (Dn,k , N (0, Ik )) ≤ ζ, j,` (r) where Ai denotes the i-th column of A(r) . Then Prs = tr(U M ), where the matrix U is defined by [4] A. Andoni and H. L. Nguyen. Eigenvalues of a matrix in the streaming model. In SODA, 2013. (r) (s) (s) (r) uj` = hA` , Bj i + hA` , Bj i. [5] A. Andoni, H. L. Nguyen, Y. Polyansnkiy, and Y. Yu. Tight lower bound for linear sketches of moments. In Since ICALP, 2013. nX X [6] Z. Bar-Yossef, T. S. Jayram, R. Kumar, and D. Sivaku(r) 2 (s) 2 2 ujk ≤ 2 |Aik | |Bij | mar. An information statistics approach to data i X i X o stream and communication complexity. J. Comput. (s) 2 (r) 2 + |Aik | |Bij | Syst. Sci., 68(4):702–732, 2004. i i [7] L. Bhuvanagiri, S. Ganguly, D. Kesh, and C. Saha. and thus Simpler algorithm for estimating frequency moments X X nX (r) (s) of data streams. In SODA, pages 708–713, 2006. kU k2F ≤ 2 |Aik |2 |Bij |2 + j,k i i [8] E. J. Candès and B. Recht. 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Eldar and G. Kutyniok, editors, Compressed Sensing: Theory and Applications. Cambridge University Press, 2011. Appendices A Proof of Theorem 3.3 Proof. First we strengthen Lemma 2.6 in [26] and in this proof we shall follow Jiang’s notation that the we read the upper-left p × q block despite the usage of rn and sn in the statement of the theorem. It holds that fs0 (s, t) = −2/(1 − 2s − t) = −2 + −1/(d+2) O(n ) and ft0 (s, t) = −1 + O(n1/(d+2) ). The bounds for second-order derivatives of f (s, t) still hold. Thus ZZ 3kq 1 2 Bn = n ln(1 − 2s − t)dsdt + +O 2n n1/(d+2) ZZ = n2 ln(1 − 2s − t)dsdt + o(1). Now, we expand the Taylor series into r terms, 2 d ln(1+s+t)− (s+t)− (s+t) +· · ·+(−1)d+1 (s+t) 2 d ≤ (s + t)d+1 . d+1 So Z v Z u ln(1 + s + t)dsdt 0 = 0 d X k=1 (−1)k+1 ((u + v)k+2 − uk+2 − v k+2 ) k(k + 1)(k + 2) + O((u + v)d+3 ) as n → ∞. Substituting u = −(p + 2)/n and v = −q/n back into two integrals in (2.5), (2.7) becomes Z Z n2 v u ln(1 + s + t)dsdt 2 0 0 d =− 1X 1 (p+q+2)k+2 − (p+2)k+2 − q k+2 2 k(k+1)(k+2) nk k=1 (p + q + 2)d+3 +O nd+1 Z =− v Z 1 2 Bn + q X f (λi ) → 0 i=1 −2/n in probability as n → ∞. First, we look at the last term. It is clear from ln(1 + s + t)dsdt 0 tr((X T X)d+2 ) , nd+2 where gn ∈ [0, 2) for n sufficiently large. We want to show that (A.1) and n2 2 + g̃n 0 d X 1 (q + 2)k+2 − 2k+2 − q k+2 k(k + 1)(k + 2) nk k=1 d+3 q +O nd+1 tr((XnT Xn )m ) = q X X X xk1 ,i xk1 ,j1 xk2 ,j1 xk2 ,j2 i=1 j∈[q]m−1 k∈[p]m · · · xkm−1 ,jm−2 xkm−1 ,jm−1 xkm ,jm−1 xkm ,i that E tr((XnT Xn )m ) = O(pm n qn ), where the constant in ! O(·) notation depends on m but not p and q. For any d k+1 X k+2 pi q k+2−i 1X 1 > 0, Bn = − +o(1). 2 k(k+1)(k+2) i=1 nk i k=1 E[(tr(XnT Xn )r+2 )2 ] T r+2 r+2 Pr tr((X X ) > n ≤ i k+2−i k n n Notice that for i ≥ 2, it holds that p q /n ≤ 2 n2r+4 2r+4 2 k 2 k 2r+4 T p q /n → 0 as n → ∞. It follows that ] E[q tr(Xn Xn ) p q ≤ = O 2 2r+4 → 0 2 n2r+4 d n 1 X pk+1 q Bn = − + o(1). as n → ∞. So the last term goes to 0 in probability. 2 k(k + 1)nk k=1 For the other terms, fix > 0, we see that This is our desired form of Lemma 2.6. Pr | tr((XnT Xn )m ) − E tr((XnT Xn )m )| ≥ nm Now, following the proof of Lemma 2.7, we expand into more terms E[q(tr((XnT Xn )2m )] E[(tr((XnT Xn )m )2 ] ≤ ≤ 2 d+1 d+2 x x x 1 x x 2 n2m 2n2m ln 1 − − · . = − − 2 −· · ·− d+1 d+2 p2m q 2 n n n (d + 1)n d + 2 (ξ − n) = O 2 2m → 0 n so Hence f (x) = where p+q+1 n−p−q−1 n−p−q−1 d+1 hence x− x−· · ·− x 2 q 2n 4n 2(d+1)nd+1 X p+q+1 d+2 E tr(X TX) f (λi ) → x 2n + gn (x) d+1 , i=1 n d+1 X n−p−q−1 E tr((X TX)j ) − j 2jn d+1 n (n − p − q − 1) j=2 gn (x) = − . 2(d + 2)(ξ − n)d+2 in probability. It suffices to show that It is trivial to see that d+1 X p+q+1 n−p−q−1 1 B n + E tr(X TX)− E tr((X TX)j ) sup |gn (x)| ≤ . j 2n 2jn (1 − α)d+2 0≤x≤αn j=2 √ √ 2 The eigenvalues are bounded by O(( p + q) ) = goes to 0. Rearrange the terms in the bracket as O(n1−1/(d+2) ). Define Ω accordingly. Then d X p+q+1 1 T j T j+1 q E(tr(X X) ) − E(tr(X X) ) X jnj (j +1)nj f (λi ) j=1 i=1 = d+1 X n−p−q−1 p+q+1 tr(X T X) − 2n j=2 2jnj + tr((X T X)j ) p+q+1 E(tr(X T X)j+1 ) (d + 1)nd+1 The last term goes to 0 as E(tr(X T X)j+1 ) = O(pj+1 q) O(d/2 ) rows while roughly maintaining the singular and pd+1 /nd → 0. Hence it suffices to show that for values as follows. Call Φ a (d, δ)-subspace embedding matrix if with probability ≥ 1 − δ it holds that each k − p+q+1 pk+1 q + E(tr(X TX)k ) k(k + 1)nk knk 1 − E(tr(X T X)k+1 ) → 0, (k + 1)nk (1 − )kxk ≤ kΦxk ≤ (1 + )kxk for all x in a fixed d-dimensional subspace. In [44], it is proved that Lemma C.1. Suppose that H ⊂ Rn is a d-dimensional subspace. Let Φ be an r-by-n random matrix with −pk+1q+(k+1)(p+q+1)E(tr(X TX)k )−kE(tr(X TX)k+1 )entries i.i.d N (0, 1/r), where r = Θ(d/2 log(1/δ)). Then it holds with probability ≥ 1 − δ that nk → 0, (1 − )kxk ≤ kΦxk ≤ (1 + )kxk, ∀x ∈ H. which can be verified easily using that m E tr((XnT Xn )m ) = pm Therefore (A.1) In fact we can use more modern subspace embeddings n qn + o(n ). holds and the rest of the argument in Jiang’s paper [44, 12, 37, 39] to improve the time complexity, though since our focus is on the sketching dimension, we defer a follows. thorough study of the time complexity to future work. Now we are ready for the subspace embedding B Proof of Theorem 3.4 transform on singular values, which follows from the Proof. Choose A from D0 with probability 1/2 and from min-max principle for singular values. On with probability 1/2. Let b ∈ {0, 1} indicate which distribution A is chosen from, that is, b = 0 when A Lemma C.2. Let Φ be a (d, δ)-subspace embedding mais chosen from D0 and b = 1 otherwise. Note that trix. Then, with probability ≥ 1 − δ, it holds that both D0 and O(n) are rotationally invariant, so without (1−)σi (ΦA) ≤ σi (A) ≤ (1+)σi (ΦA) for all 1 ≤ i ≤ d. loss of generality, we can assume that S = Ir 0 Is Proof. [of Lemma C.2] The min-max principle for sinand T = . Hence SAT ∼ Zn when b = 0 and 0 gular values says that SAT ∼ Gn when b = 1. Notice that the singular σi (A) = max min kAxk, values of an othorgonal matrix are all 1s. Then with x∈Si Si kxk2 =1 probability 1 − o(1) we have that or b = 0where Si runs through all i-dimensional subspace. Observe that the range of A is a subspace of dimension at (1 − cp )n ≤ Y ≤ (1 + cp )n, b = 1 most d. It follows from Lemma C.1 that with probabilso we can recover b from Y with probability 1 − o(1), ity ≥ 1 − δ, |I −1| provided that cp ≤ 2(Ipp +1) . (1 − )kAxk ≤ kΦAxk ≤ (1 + )kAxk, ∀x ∈ Rd . Consider the event E that the algorithm’s output The claimed result follows immediately from the minindicates b = 1. Then as in the proof of Theorem 5.4, max principle for singular values. 1 dT V (Dn,k , N (0, Ik )) ≥ + o(1). 2 Let Ã = (ΦA, 0) with zero padding so that Ã is On the other hand, by Theorem 3.3, a square matrix of dimension O(d/2 ). Then by the dT V (Dn,k , N (0, Ik )) = o(1) when rs ≤ n1−η for some preceding lemma, with probability ≥ 1 − δ, kÃkp = η > 0. This contradiction implies that rs = Ω(n1−η ) kΦAkp is a (1 ± )-approximation of kAkp for all p > 0. Therefore we have reduced the problem to the case of for any η > 0. square matrices. C Reduction to Square Matrices for the upper D Proof of Theorem 3.5 bound n×d 2 Suppose that A ∈ R (n > d). When n = O(d/ ), Proof. Suppose that |S| = k and S = {(pi , qi )}ki=1 . By let Ã = (A, 0) with zero padding so that Ã is a square symmetry, without loss of generality, we can assume matrix of dimension n. Then kÃkp = kAkp for all that S does not contain a pair of symmetric entries. p > 0. Otherwise, we can sketch the matrix with Throughout this proof we rewrite L1 (S) as L1 and L2 (S) as L2 . Now, using new notation, let us denote (1 − cp )((Ip + o(1))n ≤ Y ≤ (1 + cp )((Ip + o(1))n, the marginal distribution of L2 with fixed G by L2 (G). Then dT V (L1 , L2 ) = = sup | Pr(A) − Pr(A)| sup Z Pr(A) − A⊂M(Rk ) L1 A⊂M(Rk ) L1 Z ≤ sup A⊂M(Rk ) Rn2 ≤ L2 sup A⊂M(Rk ) Pr(A|G)p(G)dG Rn2 L2 Pr(A) − Pr(A|G) p(G)dG L1 L2 2. kU −Tt U kF ≤ t0 , kV −Tt V kF ≤ t0 with probability 1 − O(1/n2 + dcd ), over the randomness of U and V . When this holds, we say that U and V are good, otherwise we say that they are bad. 3. T−t · Tt = id. Z Property 3 implies that Tt is bijective. Define Pr(A) − Pr(A|G) p(G)dG+ L1 E(x) = {(U, V ) : U V T |S = x} L2 F (δ) Z 1. ((Tt U )(Tt V )T )pq = (U V )pq + t and for all (p0 , q 0 ) ∈ S \ {(p, q)}, we have ((Tt U )(Tt V )T )p0 q0 = (U V T )p0 q0 . Pr(A) − Pr(A|G) p(G)dG L1 L2 Egood (x) = {(U, V ) ∈ E(x) : (U, V ) is good}, Ebad (x) = {(U, V ) ∈ E(x) : (U, V ) is bad} Then, using these three properties about Tt , as well as the triangle inequality, and letting p(U ), p(V ) be the (D.2) ≤ sup dT V (L1 , L2 (G)) + 2 Pr{F (δ)c }, p.d.f.’s of U and V that G∈F (δ) kU k2F 1 n×n exp − p(U ) = where F (δ) = {G ∈ R : |Gpi ,qi | ≤ δ, ∀i = 1, . . . , k} (D.6) 2 (2π)nd/2 and Pr{F (δ)c } is the probability of the complement 1 kV k2F of F (δ) under the distribution on G, and we choose p(V ) = exp − 2 (2π)nd/2 δ = n1/4 γ. Recalling the PDF of a Gaussian random variable and that k ≤ n2 , it follows from a union bound we have that that 2 2 1/2 c −δ /(2γ ) −n /2 −3 (A + te ) (A) − Pr Pr i (D.3) Pr{F (δ) } ≤ ke = ke ≤n . L1 L1 Z Now we examine dT V (L1 , L2 (G)) with G ∈ Z F (δ). For notational convenience, let ξ = = p(U )p(V )dU dV − p(U )p(V )dU dV T T (ξ1 , . . . , ξk ) = (Gp1 ,q1 , . . . , Gpk ,qk ) and define E(x) E(x−tei ) (i) T ξ = (ξ1 , . . . , ξi , 0, . . . , 0) . Applying the triangle Z Z inequality to a telescoping sum, p(U )p(V )dU dV − p(Tt U )p(Tt V )dU dV ≤ (D.4) dT V (L1 , L2 (G)) Egood (A) Egood (A) Z = sup Pr(A) − Pr (A) L2 (G) + p(U )p(V ) + p(Tt U )p(Tt V )dU dV A⊂M(Rk ) L1 Ebad (A) Z = sup Pr(A) − Pr(A − ξ) ≤ |p(U ) − p(Tt U )|p(V )dU dV L1 A⊂M(Rk ) L1 Egood (A) Z k X (i−1) (i) + p(Tt U )|p(V ) − p(Tt V )|dU dV (D.5) ≤ sup ) − Pr(A − ξ ) Pr(A − ξ F (δ)c A⊂M(Rk ) i=1 L1 k L1 where M(R ) denotes the canonical Borel algebra on Rk . To bound (D.5), we need a way of bounding | PrL1 (A) − PrL1 (A + tei )| for a value t with |t| ≤ δ. In this case, we say that we perturb a single entry (p, q) := (pi , qi ) of U V T by t while fixing the remaining k − 1 entries. We claim that there exists a mapping Tt : Rn×d → Rn×d (we defer the construction to the end of the proof) for which the following three properties hold: Egood (A) (D.7) +O 1 d + dc . n2 Using (D.6), (D.8) kU k2F − kTt U k2F |p(U ) − p(Tt U )| = p(U ) · 1 − exp . 2 Notice that kU k2F ∼ χ2 (nd), where χ2 (nd) denotes the χ-squared distribution with nd degrees of freedom, and so by a tail bound for the χ2 -distribution [31, Lemma 1], kU k2F ≤ 6nd − t0 (recall that t0 = 1/n4 ) with probability at least 1 − e−nd ≥ 1 − n−3 . When this happens, for good U it follows from the triangle inequality, the second property of Tt above, and the fact that t0 = 1/n4 that kU k2 − kTt U k2 = (kU k + kTt U k) kU k − kTt U k √ ≤ (2kU k+kU −Tt U k) kU −Tt U k ≤ 2 6nd·t0 ≤ 6n−3 . With probability 1, the matrix Ṽ := (Vq1 , Vq2 , . . . , Vqd ) has rank d. Hence we can solve Ui0 − Ui uniquely, and kUi0 − Ui k ≤ δ k∆xk ≤ . σd (Ṽ ) σd (Ṽ ) Using |1 − e|x| | ≤ 2|x| for |x| < 1 and combining with (D.8), we have Using the tail bound on the minimum √ singular value given in [42], we have Pr{σd (ṽ) ≤ / d} ≤ C + cd , where C > 0 and 0 < c < 1 are absolute constants. Choosing = n−4 and recalling that d ≤ n, we see that with probability at least 1 − Cn−4 − cd , it holds that σd (Ṽ ) ≥ 1/n9/2 and thus |p(U ) − p(Tt U )| ≤ p(U ) · 12n−3 . kUi0 − Ui k ≤ n9/2 δ = n9/2+1/4 γ ≤ n−5 . (D.9) Similarly it holds that kV k2F ≤ 6nd−t0 with probability This proves property 2 of this case. To show property ≥ 1 − n−3 and when this happens, 3 in this case, we can show something stronger, which is T−t ◦ Tt = id, namely, this step is invertible. Indeed, if −3 |p(V ) − p(Tt V )| ≤ p(V ) · 12n . we replace ∆x with −∆x in (D.13) the solution Ui0 − Ui It then follows that will be the opposite sign as well. Case 1b. Suppose that the q-th column contains s ≤ d (D.10) Z entries read. Similarly to Case 1a we have Ui0 = Ui and 1 |p(U ) − p(Tt U )|p(V )dU dV = O , kVi0 − Vi k ≤ n−5 with probability ≥ 1 − Cn−4 − cd . The n3 Egood (A) invertibility is similar as in Case 1a and therefore holds. (D.11) Case 2. Suppose that there are more than d entries Z 1 read in both the p-th row and the q-th column. Define p(Tt U )|p(V ) − p(Tt V )|dU dV = O 3 n Egood (A) J = {i ∈ [n] : i-th column has ≤ d entries contained in S} Plugging (D.10) and (D.11) into (D.7) yields that Cr = {i ∈ [n] : (r, i) ∈ S}, Rc = {i ∈ [n] : (i, c) ∈ S} Call the columns with index in J good columns and Pr(A) − Pr(A + tei ) = O 1 + dcd , (D.12) L1 L1 n2 those with index in J c bad columns. Note that |J c | ≤ d since the total number of entries which, combined with (D.5), (D.2), and (D.3), finally in S is at most d2 . Take the columns in J c ∩ Cp , and leads to notice that q ∈ J c ∩ Cp . As in Case 1, we can change c Up to Up0 such that U 0 V T agrees with the perturbed dT V (L1 , L2 ) ≤ dT V (L1 , L2 (G)) + 2 Pr{F (δ) } entry of U V T at (p, q) and keeps the entries of U V T = O k(n−2 + dcd ) . the same for all (p, q 0 ) for q 0 ∈ (J c ∩ Cp ) \ {q}, since Construction of Tt . Now we construct Tt . Sup- |(J c ∩ Cp ) \ {q}| ≤ d − 1. pose the entry to be perturbed is (p, q). However, this new choice of U 0 possibly causes 0 Case 1a. Suppose that the p-th row contains s ≤ d (U V T )p,i 6= (U V T )p,i for i ∈ Cp ∩ J. For each entries read, say at columns q1 , . . . , qs . Without loss of i ∈ Cp ∩ J, we also need to change Vi to a vector Vi0 generality, we can assume that s = d, as we can preserve without affecting the entries read in any bad column. the entries in S, perturbing one of them, and preserve Now for each good column, the matrix Ũ used in Case d − s arbitrary additional entries in the p-th row. 1 applied to each i in Cp ∩ J is no longer i.i.d. Gaussian Then we have (Ui denotes the i-th row of U and Vi because one row of Ṽ has been changed, and this change the i-th column of V ) has `2 norm at most n−5 (the guarantee on property 2 in Case 1) with probability at least 1 − 4n−3 − cd , Ui Vq1 Vq2 · · · Vqd = x and since the minimum singular value is a 1-Lipschitz 0 function of matrices, the minimum singular value is and we want to construct Ui such that perturbed by at most n−5 . Hence for each good column Ui0 Vq1 Vq2 · · · Vqd = x + ∆x i, with probability at least 1 − 4n−3 − cd , we have kVi − Vi0 k ≤ n−5 . Since there are at most d good Hence property 1 automatically holds and columns, by a union bound, with probability at least (D.13) (Ui0 − Ui ) Vq1 Vq2 · · · Vqd = ∆x 1 − 4/n2 − dcd we have kV − V 0 kF ≤ n−4 . This concludes the proof of properties 1 and 2 in Case i.e., 2. 0 Up1 The final step is to verify property 3. Suppose .. .. . . that Tt (U, V ) = (U 0 , V 0 ) and T−t (U 0 , V 0 ) = (U 00 , V 00 ), 00 00 00 we want to show that U = U and V = V . Observe Up (Vi − Vi ) = 0 , 0 00 c . . that Vi = (V )i = (V )i for i ∈ J ∩ Cp = {q1 , . . . , qd }, .. .. we have 0 Upd (Up0 − Up ) Vq1 Vq2 · · · Vqd = ∆x whence it follows immediately that Vi00 = Vi provided 00 0 that (UpT1 , . . . , UpTd ) is invertible. Together with Vi = (Up − Up ) Vq1 Vq2 · · · Vqd = −∆x Vi0 = Vi00 for all i ∈ Cp ∩ J c , we conclude that V 00 = V . Adding the two equation yields E Proof of Theorem 4.1 (U 00 − Up ) Vq1 Vq2 · · · Vq = 0, p d Up00 and thus = Up provided that (Vq1 , Vq2 , . . . , Vqd ) is invertible. Since Ui = Ui0 = Ui00 for all i 6= p, we have U = U 00 . Next we show that Vi00 = Vi for each i ∈ Cp ∩J. Suppose Ri = {p1 , . . . , pd } 3 p. Similarly to the above we have that 0 Up1 0 .. .. . . 0 0 Up (Vi − Vi ) = −hUp0 − Up , Vi i . .. .. . 0 0 Upd We say that two cycles σ = ({i}, {j}) and τ = ({i0 }, {j 0 }) are (m1 , m2 )-disjoint if |i∆i0 | = 2m1 and |j∆j 0 | = 2m2 , denoted by |σ∆τ | = (m1 , m2 ). Proof. Let X = U ΣV be the SVD of X. Let G and H be random matrices with i.i.d. N (0, 1) entries. By rotational invariance, GXH T is identically distributed as GΣH T . Let k = n1−2/p and X̃ be the upper-left k×k block of GΛH T =: X̃. It is clear that X̃s,t = n X σi Gi,s Hi,t i=1 and Up001 0 .. .. . . 00 00 Up (Vi − Vi0 ) = −hUp00 − Up0 , Vi0 i . . .. .. . 00 0 Upd Up00 = Up Adding the two equations and recalling that and Ui = Ui0 = Ui00 for all i 6= p, we obtain that 0 U p1 .. .. . 00 0 . Up (Vi − Vi ) + Up − Up (Vi0 − Vi ) . .. .. . Upd 0 0 .. . 0 0 = hUp − Up , Vi − Vi i , .. . 0 Define Y = 1 X X̃σ . |C| σ∈C Suppose that σ = ({is }, {js }) then X̃σ = q Y X σ `s σ m s `1 ,...,`q s=1 m1 ,...,mq q Y G`s ,is H`s ,js Gms ,is+1 Hms ,js s=1 P 2q It is easy to see that EX̃σ = σi = kAkpp (all {`s } and {ms } are the same) and thus EY = kXkpp . Now we compute EY 2 . EY 2 = q q 1 X X |C|2 m =0 m =0 1 2 X E(X̃σ X̃τ ). σ,τ ∈C |σ∆τ |=(m1 ,m2 ) Suppose that |σ∆τ | = (m1 , m2 ), X Yq E(X̃σ X̃τ ) = σ`i σmi σ`0i σm0i · `1 ,...,`q `01 ,...,`0q m1 ,...,mq m01 ,...,m0q E nYq i=1 o G`s ,is Gms ,is+1 G`0s ,i0s Gm0s ,i0s+1 · s=1 nYq o E H`s ,js Hms ,js H`0s ,js0 Hm0s ,js0 . s=1 Algorithm 2 The sketching algorithm for odd p ≥ 3. Input: n, > 0, odd integer p ≥ 3 and PSD A ∈ Rn×n E(X̃σ X̃τ ) .m,p 1: N ← Ω(−2 ) 1−2/p 2(2q−2m1 −1) 2(m2 +1) 4(m1−m2 ) × n matrices with kAk2 kAk2(m2 +1) kAk4 , m2 ≤ m1 ≤ q−1 2: Let {Gi } be independent n i.i.d. N (0, 1) entries. 2(2q−2m2 −1) 2(m1 +1) 4(m2−m1 ) kAk2(m1 +1) kAk4 , m1 ≤ m2 ≤ q−1 kAk2 3: Maintain each Gi XGT i , i = 1, . . . , N 4(q−m2 −1) 4(m2 +1) kAk4 kAk2(m2 +1) , m1 = q, m2 ≤ q−1 4: Compute Z as defined in (F.14) 4(q−m1 −1) 4(m +1) kAk4 kAk2(m11 +1) , m2 = q, m1 ≤ q−1 5: return Z 4q m1 = m 2 = q kAk2q , It is not difficult to see that (see Appendix G for details) 2 By Hölder’s inequality, kAk22 ≤ n1− p kAk2p and 2(m+1) kAk2(m+1) ≤ n1− 2(m+1) p dent on p only. Since kAk2(m+1) p Z= Thus, N 1 X Yi , N i=1 then Yi ’s are i.i.d. copies of Y . It follows that p−m −m −2 2p 1 2 n kAk , m , m ≤ q − 1 1 2 p EX = EY = kAkpp nq−m1 −1 kAk2p , p2 = q, p1 ≤ q − 1 p E(X̃σ X̃τ ) .m,p p1 = q, p2 ≤ q − 1 and nq−m2 −1 kAk2p p , 2p EY 2 1 V ar(Y ) m1 = m2 = q. kAkp , ≤ = 2 kAkp2p . V ar(X) = N N 4 There are Oq (k q+m1 k q+m2 ) = Op (k p+m1 +m2 ) pairs of Finally, by Chebyshev inequality, (m1 , m2 )-disjoint cycles and |C| = Θ(k p ), V ar(X) 1 X 1 Pr X − kAkpp > kAkpp ≤ 2p ≤ 4 , E(A A ) σ τ 2 kAkp |C|2 σ,τ ∈C |σ∆τ |=(m1 ,m2 ) 1 |C|2 np−m1 −m2 −2 ≤ Cm1 +m2 ,p p−m1 −m2 kAk2p p k 2p ≤ Cm1 +m2 ,p kAkp , m1 , m2 ≤ q − 1, X E(Aσ Aτ ) σ,τ ∈C |σ∆τ |=(m1 ,m2 ) nq−m1 −1 kAk2p p k q−m1 p ≤ m2 = q, m1 ≤ q − 1, ≤ Cm1 ,p kAk2p p , 2 X 1 E(Aσ Aτ ) 2 |C| ≤ Cm1 +m2 ,p F Algorithms for Odd p Given integers k and p < k, call a sequence (i1 , . . . , ip ) a cycle if ij ∈ [k] for all j and ij1 6= ij2 for all j1 6= j2 . On a k × k matrix A, each cycle σ defines a product Aσ = p Y Aσi ,σi+1 i=1 ≤ Cm1 +m2 ,p where we adopt the convention that ip+1 = i1 . Let C denote the set of cycles. Call two cycles σ, τ ∈ C kdisjoint if |σ∆τ | = 2k, where σ and τ are viewed as multisets. σ,τ ∈C |σ∆τ |=(m1 ,m2 ) Theorem F.1. With probability ≥ 3/4, the output X returned by Algorithm 2 satisfies (1 − )kAkpp ≤ X ≤ (1 + )kAkpp when A is positive semi-definite. The algorithm is a bilinear sketch with r·s = Op (−2 n2−4/p ). σ,τ ∈C |σ∆τ |=(m1 ,m2 ) 1 |C|2 which implies the correctness of the algorithm. It is easy to observe that the algorithm only reads the upper-left k × k block of each Gi XGTi , thus it can be maintained in O(N k 2 ) = Op (−2 n2−4/p ) space. np−2m2 −2 kAk2p p k q−m2 p ≤ m1 = q, m2 ≤ q − 1, ≤ Cm1 ,p kAk2p p , 2 X E(Aσ Aτ ) ≤ kAk2p m1 + m2 = p. p , Therefore EY 2 ≤ Cp kAk2p p for some constant Cp depen- Proof. Since X is symmetric it can be written as X = OΛOT , where Λ is a diagonal matrix and O an orthogonal matrix. Let G be a random matrix with i.i.d. N (0, 1) entries. By rotational invariance, GXGT is identically distributed as GΛGT . Let k = n1−2/p and X̃ be the upper-left k × k block of GΛGT := X̃. It is clear that n X X̃s,t = λi Gi,s Gi,t i=1 Define Y = 1 X X̃σ . C σ∈C 1 |C|2 n X X̃σ = λj1 · · · λjp j1 ,...,jp =1 then Yi ’s are i.i.d. copies of Y . It follows that Gj` ,i` Gj` ,i`+1 X EX = EY = kAkpp E(X̃σ X̃τ ). 2(p−m−1) 2(m+1) kAk2(m+1) 2 By Hölder’s inequality, kAk22 ≤ n1− p kAk2p and 2(m+1) p kAk2(m+1) , p 2(m + 1) ≤ p Thus, E(X̃σ X̃τ ) .m,p V ar(Y ) EY 2 1 ≤ = 2 kAkp2p . N N 4 Finally, by Chebyshev inequality, σ,τ ∈C σ,τ are m-disjoint E(X̃σ X̃τ ) .m,p kAk2 2(m+1) N N 1 X 1 X 1 X (Gi XGTi )σ =: Yi , N i=1 |C| N i=1 σ∈C It is not difficult to see that (see Appendix G for details), when m ≤ p − 2, it holds for m-disjoint σ and τ that kAk2(m+1) ≤ n1− Z= `=1 2 m = p. σ,τ ∈C |σ∆τ |=2m (F.14) P p and It is easy to see that EX̃σ = λi = kAkpp (all j` ’s are the same) and thus EY = kXkpp . Now we compute EY 2 . p 1 X EY = |C|2 m=0 E(Aσ Aτ ) ≤ kAk2p p , Therefore EY 2 ≤ Cp kAk2p p for some constant Cp dependent on p only. Hence if we take multiple copies of this distribution as stated in Algorithm 2 and define Suppose that σ = (i1 , . . . , ip ) then p Y X ( np−m−2 kAk2p p , 2(m + 1) ≤ p k p−m−1 kAk2p p , otherwise. V ar(X) = V ar(X) 1 ≤ , Pr X − kAkpp > kAkpp ≤ 4 2 kAk2p p which implies the correctness of the algorithm. It is easy to observe that the algorithm only reads the upper-left k × k block of each Gi XGTi , thus it can be maintained in O(N k 2 ) = Op (−2 n2−4/p ) space. G Omitted Details in the Proof of Theorem F.1 Suppose that σ = (i1 , . . . , ip ) and τ = (j1 , . . . , jp ) are m-disjoint. Then X E(X̃σ X̃τ ) = λ`1 · · · λ`p λ`01 · · · λ`0p · `1 ,...,`p `01 ,...,`0p And when σ, τ are (p − 1)- and p-disjoint, E(Aσ Aτ ) ≤ kAk2p p . ( E p Y ) G`s ,is G`s ,is+1 G`0s ,js G`0s ,js+1 . s=1 There are Op (k p+m ) pairs of m-disjoint cycles and |C| = For the expectation to be non-zero, we must have Θ(k p ), each appeared entry repeated an even number of times. Hence, if some is appears only once among the index X {is } and {js }, it must hold that `s = `s+1 . Thus for 1 np−m−2 E(Aσ Aτ ) ≤ Cm,p p−m kAk2p p each of the summation term the indices {`s } breaks |C|2 k σ,τ ∈C into a few blocks, in each of which all `s takes the same |σ∆τ |=2m value, the same holds for {`0s }. We also need to piece p ≤ Cm,p kAk2p , m ≤ − 1, the blocks of {`s } which those of {`0s }. Hence the whole p 2 sum breaks into sums corresponding to different block X 1 k p−m−1 E(Aσ Aτ ) ≤ Cm,p p−m kAk2p configurations. For a certain kind of configuration, p 2 |C| k σ,τ ∈C in which `1 , . . . , `w are free variables with multiplicity |σ∆τ |=2m r1 , . . . , rw respectively, the sum is bounded by p X − 1 < m ≤ p − 2, ≤ Cm,p kAk2p p , 2 C· λr`11 · · · λr`ww ≤ CkAkrr11 · · · kAkrrw w X 1 1 `1 ,...,`w 2p E(Aσ Aτ ) ≤ kAkp , m = p − 1 |C|2 k σ,τ ∈C |σ∆τ |=2m where the constant C depends on the configuration only, and thus can be made on m and p by taking the maximum constant over all possible block configurations. P Notice that in a configuration, all rw ’s are even, rw = 2p and w ≤ p − m. Note that s+1 kAkrr kAkss ≤ kAkr−1 r−1 kAks+1 , kAkr+s r+s ≤ r>s kAkrr kAkss , p ∈ (2, ∞), and Ip = Jp = 1 when p = 2. Denote by D0 the distribution of X and by D1 that of Y . Suppose that the k linear forms are a1 , . . . , ak . Choose X from D0 with probability 1/2 and from D1 with probability 1/2. Let b ∈ {0, 1} indicate that X ∼ Db . Without loss of generality, we can assume that a1 , . . . , ak are orthonormal. Hence (a1 , . . . , ak ) ∼ N (0, Ik ) when b = 0 and (a1 , . . . , ak ) ∼ Dn,k when b = 1. Then with probability 3/4 − o(1) we have that it is easy to see that the worst case of configuration is w = p − m and (r1 , . . . , rw ) = (2(m + 1), 2 . . . , 2), giving (1 − cp )((Ip + o(1))n ≤ Y ≤ (1 + cp )((Ip + o(1))n, b = 0 the bound (1 − cp )((Jp + o(1))n ≤ Y ≤ (1 + cp )((Jp + o(1))n, b = 1 2(p−m−1) CkAk2 2(m+1) kAk2(m+1) . so we can recover b from Y with probability 1 − |Ip −Jp | Finally, observe that the number of configurations is a o(1), provided that cp ≤ 2(Ip +Jp ) . Consider the constant depends on p and m only, giving the variance event E that the algorithm’s output indicates b = 1. Then Pr(E|X ∼ D0 ) ≤ 1/4 + o(1) while claim. Pr(E|X ∼ D1 ) ≥ 3/4 − o(1). By definition of total variation distance, dT V (Dn,k , N (0, Ik )) is at least H Proof of Theorem 5.4 |Pr(E|X ∼ D0 ) − Pr(E|X ∼ D1 )|, which is at least p p Proof. First we show that kXkp and kY kp differ by a 1 + o(1). On the other hand, √by Theorem 5.3, p constant factor. We have calculated kXkp = (Ip +o(1))n 2 d (D , N (0, Ik )) ≤ 14 when k ≤ c n for some c > 0. T V n,k in (3.3). Similarly, it holds that We meet √ a contradiction. Therefore it must hold that kY kpp = (Jp + o(1)) n, k > c n. where (H.15) p Z b p x2 Jp = 2 2 a p (b − x)(x − a) dx, πx a, b = 3 √ ∓ 2. 2 Extend the definition of Ip and Jp to p = 0 by Ip = 1 and Jp = 1/2. This agrees with kXk0 = rank(X). Now it suffices to show that Ip 6= Jp when p 6= 2. Indeed, let us consider the general integral √ √ Z (1+√β)2 p ((1 + β)2 − x)(x − (1 − β)2 ) γ x dx √ 2πβx (1− β)2 √ Change the variable y = (x − (1 + β))/ β, the integral above becomes Z 2 p p 1 ( βy + (1 + β))γ−1 4 − y 2 dy. 2π −2 Hence 1 Jp − Ip = 2π Z 2 p fp (x) 4 − x2 dx, −2 where √ p p fp (x) = ( 2x + 3) 2 −1 − (x + 2) 2 −1 . One can verify that fp < 0 on (−2, 2) when 0 < p < 2, fp > 0 on (−2, 2) when p > 2 and fp = 0 when p = 2. It is easy to compute that I2 = 1. The case p = 0 is trivial. Therefore, Ip > Jp for p ∈ [0, 2), Ip < Jp for

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