195 research outputs found

    Spectral norm of random tensors

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    We show that the spectral norm of a random n1×n2××nKn_1\times n_2\times \cdots \times n_K tensor (or higher-order array) scales as O((k=1Knk)log(K))O\left(\sqrt{(\sum_{k=1}^{K}n_k)\log(K)}\right) under some sub-Gaussian assumption on the entries. The proof is based on a covering number argument. Since the spectral norm is dual to the tensor nuclear norm (the tightest convex relaxation of the set of rank one tensors), the bound implies that the convex relaxation yields sample complexity that is linear in (the sum of) the number of dimensions, which is much smaller than other recently proposed convex relaxations of tensor rank that use unfolding.Comment: 5 page

    A Unified View of Graph Regularity via Matrix Decompositions

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    We prove algorithmic weak and \Szemeredi{} regularity lemmas for several classes of sparse graphs in the literature, for which only weak regularity lemmas were previously known. These include core-dense graphs, low threshold rank graphs, and (a version of) LpL^p upper regular graphs. More precisely, we define \emph{cut pseudorandom graphs}, we prove our regularity lemmas for these graphs, and then we show that cut pseudorandomness captures all of the above graph classes as special cases. The core of our approach is an abstracted matrix decomposition, roughly following Frieze and Kannan [Combinatorica '99] and \Lovasz{} and Szegedy [Geom.\ Func.\ Anal.\ '07], which can be computed by a simple algorithm by Charikar [AAC0 '00]. This gives rise to the class of cut pseudorandom graphs, and using work of Oveis Gharan and Trevisan [TOC '15], it also implies new PTASes for MAX-CUT, MAX-BISECTION, MIN-BISECTION for a significantly expanded class of input graphs. (It is NP Hard to get PTASes for these graphs in general.

    On orthogonal tensors and best rank-one approximation ratio

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    As is well known, the smallest possible ratio between the spectral norm and the Frobenius norm of an m×nm \times n matrix with mnm \le n is 1/m1/\sqrt{m} and is (up to scalar scaling) attained only by matrices having pairwise orthonormal rows. In the present paper, the smallest possible ratio between spectral and Frobenius norms of n1××ndn_1 \times \dots \times n_d tensors of order dd, also called the best rank-one approximation ratio in the literature, is investigated. The exact value is not known for most configurations of n1ndn_1 \le \dots \le n_d. Using a natural definition of orthogonal tensors over the real field (resp., unitary tensors over the complex field), it is shown that the obvious lower bound 1/n1nd11/\sqrt{n_1 \cdots n_{d-1}} is attained if and only if a tensor is orthogonal (resp., unitary) up to scaling. Whether or not orthogonal or unitary tensors exist depends on the dimensions n1,,ndn_1,\dots,n_d and the field. A connection between the (non)existence of real orthogonal tensors of order three and the classical Hurwitz problem on composition algebras can be established: existence of orthogonal tensors of size ×m×n\ell \times m \times n is equivalent to the admissibility of the triple [,m,n][\ell,m,n] to the Hurwitz problem. Some implications for higher-order tensors are then given. For instance, real orthogonal n××nn \times \dots \times n tensors of order d3d \ge 3 do exist, but only when n=1,2,4,8n = 1,2,4,8. In the complex case, the situation is more drastic: unitary tensors of size ×m×n\ell \times m \times n with mn\ell \le m \le n exist only when mn\ell m \le n. Finally, some numerical illustrations for spectral norm computation are presented
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