On the singular values of matrices with high displacement rank

Townsend, Alex; Wilber, Heather

doi:10.1016/j.laa.2018.02.025

Cited by 13 publications

(5 citation statements)

References 29 publications

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“…The singular value bound in Theorem 2.1 is most useful when \nu \leq 5 as it tends to not be tight for large \nu . An extension of Theorem 2.1 has derived useful bounds when \nu is large, provided that M N \ast has rapidly decaying singular values [57]. It is also known that X can have off-diagonal low rank structure if M N \ast does [39].…”

Section: The Singular Values Of Matrices With Displacementmentioning

confidence: 99%

Bounds on the Singular Values of Matrices with Displacement Structure

Beckermann¹,

Townsend²

2019

SIAM Rev.

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Matrices with displacement structure, such as Pick, Vandermonde, and Hankel matrices, appear in a diverse range of applications. In this paper, we use an extremal problem involving rational functions to derive explicit bounds on the singular values of such matrices. For example, we show that the kth singular value of a real n \times n positive definite Hankel matrix, Hn, is bounded by C\rho - k/ log n \| Hn\| 2 with explicitly given constants C > 0 and \rho > 1, where \| Hn\| 2 is the spectral norm. This means that a real n \times n positive definite Hankel matrix can be approximated, up to an accuracy of \epsilon \| Hn\| 2 with 0 < \epsilon < 1, by a rank \scrO (log n log(1/\epsilon )) matrix. Analogous results are obtained for Pick, Cauchy, real Vandermonde, L\" owner, and certain Krylov matrices.

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Section: The Singular Values Of Matrices With Displacementmentioning

confidence: 99%

Bounds on the Singular Values of Matrices with Displacement Structure

Beckermann¹,

Townsend²

2019

SIAM Rev.

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“…For example, [6,Thm. 3.1] showed that all n × n positive-definite Hankel matrices, (H n ) ij = h i+j , have an -rank that grows logarithmically in n. These results were later extended to include a broader class of matrices [36]. These results from linear algebra are considering matrices that have more rapidly decaying singular values than the LVMs we study in this paper.…”

Section: The Johnson-lindenstrauss Lemmamentioning

confidence: 80%

Why Are Big Data Matrices Approximately Low Rank?

Udell¹,

Townsend²

2019

SIAM Journal on Mathematics of Data Science

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Matrices of (approximate) low rank are pervasive in data science, appearing in recommender systems, movie preferences, topic models, medical records, and genomics. While there is a vast literature on how to exploit low rank structure in these datasets, there is less attention on explaining why the low rank structure appears in the first place. Here, we explain the effectiveness of low rank models in data science by considering a simple generative model for these matrices: we suppose that each row or column is associated to a (possibly high dimensional) bounded latent variable, and entries of the matrix are generated by applying a piecewise analytic function to these latent variables. These matrices are in general full rank. However, we show that we can approximate every entry of an m × n matrix drawn from this model to within a fixed absolute error by a low rank matrix whose rank grows as O(log(m + n)). Hence any sufficiently large matrix from such a latent variable model can be approximated, up to a small entrywise error, by a low rank matrix.

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“…The main takeaway from the fADI method is that it solves for the column space and row space of W independently. The shifts used in the iterations are known in many situations [16,41]. For example, one set of shift parameters p p p and q q q can be chosen as the zeros and poles of a rational function r ∈ R k,k , that can achieve a quasi-optimal Zolotarev number [45]…”

Section: Sylvester Matrix Equations and Fadimentioning

confidence: 99%

Parallel algorithms for computing the tensor-train decomposition

Shi

Ruth²,

Townsend³

2021

Preprint

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The tensor-train (TT) decomposition expresses a tensor in a data-sparse format used in molecular simulations, high-order correlation functions, and optimization. In this paper, we propose four parallelizable algorithms that compute the TT format from various tensor inputs: (1) Parallel-TTSVD for traditional format, (2) PSTT and its variants for streaming data, (3) Tucker2TT for Tucker format, and (4) TT-fADI for solutions of Sylvester tensor equations. We provide theoretical guarantees of accuracy, parallelization methods, scaling analysis, and numerical results. For example, for a d-dimension tensor in R n×•••×n , a two-sided sketching algorithm PSTT2 is shown to have a memory complexity of O(n d/2 ), improving upon O(n d−1 ) from previous algorithms.

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On the singular values of matrices with high displacement rank

Cited by 13 publications

References 29 publications

Bounds on the Singular Values of Matrices with Displacement Structure

Bounds on the Singular Values of Matrices with Displacement Structure

Why Are Big Data Matrices Approximately Low Rank?

Parallel algorithms for computing the tensor-train decomposition

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