On Approximating the Eigenvalues of Stochastic Matrices in Probabilistic Logspace

Doron, Dean; Sarid, Amir; Ta-Shma, Amnon

doi:10.1007/s00037-016-0150-y

Cited by 12 publications

(11 citation statements)

References 26 publications

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“…• if A is a poly(n)-conditioned Hermitian stochastic matrix, then |Det(A)| can be approximated in BPL. This follows by combining the BPL-algorithm of Doron et al [10] for approximating powers of A with a parallelized version of the gradient descent algorithm for solving Ax = b.…”

Section: Approximate Computation Of the Determinantmentioning

confidence: 99%

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Approximating the Determinant of Well-Conditioned Matrices by Shallow Circuits

Boix-Adserà,

Eldar,

Mehraban

2019

Preprint

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The determinant can be computed by classical circuits of depth O(log 2 n), and therefore it can also be computed in classical space O(log 2 n). Recent progress by [24] implies a method to approximate the determinant of Hermitian matrices with condition number κ in quantum space O(logn + logκ). However, it is not known how to perform the task in less than O(log 2 n) space using classical resources only. In this work, we show that the condition number of a matrix implies an upper bound on the depth complexity (and therefore also on the space complexity) for this task: the determinant of Hermitian matrices with condition number κ can be approximated to inverse polynomial relative error with classical circuits of depth Õ(logn•logκ), and in particular one can approximate the determinant for sufficiently well-conditioned matrices in depth Õ(logn). Our algorithm combines Barvinok's recent complex-analytic approach for approximating combinatorial counting problems [2] with the depth-reduction theorem for low-degree arithmetic circuits [26].

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Section: Approximate Computation Of the Determinantmentioning

confidence: 99%

“…) For such PSD matrices B with B ∞ ≤ 1 the powers B k for k = poly(n) can be approximated in BPL [10]. Therefore the gradient descent algorithm for solving Bx = b can be run in BPL.…”

Section: A1 Example Applications Of Proposition 30mentioning

confidence: 99%

Approximating the Determinant of Well-Conditioned Matrices by Shallow Circuits

Boix-Adserà,

Eldar,

Mehraban

2019

Preprint

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“…The idea of estimating powers of a normalized adjacency matrix 1 d A (or more generally, a stochastic matrix) by performing random walks is well known, and was used also in [DGT17] mentioned above, and in [DSTS17]. Chung and Simpson [CS15] used it in a context that is related to ours, of solving a Laplacian system L G x = b but with a boundary condition, namely, a constraint that x i = b i for all i in the support of b.…”

Section: Related Workmentioning

confidence: 99%

On Solving Linear Systems in Sublinear Time

Andoni¹,

Krauthgamer²,

Pogrow³

2018

Preprint

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We study sublinear algorithms that solve linear systems locally. In the classical version of this problem the input is a matrix S ∈ R n×n and a vector b ∈ R n in the range of S, and the goal is to output x ∈ R n satisfying Sx = b. For the case when the matrix S is symmetric diagonally dominant (SDD), the breakthrough algorithm of Spielman and Teng [STOC 2004] approximately solves this problem in near-linear time (in the input size which is the number of non-zeros in S), and subsequent papers have further simplified, improved, and generalized the algorithms for this setting.Here we focus on computing one (or a few) coordinates of x, which potentially allows for sublinear algorithms. Formally, given an index u ∈ [n] together with S and b as above, the goal is to output an approximation xu for x * u , where x * is a fixed solution to Sx = b. Our results show that there is a qualitative gap between SDD matrices and the more general class of positive semidefinite (PSD) matrices. For SDD matrices, we develop an algorithm that approximates a single coordinate x u in time that is polylogarithmic in n, provided that S is sparse and has a small condition number (e.g., Laplacian of an expander graph). The approximation guarantee is additive |x u − x * u | ≤ ǫ x * ∞ for accuracy parameter ǫ > 0. We further prove that the condition-number assumption is necessary and tight.In contrast to the SDD matrices, we prove that for certain PSD matrices S, the running time must be at least polynomial in n. This holds even when one wants to obtain the same additive approximation, and S has bounded sparsity and condition number.

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Section: Introductionmentioning

confidence: 99%

“…Other kinds of problems of spectral analysis of operators involving stochastic components are the ones in which the operators depend on some random variables, and hence, the problem is inherently stochastic. 11 Although we are interested in introducing probabilistic elements for the spectral analysis of deterministic operators-and so the problem is different-there are some similarities with this approach. Thus, some of the main tools of our methodological construction can also be found in the classical approach for the analysis of stochastic differential equations-for example, metric probability spaces, conditional expectations associated to martingales.…”

Section: Introductionmentioning

confidence: 99%

Eigenmeasures and stochastic diagonalization of bilinear maps

Erdoğan

Pérez

2020

Math Methods in App Sciences

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A new stochastic approach is presented to understand general spectral type problems for (not necessarily linear) functions between topological spaces. In order to show its potential applications, we construct the theory for the case of bilinear forms acting in couples of a Banach space and its dual. Our method consists of using integral representations of bilinear maps that satisfy particular domination properties, which is shown to be equivalent to having a certain spectral structure. Thus, we develop a measure‐based technique for the characterization of bilinear operators having a spectral representation, introducing the notion of eigenmeasure, which will become the central tool of our formalism. Specific applications are provided for operators between finite and infinite dimensional linear spaces.

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On Approximating the Eigenvalues of Stochastic Matrices in Probabilistic Logspace

Cited by 12 publications

References 26 publications

Approximating the Determinant of Well-Conditioned Matrices by Shallow Circuits

Approximating the Determinant of Well-Conditioned Matrices by Shallow Circuits

On Solving Linear Systems in Sublinear Time

Eigenmeasures and stochastic diagonalization of bilinear maps

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