Sublinear time spectral density estimation

Braverman, Vladimir; Krishnan, Aditya; Musco, Christopher

doi:10.1145/3519935.3520009

Cited by 6 publications

(3 citation statements)

References 21 publications

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“…We note that the additive error eigenvalue approximation result of Theorem 1 (analogously Theorems 3 and 4) directly gives an ϵn approximation to the spectral density in the Wasserstein distance -extending the above results to a much broader class of matrices. When ∥A∥ ∞ ≤ 1, A can have eigenvalues as large as n, while the normalized adjacency matrices studied in [13,11] have eigenvalues in [−1, 1]. So, while the results are not directly comparable, our Wasserstein error can be thought as on order of their error of ϵ after scaling.…”

Section: Related Workmentioning

confidence: 95%

“…Recent work has studied sublinear time spectral density estimation for graph structured matrices -Braverman, Krishnan, and Musco [11] show that the spectral density of a normalized graph adjacency or Laplacian matrix can be estimated to ϵ error in the Wasserstein distance in Õ(n/ poly(ϵ)) time. Cohen-Steiner, Kong, Sohler, and Valiant study a similar setting, giving runtime 2 O(1/ϵ) [13].…”

Section: Related Workmentioning

confidence: 99%

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Sublinear Time Eigenvalue Approximation via Random Sampling

Bhattacharjee,

Dexter,

Drineas

et al. 2024

Algorithmica

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We study the problem of approximating the eigenspectrum of a symmetric matrix A ∈ R n×n with bounded entries (i.e., ∥A∥∞ ≤ 1). We present a simple sublinear time algorithm that approximates all eigenvalues of A up to additive error ±ϵn using those of a randomly sampledprincipal submatrix. Our result can be viewed as a concentration bound on the complete eigenspectrum of a random submatrix, significantly extending known bounds on just the singular values (the magnitudes of the eigenvalues). We give improved error bounds of ±ϵ nnz(A) and ±ϵ∥A∥F when the rows of A can be sampled with probabilities proportional to their sparsities or their squared ℓ2 norms respectively. Here nnz(A) is the number of non-zero entries in A and ∥A∥F is its Frobenius norm. Even for the strictly easier problems of approximating the singular values or testing the existence of large negative eigenvalues (Bakshi, Chepurko, and Jayaram, FOCS '20), our results are the first that take advantage of non-uniform sampling to give improved error bounds. From a technical perspective, our results require several new eigenvalue concentration and perturbation bounds for matrices with bounded entries. Our non-uniform sampling bounds require a new algorithmic approach, which judiciously zeroes out entries of a randomly sampled submatrix to reduce variance, before computing the eigenvalues of that submatrix as estimates for those of A. We complement our theoretical results with numerical simulations, which demonstrate the effectiveness of our algorithms in practice.

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Section: Related Workmentioning

confidence: 95%

Section: Related Workmentioning

confidence: 99%

Sublinear Time Eigenvalue Approximation via Random Sampling

Bhattacharjee,

Dexter,

Drineas

et al. 2024

Algorithmica

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“…The standard literature either considers an additive error (where, unlike usual models like floating-point arithmetic, each multiplication incurs identical error regardless of magnitude) [Cle55,FP68,MH02] or eventually boils down to bounding |a i | (since their main concern is dependence on x) [Oli77,Oli79], which is insufficient to get our d 2 log(d) stability bound. The modern work we are aware of shows a O(d 2 ) bound only for Chebyshev polynomials [BKM22], sometimes used to give a O(d 3 ) bound for computing generic bounded polynomials [MMS18], since a degree-d polynomial can be written as a linear combination of T k (x) with bounded coefficients.…”

Section: Technical Overviewmentioning

confidence: 99%

An Improved Classical Singular Value Transformation for Quantum Machine Learning

Bakshi¹,

Tang²

2023

Preprint

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Quantum machine learning (QML) has shown great potential to produce large quantum speedups for computationally intensive linear algebra tasks. The quantum singular value transformation (QSVT), introduced by Gilyén, Su, Low and Wiebe [GSLW19], is a unifying framework to obtain QML algorithms. We provide a classical algorithm that matches the performance of QSVT on low-rank inputs, up to a small polynomial overhead. Under efficient quantum-accessible memory assumptions, given a bounded matrix A ∈ C m×n , a vector b ∈ C n , and a bounded degree-d polynomial p, QSVT can output a measurement from the state |p(A)b in O(d A F ) time after linear-time pre-processing. We show that, in the same setting, for any ε > 0, we can output a vectorF /ε 2 ) time after linear-time pre-processing. This improves upon the best known classical algorithm [CGL + 20a], which requires O(d 22 A 6 F /ε 6 ) time. Instantiating the aforementioned algorithm with different polynomials, we obtain fast quantum-inspired algorithms for regression, recommendation systems, and Hamiltonian simulation. We improve in numerous parameter settings on prior work, including those that use problem-specialized approaches, for quantum-inspired regression [CGL + 20b, CGL + 20a, SM21, GST22, CCH + 22] and quantum-inspired recommendation systems [Tan19, CGL + 20a, CCH + 22].Our key insight is to combine the Clenshaw recurrence, an iterative method for computing matrix polynomials, with sketching techniques to simulate QSVT classically. The tools we introduce in this work include (a) a non-oblivious matrix sketch for approximately preserving bi-linear forms, (b) a non-oblivious asymmetric approximate matrix product sketch based on 2 2 sampling, (c) a new stability analysis for the Clenshaw recurrence, and (d) a new technique to bound arithmetic progressions of the coefficients appearing in the Chebyshev series expansion of bounded functions, each of which may be of independent interest.

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Query lower bounds for log-concave sampling

Chewi,

De Dios Pont,

et al. 2023

2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)

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Sublinear time spectral density estimation

Cited by 6 publications

References 21 publications

Sublinear Time Eigenvalue Approximation via Random Sampling

Sublinear Time Eigenvalue Approximation via Random Sampling

An Improved Classical Singular Value Transformation for Quantum Machine Learning

Query lower bounds for log-concave sampling

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