9 research outputs found
Optimal Eigenvalue Approximation via Sketching
Given a symmetric matrix , we show from the simple sketch , where
is a Gaussian matrix with rows, that there is a
procedure for approximating all eigenvalues of simultaneously to within
additive error with large probability. Unlike the work of
(Andoni, Nguyen, SODA, 2013), we do not require that is positive
semidefinite and therefore we can recover sign information about the spectrum
as well. Our result also significantly improves upon the sketching dimension of
recent work for this problem (Needell, Swartworth, Woodruff FOCS 2022), and in
fact gives optimal sketching dimension. Our proof develops new properties of
singular values of for a Gaussian matrix and an matrix which may be of independent interest. Additionally we achieve
tight bounds in terms of matrix-vector queries. Our sketch can be computed
using matrix-vector multiplies, and by improving on lower
bounds for the so-called rank estimation problem, we show that this number is
optimal even for adaptive matrix-vector queries
Counting and Sampling from Substructures Using Linear Algebraic Queries
For an unknown n × n matrix A having non-negative entries, the inner product (IP) oracle takes as inputs a specified row (or a column) of A and a vector v E Rⁿ with non-negative entries, and returns their inner product. Given two input vectors x and y in Rⁿ with non-negative entries, and an unknown matrix A with non-negative entries with IP oracle access, we design almost optimal sublinear time algorithms for the following two fundamental matrix problems:
- Find an estimate X for the bilinear form x^T A y such that X ≈ x^T A y.
- Designing a sampler Z for the entries of the matrix A such that P(Z = (i,j)) ≈ x_i A_{ij} y_j /(x^T A y), where x_i and y_j are i-th and j-th coordinate of x and y respectively. As special cases of the above results, for any submatrix of an unknown matrix with non-negative entries and IP oracle access, we can efficiently estimate the sum of the entries of any submatrix, and also sample a random entry from the submatrix with probability proportional to its weight. We will show that the above results imply that if we are given IP oracle access to the adjacency matrix of a graph, with non-negative weights on the edges, then we can design sublinear time algorithms for the following two fundamental graph problems:
- Estimating the sum of the weights of the edges of an induced subgraph, and
- Sampling edges proportional to their weights from an induced subgraph. We show that compared to the classical local queries (degree, adjacency, and neighbor queries) on graphs, we can get a quadratic speedup if we use IP oracle access for the above two problems.
Apart from the above, we study several matrix problems through the lens of IP oracle, like testing if the matrix is diagonal, symmetric, doubly stochastic, etc. Note that IP oracle is in the class of linear algebraic queries used lately in a series of works by Ben-Eliezer et al. [SODA'08], Nisan [SODA'21], Rashtchian et al. [RANDOM'20], Sun et al. [ICALP'19], and Shi and Woodruff [AAAI'19]. Recently, IP oracle was used by Bishnu et al. [RANDOM'21] to estimate dissimilarities between two matrices
Testing Positive Semidefiniteness Using Linear Measurements
We study the problem of testing whether a symmetric input matrix
is symmetric positive semidefinite (PSD), or is -far from the PSD
cone, meaning that , where
is the Schatten- norm of . In applications one often needs to quickly
tell if an input matrix is PSD, and a small distance from the PSD cone may be
tolerable. We consider two well-studied query models for measuring efficiency,
namely, the matrix-vector and vector-matrix-vector query models. We first
consider one-sided testers, which are testers that correctly classify any PSD
input, but may fail on a non-PSD input with a tiny failure probability. Up to
logarithmic factors, in the matrix-vector query model we show a tight
bound, while in the
vector-matrix-vector query model we show a tight
bound, for every . We also
show a strong separation between one-sided and two-sided testers in the
vector-matrix-vector model, where a two-sided tester can fail on both PSD and
non-PSD inputs with a tiny failure probability. In particular, for the
important case of the Frobenius norm, we show that any one-sided tester
requires queries. However we introduce
a bilinear sketch for two-sided testing from which we construct a Frobenius
norm tester achieving the optimal queries. We
also give a number of additional separations between adaptive and non-adaptive
testers. Our techniques have implications beyond testing, providing new methods
to approximate the spectrum of a matrix with Frobenius norm error using
dimensionality reduction in a way that preserves the signs of eigenvalues
Quantum-Inspired Algorithms from Randomized Numerical Linear Algebra
We create classical (non-quantum) dynamic data structures supporting queries
for recommender systems and least-squares regression that are comparable to
their quantum analogues. De-quantizing such algorithms has received a flurry of
attention in recent years; we obtain sharper bounds for these problems. More
significantly, we achieve these improvements by arguing that the previous
quantum-inspired algorithms for these problems are doing leverage or
ridge-leverage score sampling in disguise; these are powerful and standard
techniques in randomized numerical linear algebra. With this recognition, we
are able to employ the large body of work in numerical linear algebra to obtain
algorithms for these problems that are simpler or faster (or both) than
existing approaches.Comment: Adding new numerical experiment
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum