Corpus ID: 237532259

Sublinear Time Eigenvalue Approximation via Random Sampling

  title={Sublinear Time Eigenvalue Approximation via Random Sampling},
  author={Rajarshi Bhattacharjee and Cameron Musco and Archan Ray},
We study the problem of approximating the eigenspectrum of a symmetric matrix A ∈ Rn×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 ± n using those of a randomly sampled Õ( 1 4 )× Õ( 1 4 ) principal submatrix. Our result can be viewed as a concentration bound on the full eigenspectrum of a random principal submatrix. It significantly extends existing work which shows concentration of just the… Expand

Figures from this paper


Eigenvalues of a matrix in the streaming model
What is shown can be seen as a form of a bi-linear dimensionality reduction: if the authors multiply an input matrix with projection matrices on both sides, the resulting matrix preserves the top eigenvalues and the residual Frobenius norm. Expand
Approximating Spectral Densities of Large Matrices
The problem of estimating the spectral density carefully is defined and how to measure the accuracy of an approximate spectral density is discussed, which is generally costly and wasteful, especially for matrices of large dimension. Expand
Fast matrix multiplication is stable
It is shown that the exponent of matrix multiplication (the optimal running time) can be achieved by numerically stable algorithms, and new group-theoretic algorithms proposed in Cohn and Umans, and Cohn et al. are all included in the class of algorithms to which the analysis applies. Expand
Fast and stable deterministic approximation of general symmetric kernel matrices in high dimensions
This paper develops a unified theoretical framework for analyzing Nyström approximations, which is valid for both SPSD and indefinite kernels and is independent of the specific scheme for selecting landmark points, and proposes the anchor net method, which operates entirely on the dataset without requiring the access to K or its matrix-vector product. Expand
The Noisy Power Method: A Meta Algorithm with Applications
A new robust convergence analysis of the well-known power method for computing the dominant singular vectors of a matrix that is called the noisy power method is provided and shows that the error dependence of the algorithm on the matrix dimension can be replaced by an essentially tight dependence on the coherence of the matrix. Expand
Improved testing of low rank matrices
This work studies the problem of quickly estimating the rank or stable rank of A, the latter often providing a more robust measure of the rank, and proves an Ω(d/ε) lower bound on the number of rows that need to be read, even for adaptive algorithms. Expand
On approximating functions of the singular values in a stream
The results considerably strengthen lower bounds in previous work for arbitrary (not necessarily sparse) matrices A and obtain similar near-linear lower bounds for Ky-Fan norms, eigenvalue shrinkers, and M-estimators, many of which could have been solvable in logarithmic space prior to this work. Expand
Fast Monte-Carlo algorithms for approximate matrix multiplication
Given an m ? n matrix A and an n ? p matrix B, we present 2 simple and intuitive algorithms to compute an approximation P to the product A ? B, with provable bounds for the norm of the "error matrix"Expand
We propose a statistical method to estimate densities of states (DOS) and thermodynamic functions of very large Hamiltonian matrices. Orthogonal polynomials are defined on the interval between lowerExpand
Norms of Random Submatrices and Sparse Approximation
Many problems in the theory of sparse approximation require bounds on operator norms of a random submatrix drawn from a fixed matrix. The purpose of this Note is to collect estimates for severalExpand