# Spectrum Approximation Beyond Fast Matrix Multiplication: Algorithms and Hardness

Understanding the singular value spectrum of a matrix $A \in \mathbb{R}^{n \times n}$ is a fundamental task in countless applications. In matrix multiplication time, it is possible to perform a full SVD and directly compute the singular values $\sigma_1,...,\sigma_n$. However, little is known about algorithms that break this runtime barrier. Using tools from stochastic trace estimation, polynomial approximation, and fast system solvers, we show how to efficiently isolate different ranges of $A$'s spectrum and approximate the number of singular values in these ranges. We thus effectively compute a histogram of the spectrum, which can stand in for the true singular values in many applications. We use this primitive to give the first algorithms for approximating a wide class of symmetric matrix norms in faster than matrix multiplication time. For example, we give a $(1 + \epsilon)$ approximation algorithm for the Schatten-$1$ norm (the nuclear norm) running in just $\tilde O((nnz(A)n^{1/3} + n^2)\epsilon^{-3})$ time for $A$ with uniform row sparsity or $\tilde O(n^{2.18} \epsilon^{-3})$ time for dense matrices. The runtime scales smoothly for general Schatten-$p$ norms, notably becoming $\tilde O (p \cdot nnz(A) \epsilon^{-3})$ for any $p \ge 2$. At the same time, we show that the complexity of spectrum approximation is inherently tied to fast matrix multiplication in the small $\epsilon$ regime. We prove that achieving milder $\epsilon$ dependencies in our algorithms would imply faster than matrix multiplication time triangle detection for general graphs. This further implies that highly accurate algorithms running in subcubic time yield subcubic time matrix multiplication. As an application of our bounds, we show that precisely computing all effective resistances in a graph in less than matrix multiplication time is likely difficult, barring a major algorithmic breakthrough.

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