# Sublinear Time Low-Rank Approximation of Positive Semidefinite Matrices

## TL;DR

This paper presents a novel sublinear time algorithm for computing relative-error low-rank approximations of positive semidefinite matrices, significantly improving efficiency over previous methods and nearly matching theoretical lower bounds.

## Contribution

The authors introduce a sublinear time algorithm for PSD matrix low-rank approximation that surpasses prior approaches and approaches optimality based on proven lower bounds.

## Key findings

- Achieves relative-error low-rank approximation in sublinear time.
- Outperforms previous $nnz(A)$ time algorithms based on oblivious subspace embeddings.
- Extends techniques to spectral norm approximation and ridge regression for PSD matrices.

## Abstract

We show how to compute a relative-error low-rank approximation to any positive semidefinite (PSD) matrix in sublinear time, i.e., for any $n \times n$ PSD matrix $A$, in $\tilde O(n \cdot poly(k/\epsilon))$ time we output a rank-$k$ matrix $B$, in factored form, for which $\|A-B\|_F^2 \leq (1+\epsilon)\|A-A_k\|_F^2$, where $A_k$ is the best rank-$k$ approximation to $A$. When $k$ and $1/\epsilon$ are not too large compared to the sparsity of $A$, our algorithm does not need to read all entries of the matrix. Hence, we significantly improve upon previous $nnz(A)$ time algorithms based on oblivious subspace embeddings, and bypass an $nnz(A)$ time lower bound for general matrices (where $nnz(A)$ denotes the number of non-zero entries in the matrix). We prove time lower bounds for low-rank approximation of PSD matrices, showing that our algorithm is close to optimal. Finally, we extend our techniques to give sublinear time algorithms for low-rank approximation of $A$ in the (often stronger) spectral norm metric $\|A-B\|_2^2$ and for ridge regression on PSD matrices.