We show how to compute a {\it relative-error} low-rank approximation to any positive semidefinite (PSD) matrix in {\it 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 $\norm{A-B}_2^2$ and for ridge regression on PSD matrices.

Joint work with Cameron Musco.