Leverage Score Sampling for Tensor Product Matrices in Input Sparsity Time
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We give an input sparsity time sampling algorithm for spectrally approximating the Gram matrix corresponding to the q-fold column-wise tensor product of q matrices using a nearly optimal number of samples, improving upon all previously known methods by poly(q) factors. Furthermore, for the important special care of the q-fold self-tensoring of a dataset, which is the feature matrix of the degree-q polynomial kernel, the leading term of our method's runtime is proportional to the size of the dataset and has no dependence on q. Previous techniques either incur a poly(q) factor slowdown in their runtime or remove the dependence on q at the expense of having sub-optimal target dimension, and depend quadratically on the number of data-points in their runtime. Our sampling technique relies on a collection of q partially correlated random projections which can be simultaneously applied to a dataset X in total time that only depends on the size of X, and at the same time their q-fold Kronecker product acts as a near-isometry for any fixed vector in the column span of X^⊗ q. We show that our sampling methods generalize to other classes of kernels beyond polynomial, such as Gaussian and Neural Tangent kernels.
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