Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Leverage Score Sampling for Tensor Product Matrices in Input Sparsity
  Time   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 $X^{\otimes q}$. 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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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: nlin
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
2
The categories of this paper are: math
10
The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: eess
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The categories of this paper are: dg-ga
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The categories of this paper are: patt-sol
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org