Tensor sparsification via a bound on the spectral norm of random tensors

Details Tensor sparsification via a bound on the spectral norm of random tensors Tensor sparsification via a bound on the spectral norm of random tensors

2010-05-26

arxiv.org/abs/1005.4732v1

Mathematics

Numerical Analysis

32 pages

Scientific paper

Given an order-$d$ tensor $\tensor A \in \R^{n \times n \times \ldots \times n}$, we present a simple, element-wise sparsification algorithm that zeroes out all sufficiently small elements of $\tensor A$, keeps all sufficiently large elements of $\tensor A$, and retains some of the remaining elements with probabilities proportional to the square of their magnitudes. We analyze the approximation accuracy of the proposed algorithm using a powerful inequality that we derive. This inequality bounds the spectral norm of a random tensor and is of independent interest. As a result, we obtain novel bounds for the tensor sparsification problem. As an added bonus, we obtain improved bounds for the matrix ($d=2$) sparsification problem.

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