Effective Tensor Sketching via Sparsification
This work addresses the computational bottleneck of tensor approximation for applications in data analysis and machine learning, offering a more efficient method with proven sample complexity bounds.
The paper tackles the problem of approximating high-dimensional tensors by proposing a tensor sparsification algorithm that selectively retains entries, achieving a given approximation accuracy with significantly lower sample complexity than existing methods. For a k-th order tensor with stable rank r_s, the sample size scales as r_s^{1/2} d^{k/2}/ε for large ε and r_s d/ε^2 for small ε, with the latter being independent of k.
In this paper, we investigate effective sketching schemes via sparsification for high dimensional multilinear arrays or tensors. More specifically, we propose a novel tensor sparsification algorithm that retains a subset of the entries of a tensor in a judicious way, and prove that it can attain a given level of approximation accuracy in terms of tensor spectral norm with a much smaller sample complexity when compared with existing approaches. In particular, we show that for a $k$th order $d\times\cdots\times d$ cubic tensor of {\it stable rank} $r_s$, the sample size requirement for achieving a relative error $\varepsilon$ is, up to a logarithmic factor, of the order $r_s^{1/2} d^{k/2} /\varepsilon$ when $\varepsilon$ is relatively large, and $r_s d /\varepsilon^2$ and essentially optimal when $\varepsilon$ is sufficiently small. It is especially noteworthy that the sample size requirement for achieving a high accuracy is of an order independent of $k$. To further demonstrate the utility of our techniques, we also study how higher order singular value decomposition (HOSVD) of large tensors can be efficiently approximated via sparsification.