We consider the problem of decomposing higher-order moment tensors, i.e., the sum of symmetric outer products of data vectors. Such a decomposition can be used to estimate the means in a Gaussian mixture model and for other applications in machine learning. The $d$th-order empirical moment tensor of a set of $p$ observations of $n$ variables is a symmetric $d$-way tensor. Our goal is to find a low-rank tensor approximation comprising $r \ll p$ symmetric outer products. The challenge is that forming the empirical moment tensors costs $O(pn^d)$ operations and $O(n^d)$ storage, which may be prohibitively expensive; additionally, the algorithm to compute the low-rank approximation costs $O(n^d)$ per iteration. Our contribution is avoiding formation of the moment tensor, computing the low-rank tensor approximation of the moment tensor implicitly using $O(pnr)$ operations per iteration and no extra memory. This advance opens the door to more applications of higher-order moments since they can now be efficiently computed. We present numerical evidence of the computational savings and show an example of estimating the means for higher-order moments.

Type

Publication

arXiv

Date

Nov 2019

Tags

Citation

S. Sherman, T. G. Kolda.
**Estimating Higher-Order Moments Using Symmetric Tensor Decomposition**.
arXiv:1911.03813, submitted for publication,
2019.
http://arxiv.org/abs/1911.03813

math.NA, cs.NA

```
@misc{ShKo19,
author = {Samantha Sherman and Tamara G. Kolda},
title = {Estimating Higher-Order Moments Using Symmetric Tensor Decomposition},
howpublished = {arXiv},
month = {November},
year = {2019},
note = {submitted for publication},
eprint = {1911.03813},
eprintclass = {math.NA},
}
```