Tensor Train Random Projection

TL;DR

This paper introduces a tensor train-based random projection method that efficiently reduces dimensions of high-dimensional data while approximately preserving pairwise distances, with theoretical guarantees and practical validation.

Contribution

It presents a novel tensor train random projection technique with TT-ranks equal to one, improving speed and storage efficiency over existing methods.

Findings

TTRP effectively preserves pairwise distances in high-dimensional data.

Theoretical analysis confirms TTRP as an expected isometric projection with bounded variance.

Numerical experiments demonstrate TTRP's efficiency on synthetic and real datasets.

Abstract

This work proposes a novel tensor train random projection (TTRP) method for dimension reduction, where pairwise distances can be approximately preserved. Our TTRP is systematically constructed through a tensor train (TT) representation with TT-ranks equal to one. Based on the tensor train format, this new random projection method can speed up the dimension reduction procedure for high-dimensional datasets and requires less storage costs with little loss in accuracy, compared with existing methods. We provide a theoretical analysis of the bias and the variance of TTRP, which shows that this approach is an expected isometric projection with bounded variance, and we show that the Rademacher distribution is an optimal choice for generating the corresponding TT-cores. Detailed numerical experiments with synthetic datasets and the MNIST dataset are conducted to demonstrate the efficiency of TTRP.