# Tensor-Train Parameterization for Ultra Dimensionality Reduction

## TL;DR

This paper introduces TTPUDR, a robust tensor-train based dimensionality reduction method that outperforms existing techniques on high-dimensional classification tasks by effectively capturing spatial relations.

## Contribution

It proposes a novel tensor-train parameterization for ultra dimensionality reduction, replacing traditional LPP with a Frobenius norm-based objective for improved robustness.

## Key findings

- TTPUDR outperforms previous methods in classification accuracy.
- The tensor-train approach effectively captures spatial relations in high-dimensional data.
- The model demonstrates robustness against outliers.

## Abstract

Locality preserving projections (LPP) are a classical dimensionality reduction method based on data graph information. However, LPP is still responsive to extreme outliers. LPP aiming for vectorial data may undermine data structural information when it is applied to multidimensional data. Besides, it assumes the dimension of data to be smaller than the number of instances, which is not suitable for high-dimensional data. For high-dimensional data analysis, the tensor-train decomposition is proved to be able to efficiently and effectively capture the spatial relations. Thus, we propose a tensor-train parameterization for ultra dimensionality reduction (TTPUDR) in which the traditional LPP mapping is tensorized in terms of tensor-trains and the LPP objective is replaced with the Frobenius norm to increase the robustness of the model. The manifold optimization technique is utilized to solve the new model. The performance of TTPUDR is assessed on classification problems and TTPUDR significantly outperforms the past methods and the several state-of-the-art methods.