# Reducing the dimensionality of data using tempered distributions

## TL;DR

This paper introduces a novel approach to unsupervised dimension reduction by reformulating it with tempered distributions, establishing a connection with sufficient dimension reduction, and proposing algorithms for optimization and practical implementation.

## Contribution

It presents a new formulation of dimension reduction using tempered distributions, linking it to SDR, and develops algorithms including an efficient approximate method for practical use.

## Key findings

- Algorithms successfully reduce data dimensionality in synthetic and real datasets.
- The approach effectively approximates empirical distributions with low-dimensional support.
- The method demonstrates comparable or improved performance over existing techniques.

## Abstract

We reformulate unsupervised dimension reduction problem (UDR) in the language of tempered distributions, i.e. as a problem of approximating an empirical probability density function by another tempered distribution, supported in a $k$-dimensional subspace. We show that this task is connected with another classical problem of data science -- the sufficient dimension reduction problem (SDR). In fact, an algorithm for the first problem induces an algorithm for the second and vice versa. In order to reduce an optimization problem over distributions to an optimization problem over ordinary functions we introduce a nonnegative penalty function that ``forces'' the support of the model distribution to be $k$-dimensional. Then we present an algorithm for the minimization of the penalized objective, based on the infinite-dimensional low-rank optimization, which we call the alternating scheme. Also, we design an efficient approximate algorithm for a special case of the problem, where the distance between the empirical distribution and the model distribution is measured by Maximum Mean Discrepancy defined by a Mercer kernel of a certain type. We test our methods on four examples (three UDR and one SDR) using synthetic data and standard datasets.