# Representation of big data by dimension reduction

## TL;DR

This paper introduces a simple, effective algorithm for detecting and approximating low-dimensional manifolds within high-dimensional data, outperforming classical methods like PCA and Isomap in certain scenarios.

## Contribution

The paper proposes a new, easy-to-implement algorithm for identifying low-dimensional manifolds in large datasets, with demonstrated advantages over existing algorithms.

## Key findings

- The algorithm successfully detects low-dimensional manifolds in high-dimensional data.
- Numerical results show improved performance compared to PCA and Isomap.
- The method is simple to implement and effective in practical scenarios.

## Abstract

Suppose the data consist of a set $S$ of points $x_j, 1 \leq j \leq J$, distributed in a bounded domain $D \subset R^N$, where $N$ and $J$ are large numbers. In this paper an algorithm is proposed for checking whether there exists a manifold $\mathbb{M}$ of low dimension near which many of the points of $S$ lie and finding such $\mathbb{M}$ if it exists. There are many dimension reduction algorithms, both linear and non-linear. Our algorithm is simple to implement and has some advantages compared with the known algorithms. If there is a manifold of low dimension near which most of the data points lie, the proposed algorithm will find it. Some numerical results are presented illustrating the algorithm and analyzing its performance compared to the classical PCA (principal component analysis) and Isomap.