Representation of big data by dimension reduction
This work addresses the challenge of dimension reduction for large datasets, offering a simple-to-implement algorithm that could benefit data analysis in fields like machine learning and statistics, though it appears incremental compared to existing methods.
The paper tackles the problem of identifying low-dimensional manifolds in high-dimensional big data, proposing an algorithm that finds such manifolds if they exist and demonstrating its performance through numerical comparisons with PCA and Isomap.
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.