A Dimension Reduction Scheme for the Computation of Optimal Unions of Subspaces

TL;DR

This paper proposes a dimension reduction approach to efficiently compute optimal unions of subspaces for high-dimensional data, preserving accuracy while reducing computational complexity.

Contribution

It introduces a transformation method that maps high-dimensional subspace clustering problems into lower dimensions, enabling effective approximation of the optimal solution.

Findings

The method reduces computational complexity significantly.

The approximation error between reduced and original solutions is estimated.

The approach maintains high accuracy in subspace identification.

Abstract

Given a set of points \F in a high dimensional space, the problem of finding a union of subspaces \cup_i V_i\subset \R^N that best explains the data \F increases dramatically with the dimension of \R^N. In this article, we study a class of transformations that map the problem into another one in lower dimension. We use the best model in the low dimensional space to approximate the best solution in the original high dimensional space. We then estimate the error produced between this solution and the optimal solution in the high dimensional space.