Nearness to Local Subspace Algorithm for Subspace and Motion Segmentation

There is a growing interest in computer science, engineering, and mathematics for modeling signals in terms of union of subspaces and manifolds. Subspace segmentation and clustering of high dimensional data drawn from a union of subspaces are especially important with many practical applications in computer vision, image and signal processing, communications, and information theory. This paper presents a clustering algorithm for high dimensional data that comes from a union of lower dimensional subspaces of equal and known dimensions. Such cases occur in many data clustering problems, such as motion segmentation and face recognition. The algorithm is reliable in the presence of noise, and applied to the Hopkins 155 Dataset, it generates the best results to date for motion segmentation. The two motion, three motion, and overall segmentation rates for the video sequences are 99.43%, 98.69%, and 99.24%, respectively

Dynamical Sampling and Systems of Iterative Action of Operators

We consider frames and Bessel systems generated by iterations of the form $\{A^ng: g\in \G,\, 0\le n

Reduced row echelon form and non-linear approximation for subspace segmentation and high-dimensional data clustering

Given a set of data W={w1,…,wN}∈RD drawn from a union of subspaces, we focus on determining a nonlinear model of the form U=⋃i∈ISi, where {Si⊂RD}i∈I is a set of subspaces, that is nearest to W. The model is then used to classify W into clusters. Our approach is based on the binary reduced row echelon form of data matrix, combined with an iterative scheme based on a non-linear approximation method. We prove that, in absence of noise, our approach can find the number of subspaces, their dimensions, and an orthonormal basis for each subspace Si. We provide a comprehensive analysis of our theory and determine its limitations and strengths in presence of outliers and noise

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

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.Comment: 15 pages. Some corrections were added, in particular the title was changed. It will appear in "Sampling Theory in Signal and Image Processing