580 research outputs found
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
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
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
Riemannian Multi-Manifold Modeling
This paper advocates a novel framework for segmenting a dataset in a
Riemannian manifold into clusters lying around low-dimensional submanifolds
of . Important examples of , for which the proposed clustering algorithm
is computationally efficient, are the sphere, the set of positive definite
matrices, and the Grassmannian. The clustering problem with these examples of
is already useful for numerous application domains such as action
identification in video sequences, dynamic texture clustering, brain fiber
segmentation in medical imaging, and clustering of deformed images. The
proposed clustering algorithm constructs a data-affinity matrix by thoroughly
exploiting the intrinsic geometry and then applies spectral clustering. The
intrinsic local geometry is encoded by local sparse coding and more importantly
by directional information of local tangent spaces and geodesics. Theoretical
guarantees are established for a simplified variant of the algorithm even when
the clusters intersect. To avoid complication, these guarantees assume that the
underlying submanifolds are geodesic. Extensive validation on synthetic and
real data demonstrates the resiliency of the proposed method against deviations
from the theoretical model as well as its superior performance over
state-of-the-art techniques
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