15,337 research outputs found
Kernel Truncated Regression Representation for Robust Subspace Clustering
Subspace clustering aims to group data points into multiple clusters of which
each corresponds to one subspace. Most existing subspace clustering approaches
assume that input data lie on linear subspaces. In practice, however, this
assumption usually does not hold. To achieve nonlinear subspace clustering, we
propose a novel method, called kernel truncated regression representation. Our
method consists of the following four steps: 1) projecting the input data into
a hidden space, where each data point can be linearly represented by other data
points; 2) calculating the linear representation coefficients of the data
representations in the hidden space; 3) truncating the trivial coefficients to
achieve robustness and block-diagonality; and 4) executing the graph cutting
operation on the coefficient matrix by solving a graph Laplacian problem. Our
method has the advantages of a closed-form solution and the capacity of
clustering data points that lie on nonlinear subspaces. The first advantage
makes our method efficient in handling large-scale datasets, and the second one
enables the proposed method to conquer the nonlinear subspace clustering
challenge. Extensive experiments on six benchmarks demonstrate the
effectiveness and the efficiency of the proposed method in comparison with
current state-of-the-art approaches.Comment: 14 page
Subspace clustering of dimensionality-reduced data
Subspace clustering refers to the problem of clustering unlabeled
high-dimensional data points into a union of low-dimensional linear subspaces,
assumed unknown. In practice one may have access to dimensionality-reduced
observations of the data only, resulting, e.g., from "undersampling" due to
complexity and speed constraints on the acquisition device. More pertinently,
even if one has access to the high-dimensional data set it is often desirable
to first project the data points into a lower-dimensional space and to perform
the clustering task there; this reduces storage requirements and computational
cost. The purpose of this paper is to quantify the impact of
dimensionality-reduction through random projection on the performance of the
sparse subspace clustering (SSC) and the thresholding based subspace clustering
(TSC) algorithms. We find that for both algorithms dimensionality reduction
down to the order of the subspace dimensions is possible without incurring
significant performance degradation. The mathematical engine behind our
theorems is a result quantifying how the affinities between subspaces change
under random dimensionality reducing projections.Comment: ISIT 201
Innovation Pursuit: A New Approach to Subspace Clustering
In subspace clustering, a group of data points belonging to a union of
subspaces are assigned membership to their respective subspaces. This paper
presents a new approach dubbed Innovation Pursuit (iPursuit) to the problem of
subspace clustering using a new geometrical idea whereby subspaces are
identified based on their relative novelties. We present two frameworks in
which the idea of innovation pursuit is used to distinguish the subspaces.
Underlying the first framework is an iterative method that finds the subspaces
consecutively by solving a series of simple linear optimization problems, each
searching for a direction of innovation in the span of the data potentially
orthogonal to all subspaces except for the one to be identified in one step of
the algorithm. A detailed mathematical analysis is provided establishing
sufficient conditions for iPursuit to correctly cluster the data. The proposed
approach can provably yield exact clustering even when the subspaces have
significant intersections. It is shown that the complexity of the iterative
approach scales only linearly in the number of data points and subspaces, and
quadratically in the dimension of the subspaces. The second framework
integrates iPursuit with spectral clustering to yield a new variant of
spectral-clustering-based algorithms. The numerical simulations with both real
and synthetic data demonstrate that iPursuit can often outperform the
state-of-the-art subspace clustering algorithms, more so for subspaces with
significant intersections, and that it significantly improves the
state-of-the-art result for subspace-segmentation-based face clustering
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