1,643 research outputs found
Spectral clustering of linear subspaces for motion segmentation
International audienceThis paper studies automatic segmentation of multiple motions from tracked feature points through spectral embedding and clustering of linear subspaces. We show that the dimension of the ambient space is crucial for separability, and that low dimensions chosen in prior work are not optimal. We suggest lower and upper bounds together with a data-driven procedure for choosing the optimal ambient dimension. Application of our approach to the Hopkins155 video benchmark database uniformly outperforms a range of state-of-the-art methods both in terms of segmentation accuracy and computational speed
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
Sparse Subspace Clustering: Algorithm, Theory, and Applications
In many real-world problems, we are dealing with collections of
high-dimensional data, such as images, videos, text and web documents, DNA
microarray data, and more. Often, high-dimensional data lie close to
low-dimensional structures corresponding to several classes or categories the
data belongs to. In this paper, we propose and study an algorithm, called
Sparse Subspace Clustering (SSC), to cluster data points that lie in a union of
low-dimensional subspaces. The key idea is that, among infinitely many possible
representations of a data point in terms of other points, a sparse
representation corresponds to selecting a few points from the same subspace.
This motivates solving a sparse optimization program whose solution is used in
a spectral clustering framework to infer the clustering of data into subspaces.
Since solving the sparse optimization program is in general NP-hard, we
consider a convex relaxation and show that, under appropriate conditions on the
arrangement of subspaces and the distribution of data, the proposed
minimization program succeeds in recovering the desired sparse representations.
The proposed algorithm can be solved efficiently and can handle data points
near the intersections of subspaces. Another key advantage of the proposed
algorithm with respect to the state of the art is that it can deal with data
nuisances, such as noise, sparse outlying entries, and missing entries,
directly by incorporating the model of the data into the sparse optimization
program. We demonstrate the effectiveness of the proposed algorithm through
experiments on synthetic data as well as the two real-world problems of motion
segmentation and face clustering
Probabilistic Sparse Subspace Clustering Using Delayed Association
Discovering and clustering subspaces in high-dimensional data is a
fundamental problem of machine learning with a wide range of applications in
data mining, computer vision, and pattern recognition. Earlier methods divided
the problem into two separate stages of finding the similarity matrix and
finding clusters. Similar to some recent works, we integrate these two steps
using a joint optimization approach. We make the following contributions: (i)
we estimate the reliability of the cluster assignment for each point before
assigning a point to a subspace. We group the data points into two groups of
"certain" and "uncertain", with the assignment of latter group delayed until
their subspace association certainty improves. (ii) We demonstrate that delayed
association is better suited for clustering subspaces that have ambiguities,
i.e. when subspaces intersect or data are contaminated with outliers/noise.
(iii) We demonstrate experimentally that such delayed probabilistic association
leads to a more accurate self-representation and final clusters. The proposed
method has higher accuracy both for points that exclusively lie in one
subspace, and those that are on the intersection of subspaces. (iv) We show
that delayed association leads to huge reduction of computational cost, since
it allows for incremental spectral clustering
Robust Recovery of Subspace Structures by Low-Rank Representation
In this work we address the subspace recovery problem. Given a set of data
samples (vectors) approximately drawn from a union of multiple subspaces, our
goal is to segment the samples into their respective subspaces and correct the
possible errors as well. To this end, we propose a novel method termed Low-Rank
Representation (LRR), which seeks the lowest-rank representation among all the
candidates that can represent the data samples as linear combinations of the
bases in a given dictionary. It is shown that LRR well solves the subspace
recovery problem: when the data is clean, we prove that LRR exactly captures
the true subspace structures; for the data contaminated by outliers, we prove
that under certain conditions LRR can exactly recover the row space of the
original data and detect the outlier as well; for the data corrupted by
arbitrary errors, LRR can also approximately recover the row space with
theoretical guarantees. Since the subspace membership is provably determined by
the row space, these further imply that LRR can perform robust subspace
segmentation and error correction, in an efficient way.Comment: IEEE Trans. Pattern Analysis and Machine Intelligenc
CUR Decompositions, Similarity Matrices, and Subspace Clustering
A general framework for solving the subspace clustering problem using the CUR
decomposition is presented. The CUR decomposition provides a natural way to
construct similarity matrices for data that come from a union of unknown
subspaces . The similarity
matrices thus constructed give the exact clustering in the noise-free case.
Additionally, this decomposition gives rise to many distinct similarity
matrices from a given set of data, which allow enough flexibility to perform
accurate clustering of noisy data. We also show that two known methods for
subspace clustering can be derived from the CUR decomposition. An algorithm
based on the theoretical construction of similarity matrices is presented, and
experiments on synthetic and real data are presented to test the method.
Additionally, an adaptation of our CUR based similarity matrices is utilized
to provide a heuristic algorithm for subspace clustering; this algorithm yields
the best overall performance to date for clustering the Hopkins155 motion
segmentation dataset.Comment: Approximately 30 pages. Current version contains improved algorithm
and numerical experiments from the previous versio
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
Shape Interaction Matrix Revisited and Robustified: Efficient Subspace Clustering with Corrupted and Incomplete Data
The Shape Interaction Matrix (SIM) is one of the earliest approaches to
performing subspace clustering (i.e., separating points drawn from a union of
subspaces). In this paper, we revisit the SIM and reveal its connections to
several recent subspace clustering methods. Our analysis lets us derive a
simple, yet effective algorithm to robustify the SIM and make it applicable to
realistic scenarios where the data is corrupted by noise. We justify our method
by intuitive examples and the matrix perturbation theory. We then show how this
approach can be extended to handle missing data, thus yielding an efficient and
general subspace clustering algorithm. We demonstrate the benefits of our
approach over state-of-the-art subspace clustering methods on several
challenging motion segmentation and face clustering problems, where the data
includes corrupted and missing measurements.Comment: This is an extended version of our iccv15 pape
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