198 research outputs found
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
Robust Low-Rank Subspace Segmentation with Semidefinite Guarantees
Recently there is a line of research work proposing to employ Spectral
Clustering (SC) to segment (group){Throughout the paper, we use segmentation,
clustering, and grouping, and their verb forms, interchangeably.}
high-dimensional structural data such as those (approximately) lying on
subspaces {We follow {liu2010robust} and use the term "subspace" to denote both
linear subspaces and affine subspaces. There is a trivial conversion between
linear subspaces and affine subspaces as mentioned therein.} or low-dimensional
manifolds. By learning the affinity matrix in the form of sparse
reconstruction, techniques proposed in this vein often considerably boost the
performance in subspace settings where traditional SC can fail. Despite the
success, there are fundamental problems that have been left unsolved: the
spectrum property of the learned affinity matrix cannot be gauged in advance,
and there is often one ugly symmetrization step that post-processes the
affinity for SC input. Hence we advocate to enforce the symmetric positive
semidefinite constraint explicitly during learning (Low-Rank Representation
with Positive SemiDefinite constraint, or LRR-PSD), and show that factually it
can be solved in an exquisite scheme efficiently instead of general-purpose SDP
solvers that usually scale up poorly. We provide rigorous mathematical
derivations to show that, in its canonical form, LRR-PSD is equivalent to the
recently proposed Low-Rank Representation (LRR) scheme {liu2010robust}, and
hence offer theoretic and practical insights to both LRR-PSD and LRR, inviting
future research. As per the computational cost, our proposal is at most
comparable to that of LRR, if not less. We validate our theoretic analysis and
optimization scheme by experiments on both synthetic and real data sets.Comment: 10 pages, 4 figures. Accepted by ICDM Workshop on Optimization Based
Methods for Emerging Data Mining Problems (OEDM), 2010. Main proof simplified
and typos corrected. Experimental data slightly adde
Improved Multistage Learning for Multibody Motion Segmentation
We present an improved version of the MSL method of Sugaya and Kanatani for multibody motion segmentation. We replace their initial segmentation based on heuristic clustering by an analytical computation based on GPCA, fitting two 2-D affine spaces in 3-D by the Taubin method. This initial segmentation alone can segment most of the motions in natural scenes fairly correctly, and the result is successively optimized by the EM algorithm in 3-D, 5-D, and 7-D. Using simulated and real videos, we demonstrate that our method outperforms the previous MSL and other existing methods. We also illustrate its mechanism by our visualization technique
Automatic Camera Model Selection for Multibody Motion Segmentation
We study the problem of segmenting independently moving objects in a video sequence. Several algorithms exist for classifying the trajectories of the feature points into independent motions, but the performance depends on the validity of the underlying camera imaging model. In this paper, we present a scheme for automatically selecting the best model using the geometric AIC before the segmentation stage, Using real video sequences,
we confirm that the segmentation accuracy indeed improves if the segmentation is based on the selected model. We also show that the trajectory data can be compressed into low-dimensional vectors using the selected model. This is very effective in reducing the computation time for a long video sequence
Distributed Low-rank Subspace Segmentation
Vision problems ranging from image clustering to motion segmentation to
semi-supervised learning can naturally be framed as subspace segmentation
problems, in which one aims to recover multiple low-dimensional subspaces from
noisy and corrupted input data. Low-Rank Representation (LRR), a convex
formulation of the subspace segmentation problem, is provably and empirically
accurate on small problems but does not scale to the massive sizes of modern
vision datasets. Moreover, past work aimed at scaling up low-rank matrix
factorization is not applicable to LRR given its non-decomposable constraints.
In this work, we propose a novel divide-and-conquer algorithm for large-scale
subspace segmentation that can cope with LRR's non-decomposable constraints and
maintains LRR's strong recovery guarantees. This has immediate implications for
the scalability of subspace segmentation, which we demonstrate on a benchmark
face recognition dataset and in simulations. We then introduce novel
applications of LRR-based subspace segmentation to large-scale semi-supervised
learning for multimedia event detection, concept detection, and image tagging.
In each case, we obtain state-of-the-art results and order-of-magnitude speed
ups
Low-Rank Clustering via LP1-PCA
In recent years, subspace clustering has found many practical use cases which include, for example, image segmentation, motion segmentation, and facial clustering. The image and video data that is common to these types of applications often has high dimensionality. Rather than viewing high dimensionality as a drawback, we propose a novel algorithm for subspace clustering that takes advantage of the high dimensional nature of such data. We call this algorithm LP1-PCA Spectral Clustering. Specifically, we introduce a concept that we call cluster-ID sparsity, and we propose an algorithm called LP1-PCA to attain this in low data dimensions. Our novel LP1-PCA algorithm is simple to implement and typically converges after only a few iterations. Conditions for which our algorithm performs well are discussed both theoretically and empirically, and we show that our method often attains superior clustering performance when compared to other common clustering algorithms on synthetic and real world datasets
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
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