10,973 research outputs found
Kernelized Low Rank Representation on Grassmann Manifolds
Low rank representation (LRR) has recently attracted great interest due to
its pleasing efficacy in exploring low-dimensional subspace structures embedded
in data. One of its successful applications is subspace clustering which means
data are clustered according to the subspaces they belong to. In this paper, at
a higher level, we intend to cluster subspaces into classes of subspaces. This
is naturally described as a clustering problem on Grassmann manifold. The
novelty of this paper is to generalize LRR on Euclidean space onto an LRR model
on Grassmann manifold in a uniform kernelized framework. The new methods have
many applications in computer vision tasks. Several clustering experiments are
conducted on handwritten digit images, dynamic textures, human face clips and
traffic scene sequences. The experimental results show that the proposed
methods outperform a number of state-of-the-art subspace clustering methods.Comment: 13 page
Compressive Principal Component Pursuit
We consider the problem of recovering a target matrix that is a superposition
of low-rank and sparse components, from a small set of linear measurements.
This problem arises in compressed sensing of structured high-dimensional
signals such as videos and hyperspectral images, as well as in the analysis of
transformation invariant low-rank recovery. We analyze the performance of the
natural convex heuristic for solving this problem, under the assumption that
measurements are chosen uniformly at random. We prove that this heuristic
exactly recovers low-rank and sparse terms, provided the number of observations
exceeds the number of intrinsic degrees of freedom of the component signals by
a polylogarithmic factor. Our analysis introduces several ideas that may be of
independent interest for the more general problem of compressed sensing and
decomposing superpositions of multiple structured signals.Comment: 30 pages, 1 figure, preliminary version submitted to ISIT'1
Kernelized LRR on Grassmann Manifolds for Subspace Clustering
Low rank representation (LRR) has recently attracted great interest due to
its pleasing efficacy in exploring low-dimensional sub- space structures
embedded in data. One of its successful applications is subspace clustering, by
which data are clustered according to the subspaces they belong to. In this
paper, at a higher level, we intend to cluster subspaces into classes of
subspaces. This is naturally described as a clustering problem on Grassmann
manifold. The novelty of this paper is to generalize LRR on Euclidean space
onto an LRR model on Grassmann manifold in a uniform kernelized LRR framework.
The new method has many applications in data analysis in computer vision tasks.
The proposed models have been evaluated on a number of practical data analysis
applications. The experimental results show that the proposed models outperform
a number of state-of-the-art subspace clustering methods
Visual Tracking via Dynamic Graph Learning
Existing visual tracking methods usually localize a target object with a
bounding box, in which the performance of the foreground object trackers or
detectors is often affected by the inclusion of background clutter. To handle
this problem, we learn a patch-based graph representation for visual tracking.
The tracked object is modeled by with a graph by taking a set of
non-overlapping image patches as nodes, in which the weight of each node
indicates how likely it belongs to the foreground and edges are weighted for
indicating the appearance compatibility of two neighboring nodes. This graph is
dynamically learned and applied in object tracking and model updating. During
the tracking process, the proposed algorithm performs three main steps in each
frame. First, the graph is initialized by assigning binary weights of some
image patches to indicate the object and background patches according to the
predicted bounding box. Second, the graph is optimized to refine the patch
weights by using a novel alternating direction method of multipliers. Third,
the object feature representation is updated by imposing the weights of patches
on the extracted image features. The object location is predicted by maximizing
the classification score in the structured support vector machine. Extensive
experiments show that the proposed tracking algorithm performs well against the
state-of-the-art methods on large-scale benchmark datasets.Comment: Submitted to TPAMI 201
Value function approximation via low-rank models
We propose a novel value function approximation technique for Markov decision
processes. We consider the problem of compactly representing the state-action
value function using a low-rank and sparse matrix model. The problem is to
decompose a matrix that encodes the true value function into low-rank and
sparse components, and we achieve this using Robust Principal Component
Analysis (PCA). Under minimal assumptions, this Robust PCA problem can be
solved exactly via the Principal Component Pursuit convex optimization problem.
We experiment the procedure on several examples and demonstrate that our method
yields approximations essentially identical to the true function.Comment: arXiv admin note: substantial text overlap with arXiv:0912.3599 by
other author
Alternating proximal gradient method for sparse nonnegative Tucker decomposition
Multi-way data arises in many applications such as electroencephalography
(EEG) classification, face recognition, text mining and hyperspectral data
analysis. Tensor decomposition has been commonly used to find the hidden
factors and elicit the intrinsic structures of the multi-way data. This paper
considers sparse nonnegative Tucker decomposition (NTD), which is to decompose
a given tensor into the product of a core tensor and several factor matrices
with sparsity and nonnegativity constraints. An alternating proximal gradient
method (APG) is applied to solve the problem. The algorithm is then modified to
sparse NTD with missing values. Per-iteration cost of the algorithm is
estimated scalable about the data size, and global convergence is established
under fairly loose conditions. Numerical experiments on both synthetic and real
world data demonstrate its superiority over a few state-of-the-art methods for
(sparse) NTD from partial and/or full observations. The MATLAB code along with
demos are accessible from the author's homepage
Decomposition into Low-rank plus Additive Matrices for Background/Foreground Separation: A Review for a Comparative Evaluation with a Large-Scale Dataset
Recent research on problem formulations based on decomposition into low-rank
plus sparse matrices shows a suitable framework to separate moving objects from
the background. The most representative problem formulation is the Robust
Principal Component Analysis (RPCA) solved via Principal Component Pursuit
(PCP) which decomposes a data matrix in a low-rank matrix and a sparse matrix.
However, similar robust implicit or explicit decompositions can be made in the
following problem formulations: Robust Non-negative Matrix Factorization
(RNMF), Robust Matrix Completion (RMC), Robust Subspace Recovery (RSR), Robust
Subspace Tracking (RST) and Robust Low-Rank Minimization (RLRM). The main goal
of these similar problem formulations is to obtain explicitly or implicitly a
decomposition into low-rank matrix plus additive matrices. In this context,
this work aims to initiate a rigorous and comprehensive review of the similar
problem formulations in robust subspace learning and tracking based on
decomposition into low-rank plus additive matrices for testing and ranking
existing algorithms for background/foreground separation. For this, we first
provide a preliminary review of the recent developments in the different
problem formulations which allows us to define a unified view that we called
Decomposition into Low-rank plus Additive Matrices (DLAM). Then, we examine
carefully each method in each robust subspace learning/tracking frameworks with
their decomposition, their loss functions, their optimization problem and their
solvers. Furthermore, we investigate if incremental algorithms and real-time
implementations can be achieved for background/foreground separation. Finally,
experimental results on a large-scale dataset called Background Models
Challenge (BMC 2012) show the comparative performance of 32 different robust
subspace learning/tracking methods.Comment: 121 pages, 5 figures, submitted to Computer Science Review. arXiv
admin note: text overlap with arXiv:1312.7167, arXiv:1109.6297,
arXiv:1207.3438, arXiv:1105.2126, arXiv:1404.7592, arXiv:1210.0805,
arXiv:1403.8067 by other authors, Computer Science Review, November 201
Sparse Model Uncertainties in Compressed Sensing with Application to Convolutions and Sporadic Communication
The success of the compressed sensing paradigm has shown that a substantial
reduction in sampling and storage complexity can be achieved in certain linear
and non-adaptive estimation problems. It is therefore an advisable strategy for
noncoherent information retrieval in, for example, sporadic blind and
semi-blind communication and sampling problems. But, the conventional model is
not practical here since the compressible signals have to be estimated from
samples taken solely on the output of an un-calibrated system which is unknown
during measurement but often compressible. Conventionally, one has either to
operate at suboptimal sampling rates or the recovery performance substantially
suffers from the dominance of model mismatch. In this work we discuss such type
of estimation problems and we focus on bilinear inverse problems. We link this
problem to the recovery of low-rank and sparse matrices and establish stable
low-dimensional embeddings of the uncalibrated receive signals whereby
addressing also efficient communication-oriented methods like universal random
demodulation. Exemplary, we investigate in more detail sparse convolutions
serving as a basic communication channel model. In using some recent results
from additive combinatorics we show that such type of signals can be
efficiently low-rate sampled by semi-blind methods. Finally, we present a
further application of these results in the field of phase retrieval from
intensity Fourier measurements.Comment: Book chapter, submitted to "Compressed Sensing and its Applications",
31 pages, revised versio
Partial Sum Minimization of Singular Values Representation on Grassmann Manifolds
As a significant subspace clustering method, low rank representation (LRR)
has attracted great attention in recent years. To further improve the
performance of LRR and extend its applications, there are several issues to be
resolved. The nuclear norm in LRR does not sufficiently use the prior knowledge
of the rank which is known in many practical problems. The LRR is designed for
vectorial data from linear spaces, thus not suitable for high dimensional data
with intrinsic non-linear manifold structure. This paper proposes an extended
LRR model for manifold-valued Grassmann data which incorporates prior knowledge
by minimizing partial sum of singular values instead of the nuclear norm,
namely Partial Sum minimization of Singular Values Representation (GPSSVR). The
new model not only enforces the global structure of data in low rank, but also
retains important information by minimizing only smaller singular values. To
further maintain the local structures among Grassmann points, we also integrate
the Laplacian penalty with GPSSVR. An effective algorithm is proposed to solve
the optimization problem based on the GPSSVR model. The proposed model and
algorithms are assessed on some widely used human action video datasets and a
real scenery dataset. The experimental results show that the proposed methods
obviously outperform other state-of-the-art methods.Comment: Submitting to ACM Transactions on Knowledge Discovery from Data with
minor revisio
Structured Low-Rank Matrix Factorization with Missing and Grossly Corrupted Observations
Recovering low-rank and sparse matrices from incomplete or corrupted
observations is an important problem in machine learning, statistics,
bioinformatics, computer vision, as well as signal and image processing. In
theory, this problem can be solved by the natural convex joint/mixed
relaxations (i.e., l_{1}-norm and trace norm) under certain conditions.
However, all current provable algorithms suffer from superlinear per-iteration
cost, which severely limits their applicability to large-scale problems. In
this paper, we propose a scalable, provable structured low-rank matrix
factorization method to recover low-rank and sparse matrices from missing and
grossly corrupted data, i.e., robust matrix completion (RMC) problems, or
incomplete and grossly corrupted measurements, i.e., compressive principal
component pursuit (CPCP) problems. Specifically, we first present two
small-scale matrix trace norm regularized bilinear structured factorization
models for RMC and CPCP problems, in which repetitively calculating SVD of a
large-scale matrix is replaced by updating two much smaller factor matrices.
Then, we apply the alternating direction method of multipliers (ADMM) to
efficiently solve the RMC problems. Finally, we provide the convergence
analysis of our algorithm, and extend it to address general CPCP problems.
Experimental results verified both the efficiency and effectiveness of our
method compared with the state-of-the-art methods.Comment: 28 pages, 9 figure
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