13,870 research outputs found
Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)
The implicit objective of the biennial "international - Traveling Workshop on
Interactions between Sparse models and Technology" (iTWIST) is to foster
collaboration between international scientific teams by disseminating ideas
through both specific oral/poster presentations and free discussions. For its
second edition, the iTWIST workshop took place in the medieval and picturesque
town of Namur in Belgium, from Wednesday August 27th till Friday August 29th,
2014. The workshop was conveniently located in "The Arsenal" building within
walking distance of both hotels and town center. iTWIST'14 has gathered about
70 international participants and has featured 9 invited talks, 10 oral
presentations, and 14 posters on the following themes, all related to the
theory, application and generalization of the "sparsity paradigm":
Sparsity-driven data sensing and processing; Union of low dimensional
subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph
sensing/processing; Blind inverse problems and dictionary learning; Sparsity
and computational neuroscience; Information theory, geometry and randomness;
Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?;
Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website:
http://sites.google.com/site/itwist1
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
Structured random measurements in signal processing
Compressed sensing and its extensions have recently triggered interest in
randomized signal acquisition. A key finding is that random measurements
provide sparse signal reconstruction guarantees for efficient and stable
algorithms with a minimal number of samples. While this was first shown for
(unstructured) Gaussian random measurement matrices, applications require
certain structure of the measurements leading to structured random measurement
matrices. Near optimal recovery guarantees for such structured measurements
have been developed over the past years in a variety of contexts. This article
surveys the theory in three scenarios: compressed sensing (sparse recovery),
low rank matrix recovery, and phaseless estimation. The random measurement
matrices to be considered include random partial Fourier matrices, partial
random circulant matrices (subsampled convolutions), matrix completion, and
phase estimation from magnitudes of Fourier type measurements. The article
concludes with a brief discussion of the mathematical techniques for the
analysis of such structured random measurements.Comment: 22 pages, 2 figure
Partial Sum Minimization of Singular Values in Robust PCA: Algorithm and Applications
Robust Principal Component Analysis (RPCA) via rank minimization is a
powerful tool for recovering underlying low-rank structure of clean data
corrupted with sparse noise/outliers. In many low-level vision problems, not
only it is known that the underlying structure of clean data is low-rank, but
the exact rank of clean data is also known. Yet, when applying conventional
rank minimization for those problems, the objective function is formulated in a
way that does not fully utilize a priori target rank information about the
problems. This observation motivates us to investigate whether there is a
better alternative solution when using rank minimization. In this paper,
instead of minimizing the nuclear norm, we propose to minimize the partial sum
of singular values, which implicitly encourages the target rank constraint. Our
experimental analyses show that, when the number of samples is deficient, our
approach leads to a higher success rate than conventional rank minimization,
while the solutions obtained by the two approaches are almost identical when
the number of samples is more than sufficient. We apply our approach to various
low-level vision problems, e.g. high dynamic range imaging, motion edge
detection, photometric stereo, image alignment and recovery, and show that our
results outperform those obtained by the conventional nuclear norm rank
minimization method.Comment: Accepted in Transactions on Pattern Analysis and Machine Intelligence
(TPAMI). To appea
Non-convex Optimization for Machine Learning
A vast majority of machine learning algorithms train their models and perform
inference by solving optimization problems. In order to capture the learning
and prediction problems accurately, structural constraints such as sparsity or
low rank are frequently imposed or else the objective itself is designed to be
a non-convex function. This is especially true of algorithms that operate in
high-dimensional spaces or that train non-linear models such as tensor models
and deep networks.
The freedom to express the learning problem as a non-convex optimization
problem gives immense modeling power to the algorithm designer, but often such
problems are NP-hard to solve. A popular workaround to this has been to relax
non-convex problems to convex ones and use traditional methods to solve the
(convex) relaxed optimization problems. However this approach may be lossy and
nevertheless presents significant challenges for large scale optimization.
On the other hand, direct approaches to non-convex optimization have met with
resounding success in several domains and remain the methods of choice for the
practitioner, as they frequently outperform relaxation-based techniques -
popular heuristics include projected gradient descent and alternating
minimization. However, these are often poorly understood in terms of their
convergence and other properties.
This monograph presents a selection of recent advances that bridge a
long-standing gap in our understanding of these heuristics. The monograph will
lead the reader through several widely used non-convex optimization techniques,
as well as applications thereof. The goal of this monograph is to both,
introduce the rich literature in this area, as well as equip the reader with
the tools and techniques needed to analyze these simple procedures for
non-convex problems.Comment: The official publication is available from now publishers via
http://dx.doi.org/10.1561/220000005
Fast Methods for Recovering Sparse Parameters in Linear Low Rank Models
In this paper, we investigate the recovery of a sparse weight vector
(parameters vector) from a set of noisy linear combinations. However, only
partial information about the matrix representing the linear combinations is
available. Assuming a low-rank structure for the matrix, one natural solution
would be to first apply a matrix completion on the data, and then to solve the
resulting compressed sensing problem. In big data applications such as massive
MIMO and medical data, the matrix completion step imposes a huge computational
burden. Here, we propose to reduce the computational cost of the completion
task by ignoring the columns corresponding to zero elements in the sparse
vector. To this end, we employ a technique to initially approximate the support
of the sparse vector. We further propose to unify the partial matrix completion
and sparse vector recovery into an augmented four-step problem. Simulation
results reveal that the augmented approach achieves the best performance, while
both proposed methods outperform the natural two-step technique with
substantially less computational requirements
Compressive PCA for Low-Rank Matrices on Graphs
We introduce a novel framework for an approxi- mate recovery of data matrices
which are low-rank on graphs, from sampled measurements. The rows and columns
of such matrices belong to the span of the first few eigenvectors of the graphs
constructed between their rows and columns. We leverage this property to
recover the non-linear low-rank structures efficiently from sampled data
measurements, with a low cost (linear in n). First, a Resrtricted Isometry
Property (RIP) condition is introduced for efficient uniform sampling of the
rows and columns of such matrices based on the cumulative coherence of graph
eigenvectors. Secondly, a state-of-the-art fast low-rank recovery method is
suggested for the sampled data. Finally, several efficient, parallel and
parameter-free decoders are presented along with their theoretical analysis for
decoding the low-rank and cluster indicators for the full data matrix. Thus, we
overcome the computational limitations of the standard linear low-rank recovery
methods for big datasets. Our method can also be seen as a major step towards
efficient recovery of non- linear low-rank structures. For a matrix of size n X
p, on a single core machine, our method gains a speed up of over Robust
Principal Component Analysis (RPCA), where k << p is the subspace dimension.
Numerically, we can recover a low-rank matrix of size 10304 X 1000, 100 times
faster than Robust PCA
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