2,751 research outputs found
Greed is good: algorithmic results for sparse approximation
This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho's basis pursuit (BP) paradigm can recover the optimal representation of an exactly sparse signal. It leverages this theory to show that both OMP and BP succeed for every sparse input signal from a wide class of dictionaries. These quasi-incoherent dictionaries offer a natural generalization of incoherent dictionaries, and the cumulative coherence function is introduced to quantify the level of incoherence. This analysis unifies all the recent results on BP and extends them to OMP. Furthermore, the paper develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal. From there, it argues that OMP is an approximation algorithm for the sparse problem over a quasi-incoherent dictionary. That is, for every input signal, OMP calculates a sparse approximant whose error is only a small factor worse than the minimal error that can be attained with the same number of terms
Parametric dictionary design for sparse coding
Abstract—This paper introduces a new dictionary design method for sparse coding of a class of signals. It has been shown that one can sparsely approximate some natural signals using an overcomplete set of parametric functions, e.g. [1], [2]. A problem in using these parametric dictionaries is how to choose the parameters. In practice these parameters have been chosen by an expert or through a set of experiments. In the sparse approximation context, it has been shown that an incoherent dictionary is appropriate for the sparse approximation methods. In this paper we first characterize the dictionary design problem, subject to a constraint on the dictionary. Then we briefly explain that equiangular tight frames have minimum coherence. The complexity of the problem does not allow it to be solved exactly. We introduce a practical method to approximately solve it. Some experiments show the advantages one gets by using these dictionaries
PADDLE: Proximal Algorithm for Dual Dictionaries LEarning
Recently, considerable research efforts have been devoted to the design of
methods to learn from data overcomplete dictionaries for sparse coding.
However, learned dictionaries require the solution of an optimization problem
for coding new data. In order to overcome this drawback, we propose an
algorithm aimed at learning both a dictionary and its dual: a linear mapping
directly performing the coding. By leveraging on proximal methods, our
algorithm jointly minimizes the reconstruction error of the dictionary and the
coding error of its dual; the sparsity of the representation is induced by an
-based penalty on its coefficients. The results obtained on synthetic
data and real images show that the algorithm is capable of recovering the
expected dictionaries. Furthermore, on a benchmark dataset, we show that the
image features obtained from the dual matrix yield state-of-the-art
classification performance while being much less computational intensive
Sparse Representation of Astronomical Images
Sparse representation of astronomical images is discussed. It is shown that a
significant gain in sparsity is achieved when particular mixed dictionaries are
used for approximating these types of images with greedy selection strategies.
Experiments are conducted to confirm: i)Effectiveness at producing sparse
representations. ii)Competitiveness, with respect to the time required to
process large images.The latter is a consequence of the suitability of the
proposed dictionaries for approximating images in partitions of small
blocks.This feature makes it possible to apply the effective greedy selection
technique Orthogonal Matching Pursuit, up to some block size. For blocks
exceeding that size a refinement of the original Matching Pursuit approach is
considered. The resulting method is termed Self Projected Matching Pursuit,
because is shown to be effective for implementing, via Matching Pursuit itself,
the optional back-projection intermediate steps in that approach.Comment: Software to implement the approach is available on
http://www.nonlinear-approx.info/examples/node1.htm
Exact reconstruction of gene regulatory networks using compressive sensing.
BackgroundWe consider the problem of reconstructing a gene regulatory network structure from limited time series gene expression data, without any a priori knowledge of connectivity. We assume that the network is sparse, meaning the connectivity among genes is much less than full connectivity. We develop a method for network reconstruction based on compressive sensing, which takes advantage of the network's sparseness.ResultsFor the case in which all genes are accessible for measurement, and there is no measurement noise, we show that our method can be used to exactly reconstruct the network. For the more general problem, in which hidden genes exist and all measurements are contaminated by noise, we show that our method leads to reliable reconstruction. In both cases, coherence of the model is used to assess the ability to reconstruct the network and to design new experiments. We demonstrate that it is possible to use the coherence distribution to guide biological experiment design effectively. By collecting a more informative dataset, the proposed method helps reduce the cost of experiments. For each problem, a set of numerical examples is presented.ConclusionsThe method provides a guarantee on how well the inferred graph structure represents the underlying system, reveals deficiencies in the data and model, and suggests experimental directions to remedy the deficiencies
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
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