2,295 research outputs found
A new algorithm for learning overcomplete dictionaries
International audienceIn this paper, we propose a new algorithm for learning overcomplete dictionaries. The proposed algorithm is actually a new approach for optimizing a recently proposed cost function for dictionary learning. This cost function is regularized with a term that encourages low similarity between different atoms. While the previous approach needs to run an iterative limited-memory BFGS (l-BFGS) algorithm at each iteration of another iterative algorithm, our approach uses a closedform formula. Experimental results on reconstruction of a true underlying dictionary and designing a sparsifying dictionary for a class of autoregressive signals show that our approach results in both better quality and lower computational load
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
Entropy of Overcomplete Kernel Dictionaries
In signal analysis and synthesis, linear approximation theory considers a
linear decomposition of any given signal in a set of atoms, collected into a
so-called dictionary. Relevant sparse representations are obtained by relaxing
the orthogonality condition of the atoms, yielding overcomplete dictionaries
with an extended number of atoms. More generally than the linear decomposition,
overcomplete kernel dictionaries provide an elegant nonlinear extension by
defining the atoms through a mapping kernel function (e.g., the gaussian
kernel). Models based on such kernel dictionaries are used in neural networks,
gaussian processes and online learning with kernels.
The quality of an overcomplete dictionary is evaluated with a diversity
measure the distance, the approximation, the coherence and the Babel measures.
In this paper, we develop a framework to examine overcomplete kernel
dictionaries with the entropy from information theory. Indeed, a higher value
of the entropy is associated to a further uniform spread of the atoms over the
space. For each of the aforementioned diversity measures, we derive lower
bounds on the entropy. Several definitions of the entropy are examined, with an
extensive analysis in both the input space and the mapped feature space.Comment: 10 page
New Guarantees for Blind Compressed Sensing
Blind Compressed Sensing (BCS) is an extension of Compressed Sensing (CS)
where the optimal sparsifying dictionary is assumed to be unknown and subject
to estimation (in addition to the CS sparse coefficients). Since the emergence
of BCS, dictionary learning, a.k.a. sparse coding, has been studied as a matrix
factorization problem where its sample complexity, uniqueness and
identifiability have been addressed thoroughly. However, in spite of the strong
connections between BCS and sparse coding, recent results from the sparse
coding problem area have not been exploited within the context of BCS. In
particular, prior BCS efforts have focused on learning constrained and complete
dictionaries that limit the scope and utility of these efforts. In this paper,
we develop new theoretical bounds for perfect recovery for the general
unconstrained BCS problem. These unconstrained BCS bounds cover the case of
overcomplete dictionaries, and hence, they go well beyond the existing BCS
theory. Our perfect recovery results integrate the combinatorial theories of
sparse coding with some of the recent results from low-rank matrix recovery. In
particular, we propose an efficient CS measurement scheme that results in
practical recovery bounds for BCS. Moreover, we discuss the performance of BCS
under polynomial-time sparse coding algorithms.Comment: To appear in the 53rd Annual Allerton Conference on Communication,
Control and Computing, University of Illinois at Urbana-Champaign, IL, USA,
201
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