Many natural signals exhibit a sparse representation, whenever a suitable describing model is given. Here, a linear generative model is considered, where many sparsity-based signal processing techniques rely on such a simplified model. As this model is often unknown for many classes of the signals, we need to select such a model based on the domain knowledge or using some exemplar signals. This paper presents a new exemplar based approach for the linear model (called the dictionary) selection, for such sparse inverse problems. The problem of dictionary selection, which has also been called the dictionary learning in this setting, is first reformulated as a joint sparsity model. The joint sparsity model here differs from the standard joint sparsity model as it considers an overcompleteness in the representation of each signal, within the range of selected subspaces. The new dictionary selection paradigm is examined with some synthetic and realistic simulations.
The problem of dictionary design by combining the atoms of a mother dictionary was considered in @cite_3 @cite_10 @cite_15 . In this setting, an auxiliary sparse matrix combines the mother atoms, to generate a dictionary which fits the given learning samples. The size of dictionary is fixed here and as the learned dictionary is the multiplication of a sparse matrix and a structured matrix (with a possibly fast multiplication), we can implement such a dictionary in two steps, where each of them are cheaper than @math . The dictionary selection problem can be interpreted as a particular case of sparse dictionary learning, when the sparse matrix can have only @math non-zero elements, with one non-zero on each row.
@cite_15 @cite_10 @cite_3
2086522948 2025948329 2088843485 2127114523
By solving a linear inverse problem under a sparsity constraint, one can successfully recover the coefficients, if there exists such a sparse approximation for the proposed class of signals. In this framework the dictionary can be adapted to a given set of signals using dictionary learning methods. The learned dictionary often does not have useful structures for a fast implementation, i.e. fast matrix-vector multiplication. This prevents such a dictionary being used for the real applications or large scale problems. The structure can be induced on the dictionary throughout the learning progress. Examples of such structures are shift-invariance and being multi-scale. These dictionaries can be efficiently implemented using a filter bank. In this paper a well-known structure, called compressibility, is adapted to be used in the dictionary learning problem. As a result, the complexity of the implementation of a compressible dictionary can be reduced by wisely choosing a generative model. By some simulations, it has been shown that the learned dictionary provides sparser approximations, while it does not increase the computational complexity of the algorithms, with respect to the pre-designed fast structured dictionaries. We consider the dictionary learning problem for the analysis model based sparse representation. A novel algorithm is proposed by adapting the synthesis model based simultaneous codeword optimisation (SimCO) algorithm to the analysis model. This algorithm assumes that the analysis dictionary contains unit Ł 2 -norm atoms and trains the dictionary by the optimisation on manifolds. This framework allows one to update multiple dictionary atoms in each iteration, leading to a computationally efficient optimisation process. We demonstrate the competitive performance of the proposed algorithm using experiments on both synthetic and real data, as compared with three baseline algorithms, Analysis K-SVD, analysis operator learning (AOL) and learning overcomplete sparsifying transforms (LOST), respectively. Dictionary learning (DL) for sparse coding has shown promising results in classification tasks, while how to adaptively build the relationship between dictionary atoms and class labels is still an important open question. The existing dictionary learning approaches simply fix a dictionary atom to be either class-specific or shared by all classes beforehand, ignoring that the relationship needs to be updated during DL. To address this issue, in this paper we propose a novel latent dictionary learning (LDL) method to learn a discriminative dictionary and build its relationship to class labels adaptively. Each dictionary atom is jointly learned with a latent vector, which associates this atom to the representation of different classes. More specifically, we introduce a latent representation model, in which discrimination of the learned dictionary is exploited via minimizing the within-class scatter of coding coefficients and the latent-value weighted dictionary coherence. The optimal solution is efficiently obtained by the proposed solving algorithm. Correspondingly, a latent sparse representation based classifier is also presented. Experimental results demonstrate that our algorithm outperforms many recently proposed sparse representation and dictionary learning approaches for action, gender and face recognition. This paper treats the problem of learning a dictionary providing sparse representations for a given signal class, via l1-minimization. The problem can also be seen as factorizing a d × N matrix Y = (y1 . . . yN), yn ∈ ℝd of training signals into a d × K dictionary matrix Φ and a K × N coefficient matrix X = (x1 . . . xN), xn ∈ ℝK, which is sparse. The exact question studied here is when a dictionary coefficient pair (Φ, X) can be recovered as local minimum of a (nonconvex) l1-criterion with input Y = Φ X. First, for general dictionaries and coefficient matrices, algebraic conditions ensuring local identifiability are derived, which are then specialized to the case when the dictionary is a basis. Finally, assuming a random Bernoulli-Gaussian sparse model on the coefficient matrix, it is shown that sufficiently incoherent bases are locally identifiable with high probability. The perhaps surprising result is that the typically sufficient number of training samples N grows up to a logarithmic factor only linearly with the signal dimension, i.e., N ≈ CK log K, in contrast to previous approaches requiring combinatorially many samples.