Abstract
Sparse coding aims to find a parsimonious representation of an example given an observation matrix or dictionary. In this regard, Orthogonal Matching Pursuit (OMP) provides an intuitive, simple and fast approximation of the optimal solution. However, its main building block is anchored on the minimization of the Mean Squared Error cost function (MSE). This approach is only optimal if the errors are distributed according to a Gaussian distribution without samples that strongly deviate from the main mode, i.e. outliers. If such assumption is violated, the sparse code will likely be biased and performance will degrade accordingly. In this paper, we introduce five robust variants of OMP (RobOMP) fully based on the theory of M-Estimators under a linear model. The proposed framework exploits efficient Iteratively Reweighted Least Squares (IRLS) techniques to mitigate the effect of outliers and emphasize the samples corresponding to the main mode of the data. This is done adaptively via a learned weight vector that models the distribution of the data in a robust manner. Experiments on synthetic data under several noise distributions and image recognition under different combinations of occlusion and missing pixels thoroughly detail the superiority of RobOMP over MSE-based approaches and similar robust alternatives. We also introduce a denoising framework based on robust, sparse and redundant representations that open the door to potential further applications of the proposed techniques. The five different variants of RobOMP do not require parameter tuning from the user and, hence, constitute principled alternatives to OMP.
Highlights
Sparse modeling is a learning framework with relevant applications in areas where parsimonious representations are considered advantageous, such as signal processing, machine learning, and computer vision
We propose a generalization of Correntropy Matching Pursuit (CMP) by reformulating the active set update under the lens of robust linear regression; we exploit the well known and developed theory of M–Estimators (Andersen, 2008; Huber, 2011) to devise five different robust variants of Orthogonal Matching Pursuit (OMP): RobOMP
The resulting observation vector y ∈ IR100 is computed as the linear combination of the dictionary with weights from the ideal sparse code plus a noise component n ∈ IR100: y = Dx0 + n
Summary
Sparse modeling is a learning framework with relevant applications in areas where parsimonious representations are considered advantageous, such as signal processing, machine learning, and computer vision. Sparse modeling refers to the entire process of designing and learning a model, while sparse coding, sparse representation, or sparse decomposition is. The latter formally emerged as a machine learning adaptation of the sparse coding scheme found in the mammalian primary visual cortex (Olshausen & Field, 1996)
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