G. Madhuri

G. Madhuri, Atul Negi, Kaluri V. Rangarao

Dimensionality reduction-based dictionary learning methods in the literature have often used iterative random projections. The dimensionality of such a random projection matrix is a random number that might not lead to a separable subspace structure in the transformed space. The convergence of such methods highly depends on the initial seed values used. Also, gradient descent-based updates might result in local minima. This paper proposes a constructive approach to derandomize the projection matrix using the Johnson-Lindenstrauss lemma. Rather than reducing dimensionality via random projections, a projection matrix derived from the proposed Modified Supervised PC analysis is used. A heuristic is proposed to decide the data perturbation levels and the dictionary atom's corresponding suitable description length. The projection matrix is derived in a single step, provides maximum feature-label consistency of the transformed space, and preserves the geometry of the original data. The projection matrix thus constructed is proved to be a JL-embedding. Despite confusing classes in the OCR datasets, the dictionary trained in the transformed space generates discriminative sparse coefficients with reduced complexity. Empirical study demonstrates that the proposed method performs well even when the number of classes and dimensionality increase. Experimentation on OCR and face recognition datasets shows better classification performance than other algorithms.

G. Madhuri, Atul Negi

Kernel-based nonlinear dictionary learning methods operate in a feature space obtained by an implicit feature map, and they are not independent of computationally expensive operations like Singular Value Decomposition (SVD). This paper presents an SVD-free lightweight approach to learning a nonlinear dictionary using a randomized functional link called a Random Vector Functional Link (RVFL). The proposed RVFL-based nonlinear Dictionary Learning (RVFLDL) learns a dictionary as a sparse-to-dense feature map from nonlinear sparse coefficients to the dense input features. Sparse coefficients w.r.t an initial random dictionary are derived by assuming Horseshoe prior are used as inputs making it a lightweight network. Training the RVFL-based dictionary is free from SVD computation as RVFL generates weights from the input to the output layer analytically. Higher-order dependencies between the input sparse coefficients and the dictionary atoms are incorporated into the training process by nonlinearly transforming the sparse coefficients and adding them as enhanced features. Thus the method projects sparse coefficients to a higher dimensional space while inducing nonlinearities into the dictionary. For classification using RVFL-net, a classifier matrix is learned as a transform that maps nonlinear sparse coefficients to the labels. The empirical evidence of the method illustrated in image classification and reconstruction applications shows that RVFLDL is scalable and provides a solution better than those obtained using other nonlinear dictionary learning methods.

G. Madhuri, Atul Negi

Sparse Representation (SR) of signals or data has a well founded theory with rigorous mathematical error bounds and proofs. SR of a signal is given by superposition of very few columns of a matrix called Dictionary, implicitly reducing dimensionality. Training dictionaries such that they represent each class of signals with minimal loss is called Dictionary Learning (DL). Dictionary learning methods like Method of Optimal Directions (MOD) and K-SVD have been successfully used in reconstruction based applications in image processing like image "denoising", "inpainting" and others. Other dictionary learning algorithms such as Discriminative K-SVD and Label Consistent K-SVD are supervised learning methods based on K-SVD. In our experience, one of the drawbacks of current methods is that the classification performance is not impressive on datasets like Telugu OCR datasets, with large number of classes and high dimensionality. There is scope for improvement in this direction and many researchers have used statistical methods to design dictionaries for classification. This chapter presents a review of statistical techniques and their application to learning discriminative dictionaries. The objective of the methods described here is to improve classification using sparse representation. In this chapter a hybrid approach is described, where sparse coefficients of input data are generated. We use a simple three layer Multi Layer Perceptron with back-propagation training as a classifier with those sparse codes as input. The results are quite comparable with other computation intensive methods. Keywords: Statistical modeling, Dictionary Learning, Discriminative Dictionary, Sparse representation, Gaussian prior, Cauchy prior, Entropy, Hidden Markov model, Hybrid Dictionary Learning