Sparse Coding on Symmetric Positive Definite Manifolds using Bregman Divergences

This paper introduces sparse coding and dictionary learning for Symmetric Positive Definite (SPD) matrices, which are often used in machine learning, computer vision and related areas. Unlike traditional sparse coding schemes that work in vector spaces, in this paper we discuss how SPD matrices can be described by sparse combination of dictionary atoms, where the atoms are also SPD matrices. We propose to seek sparse coding by embedding the space of SPD matrices into Hilbert spaces through two types of Bregman matrix divergences. This not only leads to an efficient way of performing sparse coding, but also an online and iterative scheme for dictionary learning. We apply the proposed methods to several computer vision tasks where images are represented by region covariance matrices. Our proposed algorithms outperform state-of-the-art methods on a wide range of classification tasks, including face recognition, action recognition, material classification and texture categorization

Building efficient deep Hebbian networks for image classification tasks

Multi-layer models of sparse coding (deep dictionary learning) and dimensionality reduction (PCANet) have shown promise as unsupervised learning models for image classification tasks. However, the pure implementations of these models have limited generalisation capabilities and high computational cost. This work introduces the Deep Hebbian Network (DHN), which combines the advantages of sparse coding, dimensionality reduction, and convolutional neural networks for learning features from images. Unlike in other deep neural networks, in this model, both the learning rules and neural architectures are derived from cost-function minimizations. Moreover, the DHN model can be trained online due to its Hebbian components. Different configurations of the DHN have been tested on scene and image classification tasks. Experiments show that the DHN model can automatically discover highly discriminative features directly from image pixels without using any data augmentation or semi-labeling

Learning-based Optimization for Signal and Image Processing

Incorporating machine learning techniques into optimization problems and solvers attracts increasing attention. Given a particular type of optimization problem that needs to be solved repeatedly, machine learning techniques can find some features for this category of optimization and develop algorithms with excellent performance. This thesis deals with algorithms and convergence analysis in learning-based optimization in three aspects: learning dictionaries, learning optimization solvers and learning regularizers.Learning dictionaries for sparse coding is significant for signal processing. Convolutional sparse coding is a form of sparse coding with a structured, translation invariant dictionary. Most convolutional dictionary learning algorithms to date operate in the batch mode, requiring simultaneous access to all training images during the learning process, which results in very high memory usage, and severely limits the training data size that can be used. I proposed two online convolutional dictionary learning algorithms that offered far better scaling of memory and computational cost than batch methods and provided a rigorous theoretical analysis of these methods.Learning fast solvers for optimization is a rising research topic. In recent years, unfolding iterative algorithms as neural networks has become an empirical success in solving sparse recovery problems. However, its theoretical understanding is still immature, which prevents us from fully utilizing the power of neural networks. I studied unfolded ISTA (Iterative Shrinkage Thresholding Algorithm) for sparse signal recovery and established its convergence. Based on the properties of parameters required by convergence, the model can be significantly simplified and, consequently, has much less training cost and better recovery performance.Learning regularizers or priors improves the performance of optimization solvers, especially for signal and image processing tasks. Plug-and-play (PnP) is a non-convex framework that integrates modern priors, such as BM3D or deep learning-based denoisers, into ADMM or other proximal algorithms. Although PnP has been recently studied extensively with great empirical success, theoretical analysis addressing even the most basic question of convergence has been insufficient. In this thesis, the theoretical convergence of PnP-FBS and PnP-ADMM was established, without using diminishing stepsizes, under a certain Lipschitz condition on the denoisers. Furthermore, real spectral normalization was proposed for training deep learning-based denoisers to satisfy the proposed Lipschitz condition

Transform learning based image and video processing

In recent years, sparse signal modeling, especially using the synthesis dictionary model, has received much attention. Sparse coding in the synthesis model is, however, NP-hard. Various methods have been proposed to learn such synthesis dictionaries from data. Numerous applications such as image denoising, magnetic resonance image (MRI), and computed tomography (CT) reconstruction have been shown to benefit from a good adaptive sparse model. Recently, the sparsifying transform model has received interest, for which sparse coding is cheap and exact, and learning, or data-driven adaptation admits computationally efficient solutions. In this thesis, we present two extensions to the transform learning framework, and some applications. In the first part of this thesis, we propose a union of sparsifying transforms model. Sparse coding in this model reduces to a form of clustering. The proposed model is also equivalent to a structured overcomplete sparsifying transform model with block cosparsity, dubbed OCTOBOS. The alternating algorithm introduced for learning such transforms involves simple closed-form solutions. Theoretical analysis provides a convergence guarantee for this algorithm. It is shown to be globally convergent to the set of partial minimizers of the non-convex learning problem. When applied to images, the algorithm learns a collection of well-conditioned square transforms, and a good clustering of patches or textures. The resulting sparse representations for the images are better than those obtained with a single learned transform, or with analytical transforms. We show the promising performance of the proposed approach in image denoising, which compares quite favorably with approaches involving a single learned square transform or an overcomplete synthesis dictionary, or Gaussian mixture models. The proposed denoising method is also faster than the synthesis dictionary based approach. Next, we develop a methodology for online learning of square sparsifying transforms. Such online learning can be particularly useful when dealing with big data, and for signal processing applications such as real-time sparse representation and denoising. The proposed transform learning algorithms are shown to have a significantly lower computational cost than online synthesis dictionary learning. In practice, the sequential learning of a sparsifying transform typically converges faster than batch mode transform learning. Preliminary experiments show the usefulness of the proposed schemes for sparse representation, and denoising. In the third part, we present a video denoising framework based on online 3D sparsifying transform learning. The proposed scheme has low computational and memory costs, and can handle streaming video. Our numerical experiments show promising performance for the proposed video denoising method compared to popular prior or state-of-the-art methods

Online Learning Discriminative Dictionary with Label Information for Robust Object Tracking

A supervised approach to online-learn a structured sparse and discriminative representation for object tracking is presented. Label information from training data is incorporated into the dictionary learning process to construct a robust and discriminative dictionary. This is accomplished by adding an ideal-code regularization term and classification error term to the total objective function. By minimizing the total objective function, we learn the high quality dictionary and optimal linear multiclassifier jointly using iterative reweighed least squares algorithm. Combined with robust sparse coding, the learned classifier is employed directly to separate the object from background. As the tracking continues, the proposed algorithm alternates between robust sparse coding and dictionary updating. Experimental evaluations on the challenging sequences show that the proposed algorithm performs favorably against state-of-the-art methods in terms of effectiveness, accuracy, and robustness

Orthogonal procrustes analysis for dictionary learning in sparse linear representation

In the sparse representation model, the design of overcomplete dictionaries plays a key role for the effectiveness and applicability in different domains. Recent research has produced several dictionary learning approaches, being proven that dictionaries learnt by data examples significantly outperform structured ones, e.g. wavelet transforms. In this context, learning consists in adapting the dictionary atoms to a set of training signals in order to promote a sparse representation that minimizes the reconstruction error. Finding the best fitting dictionary remains a very difficult task, leaving the question still open. A well-established heuristic method for tackling this problem is an iterative alternating scheme, adopted for instance in the well-known K-SVD algorithm. Essentially, it consists in repeating two stages; the former promotes sparse coding of the training set and the latter adapts the dictionary to reduce the error. In this paper we present R-SVD, a new method that, while maintaining the alternating scheme, adopts the Orthogonal Procrustes analysis to update the dictionary atoms suitably arranged into groups. Comparative experiments on synthetic data prove the effectiveness of R-SVD with respect to well known dictionary learning algorithms such as K-SVD, ILS-DLA and the online method OSDL. Moreover, experiments on natural data such as ECG compression, EEG sparse representation, and image modeling confirm R-SVD's robustness and wide applicability