Analysis of Sparse Representations Using Bi-Orthogonal Dictionaries

The sparse representation problem of recovering an N dimensional sparse vector x from M < N linear observations y = Dx given dictionary D is considered. The standard approach is to let the elements of the dictionary be independent and identically distributed (IID) zero-mean Gaussian and minimize the l1-norm of x under the constraint y = Dx. In this paper, the performance of l1-reconstruction is analyzed, when the dictionary is bi-orthogonal D = [O1 O2], where O1,O2 are independent and drawn uniformly according to the Haar measure on the group of orthogonal M x M matrices. By an application of the replica method, we obtain the critical conditions under which perfect l1-recovery is possible with bi-orthogonal dictionaries.Comment: 5 pages, 2 figures. The main result and numerical examples have been revise

On Sparse Vector Recovery Performance in Structurally Orthogonal Matrices via LASSO

In this paper, we consider the compressed sensing problem of reconstructing a sparse signal from an undersampled set of noisy linear measurements. The regularized least squares or least absolute shrinkage and selection operator (LASSO) formulation is used for signal estimation. The measurement matrix is assumed to be constructed by concatenating several randomly orthogonal bases, which we refer to as structurally orthogonal matrices. Such measurement matrix is highly relevant to large-scale compressive sensing applications because it facilitates rapid computation and parallel processing. Using the replica method in statistical physics, we derive the mean-squared-error (MSE) formula of reconstruction over the structurally orthogonal matrix in the large-system regime. Extensive numerical experiments are provided to verify the analytical result. We then consider the analytical result to investigate the MSE behaviors of the LASSO over the structurally orthogonal matrix, with an emphasis on performance comparisons with matrices with independent and identically distributed (i.i.d.) Gaussian entries. We find that structurally orthogonal matrices are at least as good as their i.i.d. Gaussian counterparts. Thus, the use of structurally orthogonal matrices is attractive in practical applications

Cognitive Robots for Social Interactions

One of my goals is to work towards developing Cognitive Robots, especially with regard to improving the functionalities that facilitate the interaction with human beings and their surrounding objects. Any cognitive system designated for serving human beings must be capable of processing the social signals and eventually enable efficient prediction and planning of appropriate responses. My main focus during my PhD study is to bridge the gap between the motoric space and the visual space. The discovery of the mirror neurons ([RC04]) shows that the visual perception of human motion (visual space) is directly associated to the motor control of the human body (motor space). This discovery poses a large number of challenges in different fields such as computer vision, robotics and neuroscience. One of the fundamental challenges is the understanding of the mapping between 2D visual space and 3D motoric control, and further developing building blocks (primitives) of human motion in the visual space as well as in the motor space. First, I present my study on the visual-motoric mapping of human actions. This study aims at mapping human actions in 2D videos to 3D skeletal representation. Second, I present an automatic algorithm to decompose motion capture (MoCap) sequences into synergies along with the times at which they are executed (or "activated") for each joint. Third, I proposed to use the Granger Causality as a tool to study the coordinated actions performed by at least two units. Recent scientific studies suggest that the above "action mirroring circuit" might be tuned to action coordination rather than single action mirroring. Fourth, I present the extraction of key poses in visual space. These key poses facilitate the further study of the "action mirroring circuit". I conclude the dissertation by describing the future of cognitive robotics study

Advances in Spectral Learning with Applications to Text Analysis and Brain Imaging

Spectral learning algorithms are becoming increasingly popular in data-rich domains, driven in part by recent advances in large scale randomized SVD, and in spectral estimation of Hidden Markov Models. Extensions of these methods lead to statistical estimation algorithms which are not only fast, scalable, and useful on real data sets, but are also provably correct. Following this line of research, we make two contributions. First, we propose a set of spectral algorithms for text analysis and natural language processing. In particular, we propose fast and scalable spectral algorithms for learning word embeddings -- low dimensional real vectors (called Eigenwords) that capture the “meaning” of words from their context. Second, we show how similar spectral methods can be applied to analyzing brain images. State-of-the-art approaches to learning word embeddings are slow to train or lack theoretical grounding; We propose three spectral algorithms that overcome these limitations. All three algorithms harness the multi-view nature of text data i.e. the left and right context of each word, and share three characteristics: 1). They are fast to train and are scalable. 2). They have strong theoretical properties. 3). They can induce context-specific embeddings i.e. different embedding for “river bank” or “Bank of America”. \end{enumerate} They also have lower sample complexity and hence higher statistical power for rare words. We provide theory which establishes relationships between these algorithms and optimality criteria for the estimates they provide. We also perform thorough qualitative and quantitative evaluation of Eigenwords and demonstrate their superior performance over state-of-the-art approaches. Next, we turn to the task of using spectral learning methods for brain imaging data. Methods like Sparse Principal Component Analysis (SPCA), Non-negative Matrix Factorization (NMF) and Independent Component Analysis (ICA) have been used to obtain state-of-the-art accuracies in a variety of problems in machine learning. However, their usage in brain imaging, though increasing, is limited by the fact that they are used as out-of-the-box techniques and are seldom tailored to the domain specific constraints and knowledge pertaining to medical imaging, which leads to difficulties in interpretation of results. In order to address the above shortcomings, we propose Eigenanatomy (EANAT), a general framework for sparse matrix factorization. Its goal is to statistically learn the boundaries of and connections between brain regions by weighing both the data and prior neuroanatomical knowledge. Although EANAT incorporates some neuroanatomical prior knowledge in the form of connectedness and smoothness constraints, it can still be difficult for clinicians to interpret the results in specific domains where network-specific hypotheses exist. We thus extend EANAT and present a novel framework for prior-constrained sparse decomposition of matrices derived from brain imaging data, called Prior Based Eigenanatomy (p-Eigen). We formulate our solution in terms of a prior-constrained l1 penalized (sparse) principal component analysis. Experimental evaluation confirms that p-Eigen extracts biologically-relevant, patient-specific functional parcels and that it significantly aids classification of Mild Cognitive Impairment when compared to state-of-the-art competing approaches

Dictionaries for fast and informative dynamic MRI acquisition

Magnetic resonance (MR) imaging is an invaluable tool for medical research and diagnosis but suffers from inefficiencies. The speed of its acquisition mechanism, based on sequentially probing the interactions between nuclear atom spins and a changing magnetic field, is limited by atomic properties and scanner physics. Modern sampling techniques termed compressed sensing have nevertheless demonstrated how near perfect reconstructions are possible from undersampled, accelerated acquisitions, showing promise for more efficient MR acquisition paradigms. At the same time, information extraction from MR images through image analysis implies a considerable dimensionality reduction, in which an image is processed for the extraction of a few clinically useful parameters. This signals an inefficient handling of information in the separated treatment of acquisition and analysis that could be tackled by joining these two essential stages of the imaging pipeline. In this thesis, we explore the use of adaptive sparse modelling for novel acquisition strategies of cardiac cine MR data. Conventional compressed sensing MR acquisition relies on fixed basis transforms for sparse modelling, which are only able to guarantee suboptimal sparse modelling. We introduce spatio-temporal dictionaries that are able to optimally adapt sparse modelling by absorbing salient features of cardiac cine data, and demonstrate how they can outperform sampling methods based on fixed basis transforms. Additionally, we extend the framework introduced to handle parallel data acquisition. Given the flexibility of the formulation, we show how it can be combined with a labelling model that provides a segmentation of the image as a by-product of the reconstruction, hence performing joint reconstruction and analysis.Open Acces