A divide-and-conquer algorithm for binary matrix completion

We propose an algorithm for low rank matrix completion for matrices with binary entries which obtains explicit binary factors. Our algorithm, which we call TBMC (\emph{Tiling for Binary Matrix Completion}), gives interpretable output in the form of binary factors which represent a decomposition of the matrix into tiles. Our approach is inspired by a popular algorithm from the data mining community called PROXIMUS: it adopts the same recursive partitioning approach while extending to missing data. The algorithm relies upon rank-one approximations of incomplete binary matrices, and we propose a linear programming (LP) approach for solving this subproblem. We also prove a $2$-approximation result for the LP approach which holds for any level of subsampling and for any subsampling pattern. Our numerical experiments show that TBMC outperforms existing methods on recommender systems arising in the context of real datasets.Comment: 14 pages,4 figure

Multispectral snapshot demosaicing via non-convex matrix completion

Snapshot mosaic multispectral imagery acquires an undersampled data cube by acquiring a single spectral measurement per spatial pixel. Sensors which acquire $p$ frequencies, therefore, suffer from severe $1/p$ undersampling of the full data cube. We show that the missing entries can be accurately imputed using non-convex techniques from sparse approximation and matrix completion initialised with traditional demosaicing algorithms. In particular, we observe the peak signal-to-noise ratio can typically be improved by 2 to 5 dB over current state-of-the-art methods when simulating a $p=16$ mosaic sensor measuring both high and low altitude urban and rural scenes as well as ground-based scenes.Comment: 5 pages, 2 figures, 1 tabl

Sparse and low rank approximations for action recognition

Action recognition is crucial area of research in computer vision with wide range of applications in surveillance, patient-monitoring systems, video indexing, Human- Computer Interaction and many more. These applications require automated action recognition. Robust classification methods are sought-after despite influential research in this field over past decade. The data resources have grown tremendously owing to the advances in the digital revolution which cannot be compared to the meagre resources in the past. The main limitation on a system when dealing with video data is the computational burden due to large dimensions and data redundancy. Sparse and low rank approximation methods have evolved recently which aim at concise and meaningful representation of data. This thesis explores the application of sparse and low rank approximation methods in the context of video data classification with the following contributions. 1. An approach for solving the problem of action and gesture classification is proposed within the sparse representation domain, effectively dealing with large feature dimensions, 2. Low rank matrix completion approach is proposed to jointly classify more than one action 3. Deep features are proposed for robust classification of multiple actions within matrix completion framework which can handle data deficiencies. This thesis starts with the applicability of sparse representations based classifi- cation methods to the problem of action and gesture recognition. Random projection is used to reduce the dimensionality of the features. These are referred to as compressed features in this thesis. The dictionary formed with compressed features has proved to be efficient for the classification task achieving comparable results to the state of the art. Next, this thesis addresses the more promising problem of simultaneous classifi- cation of multiple actions. This is treated as matrix completion problem under transduction setting. Matrix completion methods are considered as the generic extension to the sparse representation methods from compressed sensing point of view. The features and corresponding labels of the training and test data are concatenated and placed as columns of a matrix. The unknown test labels would be the missing entries in that matrix. This is solved using rank minimization techniques based on the assumption that the underlying complete matrix would be a low rank one. This approach has achieved results better than the state of the art on datasets with varying complexities. This thesis then extends the matrix completion framework for joint classification of actions to handle the missing features besides missing test labels. In this context, deep features from a convolutional neural network are proposed. A convolutional neural network is trained on the training data and features are extracted from train and test data from the trained network. The performance of the deep features has proved to be promising when compared to the state of the art hand-crafted features

On the Power of Adaptivity in Matrix Completion and Approximation

We consider the related tasks of matrix completion and matrix approximation from missing data and propose adaptive sampling procedures for both problems. We show that adaptive sampling allows one to eliminate standard incoherence assumptions on the matrix row space that are necessary for passive sampling procedures. For exact recovery of a low-rank matrix, our algorithm judiciously selects a few columns to observe in full and, with few additional measurements, projects the remaining columns onto their span. This algorithm exactly recovers an $n \times n$ rank $r$ matrix using $O(nr\mu_0 \log^2(r))$ observations, where $\mu_0$ is a coherence parameter on the column space of the matrix. In addition to completely eliminating any row space assumptions that have pervaded the literature, this algorithm enjoys a better sample complexity than any existing matrix completion algorithm. To certify that this improvement is due to adaptive sampling, we establish that row space coherence is necessary for passive sampling algorithms to achieve non-trivial sample complexity bounds. For constructing a low-rank approximation to a high-rank input matrix, we propose a simple algorithm that thresholds the singular values of a zero-filled version of the input matrix. The algorithm computes an approximation that is nearly as good as the best rank-$r$ approximation using $O(nr\mu \log^2(n))$ samples, where $\mu$ is a slightly different coherence parameter on the matrix columns. Again we eliminate assumptions on the row space

Low-Rank Matrix Approximation with Weights or Missing Data is NP-hard

Weighted low-rank approximation (WLRA), a dimensionality reduction technique for data analysis, has been successfully used in several applications, such as in collaborative filtering to design recommender systems or in computer vision to recover structure from motion. In this paper, we study the computational complexity of WLRA and prove that it is NP-hard to find an approximate solution, even when a rank-one approximation is sought. Our proofs are based on a reduction from the maximum-edge biclique problem, and apply to strictly positive weights as well as binary weights (the latter corresponding to low-rank matrix approximation with missing data).Comment: Proof of Lemma 4 (Lemma 3 in v1) has been corrected. Some remarks and comments have been added. Accepted in SIAM Journal on Matrix Analysis and Application