Simultaneously Sparse Solutions to Linear Inverse Problems with Multiple System Matrices and a Single Observation Vector

A linear inverse problem is proposed that requires the determination of multiple unknown signal vectors. Each unknown vector passes through a different system matrix and the results are added to yield a single observation vector. Given the matrices and lone observation, the objective is to find a simultaneously sparse set of unknown vectors that solves the system. We will refer to this as the multiple-system single-output (MSSO) simultaneous sparsity problem. This manuscript contrasts the MSSO problem with other simultaneous sparsity problems and conducts a thorough initial exploration of algorithms with which to solve it. Seven algorithms are formulated that approximately solve this NP-Hard problem. Three greedy techniques are developed (matching pursuit, orthogonal matching pursuit, and least squares matching pursuit) along with four methods based on a convex relaxation (iteratively reweighted least squares, two forms of iterative shrinkage, and formulation as a second-order cone program). The algorithms are evaluated across three experiments: the first and second involve sparsity profile recovery in noiseless and noisy scenarios, respectively, while the third deals with magnetic resonance imaging radio-frequency excitation pulse design.Comment: 36 pages; manuscript unchanged from July 21, 2008, except for updated references; content appears in September 2008 PhD thesi

Incremental Training of a Detector Using Online Sparse Eigen-decomposition

The ability to efficiently and accurately detect objects plays a very crucial role for many computer vision tasks. Recently, offline object detectors have shown a tremendous success. However, one major drawback of offline techniques is that a complete set of training data has to be collected beforehand. In addition, once learned, an offline detector can not make use of newly arriving data. To alleviate these drawbacks, online learning has been adopted with the following objectives: (1) the technique should be computationally and storage efficient; (2) the updated classifier must maintain its high classification accuracy. In this paper, we propose an effective and efficient framework for learning an adaptive online greedy sparse linear discriminant analysis (GSLDA) model. Unlike many existing online boosting detectors, which usually apply exponential or logistic loss, our online algorithm makes use of LDA's learning criterion that not only aims to maximize the class-separation criterion but also incorporates the asymmetrical property of training data distributions. We provide a better alternative for online boosting algorithms in the context of training a visual object detector. We demonstrate the robustness and efficiency of our methods on handwriting digit and face data sets. Our results confirm that object detection tasks benefit significantly when trained in an online manner.Comment: 14 page

Pattern Matching for sets of segments

In this paper we present algorithms for a number of problems in geometric pattern matching where the input consist of a collections of segments in the plane. Our work consists of two main parts. In the first, we address problems and measures that relate to collections of orthogonal line segments in the plane. Such collections arise naturally from problems in mapping buildings and robot exploration. We propose a new measure of segment similarity called a \emph{coverage measure}, and present efficient algorithms for maximising this measure between sets of axis-parallel segments under translations. Our algorithms run in time O(n^3\polylog n) in the general case, and run in time O(n^2\polylog n) for the case when all segments are horizontal. In addition, we show that when restricted to translations that are only vertical, the Hausdorff distance between two sets of horizontal segments can be computed in time roughly O(n^{3/2}{\sl polylog}n). These algorithms form significant improvements over the general algorithm of Chew et al. that takes time $O(n^4 \log^2 n)$. In the second part of this paper we address the problem of matching polygonal chains. We study the well known \Frd, and present the first algorithm for computing the \Frd under general translations. Our methods also yield algorithms for computing a generalization of the \Fr distance, and we also present a simple approximation algorithm for the \Frd that runs in time O(n^2\polylog n).Comment: To appear in the 12 ACM Symposium on Discrete Algorithms, Jan 200