Sparse approximation

About: Sparse approximation is a research topic. Over the lifetime, 18037 publications have been published within this topic receiving 497739 citations. The topic is also known as: Sparse approximation.

Sparse model selection via integral terms.

Abstract: Model selection and parameter estimation are important for the effective integration of experimental data, scientific theory, and precise simulations. In this work, we develop a learning approach for the selection and identification of a dynamical system directly from noisy data. The learning is performed by extracting a small subset of important features from an overdetermined set of possible features using a nonconvex sparse regression model. The sparse regression model is constructed to fit the noisy data to the trajectory of the dynamical system while using the smallest number of active terms. Computational experiments detail the model's stability, robustness to noise, and recovery accuracy. Examples include nonlinear equations, population dynamics, chaotic systems, and fast-slow systems.

A single-letter characterization of optimal noisy compressed sensing

Abstract: Compressed sensing deals with the reconstruction of a high-dimensional signal from far fewer linear measurements, where the signal is known to admit a sparse representation in a certain linear space. The asymptotic scaling of the number of measurements needed for reconstruction as the dimension of the signal increases has been studied extensively. This work takes a fundamental perspective on the problem of inferring about individual elements of the sparse signal given the measurements, where the dimensions of the system become increasingly large. Using the replica method, the outcome of inferring about any fixed collection of signal elements is shown to be asymptotically decoupled, i.e., those elements become independent conditioned on the measurements. Furthermore, the problem of inferring about each signal element admits a single-letter characterization in the sense that the posterior distribution of the element, which is a sufficient statistic, becomes asymptotically identical to the posterior of inferring about the same element in scalar Gaussian noise. The result leads to simple characterization of all other elemental metrics of the compressed sensing problem, such as the mean squared error and the error probability for reconstructing the support set of the sparse signal. Finally, the single-letter characterization is rigorously justified in the special case of sparse measurement matrices where belief propagation becomes asymptotically optimal.

Robust principal component analysis-based four-dimensional computed tomography

Abstract: The purpose of this paper for four-dimensional (4D) computed tomography (CT) is threefold. (1) A new spatiotemporal model is presented from the matrix perspective with the row dimension in space and the column dimension in time, namely the robust PCA (principal component analysis)-based 4D CT model. That is, instead of viewing the 4D object as a temporal collection of three-dimensional (3D) images and looking for local coherence in time or space independently, we perceive it as a mixture of low-rank matrix and sparse matrix to explore the maximum temporal coherence of the spatial structure among phases. Here the low-rank matrix corresponds to the 'background' or reference state, which is stationary over time or similar in structure; the sparse matrix stands for the 'motion' or time-varying component, e.g., heart motion in cardiac imaging, which is often either approximately sparse itself or can be sparsified in the proper basis. Besides 4D CT, this robust PCA-based 4D CT model should be applicable in other imaging problems for motion reduction or/and change detection with the least amount of data, such as multi-energy CT, cardiac MRI, and hyperspectral imaging. (2) A dynamic strategy for data acquisition, i.e. a temporally spiral scheme, is proposed that can potentially maintain similar reconstruction accuracy with far fewer projections of the data. The key point of this dynamic scheme is to reduce the total number of measurements, and hence the radiation dose, by acquiring complementary data in different phases while reducing redundant measurements of the common background structure. (3) An accurate, efficient, yet simple-to-implement algorithm based on the split Bregman method is developed for solving the model problem with sparse representation in tight frames.

Accelerometer-Based Gait Recognition by Sparse Representation of Signature Points With Clusters

Abstract: Gait, as a promising biometric for recognizing human identities, can be nonintrusively captured as a series of acceleration signals using wearable or portable smart devices. It can be used for access control. Most existing methods on accelerometer-based gait recognition require explicit step-cycle detection, suffering from cycle detection failures and intercycle phase misalignment. We propose a novel algorithm that avoids both the above two problems. It makes use of a type of salient points termed signature points (SPs), and has three components: 1) a multiscale SP extraction method, including the localization and SP descriptors; 2) a sparse representation scheme for encoding newly emerged SPs with known ones in terms of their descriptors, where the phase propinquity of the SPs in a cluster is leveraged to ensure the physical meaningfulness of the codes; and 3) a classifier for the sparse-code collections associated with the SPs of a series. Experimental results on our publicly available dataset of 175 subjects showed that our algorithm outperformed existing methods, even if the step cycles were perfectly detected for them. When the accelerometers at five different body locations were used together, it achieved the rank-1 accuracy of 95.8% for identification, and the equal error rate of 2.2% for verification.