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Improved Support Recovery Guarantees for the Group Lasso With Applications to Structural Health Monitoring
May 19, 2018
This paper considers the problem of estimating an unknown high dimensional signal from noisy linear measurements, {when} the signal is assumed to possess a \emph{group-sparse} structure in a {known,} fixed dictionary. We consider signals generated according to a natural probabilistic model, and establish new conditions under which the set of indices of the non-zero groups of the signal (called the group-level support) may be accurately estimated via the group Lasso. Our results strengthen existing coherence-based analyses that exhibit the well-known "square root" bottleneck, allowing for the number of recoverable nonzero groups to be nearly as large as the total number of groups. We also establish a sufficient recovery condition relating the number of nonzero groups and the signal to noise ratio (quantified in terms of the ratio of the squared Euclidean norms of nonzero groups and the variance of the random additive {measurement} noise), and validate this trend empirically. Finally, we examine the implications of our results in the context of a structural health monitoring application, where the group Lasso approach facilitates demixing of a propagating acoustic wavefield, acquired on the material surface by a scanning laser Doppler vibrometer, into antithetical components, one of which indicates the locations of internal material defects.
Noisy Matrix Completion under Sparse Factor Models
Nov 02, 2014
This paper examines a general class of noisy matrix completion tasks where the goal is to estimate a matrix from observations obtained at a subset of its entries, each of which is subject to random noise or corruption. Our specific focus is on settings where the matrix to be estimated is well-approximated by a product of two (a priori unknown) matrices, one of which is sparse. Such structural models - referred to here as "sparse factor models" - have been widely used, for example, in subspace clustering applications, as well as in contemporary sparse modeling and dictionary learning tasks. Our main theoretical contributions are estimation error bounds for sparsity-regularized maximum likelihood estimators for problems of this form, which are applicable to a number of different observation noise or corruption models. Several specific implications are examined, including scenarios where observations are corrupted by additive Gaussian noise or additive heavier-tailed (Laplace) noise, Poisson-distributed observations, and highly-quantized (e.g., one-bit) observations. We also propose a simple algorithmic approach based on the alternating direction method of multipliers for these tasks, and provide experimental evidence to support our error analyses.
Anomaly-Sensitive Dictionary Learning for Unsupervised Diagnostics of Solid Media
May 11, 2014
This paper proposes a strategy for the detection and triangulation of structural anomalies in solid media. The method revolves around the construction of sparse representations of the medium's dynamic response, obtained by learning instructive dictionaries which form a suitable basis for the response data. The resulting sparse coding problem is recast as a modified dictionary learning task with additional spatial sparsity constraints enforced on the atoms of the learned dictionaries, which provides them with a prescribed spatial topology that is designed to unveil anomalous regions in the physical domain. The proposed methodology is model agnostic, i.e., it forsakes the need for a physical model and requires virtually no a priori knowledge of the structure's material properties, as all the inferences are exclusively informed by the data through the layers of information that are available in the intrinsic salient structure of the material's dynamic response. This characteristic makes the approach powerful for anomaly identification in systems with unknown or heterogeneous property distribution, for which a model is unsuitable or unreliable. The method is validated using both synthetically
Automated Defect Localization via Low Rank Plus Outlier Modeling of Propagating Wavefield Data
Jul 19, 2013
This work proposes an agnostic inference strategy for material diagnostics, conceived within the context of laser-based non-destructive evaluation methods, which extract information about structural anomalies from the analysis of acoustic wavefields measured on the structure's surface by means of a scanning laser interferometer. The proposed approach couples spatiotemporal windowing with low rank plus outlier modeling, to identify a priori unknown deviations in the propagating wavefields caused by material inhomogeneities or defects, using virtually no knowledge of the structural and material properties of the medium. This characteristic makes the approach particularly suitable for diagnostics scenarios where the mechanical and material models are complex, unknown, or unreliable. We demonstrate our approach in a simulated environment using benchmark point and line defect localization problems based on propagating flexural waves in a thin plate.
* 16 pages, 9 figures, Submitted to the IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control on August 30th 2012