Friday, December 11, 2015

Perfect Recovery Conditions For Non-Negative Sparse Modeling / Compressive hyperspectral imaging via adaptive sampling and dictionary learning


Perfect Recovery Conditions For Non-Negative Sparse Modeling by Yuki Itoh, Marco F. Duarte, Mario Parente

Sparse modeling has been widely and successfully used in many applications such as computer vision, machine learning, and pattern recognition and, accompanied with those applications, significant research has studied the theoretical limits and algorithm design for convex relaxations in sparse modeling. However, only little has been done for theoretical limits of non-negative versions of sparse modeling. The behavior is expected to be similar as the general sparse modeling, but a precise analysis has not been explored. This paper studies the performance of non-negative sparse modeling, especially for non-negativity constrained and 1-penalized least squares, and gives an exact bound for which this problem can recover the correct signal elements. We pose two conditions to guarantee the correct signal recovery: minimum coefficient condition (MCC) and non-linearity vs. subset coherence condition (NSCC). The former defines the minimum weight for each of the correct atoms present in the signal and the latter defines the tolerable deviation from the linear model relative to the positive subset coherence (PSC), a novel type of "coherence" metric. We provide rigorous performance guarantees based on these conditions and experimentally verify their precise predictive power in a hyperspectral data unmixing application.

In this paper, we propose a new sampling strategy for hyperspectral signals that is based on dictionary learning and singular value decomposition (SVD). Specifically, we first learn a sparsifying dictionary from training spectral data using dictionary learning. We then perform an SVD on the dictionary and use the first few left singular vectors as the rows of the measurement matrix to obtain the compressive measurements for reconstruction. The proposed method provides significant improvement over the conventional compressive sensing approaches. The reconstruction performance is further improved by reconditioning the sensing matrix using matrix balancing. We also demonstrate that the combination of dictionary learning and SVD is robust by applying them to different datasets.
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