Corrections to “Compressive Sensing on Manifolds Using a Nonparametric Mixture of Factor Analyzers: Algorithm and Performance Bounds” [Dec 10 6140-6155]

Electr. & Comput. Eng. Dept., Duke Univ., Durham, NC, USA
IEEE Transactions on Signal Processing (Impact Factor: 2.79). 03/2011; 59(3):1329. DOI: 10.1109/TSP.2010.2070796
Source: DBLP

ABSTRACT Nonparametric Bayesian methods are employed to constitute a mixture of low-rank Gaussians, for data x ∈ RN that are of high dimension N but are constrained to reside in a low-dimensional subregion of RN. The number of mixture components and their rank are inferred automatically from the data. The resulting algorithm can be used for learning manifolds and for reconstructing signals from manifolds, based on compressive sensing (CS) projection measurements. The statistical CS inversion is performed analytically. We derive the required number of CS random measurements needed for successful reconstruction, based on easily-computed quantities, drawing on block-sparsity properties. The proposed methodology is validated on several synthetic and real datasets.

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Available from: David B Dunson, Sep 27, 2015
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    • "This framework can be generalized to other applications and algorithms. For instance, Gaussian mixture model based dictionary learning approaches [49]–[53] that learn a union of subspaces can also benefit from side information. "
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    ABSTRACT: A blind compressive sensing algorithm is proposed to reconstruct hyperspectral images from spectrally-compressed measurements. The wavelength-dependent data are coded and then superposed, mapping the three-dimensional hyperspectral datacube to a two-dimensional image. The inversion algorithm learns a dictionary in situ from the measurements via globallocal shrinkage priors. By using RGB images as side information of the compressive sensing system, the proposed approach is extended to learn a coupled dictionary from the joint dataset of the compressed measurements and the corresponding RGB images, to improve reconstruction quality. A prototype camera is built using a liquid-crystal-on-silicon modulator. Experimental reconstructions of hyperspectral datacubes from both simulated and real compressed measurements demonstrate the efficacy of the proposed inversion algorithm, the feasibility of the camera and the benefit of side information.
    IEEE Journal of Selected Topics in Signal Processing 09/2015; 9(6). DOI:10.1109/JSTSP.2015.2411575 · 2.37 Impact Factor
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    • "which is an analytical solution with [44] "
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    ABSTRACT: We develop a new compressive sensing (CS) inversion algorithm by utilizing the Gaussian mixture model (GMM). While the compressive sensing is performed globally on the entire image as implemented in our lensless camera, a low-rank GMM is imposed on the local image patches. This low-rank GMM is derived via eigenvalue thresholding of the GMM trained on the projection of the measurement data, thus learned {\em in situ}. The GMM and the projection of the measurement data are updated iteratively during the reconstruction. Our GMM algorithm degrades to the piecewise linear estimator (PLE) if each patch is represented by a single Gaussian model. Inspired by this, a low-rank PLE algorithm is also developed for CS inversion, constituting an additional contribution of this paper. Extensive results on both simulation data and real data captured by the lensless camera demonstrate the efficacy of the proposed algorithm. Furthermore, we compare the CS reconstruction results using our algorithm with the JPEG compression. Simulation results demonstrate that when limited bandwidth is available (a small number of measurements), our algorithm can achieve comparable results as JPEG.
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    • "This is leveraged in [10] to design subspace clustering algorithms that can perform motion segmentation. • In general, (affine) subspaces or unions of (affine) subspaces can also be used to model other data such as images of handwritten digits [13]. classification in a semi-supervised motion segmentation application. "
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    ABSTRACT: This paper considers the classification of linear subspaces with mismatched classifiers. In particular, we assume a model where one observes signals in the presence of isotropic Gaussian noise and the distribution of the signals conditioned on a given class is Gaussian with a zero mean and a low-rank covariance matrix. We also assume that the classifier knows only a mismatched version of the parameters of input distribution in lieu of the true parameters. By constructing an asymptotic low-noise expansion of an upper bound to the error probability of such a mismatched classifier, we provide sufficient conditions for reliable classification in the low-noise regime that are able to sharply predict the absence of a classification error floor. Such conditions are a function of the geometry of the true signal distribution, the geometry of the mismatched signal distributions as well as the interplay between such geometries, namely, the principal angles and the overlap between the true and the mismatched signal subspaces. Numerical results demonstrate that our conditions for reliable classification can sharply predict the behavior of a mismatched classifier both with synthetic data and in a real semi-supervised motion segmentation application.
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