Spectral clustering is a tool for extracting meaningful information from data by grouping similar objectsDtogether [1]. The method uses the eigenvector of an adjacency matrix for embedding the data into a space that captures the underlying group structure [2]. High-dimensional signals, magnetic resonance images, and hyperspectral images can be costly to acquire; even simple direct comparisons could be infeasible among such data sets. Our work shows that the meaningful organization extracted from spectral clustering is preserved under the perturbation from making compressed, incomplete and inaccurate measurements. Using bounds on the perturbation of eigenvectors, we establish error bounds of the spectral embedding when matrix completion and compressed sensing measurements are used. Given some error Nǫ in the entries of an affinity matrix A RN N, we show that the space spanned by the first k eigenvector are all within O(Nǫ) of the span of the unperturbed eigenvectors. We prove that the perturbed spectral coordinates are within O(Nǫ)of a unitary transform of the unperturbed coordinates and can give k-means cluster assignments within O(Nǫ) of the unperturbed case. This analysis holds true when the error perturbation in the entries of an affinity matrix |A(i,j) A (i,j)| ǫ is caused from making compressed arXiv:1011.0997v1

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Mirror descent [37] is becoming increasingly popular in a variety of settings in optimization and machine learning. One reason for its success is the fact that mirror descent can be adapted to fit the geometry ofthe optimization problem athand bychoosing asuitable strictly convexpotential function,theso-calledmirrormap.

We study the robust recovery of a low-rank matrix from sparsely and grossly corrupted Gaussian measurements, with no prior knowledge on the intrinsic rank. We consider the robust matrix factorization approach. We employ a robust $\ell_1$ loss function and deal with the challenge of the unknown rank by using an overspecified factored representation of the matrix variable. We then solve the associated nonconvex nonsmooth problem using a subgradient method with diminishing stepsizes. We show that under a regularity condition on the sensing matrices and corruption, which we call restricted direction preserving property (RDPP), even with rank overspecified, the subgradient method converges to the exact low-rank solution at a sublinear rate. Moreover, our result is more general in the sense that it automatically speeds up to a linear rate once the factor rank matches the unknown rank.

Compressive image recovery is a challenging problem that requires fast and accurate algorithms.

One-bit compressed sensing (1bCS) is a method of signal acquisition under extreme measurement quantization that gives important insights on the limits of signal compression and analog-to-digital conversion. The setting is also equivalent to the problem of learning a sparse hyperplane-classifier. In this paper, we propose a generic approach for signal recovery in nonadaptive 1bCS that leads to improved sample complexity for approximate recovery for a variety of signal models, including nonnegative signals and binary signals. We construct 1bCS matrices that are universal - i.e. work for all signals under a model - and at the same time recover very general random sparse signals with high probability. In our approach, we divide the set of samples (measurements) into two parts, and use the first part to recover the superset of the support of a sparse vector. The second set of measurements is then used to approximate the signal within the superset.

For a full version of this paper see [53].

We consider the problem of signal reconstruction for computed tomography (CT) under a nonlinear forward model that accounts for exponential signal attenuation, a polychromatic X-ray source, general measurement noise (e.g. Poisson shot noise), and observations acquired over multiple wavelength windows. We develop a simple iterative algorithm for single-material reconstruction, which we call EXACT (EXtragradient Algorithm for Computed Tomography), based on formulating our estimate as the fixed point of a monotone variational inequality. We prove guarantees on the statistical and computational performance of EXACT under practical assumptions on the measurement process. We also consider a recently introduced variant of this model with Gaussian measurements, and present sample and iteration complexity bounds for EXACT that improve upon those of existing algorithms. We apply our EXACT algorithm to a CT phantom image recovery task and show that it often requires fewer X-ray projection exposures, lower source intensity, and less computation time to achieve similar reconstruction quality to existing methods.

We study the robust recovery of a low-rank matrix from sparsely and grossly corrupted Gaussian measurements, with no prior knowledge on the intrinsic rank. We consider the robust matrix factorization approach. We employ a robust \ell_1 loss function and deal with the challenge of the unknown rank by using an overspecified factored representation of the matrix variable. We then solve the associated nonconvex nonsmooth problem using a subgradient method with diminishing stepsizes. We show that under a regularity condition on the sensing matrices and corruption, which we call restricted direction preserving property (RDPP), even with rank overspecified, the subgradient method converges to the exact low-rank solution at a sublinear rate. Moreover, our result is more general in the sense that it automatically speeds up to a linear rate once the factor rank matches the unknown rank.

Compressive image recovery is a challenging problem that requires fast and accurate algorithms. Recently, neural networks have been applied to this problem with promising results. By exploiting massively parallel GPU processing architectures and oodles of training data, they can run orders of magnitude faster than existing techniques. However, these methods are largely unprincipled black boxes that are difficult to train and often-times specific to a single measurement matrix. It was recently demonstrated that iterative sparse-signal-recovery algorithms can be "unrolled" to form interpretable deep networks.