A Novel Enriched Version of Truncated Nuclear Norm Regularization for Matrix Completion of Inexact Observed Data (original) (raw)
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IEEE Transactions on Information Theory, 2000
The mutual information between a complex-valued channel input and its complex-valued output is decomposed into four parts based on polar coordinates: an amplitude term, a phase term, and two mixed terms. Numerical results for the additive white Gaussian noise (AWGN) channel with various inputs show that, at high signal-to-noise ratio (SNR), the amplitude and phase terms dominate the mixed terms. For the AWGN channel with a Gaussian input, analytical expressions are derived for high SNR. The decomposition method is applied to partially coherent channels and a property of such channels called "spectral loss" is developed. Spectral loss is used to explain the behavior of the capacity of nonlinear fiber-optic channels presented in recent studies, and is applied to simplify a recently published phenomenological channel model.
Information-theoretic limits of matrix completion
2015 IEEE International Symposium on Information Theory (ISIT), 2015
We propose an information-theoretic framework for matrix completion. The theory goes beyond the low-rank structure and applies to general matrices of "low description complexity". Specifically, we consider random matrices X ∈ R m×n of arbitrary distribution (continuous, discrete, discretecontinuous mixture, or even singular). With S ⊆ R m×n an ε-support set of X, i.e., P[X ∈ S] ≥ 1 − ε, and dim B (S) denoting the lower Minkowski dimension of S, we show that k > dim B (S) measurements of the form Ai, X , with Ai denoting the measurement matrices, suffice to recover X with probability of error at most ε. The result holds for Lebesgue a.a. Ai and does not need incoherence between the Ai and the unknown matrix X. We furthermore show that k > dim B (S) measurements also suffice to recover the unknown matrix X from measurements taken with rank-one Ai, again this applies to a.a. rank-one Ai. Rank-one measurement matrices are attractive as they require less storage space than general measurement matrices and can be applied faster. Particularizing our results to the recovery of low-rank matrices, we find that k > (m + n − r)r measurements are sufficient to recover matrices of rank at most r. Finally, we construct a class of rank-r matrices that can be recovered with arbitrarily small probability of error from k < (m + n − r)r measurements.
An Optimal Hybrid Nuclear Norm Regularization for Matrix Sensing With Subspace Prior Information
IEEE Access, 2020
Matrix sensing refers to recovering a low-rank matrix from a few linear combinations of its entries. This problem naturally arises in many applications including recommendation systems, collaborative filtering, seismic data interpolation and wireless sensor networks. Recently, in these applications, it has been noted that exploiting additional subspace information might yield significant improvements in practical scenarios. This information is reflected by two subspaces forming angles with column and row spaces of the ground-truth matrix. Despite the importance of exploiting this information, there is limited theoretical guarantee for this feature. In this work, we aim to address this issue by proposing a novel hybrid nuclear norm regularization which besides low-rankness, encourages subspace prior information. Our proposed regularizer is a weighted combination of deformed nuclear norm functions. We derive a closed-form accurate expression for the mean squared error (MSE) of the pro...
2012
Abstract We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala et al.[11], message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of non-zero coordinates.
Polar Decomposition of Mutual Information and Applications to Partially Coherent Channels
Corr, 2010
The mutual information between a complex-valued channel input and its complex-valued output is decomposed into four parts based on polar coordinates: an amplitude term, a phase term, and two mixed terms. Numerical results for the additive white Gaussian noise (AWGN) channel with various inputs show that, at high signal-to-noise ratio (SNR), the amplitude and phase terms dominate the mixed terms. For the AWGN channel with a Gaussian input, analytical expressions are derived for high SNR. The decomposition method is applied to partially coherent channels and a property of such channels called "spectral loss" is developed. Spectral loss occurs in nonlinear fiber-optic channels and it may be one effect that needs to be taken into account to explain the behavior of the capacity of nonlinear fiber-optic channels presented in recent studies.
Sensing Matrix Sensitivity to Random Gaussian Perturbations in Compressed Sensing
2018 26th European Signal Processing Conference (EUSIPCO), 2018
In compressed sensing, the choice of the sensing matrix plays a crucial role: it defines the required hardware effort and determines the achievable recovery performance. Recent studies indicate that by optimizing a sensing matrix, one can potentially improve system performance compared to random ensembles. In this work, we analyze the sensitivity of a sensing matrix design to random perturbations, e.g., caused by hardware imperfections, with respect to the total (average) matrix coherence. We derive an exact expression for the average deterioration of the total coherence in the presence of Gaussian perturbations as a function of the perturbations' variance and the sensing matrix itself. We then numerically evaluate the impact it has on the recovery performance.
Mutual Information of IID Complex Gaussian Signals on Block Rayleigh-Faded Channels
IEEE Transactions on Information Theory, 2012
We present a method to compute, quickly and efficiently, the mutual information achieved by an IID (independent identically distributed) complex Gaussian signal on a block Rayleigh-faded channel without side information at the receiver. The method accommodates both scalar and MIMO (multiple-input multiple-output) settings. Operationally, this mutual information represents the highest spectral efficiency that can be attained using Gaussian codebooks. Examples are provided that illustrate the loss in spectral efficiency caused by fast fading and how that loss is amplified when multiple transmit antennas are used. These examples are further enriched by comparisons with the channel capacity under perfect channel-state information at the receiver, and with the spectral efficiency attained by pilot-based transmission.
Distribution of Compressive Measurements Generated by Structurally Random Matrices
Structurally random matrices (SRMs) have been proposed as a practical alternative to fully random matrices (FRMs) for generating compressive sensing measurements. If the compressive measurements are transmitted over a communication channel, they need to be efficiently quantized and coded and hence knowledge of the measurements' statistics required. In this paper we study the statistical distribution of compressive measurements generated by various types of SRMs(and FRMs), give conditions for asymptotic normality and point out the implications for the measurements' quantization and coding. Simulations on real-world video signals confirm the theoretical findings and show that the signal randomization of SRMs yields a dramatic improvement in quantization properties.
2020
Weighted nuclear norm minimization has been recently recognized as a technique for reconstruction of a low-rank matrix from compressively sampled measurements when some prior information about the column and row subspaces of the matrix is available. In this work, we study the recovery conditions and the associated recovery guarantees of weighted nuclear norm minimization when multiple weights are allowed. This setup might be used when one has access to prior subspaces forming multiple angles with the column and row subspaces of the ground-truth matrix. While existing works in this field use a single weight to penalize all the angles, we propose a multi-weight problem which is designed to penalize each angle independently using a distinct weight. Specifically, we prove that our proposed multi-weight problem is stable and robust under weaker conditions for the measurement operator than the analogous conditions for single-weight scenario and standard nuclear norm minimization. Moreover...
Hankel Matrix Nuclear Norm Regularized Tensor Completion for NNN-dimensional Exponential Signals
IEEE Transactions on Signal Processing, 2017
Signals are generally modeled as a superposition of exponential functions in spectroscopy of chemistry, biology and medical imaging. For fast data acquisition or other inevitable reasons, however, only a small amount of samples may be acquired and thus how to recover the full signal becomes an active research topic. But existing approaches can not efficiently recover N-dimensional exponential signals with N ≥ 3. In this paper, we study the problem of recovering N-dimensional (particularly N ≥ 3) exponential signals from partial observations, and formulate this problem as a low-rank tensor completion problem with exponential factor vectors. The full signal is reconstructed by simultaneously exploiting the CANDECOMP/PARAFAC structure and the exponential structure of the associated factor vectors. The latter is promoted by minimizing an objective function involving the nuclear norm of Hankel matrices. Experimental results on simulated and real magnetic resonance spectroscopy data show that the proposed approach can successfully recover full signals from very limited samples and is robust to the estimated tensor rank.