The problem of tensor completion arises often in signal processing and machine learning. It consists of recovering a tensor from a subset of its entries. The usual structural assumption on a tensor that makes the problem well posed is that the tensor has low rank in every mode. Several tensor completion methods based on minimization of nuclear norm, which is the closest convex approximation of rank, have been proposed recently, with applications mostly in image inpainting problems. It is often stated in these papers that methods based on Tucker factorization perform poorly when the true ranks are unknown. In this paper, we propose a simple algorithm for Tucker factorization of a tensor with missing data and its application to low-n-rank tensor completion. The algorithm is similar to previously proposed method for PARAFAC decomposition with missing data. We demonstrate in several numerical experiments that the proposed algorithm performs well even when the ranks are significantly overestimated. Approximate reconstruction can be obtained when the ranks are underestimated. The algorithm outperforms nuclear norm minimization methods when the fraction of known elements of a tensor is low.

The quality of speech signals recorded in an enclosure can be severely degraded by room reverberation. In this paper, we focus on a class of blind batch methods for speech dereverberation in a noiseless scenario with a single source, which are based on multi-channel linear prediction in the short-time Fourier transform domain. Dereverberation is performed by maximum-likelihood estimation of the model parameters that are subsequently used to recover the desired speech signal. Contrary to the conventional method, we propose to model the desired speech signal using a general sparse prior that can be represented in a convex form as a maximization over scaled complex Gaussian distributions. The proposed model can be interpreted as a generalization of the commonly used time-varying Gaussian model. Furthermore, we reformulate both the conventional and the proposed method as an optimization problem with an -norm cost function, emphasizing the role of sparsity in the considered speech dereverberation methods. Experimental evaluation in different acoustic scenarios show that the proposed approach results in an improved performance compared to the conventional approach in terms of instrumental measures for speech quality.Index Terms-Multi-channel linear prediction, sparse priors, speech dereverberation, speech enhancement.

This paper presents a system aiming at joint dereverberation and noise reduction by applying a combination of a beamformer with a single-channel spectral enhancement scheme. First, a minimum variance distortionless response beamformer with an online estimated noise coherence matrix is used to suppress noise and reverberation. The output of this beamformer is then processed by a single-channel spectral enhancement scheme, based on statistical room acoustics, minimum statistics, and temporal cepstrum smoothing, to suppress residual noise and reverberation. The evaluation is conducted using the REVERB challenge corpus, designed to evaluate speech enhancement algorithms in the presence of both reverberation and noise. The proposed system is evaluated using instrumental speech quality measures, the performance of an automatic speech recognition system, and a subjective evaluation of the speech quality based on a MUSHRA test. The performance achieved by beamforming, single-channel spectral enhancement, and their combination are compared, and experimental results show that the proposed system is effective in suppressing both reverberation and noise while improving the speech quality. The achieved improvements are particularly significant in conditions with high reverberation times

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