Abstract--This work proposes an efficient, robust adaptive beamforming technique to deal with steering vector (SV) est ima-tion mismatches and data covariance matrix reconstruction problems. In particular, the direction-of-arrival(DoA) of int erfering sources is estimated with available snapshots in which the a ngular sectors of the interfering signals are computed adaptively . Then, we utilize the well-known general linear combination algor ithm to reconstruct the interference-plus-noise covariance (I PNC) matrix using preprocessing-based spatial sampling (PPBSS). We demonstrate that the preprocessing matrix can be replaced b y the sample covariance matrix (SCM) in the shrinkage method. A power spectrum sampling strategy is then devised based on a preprocessing matrix computed with the estimated angular sectors' information. Moreover, the covariance matrix for the signal is formed for the angular sector of the signal-of-int erest (SOI), which allows for calculating an SV for the SOI using the power method. An analysis of the array beampattern in the proposed PPBSS technique is carried out, and a study of the computational cost of competing approaches is conducte d. Simulation results show the proposed method's effectivene ss compared to existing approaches. DAPTIVE beamforming spans across various fields, including wireless communications, radar, sonar, and medical imaging, where it significantly improves performan ce by increasing signal-to-noise ratio (SNR) and mitigating i n-terference [1].

We introduce a new family of matrix norms, the "local max" norms, generalizing existing methods such as the max norm, the trace norm (nuclear norm), and the weighted or smoothed weighted trace norms, which have been extensively used in the literature as regularizers for matrix reconstruction problems. We show that this new family can be used to interpolate between the (weighted or unweighted) trace norm and the more conservative max norm. We test this interpolation on simulated data and on the large-scale Netflix and MovieLens ratings data, and find improved accuracy relative to the existing matrix norms. We also provide theoretical results showing learning guarantees for some of the new norms.

Review of "Low-rank matrix reconstruction and clustering" This paper contributes a new algorithm for low-rank matrix reconstruction which is based on an application of Belief Propagation (BP) message-passing to a Bayesian model of the reconstruction problem. The algorithm, as described in the "Supplementary Material", incorporates two simplifying approximations, based on assuming a large number of rows and columns, respectively, in the input matrix. The algorithm is evaluated in a novel manner against Lloyd's K-means algorithm by formulating clustering as a matrix reconstruction problem. It is also compared against Variational Bayes Matrix Factorization (VBMF), which seems to be the only previous message-passing reconstruction algorithm. Cons There are some arguments against accepting the paper.

We study the problem of reconstructing low-rank matrices from their noisy observations. We formulate the problem in the Bayesian framework, which allows us to exploit structural properties of matrices in addition to low-rankedness, such as sparsity. We propose an efficient approximate message passing algorithm, derived from the belief propagation algorithm, to perform the Bayesian inference for matrix reconstruction. We have also successfully applied the proposed algorithm to a clustering problem, by reformulating it as a low-rank matrix reconstruction problem with an additional structural property. Numerical experiments show that the proposed algorithm outperforms Lloyd's K-means algorithm.

This work presents a cost-effective technique for designing robust adaptive beamforming algorithms based on efficient covariance matrix reconstruction with iterative spatial power spectrum (CMR-ISPS). The proposed CMR-ISPS approach reconstructs the interference-plus-noise covariance (INC) matrix based on a simplified maximum entropy power spectral density function that can be used to shape the directional response of the beamformer. Firstly, we estimate the directions of arrival (DoAs) of the interfering sources with the available snapshots. We then develop an algorithm to reconstruct the INC matrix using a weighted sum of outer products of steering vectors whose coefficients can be estimated in the vicinity of the DoAs of the interferences which lie in a small angular sector. We also devise a cost-effective adaptive algorithm based on conjugate gradient techniques to update the beamforming weights and a method to obtain estimates of the signal of interest (SOI) steering vector from the spatial power spectrum. The proposed CMR-ISPS beamformer can suppress interferers close to the direction of the SOI by producing notches in the directional response of the array with sufficient depths. Simulation results are provided to confirm the validity of the proposed method and make a comparison to existing approaches

Among machine learning problems, matrix factorization (MF) is significant because MF appears in many applications such as recommendation system, signal processing, etc. We restrict ourselves to sparse MF problem in this article, where either factorized matrix must be sparse. This is originally discussed as sparse coding in neuroscience [1, 2], and recognized as a significant problem in neuronal information processing in the brain. It also appears in sparse modeling in information science such as dictionary learning [3, 4] or sparse principal component analysis (sparse PCA) [5, 6]. Many attempts have been made so far for understanding theoretical aspects of MF, and analytical tools for random systems in statistical physics are found to be useful, e.g. Markov chain Monte Carlo method [7], replica analysis [8, 9, 10, 11, 12], and message passing [9, 10, 11, 12, 13, 14], where some works are not limited to sparse matrix case.

We introduce a new family of matrix norms, the ''local max'' norms, generalizing existing methods such as the max norm, the trace norm (nuclear norm), and the weighted or smoothed weighted trace norms, which have been extensively used in the literature as regularizers for matrix reconstruction problems. We show that this new family can be used to interpolate between the (weighted or unweighted) trace norm and the more conservative max norm. We test this interpolation on simulated data and on the large-scale Netflix and MovieLens ratings data, and find improved accuracy relative to the existing matrix norms. We also provide theoretical results showing learning guarantees for some of the new norms.

We study the problem of reconstructing low-rank matrices from their noisy observations. We formulate the problem in the Bayesian framework, which allows us to exploit structural properties of matrices in addition to low-rankedness, such as sparsity. We propose an efficient approximate message passing algorithm, derived from the belief propagation algorithm, to perform the Bayesian inference for matrix reconstruction. We have also successfully applied the proposed algorithm to a clustering problem, by formulating the problem of clustering as a low-rank matrix reconstruction problem with an additional structural property. Numerical experiments show that the proposed algorithm outperforms Lloyd's K-means algorithm.

We introduce a new family of matrix norms, the ''local max'' norms, generalizing existing methods such as the max norm, the trace norm (nuclear norm), and the weighted or smoothed weighted trace norms, which have been extensively used in the literature as regularizers for matrix reconstruction problems. We show that this new family can be used to interpolate between the (weighted or unweighted) trace norm and the more conservative max norm. We test this interpolation on simulated data and on the large-scale Netflix and MovieLens ratings data, and find improved accuracy relative to the existing matrix norms. We also provide theoretical results showing learning guarantees for some of the new norms. Papers published at the Neural Information Processing Systems Conference.

We study the problem of reconstructing low-rank matrices from their noisy observations. We formulate the problem in the Bayesian framework, which allows us to exploit structural properties of matrices in addition to low-rankedness, such as sparsity. We propose an efficient approximate message passing algorithm, derived from the belief propagation algorithm, to perform the Bayesian inference for matrix reconstruction. We have also successfully applied the proposed algorithm to a clustering problem, by formulating the problem of clustering as a low-rank matrix reconstruction problem with an additional structural property. Numerical experiments show that the proposed algorithm outperforms Lloyd's K-means algorithm.