Rank reduction strategies can be employed to attenuate noise and for prestack seismic data regularization. We present a fast version of Cadzow reduced-rank reconstruction method. Cadzow reconstruction is implemented by embedding 4D spatial data into a level-four block Toeplitz matrix. Rank reduction of this matrix via the Lanczos bidiagonalization algorithm is used to recover missing observations and to attenuate random noise. The computational cost of the Lanczos bidiagonalization is dominated by the cost of multiplying a level-four block Toeplitz matrix by a vector. This is efficiently implemented via the 4D fast Fourier transform. The proposed algorithm significantly decreases the computational cost of rank-reduction methods for multidimensional seismic data denoising and reconstruction. Synthetic and field prestack data examples are used to examine the effectiveness of the proposed method.

A fast reduced-rank interpolation method for prestack seismic volumes that depend on four spatial dimensions

The present paper considers a special class of vector random processes that we call multivariate asymptotically wide sense stationary (WSS) processes. A multivariate random process is said to be asymptotically WSS if it has constant mean and the sequence of its autocorrelation matrices is asymptotically equivalent (a.e.) to the sequence of autocorrelation matrices of some multivariate WSS process. It is shown that this class of processes contains meaningful processes other than multivariate WSS processes. In particular, we give sufficient conditions for multivariate moving average (MA) processes, multivariate autoregressive (AR) processes and multivariate autoregressive moving average (ARMA) processes to be asymptotically WSS. Furthermore, in order to solve multiple-input-multiple-output (MIMO) problems in communications and signal processing involving this kind of processes, we extend the Gray definition of a.e. sequences of matrices and his main results on these sequences to non-square matrices. As an example, the derived results on a.e. sequences of non-square matrices are applied to compute the differential entropy rate and the minimum mean square error (MMSE) for a linear predictor of a multivariate asymptotically WSS process.

Asymptotically Equivalent Sequences of Matrices and Multivariate ARMA Processes

The optimum sampling in the one- and two-dimensional (1-D and 2-D) wireless sensor networks (WSNs) with spatial-temporally correlated data is studied in this article. The impacts of the node density in the space domain, the sampling rate in the time domain, and the space-time data correlation on the network performance are investigated asymptotically by considering a large network with infinite area but finite node density and finite temporal sampling rate, under the constraint of fixed power per unit area. The impact of space-time sampling on network performances is investigated in two cases. The first case studies the estimations of the space-time samples collected by the sensors, and the samples are discrete in both the space and time domains. The second case estimates an arbitrary data point on the space-time hyperplane by interpolating the discrete samples collected by the sensors. Optimum space-time sampling is obtained by minimizing the mean square error distortion at the network fusion center. The interactions among the various network parameters, such as spatial node density, temporal sampling rate, measurement noise, channel fading, and their impacts on the system performance are quantitatively identified with analytical and numerical studies.

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Optimum sampling in spatial-temporally correlated wireless sensor networks

In this paper we introduce and analyze a new preconditioner for Toeplitz matrices that exhibits excellent spectral properties: the eigenvalues of the preconditioned matrix are highly clustered around the unity. As a result, it yields very rapid convergence when used to solve Toeplitz equations via the preconditioned conjugate gradient method. The new preconditioner can be regarded as a refinement of preconditioners built by embedding the Toeplitz matrix in a positive definite circulant. Necessary and sufficient conditions that ensure that the positive definite embedding is possible are given.

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A New Preconditioner for Toeplitz Matrices

By stacking a number of convolutional layers, convolutional neural networks (CNNs) have made remarkable performance boosts in many artificial intelligence applications. While the convolution operation is well-understood, it is still a mystery why repeated convolutions yield so good expressive power and generalization performance. Noting that the linear convolution operation can be represented as a matrix-vector product with the matrix being of a Toeplitz structure, we propose to inspect the individual convolutional layer through its asymptotic spectral representation - the spectral density matrix - by leveraging Toeplitz matrix theory. Thanks to such spectral representation, we are able to develop a simple singular value approximation method with improved accuracy, and spectral norm upper bounds with reduced computational complexity, compared with the state-of-the-art methods. Both the improved approximation and upper bounds can be employed as regularization techniques to further enhance the generalization performance of CNNs. By extensive experiments on well-deployed CNN models (e.g., ResNets), we also demonstrate that the approximation approach achieves higher accuracy and the upper bounds are effective spectral regularizers for generalization.

Asymptotic Spectral Representation of Linear Convolutional Layers

In many detection and estimation problems associated with processing of second-order stationary random processes, the observation data are the sum of two zero-mean second-order stationary processes: the process of interest and the noise process. In particular, the main performance criterion is the signal-to-noise ratio (SNR). After linear filtering, the optimal SNR corresponds to the maximal value of a Rayleigh quotient which can be interpreted as the largest generalized eigenvalue of the covariance matrices associated with the signal and noise processes, which are block multilevel Toeplitz structured for m-dimensional vector-valued second-order stationary p -dimensional random processes xi1,i2,......,ip isin \BBR m. In this paper, an extension of Szego's theorem to the generalized eigenvalues of Hermitian block multilevel Toeplitz matrices is given, providing information about the asymptotic distribution of those generalized eigenvalues and in particular of the optimal SNR after linear filtering. A simple proof of this theorem, under the hypothesis of absolutely summable elements is given. The proof is based on the notion of multilevel asymptotic equivalence between block multilevel matrix sequences derived from the celebrated Gray approach. Finally, a short example in wideband space-time beamforming is given to illustrate this theorem.

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M. Oudin

Papers

Asymptotic Generalized Eigenvalue Distribution of Block Multilevel Toeplitz Matrices