Rodrigo Cabral Farias

Fast Decomposition of Large Nonnegative Tensors

In signal processing, tensor decompositions have gained in popularity this last decade. In the meantime, the volume of data to be processed has drastically increased. This calls for novel methods to handle Big Data tensors. Since most of these huge data are issued from physical measurements, which are intrinsically real nonnegative, being able to compress nonnegative tensors has become mandatory. Following recent works on HOSVD compression for Big Data, we detail solutions to decompose a nonnegative tensor into decomposable terms in a compressed domain.

Nonnegative tensor CP decomposition of hyperspectral data

New hyperspectral missions will collect huge amounts of hyperspectral data. In addition, it is possible now to acquire time series and multiangular hyperspectral images. The process and analysis of these big data collections will require common hyperspectral techniques to be adapted or reformulated. The tensor decomposition, which is also known as multiway analysis, is a technique to decompose multiway arrays, i.e., hypermatrices with more than two dimensions (ways). Hyperspectral time series and multiangular acquisitions can be represented as a three-way tensor. Here, we apply canonical polyadic (CP) tensor decomposition techniques to the blind analysis ohyperspectral big data. In order to do so, we use a novel compression-based nonnegative CP decomposition. We show that the proposed methodology can be interpreted as multilinear blind spectral unmixing, i.e., a higher order extension of the widely known spectral unmixing. In the proposed approach, the big hyperspectral tensor is decomposed in three sets of factors, which can be interpreted as spectral signatures, their spatial distribution, and temporal/angular changes. We provide experimental validation using a study case of the snow coverage of the French Alps during the snow season.

Exploring Multimodal Data Fusion Through Joint Decompositions with Flexible Couplings

A Bayesian framework is proposed to define flexible coupling models for joint tensor decompositions of multiple datasets. Under this framework, a natural formulation of the data fusion problem is to cast it in terms of a joint maximum a posteriori (MAP) estimator. Data-driven scenarios of joint posterior distributions are provided, including general Gaussian priors and non Gaussian coupling priors. We present and discuss implementation issues of algorithms used to obtain the joint MAP estimator. We also show how this framework can be adapted to tackle the problem of joint decompositions of large datasets. In the case of a conditional Gaussian coupling with a linear transformation, we give theoretical bounds on the data fusion performance using the Bayesian Cramér-Rao bound. Simulations are reported for hybrid coupling models ranging from simple additive Gaussian models to Gamma-type models with positive variables and to the coupling of data sets which are inherently of different size due to different resolution of the measurement devices.

Adaptive quantizers for estimation

This paper addresses a problem of location parameter estimation from multibit quantized measurements. An adaptive estimation algorithm using an adjustable quantizer is proposed. By using general results from adaptive algorithms theory, the asymptotic estimation performance is obtained and optimized through the quantizer parameters. Despite its very low complexity, it can be shown that the proposed algorithm is asymptotically optimal for estimating a constant parameter. The asymptotic performance for optimal quantizer parameters is shown to rapidly reach real-valued based estimation performance as the number of bits increases. In practice, 4-bit quantization appears to be enough for estimation purposes. It is also shown that the performance gap between the quantized and continuous cases is even smaller when the parameter varies according to a random walk (Discrete Wiener process with or without drift).

Joint Decompositions with Flexible Couplings

A Bayesian framework is proposed to define flexible coupling models for joint decompositions of data sets. Under this framework, a solution to the joint decomposition can be cast in terms of a maximum a posteriori estimator. Examples of joint posterior distributions are provided, including general Gaussian priors and non Gaussian coupling priors. Then simulations are reported and show the effectiveness of this approach to fuse information from data sets, which are inherently of different size due to different time resolution of the measurement devices.

Wideband Multiple Diversity Tensor Array Processing

This paper establishes a tensor model for wideband coherent array processing including multiple physical diversities. A separable coherent focusing operation is proposed as a preprocessing step in order to ensure the multilinearity of the interpolated data. We propose an alternating least squares algorithm to process tensor data, taking into account the noise correlation structure introduced by the focusing operation. We show through computer simulations that the estimation of direction of arrival and polarization parameters improves compared to existing narrowband tensor processing and wideband MUltiple SIgnal Classification. The performance is also compared to the Cramér-Rao bounds of the wideband tensor model.

Optimal Asymmetric Binary Quantization for Estimation Under Symmetrically Distributed Noise

Estimation of a location parameter based on noisy and binary quantized measurements is considered in this letter. We study the behavior of the Cramér-Rao bound as a function of the quantizer threshold for different symmetric unimodal noise distributions. We show that, in some cases, the intuitive choice of threshold position given by the symmetry of the problem, placing the threshold on the true parameter value, can lead to locally worst estimation performance.

Adaptive estimation based on quantized measurements

In this paper, the tracking of a slowly varying scalar Wiener process based on quantized noisy measurements is studied. An adaptive algorithm using a quantizer with adjustable input gain and bias is presented as a low complexity solution. The mean and asymptotic mean squared error of the algorithm are derived. Simulations under Cauchy and Gaussian noise are presented to validate the results and a comparison with the optimal estimator in the Gaussian and real-valued measurement case shows that the loss of performance due to quantization is negligible using 4 or 5 bits of resolution.

Asymptotic approximation of optimal quantizers for estimation

In this paper, the asymptotic approximation of the Fisher information for the estimation of a scalar parameter based on quantized measurements is studied. As the number of quantization intervals tends to infinity, it is shown that the loss of Fisher information due to quantization decreases exponentially as a function of the number of quantization bits. The optimal quantization interval density and the corresponding maximum Fisher information are obtained. Comparison between optimal nonuniform and uniform quantization for the location estimation problem indicates that nonuniform quantization is slightly better. At the end of the paper, an adaptive algorithm for jointly estimating and setting the thresholds is used to show that the theoretical results can be approximately obtained in practice.

Joint tensor compression for coupled canonical polyadic decompositions

To deal with large multimodal datasets, coupled canonical polyadic decompositions are used as an approximation model. In this paper, a joint compression scheme is introduced to reduce the dimensions of the dataset. Joint compression allows to solve the approximation problem in a compressed domain using standard coupled decomposition algorithms. Computational complexity required to obtain the coupled decomposition is therefore reduced. Also, we propose to approximate the update of the coupled factor by a simple weighted average of the independent updates of the coupled factors. The proposed approach and its simplified version are tested with synthetic data and we show that both do not incur substantial loss in approximation performance.