Tensor-Based Multivariate Polynomial Optimization with Application in Blind Identification

Ayvaz, Muzaffer; Lathauwer, Lieven De

doi:10.23919/eusipco54536.2021.9616070

Cited by 2 publications

(4 citation statements)

References 25 publications

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“…The primary aim of the TeMPO framework is to develop efficient algorithms for modeling nonlinear phenomena commonly encountered in the areas of signal processing, machine learning, and artificial intelligence [15]. To achieve this, we assume structure in the nonlinear function f : R I → R N that maps the input data to output data.…”

Section: Tensor-based Multivariate Polynomial Optimizationmentioning

confidence: 99%

“…Blind deconvolution can be formulated as a multivariate polynomial optimization (MPO) problem and hence it fits into the TeMPO framework [15]. In this illustrative example, we limit ourselves to an autoregressive single-input single-output (SISO) system [57], given by…”

Section: Blind Deconvolution Of Constant Modulus Signalsmentioning

confidence: 99%

“…Exploiting the rank-1 Hermitian symmetric CPD structure in (36) and the structure of M, which is a special case of Lemma 1 and Lemma 2, efficient expressions for the computation of Jacobian-vector products for the problem (35) have been presented in [15]. A number of algorithms have been developed to solve (33) and (34).…”

Section: Blind Deconvolution Of Constant Modulus Signalsmentioning

confidence: 99%

“…In this study, we develop a framework called Tensor-Based Multivariate Polynomial Optimization (TeMPO) to deal with nonlinear optimization problems commonly encountered in signal processing, machine learning and artificial intelligence. A preliminary version, where only rank-1 CPD is considered with application in blind identification, appeared as the conference paper [15]. In the TeMPO framework, these nonlinear functions are approximated or modeled by multivariate polynomials.…”

Section: Introductionmentioning

confidence: 99%

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CPD-Structured Multivariate Polynomial Optimization

Ayvaz

Lathauwer

2022

Front. Appl. Math. Stat.

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We introduce the Tensor-Based Multivariate Optimization (TeMPO) framework for use in nonlinear optimization problems commonly encountered in signal processing, machine learning, and artificial intelligence. Within our framework, we model nonlinear relations by a multivariate polynomial that can be represented by low-rank symmetric tensors (multi-indexed arrays), making a compromise between model generality and efficiency of computation. Put the other way around, our approach both breaks the curse of dimensionality in the system parameters and captures the nonlinear relations with a good accuracy. Moreover, by taking advantage of the symmetric CPD format, we develop an efficient second-order Gauss–Newton algorithm for multivariate polynomial optimization. The presented algorithm has a quadratic per-iteration complexity in the number of optimization variables in the worst case scenario, and a linear per-iteration complexity in practice. We demonstrate the efficiency of our algorithm with some illustrative examples, apply it to the blind deconvolution of constant modulus signals, and the classification problem in supervised learning. We show that TeMPO achieves similar or better accuracy than multilayer perceptrons (MLPs), tensor networks with tensor trains (TT) and projected entangled pair states (PEPS) architectures for the classification of the MNIST and Fashion MNIST datasets while at the same time optimizing for fewer parameters and using less memory. Last but not least, our framework can be interpreted as an advancement of higher-order factorization machines: we introduce an efficient second-order algorithm for higher-order factorization machines.

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Section: Tensor-based Multivariate Polynomial Optimizationmentioning

confidence: 99%

Section: Blind Deconvolution Of Constant Modulus Signalsmentioning

confidence: 99%

Section: Blind Deconvolution Of Constant Modulus Signalsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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CPD-Structured Multivariate Polynomial Optimization

Ayvaz

Lathauwer

2022

Front. Appl. Math. Stat.

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Identification of Multilinear Systems: A Brief Overview

Dogariu¹,

Paleologu²,

Benesty³

et al. 2022

Advances in Principal Component Analysis

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Nonlinear systems have been studied for a long time and have applications in numerous research fields. However, there is currently no global solution for nonlinear system identification, and different used approaches depend on the type of nonlinearity. An interesting class of nonlinear systems, with a wide range of popular applications, is represented by multilinear (or multidimensional) systems. These systems exhibit a particular property that may be exploited, namely that they can be regarded as linearly separable systems and can be modeled accordingly, using tensors. Examples of well-known applications of multilinear forms are multiple-input/single-output (MISO) systems and acoustic echo cancellers, used in multi-party voice communications, such as videoconferencing. Many important fields (e.g., big data, machine learning, and source separation) can benefit from the methods employed in multidimensional system identification. In this context, this chapter aims to briefly present the recent approaches in the identification of multilinear systems. Methods relying on tensor decomposition and modeling are used to address the large parameter space of such systems.

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Tensor-Based Multivariate Polynomial Optimization with Application in Blind Identification

Cited by 2 publications

References 25 publications

CPD-Structured Multivariate Polynomial Optimization

CPD-Structured Multivariate Polynomial Optimization

Identification of Multilinear Systems: A Brief Overview

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