Tensor network subspace identification of polynomial state space models

Batselier, Kim; Ko, Ching-Yun; Wong, Ngai–Ching

doi:10.1016/j.automatica.2018.05.015

Cited by 17 publications

(20 citation statements)

References 17 publications

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“…Tensor algebra is the next evolutionary step in linear algebra since it deals primarily with the simultaneous coupling of three or more vector spaces or with vectors of three or more dimensions [34]. Also, tensors can be used in the identification of non-linear systems [35][36][37]. Tensors and their factorizations have a wide array of applications to various engineering fields.…”

Section: Previous Workmentioning

confidence: 99%

Extending Fuzzy Cognitive Maps with Tensor-Based Distance Metrics

et al. 2020

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Cognitive maps are high level representations of the key topological attributes of real or abstract spatial environments progressively built by a sequence of noisy observations. Currently such maps play a crucial role in cognitive sciences as it is believed this is how clusters of dedicated neurons at hippocampus construct internal representations. The latter include physical space and, perhaps more interestingly, abstract fields comprising of interconnected notions such as natural languages. In deep learning cognitive graphs are effective tools for simultaneous dimensionality reduction and visualization with applications among others to edge prediction, ontology alignment, and transfer learning. Fuzzy cognitive graphs have been proposed for representing maps with incomplete knowledge or errors caused by noisy or insufficient observations. The primary contribution of this article is the construction of cognitive map for the sixteen Myers-Briggs personality types with a tensor distance metric. The latter combines two categories of natural language attributes extracted from the namesake Kaggle dataset. To the best of our knowledge linguistic attributes are separated in categories. Moreover, a fuzzy variant of this map is also proposed where a certain personality may be assigned to up to two types with equal probability. The two maps were evaluated based on their topological properties, on their clustering quality, and on how well they fared against the dataset ground truth. The results indicate a superior performance of both maps with the fuzzy variant being better. Based on the findings recommendations are given for engineers and practitioners.

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Section: Previous Workmentioning

confidence: 99%

Extending Fuzzy Cognitive Maps with Tensor-Based Distance Metrics

et al. 2020

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“…1,[6][7][8][9][10][11] In case the relationship between input/output data exhibits nonlinear phenomena, a nonlinear model approximation is needed. The nonlinearities can be represented by various models such as basis function expansions, 12 Volterra models, [13][14][15] Gaussian processes, 16,17 neural networks, 18,19 trees, 20,21 and so on. The nonlinear model can also be approximated by a linear model with time-variant parameters via robust Kalman filter.…”

Section: Introductionmentioning

confidence: 99%

“…The intense storage requirement may also explain why only few contributions on the MIMO Volterra system identification can be found in the recent years. 14,15,37,38 Tensor network (TN) can relieve the curse of dimensionality by referring to techniques for multidimensional arrays, that is, tensors. For example, the storage cost of a square n d × n d matrix is typically n 2d , which can be prohibitively large as d increases.…”

Section: Introductionmentioning

confidence: 99%

“…A linear TN called tensor train (TT) or matrix product state (MPS) could reduce the storage cost to approximately dn 2 r 2 , where r is the maximal TN-rank and usually low in practice. 15,39,40 A TN representation for the MIMO Volterra system was proposed in Reference 14. The alternating linear scheme (ALS) and the modified ALS (MALS) algorithms were derived therein to compute the TN-cores with low TN-ranks.…”

Section: Introductionmentioning

confidence: 99%

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Noniterative tensor network‐based algorithm for Volterra system identification

Tan

Callafon

2022

Intl J Robust & Nonlinear

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Volterra model serves as one of the powerful alternatives to approximate nonlinear dynamics on the basis of input/output observations. The kernel representation of a Volterra model is attractive since it is linear in the kernel coefficients and always stable. Although the kernel representation suffers from the curse of dimensionality, the demand for storage requirements can be relieved via a tensor network (TN) technique. This allows one to approximate complicated coupled nonlinear dynamics with high degree and even multi-input multi-output (MIMO) Volterra models. The iterative feature of existing TN-based algorithms poses challenges for the algorithms to converge to an appropriate solution with both good prediction accuracy and minor overfitting problems. In this article, noniterative TN-based algorithms with two tuning factors to solve either a linear or ridge regression are proposed for MIMO Volterra system identification. The proposed algorithms do not suffer convergence problems due to its noniterative feature and perform active low-rank approximations to reduce overfitting problems. Simulations compare different algorithms and investigate the effects for different excitation signals.

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“…connected low-order tensors [16]. Combined with tensor algebra, tensor network structures can greatly decrease the computational complexity of many applications [17,18,19]. Due to their multilinear nature, multivariate B-splines easily admit a tensor network representation, which we call the Tensor Network B-splines (TNBS) model.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear system identification with regularized Tensor Network B-splines

Karagoz

Batselier

2020

Preprint

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This article introduces the Tensor Network B-spline model for the regularized identification of nonlinear systems using a nonlinear autoregressive exogenous (NARX) approach. Tensor network theory is used to alleviate the curse of dimensionality of multivariate B-splines by representing the high-dimensional weight tensor as a low-rank approximation. An iterative algorithm based on the alternating linear scheme is developed to directly estimate the low-rank tensor network approximation, removing the need to ever explicitly construct the exponentially large weight tensor. This reduces the computational and storage complexity significantly, allowing the identification of NARX systems with a large number of inputs and lags. The proposed algorithm is numerically stable, robust to noise, guaranteed to monotonically converge, and allows the straightforward incorporation of regularization. The TNBS-NARX model is validated through the identification of the cascaded watertank benchmark nonlinear system, on which it achieves state-of-the-art performance while identifying a 16-dimensional B-spline surface in 4 seconds on a standard desktop computer. An open-source MATLAB implementation is available on GitHub.

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Tensor network subspace identification of polynomial state space models

Cited by 17 publications

References 17 publications

Extending Fuzzy Cognitive Maps with Tensor-Based Distance Metrics

Extending Fuzzy Cognitive Maps with Tensor-Based Distance Metrics

Noniterative tensor network‐based algorithm for Volterra system identification

Nonlinear system identification with regularized Tensor Network B-splines

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