In the context of kernel machines, polynomial and Fourier features are commonly used to provide a nonlinear extension to linear models by mapping the data to a higher-dimensional space. Unless one considers the dual formulation of the learning problem, which renders exact large-scale learning unfeasible, the exponential increase of model parameters in the dimensionality of the data caused by their tensor-product structure prohibits to tackle high-dimensional problems. One of the possible approaches to circumvent this exponential scaling is to exploit the tensor structure present in the features by constraining the model weights to be an underparametrized tensor network. In this paper we quantize, i.e. further tensorize, polynomial and Fourier features. Based on this feature quantization we propose to quantize the associated model weights, yielding quantized models. We show that, for the same number of model parameters, the resulting quantized models have a higher bound on the VC-dimension as opposed to their non-quantized counterparts, at no additional computational cost while learning from identical features. We verify experimentally how this additional tensorization regularizes the learning problem by prioritizing the most salient features in the data and how it provides models with increased generalization capabilities. We finally benchmark our approach on large regression task, achieving state-of-the-art results on a laptop computer. @en

For the first time, this position paper introduces a fundamental link between tensor networks (TNs) and Green AI, highlighting their synergistic potential to enhance both the inclusivity and sustainability of AI research. We argue that TNs are valuable for Green AI due to their strong mathematical backbone and inherent logarithmic compression potential. We undertake a comprehensive review of the ongoing discussions on Green AI, emphasizing the importance of sustainability and inclusivity in AI research to demonstrate the significance of establishing the link between Green AI and TNs. To support our position, we first provide a comprehensive overview of efficiency metrics proposed in Green AI literature and then evaluate examples of TNs in the fields of kernel machines and deep learning using the proposed efficiency metrics. This position paper aims to incentivize meaningful, constructive discussions by bridging fundamental principles of Green AI and TNs. We advocate for researchers to seriously evaluate the integration of TNs into their research projects, and in alignment with the link established in this paper, we support prior calls encouraging researchers to treat Green AI principles as a research priority.

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This paper presents a method for approximate Gaussian process (GP) regression with tensor networks (TNs). A parametric approximation of a GP uses a linear combination of basis functions, where the accuracy of the approximation depends on the total number of basis functions M. We develop an approach that allows us to use an exponential amount of basis functions without the corresponding exponential computational complexity. The key idea to enable this is using low-rank TNs. We first find a suitable low-dimensional subspace from the data, described by a low-rank TN. In this low-dimensional subspace, we then infer the weights of our model by solving a Bayesian inference problem. Finally, we project the resulting weights back to the original space to make GP predictions. The benefit of our approach comes from the projection to a smaller subspace: It modifies the shape of the basis functions in a way that it sees fit based on the given data, and it allows for efficient computations in the smaller subspace. In an experiment with an 18-dimensional benchmark data set, we show the applicability of our method to an inverse dynamics problem.

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This paper proposes a Bayesian Volterra tensor network (TN) to solve high-order discrete nonlinear multiple-input multiple-output (MIMO) Volterra system identification problems. Using a low-rank tensor network to compress all Volterra kernels at once, we avoid the exponential growth of monomials with respect to the order of the Volterra kernel. Our contribution is to introduce a Bayesian framework for the low-rank Volterra TN. Compared to the least squares solution for Volterra TNs, we include prior assumptions explicitly in the model. In particular, we show for the first time how a zero-mean prior with diagonal covariance matrix corresponds to implementing a Tikhonov regularization for the MIMO Volterra TN. Furthermore, adopting a Bayesian viewpoint enables simulations with Bayesian uncertainty bounds based on noise and prior assumptions. In addition, we demonstrate via numerical experiments how Tikhonov regularization prevents overfitting in the case of higher-rank TNs.

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Kernel machines are one of the most studied family of methods in machine learning. In the exact setting, training requires to instantiate the kernel matrix, thereby prohibiting their application to large-sampled data. One popular kernel approximation strategy which allows to tackle large-sampled data consists in interpolating product kernels on a set of grid-structured inducing points. However, since the number of model parameters increases exponentially with the dimensionality of the data, these methods are limited to small-dimensional datasets. In this work we lift this limitation entirely by placing inducing points on a grid and constraining the primal weights to be a low-rank Canonical Polyadic Decomposition. We derive a block coordinate descent algorithm that efficiently exploits grid-structured inducing points. The computational complexity of the algorithm scales linearly both in the number of samples and in the dimensionality of the data for any product kernel. We demonstrate the performance of our algorithm on large-scale and high-dimensional data, achieving state-of-the art results on a laptop computer. Our results show that grid-structured approaches can work in higher-dimensional problems.

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The identification of symmetric tensor network MIMO Volterra models has been studied earlier via the computation of a Moore-Penrose pseudoinverse in tensor network form. The current state of the art requires the construction of a tensor network of a repeated Khatri-Rao product of a matrix with itself. This construction has a computational complexity that is dominated by one singular value decomposition (SVD) of an RI × IN matrix, where N is the number of measurements, I depends linearly on the number of inputs and input lags and R is the maximal tensor network rank. In this article, we prove an alternative method for constructing this tensor network without any computation whatsoever. The pseudoinverse can then be computed through an orthogonalization of the newly proposed tensor network. Furthermore, the proposed algorithm allows for the recursive identification of symmetric Volterra models of increasing degree D, which reduces the computation to one SVD of a RI × N matrix per step. Through numerical experiments we demonstrate how the proposed algorithm enables up to ten times faster identification of symmetric tensor network MIMO Volterra systems.

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Recent advancements in wearable EEG devices have highlighted the importance of accurate seizure detection algorithms, yet the ever-increasing size of the generated datasets poses a significant challenge to existing seizure detection methods based on kernel machines. Typically, this problem is mitigated by significantly undersampling the majority class, but in practice, these methods tend to suffer from too many false alarms. Recent works have proposed tensor networks to enable large-scale classification with kernel machines. In this paper, we explore the use of a probabilistic tensor method, the tensor-network Kalman filter for LS-SVMs (TNKF-LSSVM), for seizure detection, as we hypothesize that using more data will improve the detection performance. We show that the TNKF-LSSVM performs comparably to a regular LSSVM in detecting seizures when both are trained on the same dataset. Additionally, the TNKF-LSSVM can provide meaningful uncertainty quantification, and it is able to handle large-scale datasets beyond the capabilities of the LS-SVM (i.e., $N \gt 10 ^{5})$. However, for the presented model configuration detection performance does not seem to improve with more input data.

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Multiway data often naturally occurs in a tensorial format which can be approximately represented by a low-rank tensor decomposition. This is useful because complexity can be significantly reduced and the treatment of large-scale data sets can be facilitated. In this paper, we find a low-rank representation for a given tensor by solving a Bayesian inference problem. This is achieved by dividing the overall inference problem into subproblems where we sequentially infer the posterior distribution of one tensor decomposition component at a time. This leads to a probabilistic interpretation of the well-known iterative algorithm alternating linear scheme (ALS). In this way, the consideration of measurement noise is enabled, as well as the incorporation of application-specific prior knowledge and the uncertainty quantification of the low-rank tensor estimate. To compute the low-rank tensor estimate from the posterior distributions of the tensor decomposition components, we present an algorithm that performs the unscented transform in tensor train format.

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Tensor, a multi-dimensional data structure, has been exploited recently in the machine learning community. Traditional machine learning approaches are vector- or matrix-based, and cannot handle tensorial data directly. In this paper, we propose a tensor train (TT)-based kernel technique for the first time, and apply it to the conventional support vector machine (SVM) for high-dimensional image classification with very small number of training samples. Specifically, we propose a kernelized support tensor train machine that accepts tensorial input and preserves the intrinsic kernel property. The main contributions are threefold. First, we propose a TT-based feature mapping procedure that maintains the TT structure in the feature space. Second, we demonstrate two ways to construct the TT-based kernel function while considering consistency with the TT inner product and preservation of information. Third, we show that it is possible to apply different kernel functions on different data modes. In principle, our method tensorizes the standard SVM on its input structure and kernel mapping scheme. This reduces the storage and computation complexity of kernel matrix construction from exponential to polynomial. The validity proof and computation complexity of the proposed TT-based kernel functions are provided elaborately. Extensive experiments are performed on high-dimensional fMRI and color images datasets, which demonstrates the superiority of the proposed scheme compared with the state-of-the-art techniques.

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Nonlinear parametric system identification is the estimation of nonlinear models of dynamical systems from measured data. Nonlinear models are parameterized, and it is exactly these parameters that must be estimated. Extending familiar linear models to their nonlinear counterparts quickly leads to practical problems. For example, the generalization of a multivariate linear function to a multivariate polynomial implies that the number of parameters grows exponentially with the total degree of the polynomial. This exponential explosion of model parameters is an instance of the so-called curse of dimensionality. Both the storage and computational complexities are limiting factors in the development of system identification methods for such models.

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In recent years, the application of tensors has become more widespread in fields that involve data analytics and numerical computation. Due to the explosive growth of data, low-rank tensor decompositions have become a powerful tool to harness the notorious curse of dimensionality. The main forms of tensor decomposition include CP decomposition, Tucker decomposition, tensor train (TT) decomposition, etc. Each of the existing TT decomposition algorithms, including the TT-SVD and randomized TT-SVD, is successful in the field, but neither can both accurately and efficiently decompose large-scale sparse tensors. Based on previous research, this paper proposes a new quasi-optimal fast TT decomposition algorithm for large-scale sparse tensors with proven correctness and the upper bound of computational complexity derived. It can also efficiently produce sparse TT with no numerical error and slightly larger TT-ranks on demand. In numerical experiments, we verify that the proposed algorithm can decompose sparse tensors in a much faster speed than the TT-SVD, and have advantages on speed, precision and versatility over the randomized TT-SVD and TT-cross. And, with it we can realize large-scale sparse matrix TT decomposition that was previously unachievable, enabling the tensor decomposition based algorithms to be applied in more scenarios.

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In this article, two new algorithms are presented that convert a given data tensor train into either a Tucker decomposition with orthogonal matrix factors or a multi-scale entanglement renormalization ansatz (MERA). The Tucker core tensor is never explicitly computed but stored as a tensor train instead, resulting in both computationally and storage efficient algorithms. Both the multilinear Tucker-ranks as well as the MERA-ranks are automatically determined by the algorithm for a given upper bound on the relative approximation error. In addition, an iterative algorithm with low computational complexity based on solving an orthogonal Procrustes problem is proposed for the first time to retrieve optimal rank-lowering disentangler tensors, which are a crucial component in the construction of a low-rank MERA. Numerical experiments demonstrate the effectiveness of the proposed algorithms together with the potential storage benefit of a low-rank MERA over a tensor train.

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The estimation of an exponential number of model parameters in a truncated Volterra model can be circumvented by using a low-rank tensor decomposition approach. This low-rank property of the tensor decomposition can be interpreted as the assumption that all Volterra parameters are structured. In this article, we investigate whether it is possible to explicitly enforce symmetry of the Volterra kernels to the low-rank tensor decomposition. We show that low-rank symmetric Volterra identification is an ill-conditioned problem as the low-rank property of the exact symmetric kernels cannot be upheld in the presence of measurement noise. Furthermore, an algorithm is derived to compute the symmetric Volterra kernels directly in tensor network form.

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This article introduces the Tensor Network B-spline (TNBS) 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 s on a standard desktop computer. An open-source MATLAB implementation is available on GitHub.

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Sum-product networks (SPNs) constitute an emerging class of neural networks with clear probabilistic semantics and superior inference speed over other graphical models. This brief reveals an important connection between SPNs and tensor trains (TTs), leading to a new canonical form which we call tensor SPNs (tSPNs). Specifically, we demonstrate the intimate relationship between a valid SPN and a TT. For the first time, through mapping an SPN onto a tSPN and employing specially customized optimization techniques, we demonstrate improvements up to a factor of 100 on both model compression and inference speedup for various data sets with negligible loss in accuracy.

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We propose a new tensor completion method based on tensor trains. The to-be-completed tensor is modeled as a low-rank tensor train, where we use the known tensor entries and their coordinates to update the tensor train. A novel tensor train initialization procedure is proposed specifically for image and video completion, which is demonstrated to ensure fast convergence of the completion algorithm. The tensor train framework is also shown to easily accommodate Total Variation and Tikhonov regularization due to their low-rank tensor train representations. Image and video inpainting experiments verify the superiority of the proposed scheme in terms of both speed and scalability, where a speedup of up to 155\times is observed compared to state-of-the-art tensor completion methods at a similar accuracy. Moreover, we demonstrate the proposed scheme is especially advantageous over existing algorithms when only tiny portions (say, 1%) of the to-be-completed images/videos are known.

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An extension of the Tensor Network (TN) Kalman filter [2], [3] for large scale LTI systems is presented in this paper. The TN Kalman filter can handle exponentially large state vectors without constructing them explicitly. In order to have efficient algebraic operations, a low TN rank is required. We exploit the possibility to approximate the covariance matrix as a TN with a low TN rank. This reduces the computational complexity for general SISO and MIMO LTI systems with TN rank greater than one significantly while obtaining an accurate estimation. Improvements of this method in terms of computational complexity compared to the conventional Kalman filter are demonstrated in numerical simulations for large scale systems.

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A restricted Boltzmann machine (RBM) learns a probability distribution over its input samples and has numerous uses like dimensionality reduction, classification and generative modeling. Conventional RBMs accept vectorized data that dismiss potentially important structural information in the original tensor (multi-way) input. Matrix-variate and tensor-variate RBMs, named MvRBM and TvRBM, have been proposed but are all restrictive by model construction and have weak model expression power. This work presents the matrix product operator RBM (MPORBM) that utilizes a tensor network generalization of Mv/TvRBM, preserves input formats in both the visible and hidden layers, and results in higher expressive power. A novel training algorithm integrating contrastive divergence and an alternating optimization procedure is also developed. Numerical experiments compare the MPORBM with the traditional RBM and MvRBM for data classification and image completion and denoising tasks. The expressive power of the MPORBM as a function of the MPO-rank is also investigated.

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There has been growing interest in extending traditional vector-based machine learning techniques to their tensor forms. Support tensor machine (STM) and support Tucker machine (STuM) are two typical tensor generalization of the conventional support vector machine (SVM). However, the expressive power of STM is restrictive due to its rank-one tensor constraint, and STuM is not scalable because of the exponentially sized Tucker core tensor. To overcome these limitations, we introduce a novel and effective support tensor train machine (STTM) by employing a general and scalable tensor train as the parameter model. Experiments validate and confirm the superiority of the STTM over SVM, STM and STuM.

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This article reformulates the multiple-input-multiple-output Volterra system identification problem as an extended Kalman filtering problem. This reformulation has two advantages. First, it results in a simplification of the solution compared to the Tensor Network Kalman filter as no tensor filtering equations are required anymore. The second advantage is that the reformulation allows to model correlations between the parameters of different multiple-input-single-output Volterra systems, which can lead to better accuracy. The curse of dimensionality in the exponentially large parameter vector and covariance matrix is lifted through the use of low-rank tensor networks. The computational complexity of our tensor network implementation is compared to the conventional implementation and numerical experiments demonstrate the effectiveness of the proposed method.

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