To address the scalability limitations of Gaussian process (GP) regression, several approximation techniques have been proposed. One such method is based on tensor networks, which utilizes an exponential number of basis functions without incurring exponential computational cost. However, extending this model to a fully probabilistic formulation introduces several design challenges. In particular, for tensor train (TT) models, it is unclear which TT-core should be treated in a Bayesian manner. We introduce a Bayesian tensor train kernel machine that applies Laplace approximation to estimate the posterior distribution over a selected TT-core and employs variational inference (VI) for precision hyperparameters. Experiments show that core selection is largely independent of TT-ranks and feature structure, and that VI replaces cross-validation while offering up to 65x faster training. The method's effectiveness is demonstrated on an inverse dynamics problem.

Variational Quantum Computing (VQC) faces fundamental barriers in scalability, primarily due to barren plateaus and quantum noise sensitivity. To address these challenges, we introduce TensoMeta-VQC, a novel tensor-train (TT)-guided meta-learning framework designed to improve the robustness and scalability of VQC significantly. Our framework fully delegates the generation of quantum circuit parameters to a classical TT network, effectively decoupling optimization from quantum hardware. This innovative parameterization mitigates gradient vanishing, enhances noise resilience through structured low-rank representations, and facilitates efficient gradient propagation. Based on Neural Tangent Kernel and statistical learning theory, our rigorous theoretical analyses establish strong guarantees on approximation capability, optimization stability, and generalization performance. Extensive empirical results across quantum dot classification, Max-Cut optimization, and molecular quantum simulation tasks demonstrate that TensoMeta-VQC consistently achieves superior performance and robust noise tolerance, establishing it as a principled pathway toward practical and scalable VQC on near-term quantum devices.

The state-of-the-art tensor network Kalman filter lifts the curse of dimensionality for high-dimensional recursive estimation problems. However, the required rounding operation can cause filter divergence due to the loss of positive definiteness of covariance matrices. We solve this issue by developing, for the first time, a tensor network square root Kalman filter, and apply it to high-dimensional online Gaussian process regression. In our experiments, we demonstrate that our method is equivalent to the conventional Kalman filter when choosing a full-rank tensor network. Furthermore, we apply our method to a real-life system identification problem where we estimate $4^{14}$ parameters on a standard laptop. The estimated model outperforms the state-of-the-art tensor network Kalman filter in terms of prediction accuracy and uncertainty quantification.

Effective numerical techniques, such as CANDECOMP/PARAFAC (CP) decomposition [10, 11, 12] and Tucker decomposition [13, 14] are the most commonly used tensor decomposition approaches and have been proposed to compress full tensors and to obtain their low-rank representations. CP decomposition approximates a tensor by a sum of rank-one tensors, while Tucker decomposition decomposes a tensor into a core tensor and several factor matrices. Since CP decomposition can be seen as a special case of Tucker decomposition [15], Tucker decomposition is more flexible than CP decomposition. However, due to the existence of a core tensor, Tucker decomposition also brings challenges in both modeling and computation. In this paper, we mainly focus on Tensor Train (TT) decomposition [16], which combines the advantages of CP and Tucker decomposition, because it provides a space-saving model called TT format while preserving the representation power. This paper is interested in the decomposition of streaming data. Due to the stress on database capacity and privacy, streaming data is generated continuously by different of data sources and in small sizes, such as log files from web application [17] and information from social networks [18]. Recently, several works decompose fast streaming data, e.g.

Learning neural fields has been an active topic in deep learning research, focusing, among other issues, on finding more compact and easy-to-fit representations. In this paper, we introduce a novel low-rank representation termed Tensor Train Neural Fields (TT-NF) for learning neural fields on dense regular grids and efficient methods for sampling from them. Our representation is a TT parameterization of the neural field, trained with backpropagation to minimize a non-convex objective. We analyze the effect of low-rank compression on the downstream task quality metrics in two settings. First, we demonstrate the efficiency of our method in a sandbox task of tensor denoising, which admits comparison with SVD-based schemes designed to minimize reconstruction error. Furthermore, we apply the proposed approach to Neural Radiance Fields, where the low-rank structure of the field corresponding to the best quality can be discovered only through learning. Following the growing interest in deep neural networks, learning neural fields has become a promising research direction in areas concerned with structured representations. However, precision is usually at odds with the computational complexity of these representations, which makes training them and sampling from them a challenge. In this paper, we investigate interpretable low-rank neural fields defined on dense regular grids and efficient methods for learning them. Since, in extreme cases, the dimensionality of such fields can exceed the memory size of a typical computer by several orders of magnitude, we look at the problem of learning such fields from the angle of stochastic methods. Tensor decompositions have become a ubiquitous tool for dealing with structured sparsity of intractable volumes of data.

Physics-informed neural networks (PINNs) have been increasingly employed due to their capability of modeling complex physics systems. To achieve better expressiveness, increasingly larger network sizes are required in many problems. This has caused challenges when we need to train PINNs on edge devices with limited memory, computing and energy resources. To enable training PINNs on edge devices, this paper proposes an end-to-end compressed PINN based on Tensor-Train decomposition. In solving a Helmholtz equation, our proposed model significantly outperforms the original PINNs with few parameters and achieves satisfactory prediction with up to 15$\times$ overall parameter reduction.

We study low-rank parameterizations of weight matrices with embedded spectral properties in the Deep Learning context. The low-rank property leads to parameter efficiency and permits taking computational shortcuts when computing mappings. Spectral properties are often subject to constraints in optimization problems, leading to better models and stability of optimization. We start by looking at the compact SVD parameterization of weight matrices and identifying redundancy sources in the parameterization. We further apply the Tensor Train (TT) decomposition to the compact SVD components, and propose a non-redundant differentiable parameterization of fixed TT-rank tensor manifolds, termed the Spectral Tensor Train Parameterization (STTP). We demonstrate the effects of neural network compression in the image classification setting and both compression and improved training stability in the generative adversarial training setting.