Tensor Network Kernel Machines: A JAX Framework for Machine Learning and Nonlinear System Identification
Authors: Albert Saiapin, Kim Batselier
Organizations: Delft Center for Systems and Control, TU Delft, Netherlands
Abstract
Developing nonlinear models that are both expressive and computationally efficient remains a challenge in machine learning and nonlinear system identification. Tensor network kernel machines (TNKM) address this challenge by combining nonlinear feature representations with compact low-rank tensor-network parameterizations. However, practical and extensible software frameworks for developing TNKM models remain limited. In this work, we introduce "tnkm", an open-source Python library for constructing and training TNKM models using JAX. The library provides a unified interface for combining different feature maps, tensor-network architectures, and optimization strategies, including alternating least squares and gradient-based methods. We demonstrate the capabilities of "tnkm" on nonlinear benchmark problems, showing that the implemented models achieve competitive prediction accuracy while retaining compact parameterizations and efficient training. The proposed framework facilitates reproducible development and application of tensor-network-based learning methods.
TorchKM is an open-source library for kernel machines, including support vector machines, kernel logistic regression, and kernel quantile regression, with GPU acceleration. The library features a scikit-learn-style API and is designed to exploit GPU-friendly linear algebra, accelerating the full training and model-selection pipeline through intelligent reuse of matrix operations. Benchmarks show competitive predictive performance with substantial speedups over standard baselines. The efficiency and programmable design also make TorchKM a kernel-learning component for AI-driven workflows. Code and documentation are available at https://github.com/YikaiZhang95/torchkm, and the package can be easily installed via PyPI.
Uncertainty estimation is essential for robust decision-making in the presence of ambiguous or out-of-distribution inputs. Gaussian Processes (GPs) are classical kernel-based models that offer principled uncertainty quantification and perform well on small- to medium-scale datasets. Alternatively, formulating the weight space learning problem under tensor network assumptions yields scalable tensor network kernel machines. However, these assumptions break Gaussianity, complicating standard probabilistic inference. This raises a fundamental question: how can tensor network kernel machines provide principled uncertainty estimates? We propose a novel Bayesian Tensor Network Kernel Machine (LA-TNKM) that employs a (linearized) Laplace approximation for Bayesian inference. A comprehensive set of numerical experiments shows that the proposed method consistently matches or surpasses Gaussian Processes and Bayesian Neural Networks (BNNs) across diverse UCI regression benchmarks, highlighting both its effectiveness and practical relevance.
Tensor networks, originally developed for quantum many-body physics, are promising models for machine learning. We derive stochastic Riemannian optimizers for tree tensor networks (TTNs) on both their parameter and quotient manifolds, including adaptive and learning-rate-free schemes suitable for minibatch training. Using a hybrid CNN-TTN architecture, we evaluate the methods on Fashion-MNIST, CIFAR10, and Imagenette. The proposed optimizers achieve predictive performance comparable to unconstrained optimization while enabling numerically stable downstream compression.
Marius Willner, Maximilian Scharf, André Uschmajew +2