(MPO)$^2$: Multivariate Polynomial Optimization based on Matrix Product Operators

Central to machine learning and signal processing is the ability to perform universal function approximation and learn complex input-output relationships from limited numbers of observations. Multivariate polynomial models offer a natural way to express such relationships through multiplicative feature interactions, but their coefficient tensors grow exponentially in size with the polynomial degree. Existing tensorized polynomial models reduce this cost, yet canonical polyadic decompositions have rank-limited expressivity, and tensor train formulations are feature order dependent. We introduce Multivariate Polynomial Optimization based on Matrix Product Operators (MPO)$^2$, a framework that combines learned MPO feature embeddings with compact polynomial weight tensors. This yields feature order independent polynomial representations that can incorporate structured operators such as projections, convolutions, and masks for weight tensor symmetries. Across regression and classification benchmarks, (MPO)$^2$ improves over existing tensor decomposition based polynomial models and provides a flexible alternative for efficient polynomial function approximation.

All you need is SAMPAT

The current state of the art in AI/ML rests on deep neural architectures, which, in general, suffer from a lack of interpretability. Interpretability is crucial to gleaning insights while analyzing experimental data, where quantitative predictions may not be adequate for a scientist. We present a three layer neural architecture, SAMPAT (Smooth Approximation via Multivariate Polynomials and Analytic Transformations), that can provably learn a continuous, everywhere differentiable function, that can approximate any smooth function arbitrarily closely. SAMPAT's approximant can be expressed as a closed and compact algebraic, analytic expression, providing complete interpretability. Experiments on synthetic and benchmark datasets indicate that SAMPAT yields competitive performance with simpler representations. For many tasks, a two layer SAMPAT suffices. By imposing restrictions on the connectivity between neurons, SAMPAT may be used to provide a range of approximants, including regular and trigonometric polynomials, rational expressions, Gaussians, mixtures of Gaussians, as well as arbitrary combinations of the same; without restrictions, it learns a suitable structure. SAMPAT may be used to factorize polynomials and model nonlinear systems. With the addition of skip connections, a 4 to 6 layer SAMPAT is adequate to represent a substantive range of methods widely used in AI/ML, allowing the choice of a model's family, not just its parameters, to also be optimized as part of the learning process.

Hierarchical RBF-KAN and RBF-SKAN Architectures for Multidimensional Function Approximation and Random Field Learning

In this manuscript, we propose and analyze hierarchical Kolmogorov--Arnold neural network architectures employing radial basis functions as activation functions for approximating deterministic functions and random field models. Specifically, we develop a hierarchical radial-basis-function Kolmogorov--Arnold network (hierarchical RBF-KAN) for multidimensional deterministic function approximation and a hierarchical radial-basis-function stochastic Kolmogorov--Arnold network (hierarchical RBF-SKAN) for random field learning. From a theoretical perspective, we establish universal approximation results for both architectures. In particular, we derive quantitative approximation estimates for the hierarchical RBF-KAN, showing that the proposed framework has the potential to partially alleviate the curse of dimensionality in learning high-dimensional functions by reducing the effective dimensionality of the approximation problem. Furthermore, we show that the hierarchical RBF-SKAN can approximate random field models under the Wasserstein-2 metric. Empirically, we show that our proposed radial-basis-function-based neural network structure could effectively learn multivariate functions and random field models.

Holographic functions and neural networks

A fuzzy Boolean function is a map $f:\cube^n\to [0,1]$, where $n\in\mathbb N$. We introduce and compare three ways of saying that such a function has bounded complexity. The first is a sampling property: the value $f(x)$ can be recovered, up to small error and with high probability, from the values of a bounded number of randomly chosen coordinates of $x$. We call this the holographic property. The second is a structural property: $f$ is uniformly close to a bounded-degree polynomial in boundedly many bounded linear coordinate forms. The third is computational: $f$ is uniformly close to the output of a neural network with a bounded number of non-input neurons, bounded Lipschitz activation functions and bounded incoming weights. We prove that these three properties are equivalent up to quantitative changes of the parameters. The implication from holography to polynomial structure uses a variant of a weak version of hypergraph regularity.