Petr Taborsky | Searcharxiv

arXiv2026

(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.

cs.LG↗

arXiv2021

Generalization by design: Shortcuts to Generalization in Deep Learning

We take a geometrical viewpoint and present a unifying view on supervised deep learning with the Bregman divergence loss function - this entails frequent classification and prediction tasks. Motivated by simulations we suggest that there is principally no implicit bias of vanilla stochastic gradient descent training of deep models towards "simpler" functions. Instead, we show that good generalization may be instigated by bounded spectral products over layers leading to a novel geometric regularizer. It is revealed that in deep enough models such a regularizer enables both, extreme accuracy and generalization, to be reached. We associate popular regularization techniques like weight decay, drop out, batch normalization, and early stopping with this perspective. Backed up by theory we further demonstrate that "generalization by design" is practically possible and that good generalization may be encoded into the structure of the network. We design two such easy-to-use structural regularizers that insert an additional \textit{generalization layer} into a model architecture, one with a skip connection and another one with drop-out. We verify our theoretical results in experiments on various feedforward and convolutional architectures, including ResNets, and datasets (MNIST, CIFAR10, synthetic data). We believe this work opens up new avenues of research towards better generalizing architectures.

cs.LG↗