Paris A. Karakasis

Paris A. Karakasis, Nicholas D. Sidiropoulos

We study low-rank tensor-product B-spline (TPBS) models for regression tasks and investigate Dirichlet energy as a measure of smoothness. We show that TPBS models admit a closed-form expression for the Dirichlet energy, and reveal scenarios where perfect interpolation is possible with exponentially small Dirichlet energy. This renders global Dirichlet energy-based regularization ineffective. To address this limitation, we propose a novel regularization strategy based on local Dirichlet energies defined on small hypercubes centered at the training points. Leveraging pretrained TPBS models, we also introduce two estimators for inference from incomplete samples. Comparative experiments with neural networks demonstrate that TPBS models outperform neural networks in the overfitting regime for most datasets, and maintain competitive performance otherwise. Overall, TPBS models exhibit greater robustness to overfitting and consistently benefit from regularization, while neural networks are more sensitive to overfitting and less effective in leveraging regularization.

Paris A. Karakasis, Nicholas D. Sidiropoulos

Canonical correlation analysis (CCA) is a classic statistical method for discovering latent co-variation that underpins two or more observed random vectors. Several extensions and variations of CCA have been proposed that have strengthened our capabilities in terms of revealing common random factors from multiview datasets. In this work, we first revisit the most recent deterministic extensions of deep CCA and highlight the strengths and limitations of these state-of-the-art methods. Some methods allow trivial solutions, while others can miss weak common factors. Others overload the problem by also seeking to reveal what is not common among the views -- i.e., the private components that are needed to fully reconstruct each view. The latter tends to overload the problem and its computational and sample complexities. Aiming to improve upon these limitations, we design a novel and efficient formulation that alleviates some of the current restrictions. The main idea is to model the private components as conditionally independent given the common ones, which enables the proposed compact formulation. In addition, we also provide a sufficient condition for identifying the common random factors. Judicious experiments with synthetic and real datasets showcase the validity of our claims and the effectiveness of the proposed approach.

Paris A. Karakasis, Nicholas D. Sidiropoulos

We introduce a novel framework for clustering a collection of tall matrices based on their column spaces, a problem we term Subspace Clustering of Subspaces (SCoS). Unlike traditional subspace clustering methods that assume vectorized data, our formulation directly models each data sample as a matrix and clusters them according to their underlying subspaces. We establish conceptual links to Subspace Clustering and Generalized Canonical Correlation Analysis (GCCA), and clarify key differences that arise in this more general setting. Our approach is based on a Block Term Decomposition (BTD) of a third-order tensor constructed from the input matrices, enabling joint estimation of cluster memberships and partially shared subspaces. We provide the first identifiability results for this formulation and propose scalable optimization algorithms tailored to large datasets. Experiments on real-world hyperspectral imaging datasets demonstrate that our method achieves superior clustering accuracy and robustness, especially under high noise and interference, compared to existing subspace clustering techniques. These results highlight the potential of the proposed framework in challenging high-dimensional applications where structure exists beyond individual data vectors.