arXiv subjects

Uliana Parkina

Publications and source records attributed to Uliana Parkina.

4 recordsLinked to original sources

arXiv2026

Fast Differentiable SVD on GPU via Polar Decomposition

We present a fully GPU-oriented SVD pipeline based on polar decomposition, motivated by iterative methods that rely solely on matrix multiplications, such as the Newton-Schulz iteration. We show that this approach enables up to a $2\times$ speedup compared to standard implementations. Furthermore, we derive a numerically stable backward pass for the polar decomposition and leverage it to obtain a fully differentiable SVD. Our methods are released as open-source implementations in both PyTorch and JAX: https://github.com/fallnlove/cans_svd.

math.NA↗

arXiv2026

A lower bound for $\langle 3,2,m \rangle$ matrix multiplication

We prove that, over any field, the bilinear complexity of multiplying a $3\times 2$ matrix by a $2\times m$ matrix is strictly greater than $24m/5$. In particular, every exact bilinear algorithm for multiplying a $3\times 2$ matrix by a $2\times 5$ matrix requires at least $25$ multiplications. Together with the Hopcroft-Kerr upper bound, this proves that the $\langle 3,2,5\rangle$ matrix multiplication tensor has rank exactly $25$. The proof has been formally verified in Lean 4, with the formalization available at https://github.com/fallnlove/mm325_proof.

cs.CC↗

arXiv2026

Bradley-Terry Rankings for Recommender Systems Across Dataset Taxonomies

The ranking of recommendation algorithms is a challenging problem since model performance is sensitive to dataset characteristics such as sparsity, sequential structure, and scale. This drives a demand for a proper methodology for fair comparison between algorithms. Naive aggregation of performance metrics (e.g., averaging NDCG over benchmarks) can yield misleading rankings, undermining practical selection. To address this problem, we introduce a novel, data-driven ranking methodology based on Bradley-Terry (BT) model. We demonstrate that the obtained ranking depends on key dataset statistics. Additionally, we propose a novel metric for evaluating ranking consistency and demonstrate robustness of our ranking to incomplete data. Finally, we introduce a dataset-specific methodology for ranking algorithms on unseen datasets without running the models, relying on extensions of the Bradley-Terry framework, including BT trees and BT models with covariates.

cs.IR↗

arXiv2026

COALA: Numerically Stable and Efficient Framework for Context-Aware Low-Rank Approximation

Recent studies suggest that context-aware low-rank approximation is a useful tool for compression and fine-tuning of modern large-scale neural networks. In this type of approximation, a norm is weighted by a matrix of input activations, significantly improving metrics over the unweighted case. Nevertheless, existing methods for neural networks suffer from numerical instabilities due to their reliance on classical formulas involving explicit Gram matrix computation and their subsequent inversion. We demonstrate that this can degrade the approximation quality or cause numerically singular matrices. To address these limitations, we propose a novel inversion-free regularized framework that is based entirely on stable decompositions and overcomes the numerical pitfalls of prior art. Our method can handle possible challenging scenarios: (1) when calibration matrices exceed GPU memory capacity, (2) when input activation matrices are nearly singular, and even (3) when insufficient data prevents unique approximation. For the latter, we prove that our solution converges to a desired approximation and derive explicit error bounds.

cs.LG↗