arXiv subjects

Alexander Balinsky

Publications and source records attributed to Alexander Balinsky.

4 recordsLinked to original sources

arXiv2026

On closed linear subspaces embedded into Banach spaces and their finite-dimensionality

This paper studies a Grothendieck-type finite-dimensionality problem for closed linear subspaces embedded in Banach spaces. Let $S_p^{(q)} \subset L_p(M,d\mu)$ be a closed linear subspace of the Banach space $L_p(M,d\mu)$ defined with respect to a probability measure $d\mu$ on $M$. We prove that if $S_p^{(q)}$ is continuously embedded into $L_q(M,d\mu)$ for $q>p$, then its dimension $\dim S_p^{(q)} = N \in \mathbb{N}$ satisfies the estimate \[ \frac{1}{N}\left(\frac{\sqrt{\pi},\Gamma!\left(\frac{N+\tilde q}{2}\right)}{\Gamma!\left(\frac{\tilde q+1}{2}\right)\Gamma!\left(\frac{N}{2}\right)}\right)^{2/\tilde q}\le K_{p,q(m)}^2, \] where $1/\tilde q + 1/q = 1$, $q = 2 + (p-2)2^m > p$ with $p \neq 2$ and $m \in \mathbb{N}$, and $K_{p,q(m)}>0$ is a bounded constant. We also prove that certain closed linear subspaces of $L_p(M,d\mu)$ consisting of continuous functions on $M$ must be finite-dimensional.

math.FA↗

arXiv2025

Conformal Prediction for Privacy-Preserving Machine Learning

We investigate the integration of Conformal Prediction (CP) with supervised learning on deterministically encrypted data, aiming to bridge the gap between rigorous uncertainty quantification and privacy-preserving machine learning. Using AES-encrypted variants of the MNIST dataset, we demonstrate that CP methods remain effective even when applied directly in the encrypted domain, owing to the preservation of data exchangeability under fixed-key encryption. We test traditional $p$-value-based against $e$-value-based conformal predictors. Our empirical evaluation reveals that models trained on deterministically encrypted data retain the ability to extract meaningful structure, achieving 36.88\% test accuracy -- significantly above random guessing (9.56\%) observed with per-instance encryption. Moreover, $e$-value-based CP achieves predictive set coverage of over 60\% with 4.3 loss-threshold calibration, correctly capturing the true label in 4888 out of 5000 test cases. In contrast, the $p$-value-based CP yields smaller predictive sets but with reduced coverage accuracy. These findings highlight both the promise and limitations of CP in encrypted data settings and underscore critical trade-offs between prediction set compactness and reliability. %Our work sets a foundation for principled uncertainty quantification in secure, privacy-aware learning systems.

cs.LG↗

arXiv2009

Colorization of Natural Images via L1 Optimization

Natural images in the colour space YUV have been observed to have a non-Gaussian, heavy tailed distribution (called 'sparse') when the filter G(U)(r) = U(r) - sum_{s \in N(r)} w{(Y)_{rs}} U(s), is applied to the chromacity channel U (and equivalently to V), where w is a weighting function constructed from the intensity component Y [1]. In this paper we develop Bayesian analysis of the colorization problem using the filter response as a regularization term to arrive at a non-convex optimization problem. This problem is convexified using L1 optimization which often gives the same results for sparse signals [2]. It is observed that L1 optimization, in many cases, over-performs the famous colorization algorithm by Levin et al [3].

cs.CV↗

arXiv2007

Some Sharp L^2 Inequalities for Dirac Type Operators

We use the spectra of Dirac type operators on the sphere $S^{n}$ to produce sharp $L^{2}$ inequalities on the sphere. These operators include the Dirac operator on $S^{n}$, the conformal Laplacian and Paenitz operator. We use the Cayley transform, or stereographic projection, to obtain similar inequalities for powers of the Dirac operator and their inverses in $\mathbb{R}^{n}$.

math-ph↗