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A. A. Balinsky

Publications and source records attributed to A. A. Balinsky.

14 recordsLinked to original sources
arXiv2025

When Can We Reuse a Calibration Set for Multiple Conformal Predictions?

Reliable uncertainty quantification is crucial for the trustworthiness of machine learning applications. Inductive Conformal Prediction (ICP) offers a distribution-free framework for generating prediction sets or intervals with user-specified confidence. However, standard ICP guarantees are marginal and typically require a fresh calibration set for each new prediction to maintain their validity. This paper addresses this practical limitation by demonstrating how e-conformal prediction, in conjunction with Hoeffding's inequality, can enable the repeated use of a single calibration set with a high probability of preserving the desired coverage. Through a case study on the CIFAR-10 dataset, we train a deep neural network and utilise a calibration set to estimate a Hoeffding correction. This correction allows us to apply a modified Markov's inequality, leading to the construction of prediction sets with quantifiable confidence. Our results illustrate the feasibility of maintaining provable performance in conformal prediction while enhancing its practicality by reducing the need for repeated calibration. The code for this work is publicly available.

cs.LG↗
arXiv2024

Enhancing Conformal Prediction Using E-Test Statistics

Conformal Prediction (CP) serves as a robust framework that quantifies uncertainty in predictions made by Machine Learning (ML) models. Unlike traditional point predictors, CP generates statistically valid prediction regions, also known as prediction intervals, based on the assumption of data exchangeability. Typically, the construction of conformal predictions hinges on p-values. This paper, however, ventures down an alternative path, harnessing the power of e-test statistics to augment the efficacy of conformal predictions by introducing a BB-predictor (bounded from the below predictor).

cs.LG↗
arXiv2011

Hardy's inequality and curvature

A Hardy inequality of the form \[\int_{\tildeΩ} |\nabla f({\bf{x}})|^p d {\bf{x}} \ge (\frac{p-1}{p})^p \int_{\tildeΩ} \{1 + a(δ, \partial \tildeΩ)(\x)\}\frac{|f({\bf{x}})|^p}{δ({\bf{x}})^p} d{\bf{x}}, \] for all $f \in C_0^{\infty}({\tildeΩ})$, is considered for $p\in (1,\infty)$, where ${\tildeΩ}$ can be either $Ω$ or $\mathbb{R}^n \setminus Ω$ with $Ω$ a domain in $\mathbb{R}^n$, $n \ge 2$, and $δ({\bf{x}})$ is the distance from ${\bf{x}} \in {\tildeΩ} $ to the boundary $ \partial {\tildeΩ}.$ The main emphasis is on determining the dependance of $a(δ, \partial {\tildeΩ})$ on the geometric properties of $\partial {\tildeΩ}.$ A Hardy inequality is also established for any doubly connected domain $Ω$ in $\mathbb{R}^2$ in terms of a uniformisation of $Ω,$ that is, any conformal univalent map of $Ω$ onto an annulus.

math.SP↗
arXiv2011

The Dirac-Hardy and Dirac-Sobolev inequalities in $L^1$

Dirac-Sobolev and Dirac-Hardy inequalities in $L^1$ are established in which the $L^p$ spaces which feature in the classical Sobolev and Hardy inequalities are replaced by weak $L^p$ spaces. Counter examples to the analogues of the classical inequalities are shown to be provided by zero modes for appropriate Pauli operators constructed by Loss and Yau.

math.SP↗
arXiv2000

On the zero modes of Pauli operators

Two results are proved for $\mathrm{nul} \mathbb{P}_A$, the dimension of the kernel of the Pauli operator $\mathbb{P}_A = \bigl\{\bbfσ \cdotp \bigl(\frac{1}{i} \bbf{\nabla} + \vec{A} \bigr) \bigr\} ^2 $ in $[L^2 (\mathbb{R}^3)]^2$: (i) for $|\vec{B}| \in L^{3/2} (\mathbb{R}^3),$ where $\vec{B} = \mathrm{curl} \vec{A}$ is the magnetic field, $\mathrm{nul} \ \mathbb{P}_{tA} = 0$ except for a finite number of values of $t$ in any compact subset of $(0, \infty)$; (ii) $\bigl\{\vec{B}: \mathrm{nul} \mathbb{P}_{A} = 0, | \vec{B} | \in L^{3/2}(\mathbb{R}^3) \bigr\} $ contains an open dense subset of $[L^{3/2}(\mathbb{R}^3)]^3$.

math.SP↗
arXiv1999

Semi-classical asymptotics for the counting functions and Riesz means of Pauli and Dirac operators with large magnetic fields

We study the asymptotic behavior, as Planck's constant $\hbar\to 0$, of the number of discrete eigenvalues and the Riesz means of Pauli and Dirac operators with a magnetic field $μ\mathbf{B}(x)$ and an electric field. The magnetic field strength $μ$ is allowed to tend to infinity as $\hbar\to 0$. Two main types of results are established: in the first $μ\hbar\le constant$ as $\hbar\to 0$, with magnetic fields of arbitrary direction; the second results are uniform with respect to $μ\ge 0$ but the magnetic fields have constant direction. The results on the Pauli operator complement recent work of Sobolev.

math-ph↗
arXiv1998

On the virial theorem for the relativistic operator of Brown and Ravenhall, and the absence of embedded eigenvalues

A virial theorem is established for the operator proposed by Brown and Ravenhall as a model for relativistic one-electron atoms. As a consequence, it is proved that the operator has no eigenvalues greater than $\max(m c^2, 2 αZ - \frac{1}{2})$, where $α$ is the fine structure constant, for all values of the nuclear charge $Z$ below the critical value $Z_c$: in particular there are no eigenvalues embedded in the essential spectrum when $Z \leq 3/4 α$. Implications for the operators in the partial wave decomposition are also described.

math.SP↗
arXiv1995

Quadratic Poisson brackets and Drinfeld theory for associative algebras

The paper is devoted to the Poisson brackets compatible with multiplication in associative algebras. These brackets are shown to be quadratic and their relations with the classical Yang--Baxter equation are revealed. The paper also contains a description of Poisson Lie structures on Lie groups whose Lie algebras are adjacent to an associative structure.

q-alg↗
arXiv1995

Quadratic Poisson brackets and Drinfel'd theory for associative algebras

Quadratic Poisson brackets on associative algebras are studied. Such a bracket compatible with the multiplication is related to a differentiation in tensor square of the underlying algebra. Jacobi identity means that this differentiation satisfies a classical Yang--Baxter equation. Corresponding Lie groups are canonically equipped with a Poisson Lie structure. A way to quantize such structures is suggested.

q-alg↗
arXiv1994

Quadratic Poisson brackets compatible with an algebra structure

Quadratic Poisson brackets on a vector space equipped with a bilinear multiplication are studied. A notion of a bracket compatible with the multiplication is introduced and an effective criterion of such compatibility is given. Among compatible brackets, a subclass of coboundary brackets is described, and such brackets are enumerated in a number of examples.

hep-th↗