arXiv subjects

Oleg Szehr

Publications and source records attributed to Oleg Szehr.

At least 19 recordsLinked to original sources

arXiv2026

Schäffer's matrix inequality: the exact asymptotic constant

Let $S_n$ denote the smallest constant such that \[ |\det T|\|T^{-1}\| \leq S_n \|T\|^{n-1} \] for every invertible operator $T$ on every $n$-dimensional complex Banach space. In Hilbert space the optimal constant is $1$. For arbitrary Banach spaces, J. J. Schäffer proved in 1970 that \[S_n\leq \sqrt{en}. \] Subsequent work showed that $S_n$ grows like $\sqrt n$, but the sharp asymptotic constant has remained open for more than five decades. We resolve this problem by proving \[ \lim_{n\to\infty}\frac{S_n}{\sqrt n}=\sqrt e. \] Thus Schäffer's upper bound is asymptotically sharp, including its constant. Our proof is constructive, providing explicit Banach-space norms through duality and explicit matrices through the theory of model operators. At the analytic core of the argument, an extremal formulation of Schäffer's problem in the Wiener algebra reduces the matching asymptotic lower bound for $S_n$ to uniformly controlling the Taylor coefficients of products $QB_n$, where $B_n$ is a finite Blaschke product of degree $n$ and $Q$ is a polynomial factor. We optimize simultaneously the zero distribution of $B_n$ and the choice of $Q$. The resulting zeros follow a logarithmic asymptotic distribution, and a sharp uniform asymptotic analysis of the Taylor coefficients of $QB_n$ yields the constant $\sqrt e$. The corresponding model operators then yield matrices with these spectra that asymptotically attain Schäffer's bound.

math.FA↗

arXiv2026

Projected Inner-Function Dynamics and Crystalline Measures of Meyer-Blaschke type

We generalize a construction of Yves Meyer of sparse crystalline measures arising from powers of a Blaschke factor. Starting from a recursion $f_n=θ^n f_0$ on the unit circle where $θ$ is an inner function, we project the Fourier coefficient array $\widehat{f_n}(k)$ to the real line by placing its entries at the frequencies $k+αn$. We identify the role of model spaces in this construction: in Meyer's one-factor Blaschke recursion, the requirement that the coefficient array $\widehat{f_n}(k)$ vanish whenever $kn<0$ is equivalent to $f_0\in K_{zb_λ}$, and for general inner functions the condition $f_0\in K_{zθ}$ yields a purely atomic Radon measure with locally finite support and polynomial growth on the Fourier side. We also show that, when $f_0$ is holomorphic in an annulus containing the unit circle, exponential Fourier decay is sufficient to obtain a purely atomic Radon measure of polynomial growth, though not necessarily locally finite support. For finite Blaschke products, the coefficient recursion gives an explicit annihilating exponential polynomial whose zero set controls the support and separation of the inverse Fourier transform. This yields Meyer-Blaschke-type crystalline measures and Poisson identities with sampling and finite-truncation consequences.

math.CA↗

arXiv2026

Spectral Criteria for Uniqueness Pairs of Unitary Transforms

The identification of sampling sets that enable unique signal recovery is fundamental to many applications in signal processing and remains a central problem in mathematical analysis. Recent studies, particularly in the context of the Fourier transform and crystalline measures, have developed a theory of recovery from two-sided sampling, where samples are prescribed simultaneously in the physical and transformed domains. Kulikov, Nazarov, and Sodin introduced a method for identifying such uniqueness pairs based on functional inequalities of the Wirtinger-Poincaré type. In this work, we propose an alternative spectral approach motivated by quantum mechanics. The guiding observation is that zeros of a function and of its transform impose Dirichlet-type confinement in two conjugate representations, thereby converting two-sided uniqueness questions into lower-bound problems for confined Hamiltonians. For the Fourier transform, the relevant Hamiltonian is the harmonic oscillator, and the uniformly supercritical uniqueness criterion is recovered through a variational spectral argument. Our viewpoint extends to unitary transforms whose associated localization operators admit local Sturm-Liouville or Schrödinger-type confined realizations, a class that includes transforms commonly used in signal processing and mathematical physics. It abstracts the Wirtinger-Poincaré mechanism by replacing the ordinary Dirichlet-Laplacian constant with the local spectral floor of a Hamiltonian-type operator associated with the transform. We formulate this principle for Sturm-Liouville operators with weights or nontrivial coefficients, and illustrate it for the fractional Fourier transform and the Hankel transform, where phase-space rotation and singular endpoint behavior enter the uniqueness criteria.

math.CA↗

arXiv2026

Signal Recovery from Time and Frequency Samples

We analyze signal recovery when samples are taken concomitantly from a signal and its Fourier transform. This two-sided sampling framework extends classical one-sided reconstruction and is particularly useful when measurements in either domain alone are insufficient because of sensing, storage, or bandwidth constraints. We formulate the resulting recovery problem in finite-dimensional spaces and reproducing kernel Hilbert spaces, and illustrate the infinite-dimensional setting in a Fourier-symmetric Sobolev space. Numerical experiments with sinc- and Hermite-based schemes indicate that, under a fixed sampling budget, two-sided sampling often yields better conditioned systems than one-sided approaches. A simplified spectrum-monitoring example further demonstrates improved reconstruction when limited time samples are supplemented with frequency-domain information.

eess.SP↗

arXiv2026

Automatic Prompt Optimization for Dataset-Level Feature Discovery

Feature extraction from unstructured text is a critical step in many downstream classification pipelines, yet current approaches largely rely on hand-crafted prompts or fixed feature schemas. We formulate feature discovery as a dataset-level prompt optimization problem: given a labelled text corpus, the goal is to induce a global set of interpretable and discriminative feature definitions whose realizations optimize a downstream supervised learning objective. To this end, we propose a multi-agent prompt optimization framework in which language-model agents jointly propose feature definitions, extract feature values, and evaluate feature quality using dataset-level performance and interpretability feedback. Instruction prompts are iteratively refined based on this structured feedback, enabling optimization over prompts that induce shared feature sets rather than per-example predictions. This formulation departs from prior prompt optimization methods that rely on per-sample supervision and provides a principled mechanism for automatic feature discovery from unstructured text.

cs.CL↗

arXiv2025

Deep Hedging Under Non-Convexity: Limitations and a Case for AlphaZero

This paper examines replication portfolio construction in incomplete markets - a key problem in financial engineering with applications in pricing, hedging, balance sheet management, and energy storage planning. We model this as a two-player game between an investor and the market, where the investor makes strategic bets on future states while the market reveals outcomes. Inspired by the success of Monte Carlo Tree Search in stochastic games, we introduce an AlphaZero-based system and compare its performance to deep hedging - a widely used industry method based on gradient descent. Through theoretical analysis and experiments, we show that deep hedging struggles in environments where the optimal action-value function is not subject to convexity constraints - such as those involving non-convex transaction costs, capital constraints, or regulatory limitations - converging to local optima. We construct specific market environments to highlight these limitations and demonstrate that AlphaZero consistently finds near-optimal replication strategies. On the theoretical side, we establish a connection between deep hedging and convex optimization, suggesting that its effectiveness is contingent on convexity assumptions. Our experiments further suggest that AlphaZero is more sample-efficient - an important advantage in data-scarce, overfitting-prone derivative markets.

stat.ML↗

arXiv2025

On the Convergence and Stability of Upside-Down Reinforcement Learning, Goal-Conditioned Supervised Learning, and Online Decision Transformers

This article provides a rigorous analysis of convergence and stability of Episodic Upside-Down Reinforcement Learning, Goal-Conditioned Supervised Learning and Online Decision Transformers. These algorithms performed competitively across various benchmarks, from games to robotic tasks, but their theoretical understanding is limited to specific environmental conditions. This work initiates a theoretical foundation for algorithms that build on the broad paradigm of approaching reinforcement learning through supervised learning or sequence modeling. At the core of this investigation lies the analysis of conditions on the underlying environment, under which the algorithms can identify optimal solutions. We also assess whether emerging solutions remain stable in situations where the environment is subject to tiny levels of noise. Specifically, we study the continuity and asymptotic convergence of command-conditioned policies, values and the goal-reaching objective depending on the transition kernel of the underlying Markov Decision Process. We demonstrate that near-optimal behavior is achieved if the transition kernel is located in a sufficiently small neighborhood of a deterministic kernel. The mentioned quantities are continuous (with respect to a specific topology) at deterministic kernels, both asymptotically and after a finite number of learning cycles. The developed methods allow us to present the first explicit estimates on the convergence and stability of policies and values in terms of the underlying transition kernels. On the theoretical side we introduce a number of new concepts to reinforcement learning, like working in segment spaces, studying continuity in quotient topologies and the application of the fixed-point theory of dynamical systems. The theoretical study is accompanied by a detailed investigation of example environments and numerical experiments.

stat.ML↗

arXiv2025

Understanding Action Effects through Instrumental Empowerment in Multi-Agent Reinforcement Learning

To reliably deploy Multi-Agent Reinforcement Learning (MARL) systems, it is crucial to understand individual agent behaviors. While prior work typically evaluates overall team performance based on explicit reward signals, it is unclear how to infer agent contributions in the absence of any value feedback. In this work, we investigate whether meaningful insights into agent behaviors can be extracted solely by analyzing the policy distribution. Inspired by the phenomenon that intelligent agents tend to pursue convergent instrumental values, we introduce Intended Cooperation Values (ICVs), a method based on information-theoretic Shapley values for quantifying each agent's causal influence on their co-players' instrumental empowerment. Specifically, ICVs measure an agent's action effect on its teammates' policies by assessing their decision (un)certainty and preference alignment. By analyzing action effects on policies and value functions across cooperative and competitive MARL tasks, our method identifies which agent behaviors are beneficial to team success, either by fostering deterministic decisions or by preserving flexibility for future action choices, while also revealing the extent to which agents adopt similar or diverse strategies. Our proposed method offers novel insights into cooperation dynamics and enhances explainability in MARL systems.

cs.AI↗

arXiv2025

Enhancing Aerial Combat Tactics through Hierarchical Multi-Agent Reinforcement Learning

This work presents a Hierarchical Multi-Agent Reinforcement Learning framework for analyzing simulated air combat scenarios involving heterogeneous agents. The objective is to identify effective Courses of Action that lead to mission success within preset simulations, thereby enabling the exploration of real-world defense scenarios at low cost and in a safe-to-fail setting. Applying deep Reinforcement Learning in this context poses specific challenges, such as complex flight dynamics, the exponential size of the state and action spaces in multi-agent systems, and the capability to integrate real-time control of individual units with look-ahead planning. To address these challenges, the decision-making process is split into two levels of abstraction: low-level policies control individual units, while a high-level commander policy issues macro commands aligned with the overall mission targets. This hierarchical structure facilitates the training process by exploiting policy symmetries of individual agents and by separating control from command tasks. The low-level policies are trained for individual combat control in a curriculum of increasing complexity. The high-level commander is then trained on mission targets given pre-trained control policies. The empirical validation confirms the advantages of the proposed framework.

cs.AI↗

arXiv2023

Hierarchical Multi-Agent Reinforcement Learning for Air Combat Maneuvering

The application of artificial intelligence to simulate air-to-air combat scenarios is attracting increasing attention. To date the high-dimensional state and action spaces, the high complexity of situation information (such as imperfect and filtered information, stochasticity, incomplete knowledge about mission targets) and the nonlinear flight dynamics pose significant challenges for accurate air combat decision-making. These challenges are exacerbated when multiple heterogeneous agents are involved. We propose a hierarchical multi-agent reinforcement learning framework for air-to-air combat with multiple heterogeneous agents. In our framework, the decision-making process is divided into two stages of abstraction, where heterogeneous low-level policies control the action of individual units, and a high-level commander policy issues macro commands given the overall mission targets. Low-level policies are trained for accurate unit combat control. Their training is organized in a learning curriculum with increasingly complex training scenarios and league-based self-play. The commander policy is trained on mission targets given pre-trained low-level policies. The empirical validation advocates the advantages of our design choices.

cs.LG↗

arXiv2022

Hedging using reinforcement learning: Contextual $k$-Armed Bandit versus $Q$-learning

The construction of replication strategies for contingent claims in the presence of risk and market friction is a key problem of financial engineering. In real markets, continuous replication, such as in the model of Black, Scholes and Merton (BSM), is not only unrealistic but it is also undesirable due to high transaction costs. A variety of methods have been proposed to balance between effective replication and losses in the incomplete market setting. With the rise of Artificial Intelligence (AI), AI-based hedgers have attracted considerable interest, where particular attention was given to Recurrent Neural Network systems and variations of the $Q$-learning algorithm. From a practical point of view, sufficient samples for training such an AI can only be obtained from a simulator of the market environment. Yet if an agent was trained solely on simulated data, the run-time performance will primarily reflect the accuracy of the simulation, which leads to the classical problem of model choice and calibration. In this article, the hedging problem is viewed as an instance of a risk-averse contextual $k$-armed bandit problem, which is motivated by the simplicity and sample-efficiency of the architecture. This allows for realistic online model updates from real-world data. We find that the $k$-armed bandit model naturally fits to the Profit and Loss formulation of hedging, providing for a more accurate and sample efficient approach than $Q$-learning and reducing to the Black-Scholes model in the absence of transaction costs and risks.

cs.LG↗

arXiv2022

On the asymptotic behavior of Jacobi polynomials with first varying parameter

We investigate the large $n$ behavior of Jacobi polynomials with varying parameters $P_{n}^{(an+α,\,bn+β)}(1-2λ^{2})$ for $a,b >-1$ and $λ\in(0,\,1)$. This is a well-studied topic in the literature but some of the published results appear to be discordant. To address this issue we provide an in-depth investigation of the case $b = 0$, which is most relevant for our applications. Our approach is based on a new and surprisingly simple representation of $P_{n}^{(an+α,\,β)}(1-2λ^{2}),\:a>-1$ in terms of two integrals. The integrals' asymptotic behavior is studied using standard tools of asymptotic analysis: one is a Laplace integral and the other is treated via the method of stationary phase. As a consequence we prove that if $a\in(\frac{2λ}{1-λ},\infty)$ then $λ^{an}P_{n}^{(an+α,β)}(1-2λ^{2})$ shows exponential decay and we derive simple exponential upper bounds in this region. If $a\in(\frac{-2λ}{1+λ},\,\frac{2λ}{1-λ})$ then the decay of $λ^{an}P_{n}^{(an+α,β)}(1-2λ^{2})$ is $\mathcal{O}(n^{-1/2})$ and if $a\in\{\frac{-2λ}{1+λ},\,\frac{2λ}{1-λ}\}$ then $λ^{an}P_{n}^{(an+α,β)}(1-2λ^{2})$ decays as $\mathcal{O}(n^{-1/3})$. A new phenomenon occurs in the parameter range $a\in(-1,\frac{-2λ}{1+λ})$, where we find that the behavior depends on whether or not $an+α$ is an integer: If $a\in(-1,\frac{-2λ}{1+λ})$ and $an+α$ is an integer then $λ^{an}P_{n}^{(an+α,β)}(1-2λ^{2})$ decays exponentially. If $a\in(-1,\frac{-2λ}{1+λ})$ and $an+α$ is not an integer then $λ^{an}P_{n}^{(an+α,β)}(1-2λ^{2})$ may increase exponentially depending on the proximity of the sequence $(an + α)_n$ to integers.

math.CA↗

arXiv2021

Hedging of Financial Derivative Contracts via Monte Carlo Tree Search

The construction of approximate replication strategies for pricing and hedging of derivative contracts in incomplete markets is a key problem of financial engineering. Recently Reinforcement Learning algorithms for hedging under realistic market conditions have attracted significant interest. While research in the derivatives area mostly focused on variations of $Q$-learning, in artificial intelligence Monte Carlo Tree Search is the recognized state-of-the-art method for various planning problems, such as the games of Hex, Chess, Go,... This article introduces Monte Carlo Tree Search as a method to solve the stochastic optimal control problem behind the pricing and hedging tasks. As compared to $Q$-learning it combines Reinforcement Learning with tree search techniques. As a consequence Monte Carlo Tree Search has higher sample efficiency, is less prone to over-fitting to specific market models and generally learns stronger policies faster. In our experiments we find that Monte Carlo Tree Search, being the world-champion in games like Chess and Go, is easily capable of maximizing the utility of investor's terminal wealth without setting up an auxiliary mathematical framework.

cs.AI↗

arXiv2020

An exact kernel framework for spatio-temporal dynamics

A kernel-based framework for spatio-temporal data analysis is introduced that applies in situations when the underlying system dynamics are governed by a dynamic equation. The key ingredient is a representer theorem that involves time-dependent kernels. Such kernels occur commonly in the expansion of solutions of partial differential equations. The representer theorem is applied to find among all solutions of a dynamic equation the one that minimizes the error with given spatio-temporal samples. This is motivated by the fact that very often a differential equation is given a priori (e.g.~by the laws of physics) and a practitioner seeks the best solution that is compatible with her noisy measurements. Our guiding example is the Fokker-Planck equation, which describes the evolution of density in stochastic diffusion processes. A regression and density estimation framework is introduced for spatio-temporal modeling under Fokker-Planck dynamics with initial and boundary conditions.

math.ST↗

arXiv2020

Interpolation without commutants

We introduce a "dual-space approach" to mixed Nevanlinna-Pick/Carathéodory-Schur interpolation in Banach spaces X of holomorphic functions on the disk. Our approach can be viewed as complementary to the well-known commutant lifting approach of D. Sarason and B. Nagy-C.Foiaş. We compute the norm of the minimal interpolant in X by a version of the Hahn-Banach theorem, which we use to extend functionals defined on a subspace of kernels without increasing their norm. This Functional extensions lemma plays a similar role as Sarason's Commutant lifting theorem but it only involves the predual of X and no Hilbert space structure is needed. As an example, we present the respective Pick-type interpolation theorems for Beurling-Sobolev spaces.

math.FA↗

arXiv2017

A constructive approach to Schaeffer's conjecture

J.J. Schaeffer proved that for $any$ induced matrix norm and $any$ invertible $T=T(n)$ the inequality \[\left|\det T\right|\left\Vert T^{-1}\right\Vert \leq\mathcal{S}\left\Vert T\right\Vert ^{n-1}\] holds with $\mathcal{S}=\mathcal{S}(n)\leq\sqrt{en}$. He conjectured that the best $\mathcal{S}$ was actually bounded. This was rebutted by Gluskin-Meyer-Pajor and subsequent contributions by J. Bourgain and H. Queffelec that successively improved lower estimates on $\mathcal{S}$. These articles rely on a link to the theory of power sums of complex numbers. A probabilistic or number theoretic analysis of such inequalities is employed to prove the existence of $T$ with growing $\mathcal{S}$ but the explicit construction of such $T$ remains an open task. In this article we propose a constructive approach to Schaeffer's conjecture that is not related to power sum theory. As a consequence we present an explicit sequence of Toeplitz matrices with singleton spectrum $\{λ\}\subset\mathbb{D}-\{0\}$ such that $\mathcal{S}\geq c(λ)\sqrt{n}$. Our framework naturally extends to provide lower estimates on the resolvent $\left\Vert (ζ-T)^{-1}\right\Vert$ when $ζ\neq0$. We also obtain new upper estimates on the resolvent when the spectrum is given. This yields new upper bounds on $\left\Vert T^{-1}\right\Vert$ in terms of the eigenvalues of $T$ which slightly refine Schaeffer's original estimate.

math.NA↗

arXiv2017

$l_{p}$-norms of Fourier coefficients of powers of a Blaschke factor

We determine the asymptotic behavior of the $l_{p}$-norms of the sequence of Taylor coefficients of $b^{n}$, where $b=\frac{z-λ}{1-\barλz}$ is an automorphism of the unit disk, $p\in[1,\infty]$, and $n$ is large. It is known that in the parameter range $p\in[1,2]$ a sharp upper bound \begin{align*} \left|\!\left|b^{n}\right|\!\right|_{l_{p}^A}\leq C_{p}n^{\frac{2-p}{2p}} \end{align*} holds. In this article we find that this estimate is valid even when $p\in[1,4)$. We prove that \begin{align*} \left|\!\left|b^{n}\right|\!\right|_{l_{4}^A}\leq C_{4}\left(\frac{\log n}{n}\right)^{\frac{1}{4}} \end{align*} and for $p\in(4,\infty]$ that \begin{align*} \left|\!\left|b^{n}\right|\!\right|_{l_{p}^A}\leq C_{p}n^{\frac{1-p}{3p}} & . \end{align*} We prove that our upper bounds are sharp as $n$ tends to $\infty$ i.e. they have the correct asymptotic $n$ dependence.

math.CA↗

arXiv2015

Spectral-Variation Bounds in Hyperbolic Geometry

We derive new estimates for distances between optimal matchings of eigenvalues of non-normal matrices in terms of the norm of their difference. We introduce and estimate a hyperbolic metric analogue of the classical spectral-variation distance. The result yields a qualitatively new and simple characterization of the localization of eigenvalues. Our bound improves on the best classical spectral-variation bounds due to Krause if the distance of matrices is sufficiently small and is sharp for asymptotically large matrices. Our approach is based on the theory of model operators, which provides us with strong resolvent estimates. The latter naturally lead to a Chebychev-type interpolation problem with finite Blaschke products, which can be solved explicitly and gives stronger bounds than the classical Chebychev interpolation with polynomials. As compared to the classical approach our method does not rely on Hadamard's inequality and immediately generalize to algebraic operators on Hilbert space.

math.NA↗