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The use of quantum computing to accelerate complex optimization problems is a burgeoning research field. This paper applies Quantum Linear System Algorithms (QLSAs) to Newton systems within Interior Point Methods (IPMs) to take advantage of…
The rapid advancement of Large Language Models (LLMs) necessitates a deep understanding of their fundamental performance limits. This paper investigates the limits of LLM inference, focusing on hardware-imposed bottlenecks in…
We investigate the problem of best policy identification in discounted linear Markov Decision Processes in the fixed confidence setting under a generative model. We first derive an instance-specific lower bound on the expected number of…
The problem of phase synchronization is to estimate the phases (angles) of a complex unit-modulus vector $z$ from their noisy pairwise relative measurements $C = zz^* + \sigma W$, where $W$ is a complex-valued Gaussian random matrix. The…
Fairness has become increasingly pivotal in machine learning for high-risk applications such as machine learning in healthcare and facial recognition. However, we see the deficiency in the previous logits space constraint methods.…
There are two primary approaches to solving Markov decision problems (MDPs): dynamic programming based on the Bellman equation and linear programming (LP). Dynamic programming methods are the most widely used and form the foundation of both…
In this paper we provide explicit upper and lower bounds on certain $L^2$ $n$-widths, i.e., best constants in $L^2$ approximation. We further describe a numerical method to compute these $n$-widths approximately, and prove that this method…
In this paper, we target the problem of sufficient dimension reduction with symmetric positive definite matrices valued responses. We propose the intrinsic minimum average variance estimation method and the intrinsic outer product gradient…
This paper presents a new performance bound for estimation problems where the parameter to estimate lies in a Riemannian manifold (a smooth manifold endowed with a Riemannian metric) and follows a given prior distribution. In this setup,…
The rapid advancement of Large Language Models (LLMs) has introduced new possibilities and challenges in physics education, necessitating rigorous evaluation of their capabilities as both problem solvers and automated assessors. This paper…
The level set estimation problem seeks to find all points in a domain ${\cal X}$ where the value of an unknown function $f:{\cal X}\rightarrow \mathbb{R}$ exceeds a threshold $\alpha$. The estimation is based on noisy function evaluations…
Uncertainty is a fundamental aspect of real-world scenarios, where perfect information is rarely available. Humans naturally develop complex internal models to navigate incomplete data and effectively respond to unforeseen or partially…
The proximal gradient algorithm for minimizing the sum of a smooth and a nonsmooth convex function often converges linearly even without strong convexity. One common reason is that a multiple of the step length at each iteration may…
Empirically, Deep Learning (DL) has demonstrated unprecedented success in practical applications. However, DL remains by and large a mysterious "black-box", spurring recent theoretical research to build its mathematical foundations. In this…
This work provides closed-form solutions and minimum achievable errors for a large class of low-rank approximation problems in Hilbert spaces. The proposed theorem generalizes to the case of bounded linear operators the previous results…
It is now generally accepted that Euclidean-based metrics may not always adequately represent the subjective judgement of a human observer. As a result, many image processing methodologies have been recently extended to take advantage of…
In this work, we study linear error-correcting codes against adversarial insertion-deletion (indel) errors. While most constructions for the indel model are nonlinear, linear codes offer compact representations, efficient encoding, and…
We consider linear programming (LP) problems in infinite dimensional spaces that are in general computationally intractable. Under suitable assumptions, we develop an approximation bridge from the infinite-dimensional LP to tractable finite…
We consider finite Morse index solutions to semilinear elliptic questions, and we investigate their smoothness. It is well-known that: - For $n=2$, there exist Morse index $1$ solutions whose $L^\infty$ norm goes to infinity. - For $n \geq…
Lipschitz decomposition is a useful tool in the design of efficient algorithms involving metric spaces. While many bounds are known for different families of finite metrics, the optimal parameters for $n$-point subsets of $\ell_p$, for $p >…