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In this paper we propose and study a version of the Dyadic Classification and Regression Trees (DCART) estimator from Donoho (1997) for (fixed design) quantile regression in general dimensions. We refer to this proposed estimator as the…

We study theoretical properties of a broad class of regularized algorithms with vector-valued output. These spectral algorithms include kernel ridge regression, kernel principal component regression, various implementations of gradient…

For each $n$, let $\text{RD}(n)$ denote the minimum $d$ for which there exists a formula for the general polynomial of degree $n$ in algebraic functions of at most $d$ variables. In 1945, Segre called for a better understanding of the large…

This paper provides a theoretical justification of the superior classification performance of deep rectifier networks over shallow rectifier networks from the geometrical perspective of piecewise linear (PWL) classifier boundaries. We show…

We establish the first information-theoretic limits for multimodal retrieval. Casting ranking as lossy source coding, we derive a single-letter rate-distortion function $R(D)$ for reciprocal-rank distortion and prove a converse bound that…

We begin this report by describing the Probably Approximately Correct (PAC) model for learning a concept class, consisting of subsets of a domain, and a function class, consisting of functions from the domain to the unit interval. Two…

We study the problem of learning a high-density region of an arbitrary distribution over $\mathbb{R}^d$. Given a target coverage parameter $\delta$, and sample access to an arbitrary distribution $D$, we want to output a confidence set $S…

Reinforcement learning (RL) with linear function approximation has received increasing attention recently. However, existing work has focused on obtaining $\sqrt{T}$-type regret bound, where $T$ is the number of interactions with the MDP.…

Understanding the self-directed learning complexity has been an important problem that has captured the attention of the online learning theory community since the early 1990s. Within this framework, the learner is allowed to adaptively…

Conceptual Scaling is a useful standard tool in Formal Concept Analysis and beyond. Its mathematical theory, as elaborated in the last chapter of the FCA monograph, still has room for improvement. As it stands, even some of the basic…

Let U be a monster model and let D be a subset of U. Let (U,D) denote theexpansion of U with a new predicate for D. Write e(D) for the collection of all subsets C of U such that (U,C) is elementary equivalent to (U,D). We prove that if e(D)…

Universal online learning aims to achieve optimal regret guarantees without requiring prior knowledge of the curvature of online functions. Existing methods have established minimax-optimal regret bounds for universal online learning, where…

This paper develops new semidefinite programming (SDP) relaxation techniques for two classes of mixed binary quadratically constrained quadratic programs (MBQCQP) and analyzes their approximation performance. The first class of problem…

Standard Model determinations of properties of strongly interacting systems of hadrons have become possible with the powerful method of lattice quantum chromodynamics (LQCD), a method with growing applicability and reliability. While growth…

Large language models (LLMs) are shifting from answer providers to intelligent tutors in educational settings, yet current supervised fine-tuning methods only learn surface teaching patterns without dynamic adaptation capabilities. Recent…

Recursion is the fundamental paradigm to finitely describe potentially infinite objects. As state-of-the-art reinforcement learning (RL) algorithms cannot directly reason about recursion, they must rely on the practitioner's ingenuity in…

The VC-dimension of a family of sets is a measure of its combinatorial complexity used in machine learning theory, computational geometry, and even model theory. Computing the VC-dimension of the $k$-fold union of geometric set systems has…

Recent advances in Multi-modal Large Language Models (MLLMs) have showcased remarkable capabilities in vision-language understanding. However, enabling robust video spatial reasoning-the ability to comprehend object locations, orientations,…

The fundamental theorem of statistical learning states that binary PAC learning is governed by a single parameter -- the Vapnik-Chervonenkis (VC) dimension -- which determines both learnability and sample complexity. Extending this to…

The unit Euclidean distance degree and the generic Euclidean distance degree are two well-studied invariants of projective varieties. These quantities measure the algebraic complexity of nearest-point problems on a variety, and in many…