arXiv subjects

Jennifer Zhu

Publications and source records attributed to Jennifer Zhu.

4 recordsLinked to original sources

arXiv2026

Categorical (Co)Limits of Quantum Graphs

We begin with the characterization of quantum graphs as left ideals in $\mathcal M \otimes_{eh} \mathcal M$ (the extended Haagerup tensor product of $\mathcal M$ with itself) to avoid technicalities surrounding representation dependence of quantum graphs. These left ideals roughly correspond to a canonical complement of a quantum graph. Using these left ideals and some operator space theory, we find a new, representation-free characterization of a morphism of quantum graphs compatible with previous representation-dependent morphisms. A notion of categorical (co)limit of quantum graphs follows. We also briefly explore an alternative quantization of graphs as bimodules over $C^*$-algebras ($C^*$-graphs), mostly to emphasize the point that a morphism of $C^*$-graphs is not a morphism of $C^*$-correspondences.

math.OA↗

arXiv2024

Towards Improved Preference Optimization Pipeline: from Data Generation to Budget-Controlled Regularization

Direct Preference Optimization (DPO) and its variants have become the de facto standards for aligning large language models (LLMs) with human preferences or specific goals. However, DPO requires high-quality preference data and suffers from unstable preference optimization. In this work, we aim to improve the preference optimization pipeline by taking a closer look at preference data generation and training regularization techniques. For preference data generation, we demonstrate that existing scoring-based reward models produce unsatisfactory preference data and perform poorly on out-of-distribution tasks. This significantly impacts the LLM alignment performance when using these data for preference tuning. To ensure high-quality preference data generation, we propose an iterative pairwise ranking mechanism that derives preference ranking of completions using pairwise comparison signals. For training regularization, we observe that preference optimization tends to achieve better convergence when the LLM predicted likelihood of preferred samples gets slightly reduced. However, the widely used supervised next-word prediction regularization strictly prevents any likelihood reduction of preferred samples. This observation motivates our design of a budget-controlled regularization formulation. Empirically we show that combining the two designs leads to aligned models that surpass existing SOTA across two popular benchmarks.

cs.LG↗

arXiv2020

Large Sets with Small Injective Projections

Let $\ell_1,\ell_2,\dots$ be a countable collection of lines in ${\mathbb R}^d$. For any $t \in [0,1]$ we construct a compact set $Γ\subset{\mathbb R}^d$ with Hausdorff dimension $d-1+t$ which projects injectively into each $\ell_i$, such that the image of each projection has dimension $t$. This immediately implies the existence of homeomorphisms between certain Cantor-type sets whose graphs have large dimensions. As an application, we construct a collection $E$ of disjoint, non-parallel $k$-planes in $\mathbb{R}^d$, for $d \geq k+2$, whose union is a small subset of $\mathbb{R}^d$, either in Hausdorff dimension or Lebesgue measure, while $E$ itself has large dimension. As a second application, for any countable collection of vertical lines $w_i$ in the plane we construct a collection of nonvertical lines $H$, so that $F$, the union of lines in $H$, has positive Lebesgue measure, but each point of each line $w_i$ intersects at most one $h\in H$ and, for each $w_i$, the Hausdorff dimension of $F\cap w_i$ is zero.

math.MG↗

arXiv2018

The bullet problem with discrete speeds

Bullets are fired, one per second, with independent speeds sampled uniformly from a discrete set. Collisions result in mutual annihilation. We show that the second fastest bullet survives with positive probability, while a slowest bullet does not. This also holds for exponential spacings between firing times, and for certain non-uniform measures that place less probability on the second fastest bullet. Our results provide new insights into a two-sided version of the bullet process known to physicists as ballistic annihilation.

math.PR↗