arXiv subjects

Alexander Xue

Publications and source records attributed to Alexander Xue.

7 recordsLinked to original sources

arXiv2026

Numerical Instabilities in the Kaczmarz Method and Stabilization by Iterative Refinement

The randomized Kaczmarz method and its accelerated variants are a powerful class of iterative methods for solving large-scale linear systems, offering guaranteed convergence with low per-iteration cost. However, their numerical stability remains poorly understood. In this work, we investigate the stability properties of both classical and accelerated randomized Kaczmarz methods, with an emphasis on how error propagates across iterations and interacts with acceleration. We show that both classical and accelerated randomized Kaczmarz fail to be forward stable. To address this issue, we propose the integration of iterative refinement into randomized Kaczmarz frameworks. We demonstrate that refinement can effectively control error accumulation and recover high-accuracy solutions, even when the system is ill-conditioned. Numerical experiments corroborate our theoretical findings and illustrate the practical benefits of combining refinement with both classical and accelerated Kaczmarz methods.

math.NA↗

arXiv2026

Macroscopic Traffic Flow Network Modeling For Wildfire Evacuation: A Game-Theoretic Junction Optimization Approach with Application to Lahaina Fire

The 2023 Lahaina wildfire killed 102 people on a peninsula served by a single two-lane highway, making exit lane capacity the binding constraint on evacuation time. We model the evacuation as a system of hyperbolic scalar conservation laws on a directed graph with game-theoretic junction conditions that maximize total network flux, an evacuation-calibrated piecewise linear-quadratic flux function, and a loss-driven optimization framework that tunes traffic distribution toward priority corridors. Analytical results on a toy network and numerical simulations of the Lahaina road network reveal a phase transition in exit lane capacity. Additional lanes improve throughput linearly until a computable critical threshold, beyond which no route optimization yields further benefit. For Lahaina, reversing one southbound lane captures nearly all achievable improvement, and a fourth lane can be reserved for emergency vehicles with negligible impact on civilian clearance time. These results provide a rigorous mathematical basis for contraflow recommendations in wildland-urban interface evacuations.

math.NA↗

arXiv2025

Stochastic Gradient Descent for Incomplete Tensor Linear Systems

Solving large tensor linear systems poses significant challenges due to the high volume of data stored, and it only becomes more challenging when some of the data is missing. Recently, Ma et al. showed that this problem can be tackled using a stochastic gradient descent-based method, assuming that the missing data follows a uniform missing pattern. We adapt the technique by modifying the update direction, showing that the method is applicable under other missing data models. We prove convergence results and experimentally verify these results on synthetic data.

math.NA↗

arXiv2024

Differentially Private Random Feature Model

Designing privacy-preserving machine learning algorithms has received great attention in recent years, especially in the setting when the data contains sensitive information. Differential privacy (DP) is a widely used mechanism for data analysis with privacy guarantees. In this paper, we produce a differentially private random feature model. Random features, which were proposed to approximate large-scale kernel machines, have been used to study privacy-preserving kernel machines as well. We consider the over-parametrized regime (more features than samples) where the non-private random feature model is learned via solving the min-norm interpolation problem, and then we apply output perturbation techniques to produce a private model. We show that our method preserves privacy and derive a generalization error bound for the method. To the best of our knowledge, we are the first to consider privacy-preserving random feature models in the over-parametrized regime and provide theoretical guarantees. We empirically compare our method with other privacy-preserving learning methods in the literature as well. Our results show that our approach is superior to the other methods in terms of generalization performance on synthetic data and benchmark data sets. Additionally, it was recently observed that DP mechanisms may exhibit and exacerbate disparate impact, which means that the outcomes of DP learning algorithms vary significantly among different groups. We show that both theoretically and empirically, random features have the potential to reduce disparate impact, and hence achieve better fairness.

cs.LG↗

arXiv2024

Polymatroids are to finite groups as matroids are to finite fields

Given a subgroup $\mathcal{H}$ of a product of finite groups $\mathcal{G} = \displaystyle\prod^n_{i=1} \Gamma_i$ and $b>1,$ we define a polymatroid $P(\mathcal{H},b).$ If all of the $\Gamma_i$ are isomorphic to $\mathbb{Z}/p\mathbb{Z},$ $p$ a prime, and $b=p,$ then $P(\mathcal{H},b)$ is the usual matroid associated to any $\mathbb{Z}/p\mathbb{Z}$-matrix whose row space equals $\mathcal{H}.$ In general, there are many ways in which the relationship between $P(\mathcal{H},b)$ and $\mathcal{H}$ mirrors that of the relationship between a matroid and a subspace of a finite vector space. These include representability by excluded minors, the Crapo-Rota critical theorem, the existence of a concrete algebraic object representing the polymatroid dual of $P(\mathcal{H},b),$ analogs of Greene's theorem and the MacWilliams identities when $\mathcal{H}$ is a group code over a nonabelian group, and a connection to the combinatorial Laplacian of a quotient space determined by $\mathcal{G}$ and $\mathcal{H}.$ We use the group Crapo-Rota critical theorem to demonstrate an extension to hypergraphs of the classical duality between proper colorings and nowhere-zero flows on graphs.

math.CO↗

arXiv2020

Quantitative combinatorial geometry for concave functions

We prove several exact quantitative versions of Helly's and Tverberg's theorems, which guarantee that a finite family of convex sets in $R^d$ has a large intersection. Our results characterize conditions that are sufficient for the intersection of a family of convex sets to contain a "witness set" which is large under some concave or log-concave measure. The possible witness sets include ellipsoids, zonotopes, and $H$-convex sets. Our results also bound the complexity of finding the best approximation of a family of convex sets by a single zonotope or by a single $H$-convex set. We obtain colorful and fractional variants of all our Helly-type theorems.

math.CO↗

arXiv2019

Balanced convex partitions of lines in the plane

We prove an extension of a ham sandwich theorem for families of lines in the plane by Dujmović and Langerman. Given two sets $A, B$ of $n$ lines each in the plane, we prove that it is possible to partition the plane into $r$ convex regions such that the following holds. For each region $C$ of the partition there is a subset of $c_r n^{1/r}$ lines of $A$ whose pairwise intersections are in $C$, and the same holds for $B$. In this statement $c_r$ only depends on $r$. We also prove that the dependence on $n$ is optimal.

math.MG↗