arXiv subjects

Junwei Lu

Publications and source records attributed to Junwei Lu.

At least 19 recordsLinked to original sources

arXiv2026

Optimal Value Inference for Reinforcement Learning

We study offline inference for the optimal value in reinforcement learning under finite state and action spaces. Two new nuisances are derived as fixed points of a self-induced Bellman equation, in which we approximate the maximum Bellman operator by its softmax correspondence. We propose a debiased estimator through the Neyman orthogonality and establish its asymptotic normality under diverging horizons even when the behavior policy changes with time, as long as the nuisances have the statistical rates that can be achieved by many machine learning methods. We provide a concrete estimating procedure for these nuisances and show they can lead to valid inference. Synthetic experiments validate the numerical performance of our inference method, and we implement it in real-life decision-making problems, including bike repositioning and AI agentic tool use.

stat.ML↗

arXiv2026

Vector Balancing in Polynomial Time

The Komlós problem asks for signs $\varepsilon\in\{-1,1\}^n$ with $\|A\varepsilon\|_\infty$ bounded by an absolute constant whenever the columns of $A\in\mathbb R^{m\times n}$ have Euclidean norm at most $1$. We give the first polynomial-time algorithm for this problem in the unit-cost real-RAM model. Our deterministic algorithm achieves discrepancy at most $8272$. As a consequence, we obtain a deterministic $O(\sqrt t)$-discrepancy algorithm for the Beck-Fiala problem on set systems in which each element belongs to at most $t$ sets. Our algorithm rounds a fractional coloring while keeping the row sums bounded. Its key idea is to keep the number of nearly tight row constraints small relative to the number of active coordinates. We achieve this through a spectral bound based on a weighted Gram matrix of squared row entries. We maintain the bound using a cubic spectral potential and negative-curvature updates that satisfy the required linear constraints.

cs.DS↗

arXiv2026

RadonBench: A Benchmark of Classical and Learning-Guided Search Techniques for Problems in Combinatorial Geometry

Classical optimization and learning-guided search offer complementary approaches to open problems in discrete mathematics, but their outputs are difficult to compare when search spaces, validity checks, and stopping rules differ. We present \emph{RadonBench}, a benchmark for constructing maximum-cardinality Radon-free lattice configurations: finite subsets of $\mathbb{Z}^d$ admitting no integral Radon partition. It remains open whether Radon-free configurations with $11$ points in $\mathbb{Z}^3$ or $21$ points in $\mathbb{Z}^4$ are possible. If they exist, they would improve the corresponding lower bounds on the integer Radon numbers. We compare exhaustive extension, greedy local search, PatternBoost, AlphaZero, and AlphaEvolve. All outputs use one success predicate: a convex target-sized configuration succeeds exactly when its number of integral Radon points is zero. PatternBoost and AlphaZero construct verified Radon-free configurations of sizes $10$ and $20$, one point short of the respective targets; AlphaZero's best convex target terminals have two integral Radon points in both dimensions. Exhaustive extension over fixed seeds and declared finite search boxes likewise reaches a minimum of two integral Radon points. Greedy local search and AlphaEvolve each produce target-sized near misses with one integral Radon point. No audited run reaches zero. These results establish reproducible near-miss baselines over explicitly scoped search domains.

cs.LG↗

arXiv2026

Traveling Salesman Problem on Graphs with Proportional Independent Set

We give a deterministic algorithm for minimum-cost Hamiltonian cycle in undirected graphs with arbitrary real edge costs and an independent set of size at least $αn$, for any fixed $3/7<α\le1/2$. It runs in $O^*(γ(α)^n)$ time for an explicit constant $γ(α)<2$; for bipartite graphs, the bound is $O^*((1+2\sqrt2)^{n/2})\approx O^*(1.957^n)$. The algorithm is based on a restricted Held--Karp dynamic program. Its state enumeration uses structural properties of minimum-cost assignments arising from a regular triangulation of a product of simplices.

cs.DS↗

arXiv2026

Prior-Informed Generalizable Low-Rank Representation Learning via Regularized Adversarial Reweighting

Modern learning problems increasingly leverage data from heterogeneous sources to improve robustness and generalizability under distribution shift, often with high-dimensional parameters admitting low-rank structure. Existing group distributionally robust optimization (group-DRO) approaches optimize against adverse source mixtures, but extending this principle to low-rank matrix estimation with a broad class of losses leads to a challenging nonconvex-concave problem for which methods and theory remain limited. We propose Prior-Informed generalizable Low-rank representation Learning via Adversarial Reweighting (PILLAR), which smooths the group-DRO objective by regularizing adversarial source weights toward a user-specified reference distribution while learning a shared low-rank representation. By tempering extreme adversarial reweighting, PILLAR improves estimation stability and reduces the potential conservativeness of group-DRO. Using Burer-Monteiro factorization, we optimize the nonconvex-concave objective by bifactored gradient descent with mirror ascent. We establish linear convergence to a statistical neighborhood of the population saddle point and derive an error bound that separates low-rank estimation from adversarial reweighting. We further show that PILLAR improves on vanilla group DRO when the latter's excess conservativeness dominates source heterogeneity and prior misspecification. Simulations and an electronic health record study of suicide-attempt risk demonstrate improved worst-source and transported predictive accuracy.

stat.ME↗

arXiv2026

Tightness and Error Exponents of SDP with Logarithmically Many Communities

We study a semidefinite programming (SDP) relaxation for community recovery when the number of communities grows logarithmically. In the balanced stochastic block model with $n=km$ vertices, we consider the regime $k/\log m\toγ>0$, with edge probabilities $α\log m/m$ within communities and $β\log m/m$ across them, for fixed $α>β>0$. We derive the sharp asymptotic tightness boundary away from critical cases. When rare vertices cause tightness to fail while the bulk remains spectrally stable, the normalized matrix error of every near-optimal solution still vanishes. We prove matching high-probability exponents for this error and the normalized optimal objective gain, governed by the same local correction that determines tightness. Throughout this spectrally stable region, a single SDP solve followed by explicit rounding and refinement achieves exact community recovery above the information-theoretic threshold. This guarantee holds even when the planted community matrix is not an optimal solution to the SDP.

math.ST↗

arXiv2026

Channel-Preserving Representation Alignment for EEG-to-Music Reconstruction

Reconstructing naturalistic music from EEG requires extracting music-specific information from weak signals distributed across electrodes. Existing spectrogram regression and diffusion approaches provide audio synthesis mechanisms, but learning the EEG-to-music mapping faces an information--estimation tradeoff. Combining electrode measurements into fewer features can ease estimation from limited recordings while obscuring distinctions between music segments. We propose channel-preserving representation alignment, separating electrode information retention from prediction regularization. Per-electrode tokenization preserves spatial identity for learned aggregation, while multi-view self-distillation and channel dropout encourage consistency across electrode subsets during pretraining and music alignment. The aligned representation conditions a pretrained audio generator. Our theory explains these complementary choices. A finite-sample regression analysis identifies when retained task information outweighs estimation cost. For nonlinear alignment, an exact decomposition reveals a prediction-consistency penalty in the masked contrastive objective, with margin-dependent bounds connecting prediction variability to ranking errors across electrode subsets. On NMED-T/H, our method improves within-subject 50-way identification from 0.388 to 0.485 and generated-audio CLAP similarity from 0.635 to 0.683 compared with previous work.

cs.SD↗

arXiv2026

Combinatorial Inference on the Optimal Assortment in Multinomial Logit Models

Assortment optimization has received active explorations in the past few decades due to its practical importance. Despite the extensive literature dealing with optimization algorithms and latent score estimation, uncertainty quantification for the optimal assortment still needs to be explored and is of great practical significance. Instead of estimating and recovering the complete optimal offer set, decision-makers may only be interested in testing whether a given property holds true for the optimal assortment, such as whether they should include several products of interest in the optimal set, or how many categories of products the optimal set should include. This paper proposes a novel inferential framework for testing such properties. We consider the widely adopted multinomial logit (MNL) model, where we assume that each customer will purchase an item within the offered products with a probability proportional to the underlying preference score associated with the product. We reduce inferring a general optimal assortment property to quantifying the uncertainty associated with the sign change point detection of the marginal revenue gaps. We show the asymptotic normality of the marginal revenue gap estimator, and construct a maximum statistic via the gap estimators to detect the sign change point. By approximating the distribution of the maximum statistic with multiplier bootstrap techniques, we propose a valid testing procedure. We also conduct numerical experiments to assess the performance of our method.

stat.ML↗

arXiv2026

Unimodality of Independence Polynomials for Sufficiently Large Forests

We prove that the independence sequence of every sufficiently large forest is unimodal. The result follows from establishing log-concavity on a central interval of the sequence, along with monotonicity of the initial and final segments. The main analytic step is a central limit theorem for the size of a random independent set sampled with the hard-core model, uniform over all forests and over an interval of positive fugacities.

math.CO↗

arXiv2026

A Two-Stage Construction of Positive Curvature on the Gromoll-Meyer Sphere

We construct an explicit one-parameter family of smooth metrics on the Gromoll-Meyer exotic seven-sphere, converging in $C^\infty$ to a fixed further Cheeger deformation of the Eschenburg-Kerin metric and having strictly positive sectional curvature for all sufficiently small positive parameter values. The first perturbation preserves the totally geodesic flats of one zero-plane family while making curvature positive near the other; the second removes the remaining zero curvature. The metric and the proof were discovered by the Odin Automatic AI Research Agent.

math.DG↗

arXiv2026

Vector Balancing via Directional Total Variation

Our main result is a $3\sqrt{2π}$ bound for the Komlós signing problem: every finite family of real vectors of Euclidean norm at most one admits a signed sum of $\ell_\infty$-norm less than this constant, independently of the dimension and the family size. For any $κ\ge0$, if a bounded open convex set supports a probability density with directional total variation at most $κ$ in every unit direction, then its open-set Banaszczyk transform supports another such density with the same $κ$, provided the translation vector $v$ satisfies $κ\|v\|_2\le1/3$. As a consequence, every finite set system in which each element belongs to at most $t$ sets, where $t\ge1$ is an integer, admits a two-coloring whose imbalance in each set is less than $3\sqrt{2πt}$. This gives the square-root dependence predicted by the Beck-Fiala conjecture. The proof was discovered by the Odin Automatic AI Research Agent.

math.CO↗

arXiv2026

Optimal Linear Functional Estimation for Proportionally High-Dimensional Linear Models

We study estimation of a linear functional in random-design linear models when $γ=p/n \in (1, \infty)$ under the dense regime where the design covariance is unknown, and neither the regression vector nor the precision matrix is assumed sparse. Suppose the spectrum of the design covariance belongs to $[\ell,L]$, we establish the minimax mean squared rate for the linear functional has the order $n^{-α(κ,γ)}$, where the exponent $α(κ,γ)$ has an explicit elliptic-integral representation and $α(κ,γ) < 1$ when both $κ> 1$ and $γ> 1$. We propose the Bocca estimator, which combines empirically calibrated resolvents through a complex contour integral to attain the optimal rate.

math.ST↗

arXiv2026

An Improved Lower Bound for the Complex Grothendieck Constant

We prove $K_G^{\mathbb{C}}>1.35584631827168$ for the classical complex Grothendieck constant, closing more than one quarter of the gap between Davie's lower bound and Haagerup's upper bound. The numerical part of the proof is rigorously verified by interval arithmetic. The lower bound and the proof were discovered by the Odin Automatic AI Research Agent.

math.FA↗

arXiv2026

A Metric with Positive Sectional Curvature on $S^3\times S^3$

We construct a smooth Riemannian metric with strictly positive sectional curvature on $S^3\times S^3$. Since $S^3\times S^3$ is even-dimensional and has Euler characteristic zero, our result disproves the positive-curvature case of Hopf's sign conjecture. The metric and the proof were discovered by the Odin Automatic AI Research Agent.

math.DG↗

arXiv2026

On Unavoidable Faces of High-Dimensional Polytopes

Kalai's cube--simplex conjecture asserts that for all positive integers $\ell,k$, there is an integer $f(\ell,k)$ such that every polytope of dimension at least $f(\ell,k)$ has either a simplex $\ell$-face or a cube $k$-face; let $f_s(\ell,k)$ denote the threshold restricted to simple polytopes. Finiteness of $f(\ell,k)$ is known only for $\ell,k \leq 2$. In addition, Kalai proved that $f_s(2,k) \leq 2k^2$. Here we prove that $f_s(\ell,k)$ is finite for all $\ell \geq 2$ and $k \geq 3$, the first such result beyond $\ell = 2$, with $f_s(2,k) \leq 2k^2-1$ and $f_s(\ell,k) \leq \tfrac{1}{2}k^2\ell\,2^k$ for $\ell \geq 3$. In the opposite direction, we obtain the lower bounds $f(\ell,k) \geq (5\lfloor \ell/2 \rfloor + (\ell \bmod 2) - 1)(k-1)+1$ and $f_s(\ell,k) \geq \max\{4,\,2(\ell-1)\}(k-1)+1$. A companion question asks for the minimum possible size of a 3-face within a higher-dimensional polytope. Meisinger, Kleinschmidt and Kalai proved that every rational $d$-polytope with $d \geq 9$ has a $3$-face with fewer than $78$ vertices or fewer than $78$ facets. Here we improve their bound: every convex polytope of dimension at least $15$ has a $3$-face with at most $13$ facets. One step of our proof requires an explicit exact rational certificate or identity on flag numbers. This certificate is computed using linear programming.

math.CO↗

arXiv2026

A Complex Structure on $S^2\times S^4$

We show that $S^2\times S^4$ admits a complex structure. Starting from a modular family of complex two-tori associated with the $(3,4,\infty)$ triangle group and the compactification constructed in [Alpöge 2026], we replace the period lattice by its unique monodromy-invariant index-two superlattice and compactify the resulting family. In the integral Mayer--Vietoris calculation, a primitive local class whose double is the class of a cusp component and the unique nonzero torsion class contributed by the multiplicity-four fibre restrict to the same class of order two on the common boundary. We then prove that the resulting compact complex threefold is diffeomorphic to $S^2\times S^4$. The proof is discovered by the Odin Automatic AI Research Agent.

math.GM↗

arXiv2026

A Metric with Positive Sectional Curvature on $S^2\times S^3$

We prove that $S^2\times S^3$ admits a Riemannian metric with positive sectional curvature. We view it as a principal circle bundle over $S^2\times S^2$. A diagonal Cheeger deformation of the base and a connection whose curvature form vanishes on the remaining flat tori yield a nonnegatively curved connection metric whose zero-curvature planes are the horizontal lifts of the tangent planes to those tori. We then perturb this metric by the real part of a global complex-valued symmetric $2$-tensor. Differentiation along the circle fibers produces a trace-free first variation of the second fundamental form on local horizontal lifts of the flat tori. The Gauss equation converts this into a positive second-order curvature term that dominates as the fibers shrink. A quantitative lower bound for the Hessian in directions normal to the set of zero-curvature planes extends this positivity to nearby planes. The metric and the proof are discovered by the Odin Automatic AI Research Agent.

math.DG↗

arXiv2026

Contextual Online Uncertainty-Aware Preference Learning for Human Feedback

Reinforcement Learning from Human Feedback (RLHF) has become a pivotal paradigm in artificial intelligence to align large models with human preferences. In this paper, we propose a novel statistical framework to simultaneously conduct the online decision-making and statistical inference on the optimal model using human preference data based on dynamic contextual information. Our approach introduces an efficient decision strategy that achieves both the optimal regret bound and the asymptotic distribution of the estimators. A key challenge in RLHF is handling the dependent online human preference outcomes with dynamic contexts. To address this, in the methodological aspect, we propose a two-stage algorithm starting with $ε$-greedy followed by exploitations; in the theoretical aspect, we tailor anti-concentration inequalities and matrix martingale concentration techniques to derive the uniform estimation rate and asymptotic normality of the estimators using dependent samples from both stages. Extensive simulation results demonstrate that our method outperforms state-of-the-art strategies. We apply the proposed framework to analyze the human preference data for ranking large language models on the Massive Multitask Language Understanding dataset, yielding insightful results on the performance of different large language models for medical anatomy knowledge.

stat.ML↗