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arXiv2026

Product and Quotient Set Dominance in Finite Groups

A classic problem in additive number theory asks whether a given subset of integers has more sums or differences. This inquiry naturally extends to finite groups, where one compares the sizes of the product set $AA$ and the quotient set $AA^{-1}$ for a subset $A \subseteq G$. A finite group is considered product-dominant if it contains a higher proportion of subsets where $|AA| > |AA^{-1}|$, and quotient-dominant if the reverse is true. We establish statistical criteria to determine the dominance behavior of large finite groups by analyzing the moments of the random variable $d(A) := |AA| - |AA^{-1}|$. By translating product and quotient relationships into graph-theoretic constructions, we compute these moments through counting independent sets. Our main result demonstrates that for a growing sequence of finite groups $G_k$ with center size $Z_k$ and involution count $T_k$, the condition $Z_k = o(T_k)$ implies the groups are eventually product-dominant. Conversely, if $T_k \cdot \sqrt{3}^{R_k/Z_k} = o(Z_k)$ (where $R_k$ denotes the number of elements whose square is central), the groups are eventually quotient-dominant. Using these asymptotic conditions, we resolve the Miller--Vissuet conjecture by proving that all dihedral groups $D_n$ for $n \geq 3$ are product-dominant. Furthermore, we classify the dominance behavior of several major families of groups, proving that symmetric groups and non-abelian finite simple groups are product-dominant, while groups of odd order and certain abelian groups are quotient-dominant. Finally, for groups that fall outside our asymptotic conditions, we derive an explicit formula for the first moment utilizing the irreducible characters and conjugacy classes of the group.

math.NT↗

arXiv2026

Sample-Preserving Search-to-Decision Reduction for Noisy Linear Equations over Large Moduli

The Noisy Linear Equations problem involves finding the solution to a random linear system over a finite field given noisy evaluations. This is a generalisation of various learning problems widely used in cryptography, including Learning with Errors (LWE), Learning with Rounding (LWR), and Learning Parity with Noise (LPN). We present a new sample-preserving search-to-decision reduction for Noisy Linear Equations that has complexity $\mathrm{poly}(n,m,\log q, q/σ)$, where $n$ is the dimension of the secret, $m$ is the number of samples, $q$ is the modulus, and $σ$ is the magnitude of the noise (defined suitably for the kind of noise involved). In particular, this implies search-to-decision reductions for LWE and LWR that run in time $\mathrm{poly}(n)$ even if the modulus $q$ is exponentially large, as long as the noise-to-modulus ratio $σ/q$ is non-negligible. Our reduction works for prime moduli $q$. If $q$ is composite, it instead produces a list that contains a small multiple of the solution. Prior to this work, known search-to-decision reductions for LWE and LWR either did not preserve the parameters of the problem (i.e., $m$, $n$, $σ$, and $q$), or had complexity that scaled polynomially with either $σ$ or the largest prime factor of $q$.

cs.CR↗

arXiv2026

OGPO: Offline Goal-conditioned Policy Optimization for Recoverable Vision-Language-Action Models

Vision-language-action (VLA) models have shown strong promise in robotic manipulation, yet their post-training still predominantly relies on trajectory-level supervised imitation over robot demonstrations. While effective for learning feasible actions, this paradigm supervises policies to reproduce expert actions at each step, biasing them toward demonstration-specific action paths and providing limited guidance for recovering toward task-progress states once execution deviates from demonstrated trajectories. To address this limitation, we propose OGPO, an Offline Goal-conditioned Policy Optimization framework for recoverable VLA post-training. Instead of treating offline demonstrations merely as step-wise action labels, OGPO adaptively relabels trajectories with semantic key states and constructs goal-conditioned process rewards to optimize the policy for reaching these task-progress states. In this way, OGPO shifts VLA post-training from trajectory-level action imitation to key-state reachability learning, enabling policies to acquire a more robust ability to complete goal states from feasible off-demonstration states without requiring additional online interaction. Experiments on LIBERO and MetaWorld show that OGPO consistently improves VLA policy performance over standard supervised fine-tuning. Moreover, evaluations on LIBERO-Plus and our proposed LIBERO-ReAct, a mid-execution perturbation benchmark, further demonstrate that OGPO achieves stronger robustness and goal-directed recoverability under distribution shifts and recoverable off-demonstration states.

cs.AI↗

arXiv2026

Controlled-Minimax: Stabilizing Adversarial Optimization via Control Theory

Minimax optimization is fundamental to adversarial learning yet suffers from severe stability challenges, often manifesting as persistent limit cycles or divergence. This paper investigates these instabilities through the lens of continuous-time dynamical systems. By modeling training dynamics as a second-order Ordinary Differential Equation (ODE), we rigorously identify the root cause of non-convergence: the problematic eigenstructure of the loss function's Hessian near equilibrium points. To address this, we introduce Controlled-Minimax, a novel stabilization framework that reformulates the optimization trajectory as a controllable physical system. Unlike heuristic regularization techniques, our approach derives a specific feedback controller that provides theoretical guarantees of local exponential stability, regardless of the underlying Hessian spectrum. Consequently, our method stabilizes general minimax problems where traditional gradient descent-ascent methods fail. Extensive experiments on pathological benchmarks including the Dirac-GAN, 2D Gaussian mixtures, and Swiss Roll datasets, as well as high-resolution image synthesis with StyleGAN2-ada, demonstrate that Controlled-Minimax effectively suppresses oscillatory behavior while ensuring fast and robust convergence.

cs.LG↗

arXiv2026

DeFiRank: Federated Pairwise Ranking for Governance in Decentralized Financial Networks

Existing governance approaches in decentralized financial networks rely on manual review processes conducted by human committees, which introduces subjective bias, creates a scalability bottleneck under high network volume, and fails to capture complex behavioural patterns and evolving network dynamics. To address these limitations, we propose \name, a federated learning-based pairwise ranking framework in which multiple institutions collaboratively train a ranking model without sharing their private transaction data. This collaborative learning enables the model to capture network-wide behavioural patterns that signify strong network participation and directly contribute to the network effect, including diversity of counterparties, usage of multiple applications, and frequency of transactions. The proposed system ranks providers based on the authenticity of their actions, promoting genuine contributors and demoting those exhibiting suspicious behaviour. We evaluate \name\ on a real-world blockchain transaction dataset. Results demonstrate that the federated approach achieves ranking performance modestly below the centralized baseline (NDCG@15: $\approx$0.87 vs.\ $\approx$0.92) while preserving data privacy and matching its ranking stability (Kendall $τ$: $\approx$0.70 for both regimes), producing reliable governance decisions, including provider promotion and demotion (648 and 39 unique providers, respectively), without requiring centralized access to sensitive transaction data.

cs.AI↗

arXiv2026

AI-Mediation in Retail Markets: Revenue Effects and Assortment Optimization

AI shopping assistants increasingly influence which products consumers consider. We study retail assortment planning under AI mediation, where a retailer chooses a catalog, an AI intermediary forms a customer-facing shortlist, and the customer chooses from that shortlist according to a multinomial logit model. We consider both deterministic mediation, in which the AI follows a fixed segment-specific ranking, and stochastic mediation based on the Plackett-Luce model. Empirically, we construct a dataset of product rankings from leading AI models, estimate Plackett-Luce ranking models, and evaluate their predictive performance on held-out assortments. Combining these estimates with synthetic customer preferences, we find that accounting for AI mediation can materially change optimal assortments and revenue. Across the tested capacity configurations, ignoring mediation yields mean revenue losses between 2.52% and 11.37% relative to the PL optimum. A much simpler deterministic plug-in, based only on the ordering of fitted AI utilities, performs substantially better, with mean losses of at most 0.53%. Motivated by this performance, we study assortment optimization under deterministic mediation. We characterize its computational complexity and show that, for any fixed number of customer segments, the problem admits a fully polynomial-time approximation scheme.

cs.AI↗

arXiv2026

A Comparison Theorem for $p$-adic GKZ Hypergeometric $\mathcal D$-Modules and Complexes

We compare the $p$-adic GKZ hypergeometric complex with the hypergeometric arithmetic $\mathcal D$-module defined via six functors. Under the $p$-adic semi-nonresonance condition, the natural comparison morphism is a quasi-isomorphism. We prove concentration of the $p$-adic GKZ hypergeometric complex in degree $n$, establish overholonomicity under another non-Liouville hypothesis. When $(q-1)γ\in\mathbb Z^n$, we construct a Frobenius structure on its $n$-th cohomology and prove that its restriction to the nondegenerate locus is an overconvergent $F$-isocrystal of rank $n!\operatorname{vol}(Δ)$.

math.AG↗

arXiv2026

Beyond Second-Order Information: Paid Logistic Regression with Noisy Features

We consider online binary prediction with paid stochastic features, where the learner can spend resources to obtain better observations of the underlying features. This setting is motivated by applications such as medical testing, where additional cost can provide more precise measurements of patient's features for predicting a binary outcome. At each round, a latent feature is drawn i.i.d., the response is generated through a logistic model, and the learner chooses how much to pay before observing a noisy version of the feature. The goal of learner is to jointly control prediction loss and acquisition cost. In contrast to the former result on the online linear regression with paid features, covariance information alone is no longer sufficient: two noise distributions with the same covariance may induce different optimal logistic losses, and lower variance can even lead to higher loss. Under a structural ordering in which larger payments yield better observations by removing independent noise, we obtain $\widetilde O(\sqrt{T})$ regret when the noise structure is known and $\widetilde O(T^{2/3})$ regret when it is unknown, with matching lower bounds up to logarithmic factors.

stat.ML↗

arXiv2026

SCCGBench: Executability-Oriented Evaluation for Real-World Web Service-Call Code Generation

LLM-generated service-call code is increasingly important for service computing, where developers rely on large language models (LLMs) to compose services and invoke third-party services. However, current approaches have not yet evaluated the invocation and execution of services in the actual code environment. To support execution-grounded evaluation, we present the Service-Call Code Generation Benchmark (SCCGBench), a benchmark of 3,135 traceable and decontamination-filtered RapidAPI call functions from real GitHub repositories. SCCGBench uses redacted-intent generation to reduce circular leakage and introduces Mock, Robust, ParamF1, and host-, endpoint-, method-, and parameter-level diagnostics. Mock denotes a virtual-execution hit that simultaneously matches the gold host, endpoint, and HTTP method, while Robust further requires status checking and exception handling. We evaluate six code LLMs and one non-neural retrieval baseline under intent-only, retrieval-augmented, and gold-documentation Oracle settings. Results show that models often generate syntactically correct or superficially similar HTTP-request code but achieve low Mock success in the intent-only setting; the strongest evaluated model obtains only 3.76% Mock. Under the gold-documentation Oracle settings (C3 and C5), Mock reaches 88.73%--90.52% for the open-source models evaluated in RQ2 and 76.47%--81.86% for DeepSeek-V4-Pro, whereas sparse and dense retrieval remain far below this oracle level. The findings indicate that service/API grounding is the main bottleneck: Mock is dominated by host, endpoint, and method alignment, while ParamF1 separately reveals parameter-alignment errors. SCCGBench provides an execution-grounded benchmark for Web service invocation, API recommendation, service-interface retrieval, endpoint grounding, and reliable service-call automation.

cs.SE↗

arXiv2026

Teaching neural decoders with language

Brain-computer interfaces decode neural activity, yet class-label supervision (CLS) leaves relationships among classes implicit. We propose Language as Supervision (LaS), which aligns neural features with natural-language class descriptions through learned projections and predicts classes using candidate text prototypes. We evaluate seven encoder architectures with independently trained models for visual, motor, and speech decoding from scalp and intracranial recordings. LaS improves speech-decoding accuracy by up to 17.15~pp over matched CLS baselines. Its representations support candidate-set reuse and label-supervised subject and session adaptation, while LaS enables conditional unseen-frequency decoding in steady-state visual evoked potential (SSVEP) responses. Correct class-description correspondence supports this generalization, and dual-sided attribution visualization shows neural and textual contributions to the same class-matching score. These findings suggest that text descriptions can serve as general-purpose references for learning and interpretation in neural decoding.

cs.AI↗

arXiv2026

A volume-controlled level-set method for topology optimization

Topology optimization typically involves a geometric volume/mass constraint due to the limited use of material. As a popular technique for topology optimization, the level-set method typically employs the Lagrange method, the penalty method, or the augmented Lagrangian approach to address the volume constraint. This requires careful parameter tuning and may lead to non-monotonic changes in the objective. In this paper, we present a new volume-controlled level-set method for topology optimization. By imposing a scalar weighted volume?compatibility constraint in a modified Stokes-type extension-regularization problem, the velocity is made volume compatible at the beginning of each frozen transport interval. At the continuous level, we prove that the initial rate of change of the regularized volume vanishes and that the subsequent frozen-velocity drift is of order $T^2$; under a narrow-band assumption, the corresponding constant is independent of the smoothing parameter. For the discontinuous Galerkin spatial semi-discretization of the level-set transport equation, we derive a quantitative regularized-volume drift estimate that displays the dependence on the smoothing parameter, spatial mesh size, transport time, and discrete velocity. The effectiveness of the present algorithm is demonstrated through the minimum compliance problem for linear elasticity, the eigenfrequency maximization for structural vibration, and the minimum energy dissipation problem for Stokes flows.

math.OC↗

arXiv2026

Sparse Dictionary Learning for Measure-Valued Data via Linearized Optimal Transport: Efficient Algorithms and Sample Complexity

We study dictionary learning for probability measure-valued data by embedding measures into Hilbert spaces using features derived from optimal transport geometry. We learn data-adapted dictionaries through sparse autoencoder parameterizations, producing fast, scalable representations of probability measures as sparse finite-dimensional codes. The resulting framework combines the rich, geometric structure of optimal transport with the interpretability and efficiency of sparse representation learning. We demonstrate its utility across diverse data modalities, including hyperspectral images, geometric point clouds, and document embeddings in large language model latent spaces. We also prove generalization bounds for sparse dictionary learning objectives on broad classes of $L^2$-valued measure embeddings, expressed in terms of the metric complexity of the atom class. For optimal transport and entropic transport map dictionaries, we derive new covering number bounds that yield explicit rates. Together, our computational and theoretical results support a new, scalable model for representation learning in the space of probability measures via embedding into Hilbert spaces.

stat.ML↗

arXiv2026

Symmetry-Enforced Topology and Van Hove Saddle Points in Hexagonal TiAs

We report the first successful growth of stoichiometric hexagonal TiAs single crystals (space group $P6_3/mmc$) using chemical vapor transport and investigate their vibrational, electronic, topological, and magnetotransport properties. Polarized Raman spectroscopy identifies all four Raman-active phonon modes ($A_{1g} + E_{1g} + 2E_{2g}$), consistent with the hexagonal crystal symmetry. First-principles calculations reveal nontrivial band topology characterized by a symmetry indicator $\mathbb{Z}_4 = 2$ and a strong topological index $ν_0 = 1$, together with symmetry-protected fourfold band degeneracies at the nonsymmorphic Brillouin-zone boundary. The Fermi surface exhibits an orbital-dependent crossover from quasi-two-dimensional cylindrical sheets to a corrugated three-dimensional structure along $k_z$. A Van Hove saddle point approximately 0.5 eV below the Fermi level along $Γ\text{--}K$ is associated with a neck-disrupting Lifshitz transition and enhanced density of states. Magnetotransport measurements reveal a pronounced twofold angular magnetoresistance, consistent with anisotropic Fermi-surface geometry and open cyclotron orbits. Hall measurements indicate uncompensated, hole-dominated transport, accounting for the absence of extreme magnetoresistance and contributing to the field-orientation-dependent Hall response. These results establish hexagonal TiAs as a platform for exploring the interplay of nonsymmorphic band topology, Van Hove singularities, and anisotropic electronic transport in transition-metal monopnictides.

cond-mat.mtrl-sci↗

arXiv2026

What Egocentric Video Embeddings Encode and How to Combine Them with Inertial Sensors for Human Activity Recognition

Egocentric video and body-worn inertial sensors are increasingly recorded together, and the video is usually encoded by pretrained models. What these embeddings represent when the camera sits on the head, and how they should be combined with inertial sensing, has received little attention. Across VideoMAEv2, CLIP, DINOv2 and the egocentric VC-1, the leading principal directions of the embedding are dominated by where the camera looks. Measured head orientation explains them on Ego-Exo4D for egocentric (R^2 0.49-0.70) but not exocentric cameras, and the scene layout explains them on WEAR. The part that the limbs' inertial signals predict separates activities across people. In a set-level probe, the rest separates exercise sets within a participant but not activities across people. This is consistent with separate roles: the video finds the clips of the same set, and the inertial classifier labels them. On WEAR, with temporal order removed and one random limb per one-second clip, we propagate inertial predictions over a within-participant video graph whose leading components are removed. This raises subject-wise cross-validated macro-F1 from 0.648 to 0.852. In our comparisons, propagation helps only over the video graph, and fusing the modalities in one classifier, or adding inertial edges to the graph, performs worse.

cs.CV↗

arXiv2026

Collaborative Learning and Reasoning over Model Slices

Large language models (LLMs) excel across diverse tasks, benefiting from knowledge priors, in-context reasoning, and test-time computation. Despite these strengths, single-model reasoning remains vulnerable to error propagation, as later steps depend on the model's earlier outputs, leaving reasoning blind spots difficult to independently reassess. To address this limitation, we develop \textit{collaborative learning}, encompassing the construction of heterogeneous collaborators and the learning of effective cross-model collaboration. Specifically, \textit{model slicing} replaces a large model with smaller, heterogeneous models under a comparable parameter budget, enabling prior reasoning to be reassessed across distinct model parameterizations. \textit{Pyramid supervision} combines protocol distillation with step-level supervision to refine reasoning and interaction decisions throughout collaboration. We evaluate our approach across four task categories: multi-hop question answering, mathematical reasoning, code generation, and open-ended constrained generation. The trained heterogeneous models achieve comparable or better task performance than the original large model and established multi-agent baselines, including reinforcement-learning-based methods, while also achieving higher inference efficiency than the multi-agent baselines and exhibiting differentiated collaboration patterns across tasks. These findings show that learned collaboration turns model heterogeneity into complementary reasoning capabilities.

cs.AI↗

arXiv2026

On Eigenvector Statistics of Large-Dimensional Sample Covariance Matrices with Applications

We study the eigenvector-weighted spectral statistics \[ T_n(φ,g)=p^{-1}tr\{φ(\boldsymbol{S}_n)g(\boldsymbolΣ)\} \] of large-dimensional sample covariance matrices $\boldsymbol{S}_n$ with population covariance $\boldsymbolΣ$. These statistics combine functions of sample eigenvalues with sample--population eigenvector overlaps and arise in statistical losses that depend on both. As the dimension and sample size grow proportionally, we establish a joint central limit theorem for $\{T_n(φ_a,g_a)\}_{a=1}^r$ with multiple analytic test functions and population weights, centered by finite-dimensional deterministic equivalents. A key intermediate step is a joint Gaussian process limit for the weighted resolvent traces \[ m_n^g(z)=p^{-1}tr\{(\boldsymbol{S}_n-z\boldsymbol{I}_p)^{-1}g(\boldsymbolΣ)\}. \] The limiting means and cross-covariances are expressed in terms of the population spectral distribution. We apply these results in two settings: one concerns the variance component of minimum-norm prediction risk, and the other concerns the Frobenius loss of fixed rotationally invariant covariance estimators. For fixed affine shrinkage rules, the loss means, variances, and cross-covariances have explicit formulas in terms of population spectral moments.

math.ST↗

arXiv2026

An Instability of the Quantum Markov Length

The Markov length, the length scale governing the exponential decay of conditional mutual information (CMI) in a quantum many-body state, has proven useful for characterizing mixed-state phases of matter. Nevertheless, we show that a state's Markov length is not generally stable under weak local channels. The failure of stability occurs already at a pure product state. We demonstrate this explicitly in a one-dimensional fermionic chain, where a fine-tuned depth-two circuit of nearest-neighbor Gaussian channels maps the vacuum to a state whose CMI decays \emph{algebraically} and whose Markov length is therefore infinite, even when the channels are taken arbitrarily close to the identity. The algebraic decay is rooted in the nonanalyticity of the von Neumann entropy at rank-deficient density operators. We exhibit an analogous instability on an ordinary qubit chain. More generally, we show that the Markov length is unstable in any translation-invariant state with even a single rank-deficient reduced density operator. These results pose a challenge to recent definitions of mixed-state phase equivalence based on the Markov length.

quant-ph↗

arXiv2026

Direct proofs for spaces with cyclic parallel Ricci tensor

We give direct proofs of rigidity results for gradient Ricci solitons, compact quasi-Einstein manifolds with boundary, and critical spaces with cyclic parallel Ricci tensor. For solitons and quasi-Einstein manifolds, successive traces of first-order pointwise identities yield quadratic equations for the Ricci tensor and its modification. These equations imply radial flatness and radial $Q$-flatness, respectively, and the corresponding global rigidity theorems apply. For critical spaces, comparison of two expressions for the Laplacian of the Hessian gives a pointwise tensor identity whose scalar contraction yields a weighted divergence formula. Integration then gives the vacuum static, CPE, and Miao--Tam conclusions. The same comparison also proves Ricci parallelism pointwise for solitons and quasi-Einstein manifolds.

math.DG↗