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Search indexed arXiv papers on artificial intelligence, large language models, computer vision and robotics. Read source abstracts and follow links to arXiv.

At least 343 records · Page 19Linked to original sources

arXiv2026

Data-driven approximation of regions of attraction via an LP-based selection of PWA Lyapunov functions

This paper presents a method to approximate regions of attraction of unknown nonlinear dynamical systems from data. Assuming point-wise evaluations of the vector field and known Lipschitz bounds, a polyhedral uncertainty set of admissible dynamics is constructed. This uncertainty description enables the synthesis of a continuous piece-wise affine Lyapunov candidate via a linear program, enforcing a robust decrease condition for all admissible vector fields. The approach allows certification of a region of attraction consistent with the available data. Numerical examples illustrate the effectiveness of the proposed method in extracting certified regions of attraction from sparse data.

math.OC↗

arXiv2026

Pseudo-Formalization for Automatic Proof Verification

Reliable verification of proofs remains a bottleneck for training and evaluating AI systems on hard mathematical reasoning. Fully formal proofs, in languages like Lean, are easy to verify because they are unambiguous and modular. Most proofs, particularly those written by AI systems, have neither property, and translating them into formal languages remains challenging in many frontier math settings. We propose Pseudo-Formalization (PF), a proof format that captures the modularity and precision of formal proofs while retaining the flexibility of natural language. A Pseudo-Formal proof is decomposed into self-contained modules, each stating its premises, conclusion, and proof in natural language. To verify the correctness of a regular natural language proof, an LLM translates it to Pseudo-Formal and then verifies each module independently, an algorithm we call Block Verification (BV). We evaluate PF+BV on two benchmarks spanning olympiad and research-level mathematics, where it pareto-dominates LLM-as-judge baselines on error-finding precision and recall. To support future work, we release our research-level proof verification benchmark ArxivMathGradingBench.

cs.LO↗

arXiv2026

Simulations of Particle-Laden Flows with Large Dispersed-Phase Size Disparities Using Scalable Parallel Adaptive Methods

The numerical simulation of multiphase flows involving dispersed components with large scale disparities, such as the collisions between millimeter-sized bubbles and micron-sized mineral particles in flotation, poses a significant computational challenge. Accurately resolving the thin boundary layers of finite-size objects while tracking massive numbers of small particles within a large turbulent domain is often prohibitively expensive on uniform grids. To address this, we present a parallel scalable computational framework that couples the lattice Boltzmann method with the immersed boundary method on a dynamically adaptive octree grid. A key algorithm is developed for the efficient parallel host-cell searching, which significantly accelerates the tracking of Lagrangian points on distributed unstructured grids. The accuracy and robustness of the code are rigorously validated against canonical benchmarks, including the flow induced by an oscillating cylinder and the sedimentation of a sphere. The framework is applied to the multiscale problem of bubble-particle collisions. In quiescent flow, the simulations accurately capture the hydrodynamic interception mechanism, reproducing the theoretical collision efficiency scaling law proportional to the square of the particle-to-bubble size ratio. Furthermore, the framework is applied to the simulation of fully resolved bubbles interacting with inertial point particles in homogeneous isotropic turbulence.

physics.flu-dyn↗

arXiv2026

A Typed Tensor Language for Shared-State Federated Computation

Shared-state federated computations combine client-local tensor computation, mergeable aggregation into shared state, and shared-only post-processing. We introduce a typed tensor language for this class of computations. Its two tensor sorts separate client-partitioned data from globally available values, and typing tracks the partitioned axis. A virtual global tensor serves as a semantic reference for centralized evaluation. We show that typed one-round programs factor through shared tensors whose shapes depend on the program but are independent of client and sample counts. The converse applies to typed-realizable factorizations: each encoder component is represented by an allowed aggregation or contraction with its valid merge, and the decoder is shared-only. The construction extends round by round to programs whose persistent state is shared. For a loss supplied with a client-local per-sample gradient expression, summation represents the empirical gradient. This gives typed programs for server-side first-order updates and, with shared linear algebra, curvature-block updates. The language covers federated analytics and FedSGD. General multi-local-step FedAvg and persistent private client state are outside its scope.

cs.LG↗

arXiv2026

A Note on EFX Inapproximability for Chores

We study the approximability of envy-free up to any item (EFX) allocations for indivisible chores under complement-free cost functions. Our main result is an instance with $255$ agents and $764$ chores with binary XOS cost functions, in which no $α$-EFX allocation exists for any $1\leα<2$. The construction is based on a combinatorial expansion property of binary labels, obtained using Sidon sets and Reed--Solomon codes. We formalize the main result in Lean 4. We also study the special case of three agents. We construct a six-chore instance with monotone subadditive cost functions for which no $α$-EFX allocation exists for any $1\leα< 2^{1/3}$, and a six-chore instance with monotone submodular cost functions for which no $α$-EFX allocation exists for any $1\leα<20/19$. These constructions are obtained by refining the original counterexample of \cite{CS24}.

cs.GT↗

arXiv2026

Gravitational wave detectability range informed by external messengers

A rapid estimate of gravitational-wave (GW) detectability associated with astronomical transients is crucial for optimizing multi-messenger follow-up strategies and for constraining the physical origin of the transient itself. We introduce here the Targeted Detectability Range (TDR), designed to evaluate, with minimal computational effort, the detectability of compact binary coalescences under the hypothesis of association with an external messenger, such as an electromagnetic or neutrino signal. Unlike the standard GW range, which is based on averaged source parameters, the TDR incorporates prior information from observations of the external messenger, including sky localization, inclination constraints, and physically motivated bounds on component masses. We report the TDR of all short- and long-duration gamma-ray bursts, observed during the first three observing runs of Advanced LIGO and Advanced Virgo. The method is validated by performing a systematic comparison with the 90$\%$ exclusion distances provided by modeled targeted GW searches. In the absence of a coincident detection by all-sky, all-time GW searches, the TDR provides a rapid and quantitative constraint on a possible merger origin of the astrophysical source. Its low-latency implementation and public availability would enable timely prioritization of follow-up observations and optimized allocation of observational resources, with direct impact on the physical interpretation of astronomical transients.

astro-ph.HE↗

arXiv2026

On volumes of simplices in intermediate dimensions

A variant of the Falconer distance problem asks for fixed $k\geq 1$ and $d\geq k+1$, how large does the Hausdorff dimension of a Borel set $E\subset\mathbb{R}^d$ need to be to guarantee that there exist $x_0,\ldots,x_{k}\in E$ such that $\text{Vol}_{k+1}^{(x_0,\ldots,x_{k})}(E) = \lbrace \text{Vol}_{k+1}(x_0,\ldots,x_{k},x_{k+1}) : x_{k+1}\in E \rbrace$ has positive Lebesgue measure. Here $\text{Vol}_{k+1}(x_0,\ldots,x_{k},x_{k+1})$ denotes the $k+1$-volume of the $k+1$ simplex formed by $x_0,\ldots,x_{k},x_{k+1}$. Recently, Shmerkin and Yavicoli established a sharp dimensional threshold $k$ in the case when $d=k+1$. In this paper we extend their result to $k+1 \leq d \leq 2k$ and obtain a non-trivial dimensional threshold $d-k$ when $d>2k$. The result is motivated by ideas from Shmerkin and Yavicoli. A crucial part of the argument is an application of work by Bright, Ortiz and Zakharov on a continuum Beck-type theorem for hyperplanes as well as classic results of Marstrand on projections and slicing theorems. In addition, we investigate a more elementary approach under a condition called the Fubini property for Hausdorff dimension as introduced in the work of Héra, Keleti and Máthé.

math.CA↗

arXiv2026

Algebroid Desingularizable Poisson Structures

We introduce algebroid desingularizable Poisson manifolds, those Poisson manifolds induced by a symplectic Lie algebroid with an almost-injective anchor. This class coincides with that of $E$-symplectic manifolds, and contains log-symplectic manifolds and $b^m$-symplectic manifolds. We give an infinitesimal obstruction to desingularizability: at a zero of the Poisson structure the isotropy Lie algebra must contain an abelian ideal of at least half the dimension of the manifold, so no Poisson structure with a semisimple isotropy Lie algebra can be desingularized. We then characterize the linear case, showing that the dual of a real $n$-dimensional Lie algebra $\mathfrak{g}$, equipped with the KKS Poisson structure, is desingularizable if and only if $\mathfrak{g}$ possesses an abelian ideal of dimension $n-r$, where $2r$ is the maximal coadjoint orbit dimension. In this case, we construct a desingularizing algebroid explicitly.

math.DG↗

arXiv2026

Spectral Tail Auxiliary Learning for AI-Generated Image Detection

As generative image models evolve rapidly, the perceptual gap between generated and real images continues to narrow, making AI-generated image detection increasingly challenging. Many existing methods exploit frequency-domain cues for detection, typically described as frequency-domain artifacts or high-frequency discrepancies. However, the specific and recurring spectral regularities remain insufficiently understood and characterized. In this paper, we systematically analyze the one-dimensional radial log-power spectra of real and generated images. We find that generated images do not necessarily exhibit higher or lower energy across the entire spectrum or high-band range. Instead, their spectra deviate from the power-law decay and show an anomalous uplift in the ultra-high-frequency tail. We term this phenomenon spectral tail uplift. We further attribute this phenomenon to nonlinear harmonic accumulation in trained generative models, suggesting that it can serve as a structural cue across generative architectures. Based on this observation, we propose Spectral Tail Auxiliary Learning (STAL), a frequency-domain auxiliary supervision framework for generalizable AI-generated image detection. STAL transfers spectral-tail cues from a tail-aware frequency teacher to a spatial detector during training, while all frequency-domain modules are discarded at inference time. Consequently, STAL introduces no inference overhead. Extensive experiments on 9 public datasets show that STAL achieves strong generalization and stability across generators, data distributions, and real-world scenarios.

cs.CV↗

arXiv2026

HyperGuide: Hyperbolic Guidance for Efficient Multi-Step Reasoning in Large Language Models

Multi-step reasoning remains a central challenge for large language models: single-pass generation is efficient but lacks accuracy; tree-search methods explore multiple paths but are computation-heavy. We address this gap by distilling reasoning progress into a hyperbolic geometric signal that guides step-by-step generation. Our approach is motivated by a structural observation: in combinatorial reasoning trees, solution-bearing states are few while dead ends are exponentially numerous. The hyperbolic space matches this asymmetry, with compact volume near the origin and exponentially expanding capacity toward the boundary, so that distance-to-origin naturally encodes solution proximity while angular separation distinguishes branches requiring different next operations. We train a lightweight head to project LLM hidden states into this space, then fine-tune a low-rank adapter interactively on its own reasoning attempts to act on the injected signal. Across multiple benchmarks, the geometric signal yields consistent gains, with larger improvements on deeper reasoning chains. Our code is publicly available at https://github.com/yuyuliu11037/HyperGuide.

cs.AI↗

arXiv2026

Thermodynamics of classifiers

The rise of artificial intelligence has led to a growing demand for computational power. In this situation, energy-efficient computing methods are being actively investigated, which raises a fundamental question: what is the relationship between computational error and thermodynamic cost? Here, we derive the error-cost trade-off in the binary classifier by considering classification based on nonequilibrium thermodynamics. We derive trade-off relations showing that classification error cannot be lower than fundamental bounds set by thermodynamic quantities: entropy production, which measures irreversibility, and dynamical activity, which quantifies the activity of the system. Our results show that when entropy production or dynamical activity vanishes, the classification error reaches $1/2$, equivalent to random guessing, whereas greater thermodynamic costs enable lower error. The results establish a quantitative trade-off in physical information processing, potentially leading to the development of energy-efficient computing devices.

cond-mat.stat-mech↗

arXiv2026

Route What Remains: A Meta-Modal Agent for Missing-Modality Candidate Reranking in Recommender Systems

Missing-modality recommenders usually reconstruct absent representations, although the observed evidence may not determine the missing content. We formulate candidate reranking as budgeted sequential evidence acquisition. A policy queries text, image, and interaction-graph tools, incorporates \texttt{Null} returns into its observation history, and sparsely rescores a retrieved candidate pool. Our \textbf{Meta-Modal Agent} (MMA) uses PPO to optimize terminal NDCG and tool cost without explicit access to the route-availability mask or target identity. When only one evidence route is available, MMA-Auto improves NDCG@10 by $10.0$\% over the strongest completion baseline and by $9.5$\% over a fixed router with the same Llama scorer. It obtains the highest result in all nine reported combinations of dataset and available route against these comparators. MMA-Auto also reduces failed calls by 17.8 percentage points and uses 1.1 fewer turns than the fixed router. On the fixed candidate pools produced by full-catalog retrieval, MMA-Auto improves NDCG@10 by $19.7$\%. These results associate adaptive evidence routing with improved reranking under severe, constructed missingness. The code is available at: https://anonymous.4open.science/r/WSDM2027-MMA-C381.

cs.IR↗

arXiv2026

Sharp Bounds and Canonical Blow-ups for Toric Projectivization

For every dimension $n\ge3$, we construct a family $Σ_n(a,b)$ of smooth complete fans with Picard number five and fixed combinatorial type. For integers $a,b\ge1$, the minimum number of new rays in a smooth projective refinement is $\min(a,b)$, attained by ordinary invariant codimension-two blow-ups. If projective simplicial refinements with maximal-cone multiplicity at most an integer $μ\ge1$ are allowed, the minimum is $\lceil\min(a,b)/μ\rceil$. The threefold lower bound depends only on subdivisions of four two-dimensional cones. Toric wedges preserve this minimum while raising the dimension, and a selective support-function criterion gives the matching upper bound. We also construct a projectivization algorithm natural under lattice isomorphisms. For smooth input, it principalizes an intrinsic monomial ideal by ordinary blow-ups with smooth, possibly disconnected centers. Running a fixed functorial principalization algorithm over $\mathbb{Q}$ gives a sequence of fans valid over every field. An intrinsic resolution procedure treats arbitrary complete rational fans. Finally, Oda's threefold shows why symmetry can prevent a canonical ordering of the components of a center, even when their simultaneous blow-up is canonical.

math.AG↗

arXiv2026

Merge Trees of Length-Filtered Lattice Knot Spaces

We study lattice-filtered move graphs as finite-state models for knot types under a length cap. At level $N$, vertices are lattice polygons of a fixed knot type with length at most $N$, modulo orientation-preserving lattice isometries, and edges are local moves. The first level at which two initial components become connected defines a discrete merge scale; after subtracting the birth level it is an ultrapseudometric. For the standard BFACF moves on the simple cubic lattice, the theorem of Janse van Rensburg and Whittington gives connectivity without a cap; our question is the least cap connecting a prescribed pair, and explicit BFACF paths serve as finite PL isotopy certificates. We completely determine the minimal-layer merge trees of the amphichiral knots $4_1$ and $6_3$. The $152$ minimal $4_1$ classes form four components of sizes $58,58,18,18$ at $N=30$ and a single component at $N=32$, so the merge tree is $4\to1$ with barrier $2$. The $148$ minimal $6_3$ classes form twelve components at $N=40$, exchanged in pairs by reflection; at $N=42$ they merge into two mirror components, each with $74$ minimal classes and $12337$ states, and a verified path joins them at $N=44$. Hence the merge tree is $12\to2\to1$ with possible barriers $0,2,4$. Independently verified seed-to-mirror certificates realize the extremal barriers $2$ for $4_1$ and $4$ for $6_3$. Checks for the trefoil, the five-crossing prime knots and composite trefoils are included as reproducibility tests.

math.GT↗

arXiv2026

On Reliability of Membership Inference Vulnerability Evaluation

Membership inference attacks (MIAs) are popular methods for empirically assessing the leakage of sensitive information in the training data through models or statistics learned from the data. The MI vulnerability is often evaluated through a binary classifier that tries to predict whether a particular sample was in the training data. In order to evaluate the effectiveness of MIAs multiple \textit{shadow models} are trained using random partitions of a larger dataset. After training the shadow models the MI vulnerability can be evaluated for all the samples for which we obtained shadow models. In order to evaluate the MI vulnerability reliably one needs a lot of shadow models which can be computationally infeasible. Therefore instead of reporting the actual sample level vulnerabilities aggregates over multiple samples are often reported in practice. We demonstrate two key weaknesses in typical MIA evaluation pipeline. First, we show that sampling the shadow datasets from a fixed superset leads to finite sample bias inflating the vulnerability estimates. Second, we show that evaluating the true positive rate (TPR) by concatenating MIA scores across multiple individuals, commonly used in the very low false positive rate (FPR) regime, is not calibrated across the per-sample FPRs. For both weaknesses we propose fixes that in the most simple approximate form do not incur any additional computation cost. We show that with additional computation one can further improve the reliability of the vulnerability estimation.

cs.LG↗

arXiv2026

Text-Preserving Lossy Text Compression: A Study of Strategic Deletion and LLM Reconstruction

Traditional lossless text compression preserves every byte, but its gains on natural language are often modest in realistic operating regimes. We study \emph{lossy semantic text compression}, where the encoder strategically deletes parts of the text and a large language model (LLM) reconstructs the original content from the retained skeleton. We benchmark a progression of deletion strategies, including uniform step deletion, word-length-guided deletion (WordLen), word-frequency-guided deletion (WordFreq), LP-optimized deletion (Opt), entropy-based deletion using GPT-2 surprisal, and hybrid methods that combine frequency and surprisal signals. Evaluation on the BBC News dataset across retention rates $\r_{keep} \in [0.1,0.9]$ shows three main findings. First, WordFreq is a strong low-cost baseline: despite using only a static frequency lookup, it remains competitive with much more expensive semantic methods while being far faster at the encoder. Second, semantic and hybrid methods provide their clearest gains at mild-to-moderate compression, whereas word-frequency deletion is often more robust at the lowest retention rates. Third, QLoRA fine-tuning yields a strong local decoder that is competitive with Gemini 2.0 Flash and is often strongest in decoder-only comparisons. Additional English and Chinese experiments show that the overall framework transfers across domains, while the best deletion rule remains dataset-dependent.

cs.CL↗

arXiv2026

The Confidence Shortcut: A Reasoning Failure Mode of Masked Diffusion Models

Chain-of-thought reasoning helps autoregressive models solve complex problems by generating intermediate steps that support later predictions. Masked diffusion models (MDMs) offer a similar opportunity through arbitrary-order generation: they can ideally reveal intermediate results along logical dependencies. In practice, however, standard decoding simply prioritizes high-confidence tokens, which need not align with this dependency order. We identify this discrepancy as the \emph{confidence shortcut}: models commit with high certainty to plausible tokens while neglecting long-range dependencies. In multi-digit addition, models predict higher-order digits without properly tracking carries through long chains. Controlled pretraining across diverse reasoning tasks confirms that confidence-guided ordering often selects suboptimal sequences, and confidence-aligned training schemes can exacerbate these failures---for example, increasing addition error rates by an order of magnitude. Our findings caution against relying solely on confidence to choose generation orders and against training objectives that reinforce this preference. The experimental code is available at https://github.com/jinha2536/mdm-arithmetic.

cs.AI↗

arXiv2026

The Q-Calculus: A Quaternion-Based Laws of Form System

This paper introduces a Laws of Form version of the Quaternions. We call this the Q-Calculus, a 16-valued extension of Laws of Form (LoF) which is closely related to the BF Calculus (where we have a single square root of the mark) and the concept of the square root of negation (related to the the square root of minus one). We construct Q as a system of LoF mark operators acting on 4-tuples, and prove that the set of eight operators in Q is isomorphic to the quaternion group, which is non-commutative. We give a novel proof of several of Q's distribution laws using non-commutative logic gates. We indicate how to represent Q as braids by associating elementary braids to square roots of negation. This results in a very concise representation of Q as LoF braids. We end the paper with an indication of how we can represent the Artin braid group in LoF and how we can generalize our work with the quaternions to Clifford algebras.

math.LO↗