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01.
arXiv (CS.CV) 2026-06-16

CoIRL-AD: Collaborative-Competitive Imitation-Reinforcement Learning in Latent World Models for Autonomous Driving

End-to-end autonomous driving models trained with imitation learning (IL) often generalize poorly, particularly in long-tail scenarios where expert demonstrations are sparse. Reinforcement learning (RL) can provide complementary task-level supervision, but applying RL to real-world autonomous driving is challenging in offline settings without interactive simulators, where datasets are dominated by expert actions and provide limited behavioral diversity. We propose CoIRL-AD, a competitive dual-policy framework that integrates IL and RL under a unified offline training regime. CoIRL-AD decouples imitation and reward optimization into separate actors to alleviate objective conflicts, uses imagined future rollouts for long-horizon reward estimation, and introduces a competition mechanism that selectively transfers beneficial behaviors while keeping RL anchored to expert-like driving. Experiments on the nuScenes benchmark show that CoIRL-AD consistently improves robustness over strong IL-based baselines, with especially large gains in cross-city generalization and long-tail scenarios. Code is available at: https://github.com/SEU-zxj/CoIRL-AD.

02.
arXiv (math.PR) 2026-06-18

Rigidity of infinite exchangeable sequences with Gaussian marginals

arXiv:2606.18654v1 Announce Type: new Abstract: We study infinite exchangeable sequences with Gaussian one-dimensional marginals. We formulate the conjecture that joint Gaussianity of a single pair of coordinates forces the entire sequence to be a Gaussian process. Although this conjecture remains open, we prove that joint Gaussianity of the first four coordinates is sufficient. We also establish the corresponding two-point criterion under the additional assumption that the directing measure is almost surely infinitely divisible.

03.
arXiv (CS.CV) 2026-06-12

CACR:Reinforcing Temporal Answer Grounding in Instructional Video via Candidate-Aware Causal Reasoning

The task of temporal answer grounding in instructional video (TAGV), which aims to locate precise video segments that respond to natural language queries, is increasingly important for direct video answer retrieval. This task remains challenging due to the need to comprehend semantically complex questions and to address the significant length mismatch between untrimmed videos and short target moments. Existing methods often suffer from sensitivity to irrelevant content or insufficient visual reasoning capabilities. To tackle these limitations, we propose a Candidate-Aware Causal Reasoning (CACR) framework. Our approach first employs a Visual-Language Pre-training based Candidate Selection (VBCS) algorithm to efficiently generate K candidate segments, then applies a temporal logic reasoning module enhanced by a rejection reward mechanism and optimized via Group Relative Policy Optimization (GRPO) for robust inference. Extensive experiments on six benchmarks demonstrate that our method achieves state-of-the-art performance in terms of mean Intersection-over-Union (mIoU), providing a new perspective for reasoning-based retrieval in long videos.

04.
arXiv (math.PR) 2026-06-18

Cramér-Type Moderate Deviations for Engel's Series via a Martingale Approach

arXiv:2606.18866v1 Announce Type: new Abstract: Let $x$ be uniformly distributed on $(0,1)$, and let $(q_n)_{n\geq1}$ be the digits of its Engel series expansion. We establish a Cramér-type moderate deviation expansion for $(\log q_n-n)/\sqrt n$. The proof is based on a martingale decomposition and asymptotic results for martingales. As consequences, we obtain a moderate deviation principle over the full range of scales between the central limit theorem and the law of large numbers, without the additional lower rate restriction required in several earlier works. We also derive a uniform Berry–Esseen bound of order $(\log n)/\sqrt n$.

05.
arXiv (CS.AI) 2026-06-16

AgenticRec: A Recommendation-Oriented Agentic Framework with Progressive Tool-Integrated Reasoning Optimization

arXiv:2603.21613v2 Announce Type: replace-cross Abstract: Recommender agents built on Large Language Models offer a promising paradigm for personalized recommendation. However, existing agents typically suffer from a misalignment between their tool-integrated reasoning trajectories and recommendation feedback, limiting their ability to distinguish fine-grained user preferences. To address these challenges, we propose AgenticRec, an agentic recommendation framework that formulates recommendation as a tool-integrated reasoning process over a recommendation-oriented tool suite. Built upon this framework, we further develop a dedicated two-stage training paradigm tailored for recommender agents. In the first stage, we introduce Recommendation-Oriented Trajectory Activation, optimize the agentic recommendation ability under implicit feedback. In the second stage, Progressive Preference Refinement further refines the agent through bidirectional preference reasoning over self-bootstrapped hard pairs, progressively sharpening preference boundaries. Theoretical analysis and extensive experiments demonstrate the effectiveness of AgenticRec. Our code is available at https://anonymous.4open.science/r/AgenticRec-FB16.

06.
arXiv (math.PR) 2026-06-18

On a class of reflected McKean-Vlasov Stochastic Differential Equations with jumps

arXiv:2606.18433v1 Announce Type: new Abstract: This paper investigates a class of reflected McKean-Vlasov Stochastic Differential Equations driven by both Brownian motion and a compensated Poisson random measure. We establish the existence and uniqueness of solutions and provide moments estimates for the state processes.

07.
arXiv (CS.CV) 2026-06-15

Enhancing Underwater Light Field Images via Global Geometry-aware Diffusion Process

This work studies the challenging problem of acquiring high-quality underwater images via 4-D light field (LF) imaging. To this end, we propose GeoDiff-LF, a novel diffusion-based framework built upon SD-Turbo to enhance underwater 4-D LF imaging by leveraging its spatial-angular structure. GeoDiff-LF consists of three key adaptations: (1) a modified U-Net architecture with convolutional and attention adapters to model geometric cues, (2) a geometry-guided loss function using tensor decomposition and progressive weighting to regularize global structure, and (3) an optimized sampling strategy with noise prediction to improve efficiency. By integrating diffusion priors and LF geometry, GeoDiff-LF effectively mitigates color distortion in underwater scenes. Extensive experiments demonstrate that our framework outperforms existing methods across both visual fidelity and quantitative performance, advancing the state-of-the-art in enhancing underwater imaging. The code will be publicly available at https://github.com/linlos1234/GeoDiff-LF.

08.
arXiv (CS.LG) 2026-06-16

Pre-Training for Simulation-Based Science: A Study on Jet Foundation Model Training Objectives

arXiv:2606.14870v1 Announce Type: cross Abstract: Foundation models (FMs) trained on large datasets and fine-tuned on downstream tasks have emerged as a powerful paradigm in AI for science. Industrial FMs are typically trained using self-supervision with masking due to the lack of labels. In many scientific domains, accurate simulations are plentiful and facilitate large, labeled datasets. This opens up new possibilities for pre-training. We present a systematic comparison of pre-training methods using the OmniLearned High Energy Physics FM framework. We test supervised classification, flow-matching generation, and self-supervised masked particle modeling. All models are pre-trained on the JetClass dataset and fine-tuned on two representative downstream tasks, top jet classification and JetNet conditional generation. Among other observations, for classification tasks, we find that pure classifier pre-training is optimal when downstream labels and model capacity are plentiful, but combining it with self-supervised masked particle modeling (MPM) is uniquely powerful in the low-finetuning label regime. Flow matching-based generative pre-training seems to provide little benefit for downstream classification, and interestingly, for downstream generation, we find that flow matching must be in the pre-training objective to see a significant finetuning advantage, hinting at the orthogonality of classification and generation tasks. That is, for a model to transfer to both generative and classification downstream tasks, it must be pre-trained on both. This study provides a template for controlled scaling analysis of pre-training objectives for foundation models in simulation-based sciences.

10.
Nature (Science) 2026-06-18

Daily briefing: The proteins that protect us from deadly mutations

Authors:

Proteins that ‘buffer’ the effects of mutations could help to treat diseases such as cancers. Plus, goats can follow human voices and the battle over a key ocean observatory project in the United States. Proteins that ‘buffer’ the effects of mutations could help to treat diseases such as cancers. Plus, goats can follow human voices and the battle over a key ocean observatory project in the United States.

11.
arXiv (CS.LG) 2026-06-17

Verified Detection and Prevention of Concurrency Anomalies in Multi-Agent Large Language Model Systems

Authors:

arXiv:2606.17182v1 Announce Type: new Abstract: Multi-agent LLM systems share state through memory stores, vector indices, and tool registries. We model such sharing as long-running read-generate-write operations under deterministic-generation semantics – the regime durable-execution engines enforce by deterministic replay – and formalize four concurrency anomalies in TLA+: stale-generation, phantom-tool, causal-cascade, and tool-effect reordering, structural analogues of classical isolation anomalies, each with a TLC counter-example. The exclusion lattice over these anomalies is trivial; the contribution is the mechanically verified realizability and strict separation of one maximal chain within it, $L_0 \subsetneq \cdots \subsetneq L_4$, to our knowledge the first machine-checked consistency hierarchy for such runtimes. A development of 274 Verus obligations (zero assume, zero admit; trust base: two structural axioms and a mutex correspondence) proves the detectors sound and complete against the specifications and each runtime its avoidance set. Three deployed Rust runtimes realize L0-L1 (pessimistic locking, serializable snapshot isolation, default-SI), each verified against stale-generation and refined to its state machine; L2-L4 are exec-mode-verified with dependency-free prevention twins (A3, A6, A2: 0/1000 versus 1000/1000), and L2 is run live across three model families (A3 prevented in all 120 retracted sessions). We reproduce a silent lost update in ByteDance's deer-flow, formalizing its fix as a verified $L_0 \to L_1$ refinement, and exhibit tool-effect reordering in LangGraph's ToolNode on unmodified output, removed by an L3 commit-order sequencer. The verified detector, refinements, and realizability artifacts are the contribution; the phenomena and lattice are classical.

12.
arXiv (CS.CV) 2026-06-19

Can Agents Distinguish Visually Hard-to-Separate Diseases in a Zero-Shot Setting? A Pilot Study

The rapid progress of multimodal large language models (MLLMs) has led to increasing interest in agent-based systems. While most prior work in medical imaging concentrates on automating routine clinical workflows, we study an underexplored yet clinically significant setting: distinguishing visually hard-to-separate diseases in a zero-shot setting. We benchmark representative agents on two imaging-only proxy diagnostic tasks, (1) melanoma vs. atypical nevus and (2) pulmonary edema vs. pneumonia, where visual features are highly confounded despite substantial differences in clinical management. We introduce a multi-agent framework based on contrastive adjudication. Experimental results show improved diagnostic performance (an 11-percentage-point gain in accuracy on dermoscopy data) and reduced unsupported claims on qualitative samples, although overall performance remains insufficient for clinical deployment. We acknowledge the inherent uncertainty in human annotations and the absence of clinical context, which further limit the translation to real-world settings. Within this controlled setting, this pilot study provides preliminary insights into zero-shot agent performance in visually confounded scenarios.

13.
arXiv (quant-ph) 2026-06-16

Generalized Kerr-Cat Qubit Codes

arXiv:2606.14901v1 Announce Type: new Abstract: We present a systematic study of Schrödinger cat codes constructed from Kerr-type coherent states, including displaced Kerr coherent states and Barut–Girardello Kerr coherent states, each admitting two distinct families determined by the sign of the Kerr nonlinearity. By tuning the Kerr parameter and coherent-state amplitude, these states interpolate between $\mathfrak{su}(2)$, $\mathfrak{su}(1,1)$ coherent states, providing a unified and versatile foundation for this type of bosonic quantum error correction. Unlike standard two-component Schrödinger cat codes, where a single photon-loss event induces an uncorrectable bit-flip, the nonlinear phase-space structure of Kerr cat states enables simultaneous detection and correction of both photon-loss and dephasing errors within a unified recovery framework, with optimal recovery operations determined via convex optimization. We demonstrate that Kerr cat encodings significantly outperform conventional cat codes under combined loss and dephasing noise, and that judicious parameter optimization can suppress both error channels to a level that reduces the overhead of additional error correction layers. We further show that Kerr-deformed coherent-state manifolds under engineered two-photon driving emerge as effective steady states of driven-dissipative dynamics, with single-photon decoherence strongly suppressed and leakage outside the protected manifold appearing only as higher-order corrections in the deformation strength. Our extended formalism identifies generalized Kerr Schrödinger cat codes as promising candidates for fault-tolerant bosonic quantum computation in experimental platforms such as nonlinear photonics.

14.
arXiv (CS.AI) 2026-06-18

Bounded Context Management for Tabular Foundation Models on Stream Learning

arXiv:2606.18677v1 Announce Type: cross Abstract: Tabular stream learning requires predictions on sequentially arriving examples under distribution shift. While standard methods adapt by updating model states, tabular foundation models (TFMs) make predictions conditioned on a labeled context in an in-context manner, making them a natural alternative for stream learning. This shifts the challenge from how to update the model to how to manage the context. We propose a future information view that yields three practical requirements for context management: preserve recent examples, retain uncertain examples, and remove redundant examples. We instantiate these requirements as CURE (Context management via Uncertainty-aware admission and Redundancy aware Eviction), a context-managing policy with entropy-gated admission and redundancy-aware eviction. Across seven streams, CURE shows up to 27.0% relative improvement over classical stream learners, remains robust across multiple TFM backbones, and ranks first among other policy variants. Code and datasets are available at https://github.com/morcellinus/CURE-ICML-FMSD.

15.
arXiv (CS.CL) 2026-06-11

Substrate Asymmetry in User-Side Memory: A Diagnostic Framework

Authors:

User-side memory in LLMs is typically scored as a single "personalization" capability: given a user's history, is the output more user-aware? We show this aggregate metric hides opposite-direction failures. Memory factorises into at least three orthogonal axes – behavioral consistency (style, voice), factual presence (recall facts in history), and factual absence (abstain when a fact is absent) – and no single substrate wins all three. Comparing per-user gamma-LoRA (a small LoRA adapter trained on each user's history; gamma denotes per-user, not per-task) against BGE-large dense top-K retrieval on a controlled 50-user synthetic corpus and a real-data probe (LaMP-3), we find gamma-LoRA decisively wins behavioral style while RAG decisively wins factual absence – and the same query-projection cells in attention layers 21-35 causally load-bear both effects in opposite directions (zeroing those LoRA weights raises absence-probe TPR by +33 pp and drops presence-probe TPR by 20 pp). On the more heavily RLHF-tuned Llama-3.1-8B-Instruct the asymmetry strengthens, not heals: parametric memory's behavioral advantage collapses while its absence-calibration deficit against retrieval widens – an alignment tax on parametric user-memory. On real-data LaMP-3, gamma-LoRA underperforms a majority baseline; a 9-condition mitigation sweep diagnoses this as instruction-following collapse, not substrate failure (a 9x2 cross-product shows the eval-time {1..5} logit mask drives main_acc to >=0.995 on every recipe), and the best training-time fix replicates bit-identically on Llama. Finally, substrate-selection routing is question-classification, not calibration: a 110M DistilBERT on the question text alone beats every logit-based router. We contribute the diagnostic framework, the diagnosed real-data negative, the alignment-tax replication, and the routing-as-classification finding.

16.
arXiv (CS.CV) 2026-06-18

Experimental Analysis of Neural Network-Based Image Classification on the CIFAR-10 Dataset

An experimental investigation of neural image classification on the CIFAR-10 benchmark is presented through fully connected and convolutional network formulations. The analysis emphasizes the complete learning pipeline: image vectorization, normalization, one-hot class encoding, supervised loss minimization, learning-rate selection, mini-batch training, convolutional feature extraction, max-pooling, and validation-based generalization assessment. A convolutional architecture with six convolutional layers and three max-pooling stages is evaluated for ten training epochs using a batch size of 128 and an Adam optimizer with a learning rate of 0.001. The validation accuracy reaches approximately 74.77%, while the validation loss begins to increase after the middle of training despite continued reduction in training loss. The resulting behavior illustrates the practical difference between representation learning and memorization, and it provides a compact experimental baseline for future studies on regularization, data augmentation, deeper architectures, and reproducible image-classification education.

17.
arXiv (CS.AI) 2026-06-24

Random coloured digraphs defined by a Markov logic network

arXiv:2606.23715v1 Announce Type: cross Abstract: A Markov Logic Network (MLN) is a probabilistic relational model used in Statistical Relational Artificial Intelligence for defining a probability distribution on the set of possible worlds with domain $D$ for an arbitrary finite domain $D$. An MLN consists of soft constraints with associated weights which are nonnegative real numbers. In this study we consider a language speaking about a property $P(x)$ and a relation $R(x, y)$. We consider an MLN for which every Boolean combination of $P(x)$ and $R(x, y)$ is a soft constraint (with associated weight). Let $n$ denote the size (cardinality) of the domain. We show that, for every choice of weights, if the weights are scaled by $1/n$ then, for every first-order sentence $\varphi$, the probability that $\varphi$ holds tends to either 0 or 1 as $n \to \infty$; that is, a 0-1 law for first-order logic holds. Morover, the limit probability does not depend on the weights. If we instead use the standard semantics of MLNs, in the case of which the weights are not scaled, then the limit behaviour is more complicated and depends on the weights. With unscaled weights we get 7 qualitatively different cases which depend on the weights. In some cases we have a 0-1 law for first-order logic, in some cases not, but we may still have a convergence law. The influence of the weights on the asymptotic probability of a first-order sentence may be in the form of a sudden ``phase transition'' from one of the 7 cases to another. The presence of a convergence law has positive implications for inference on large domains.

18.
arXiv (CS.AI) 2026-06-19

QMFOL: Benchmarking Large Language Model Reasoning via Quantifiable Monadic First-Order Logic Test Case Generation

arXiv:2606.20227v1 Announce Type: new Abstract: Large Language Models (LLMs) have made significant progress in reasoning, particularly in deductive reasoning, which is crucial for high-stakes decision-making. As models improve, evaluation benchmarks should evolve to keep pace. However, existing benchmarks lack fine-grained control over logical complexity and struggle to balance semantic diversity with logical consistency. To address these issues, we propose QMFOL, an automated framework for generating monadic first-order logic reasoning tasks with quantifiable and controllable complexity. It constructs formal logical structures using conjunction and disjunction patterns, enabling precise control over reasoning depth, width, label types, and distractors. These structures are then translated into natural language via LLMs, with logical consistency ensured through round-trip verification using an external prover. Based on our framework, we build QMFOLBench, a benchmark comprising 2880 instances with 960 configurations across diverse logical and semantic dimensions. Evaluations on six large reasoning models (LRMs) and two LLMs show that performance degrades and computational overhead increases with rising logical complexity. Models perform better on True-labeled tasks than on False or Unknown ones, and exhibit sensitivity to semantic variation. Overall, QMFOL offers a scalable and reliable approach for constructing deductive reasoning benchmarks with controllable complexity, enabling more precise evaluation of reasoning capabilities in modern language models.

19.
medRxiv (Medicine) 2026-06-22

Body composition subphenotypes, cardiometabolic risk and incident outcomes: validation in the population-based NAKO and UK Biobank imaging cohorts

Background Anthropometric measures do not adequately capture heterogeneity in body fat distribution and corresponding cardiometabolic risk, whereas magnetic resonance imaging (MRI) enables precise differentiation and quantification of adipose tissue compartments and ectopic fat. We aimed to validate previously derived MRI-based body composition subphenotypes and their cardiometabolic risk profiles in two independent European cohorts. Methods Using deep learning-based image analysis, we quantified bone marrow, visceral, subcutaneous, cardiac, renal sinus, hepatic, skeletal muscle, and pancreatic fat in the imaging substudies of two population-based cohorts: the German National Cohort (NAKO, N=29,314, age range 19-74 years) and the UK Biobank (N=36,109, age range 40-69 years). Body composition subphenotypes, previously identified by k-means clustering, were evaluated using a rigorous statistical cluster validation framework with method-based and results-based approaches. In NAKO, cross-sectional associations between subphenotypes and estimated cardiovascular disease risk scores were examined using linear regression. In UK Biobank, longitudinal associations between subphenotypes and incident cardiometabolic outcomes, ascertained through hospital record linkage, were analysed using Cox regression. Findings All five body composition subphenotypes were robustly validated across both cohorts, and showed distinct fat distribution patterns and cardiometabolic risk profiles: I "lean", II "average adiposity", III "bone and muscle adiposity", IV "hepato-abdominal adiposity", and V "general and pancreatic adiposity". Subphenotypes I-III showed progressive adipose tissue remodelling patterns likely reflecting ageing trajectories. The "hepato-abdominal adiposity" subphenotype showed highest risk of incident diabetes, whereas the "general and pancreatic adiposity" subphenotype showed highest overall cardiovascular disease burden and metabolic impairment. Interpretation MRI-derived body composition subphenotypes represent distinct fat distribution patterns that reflect ageing- and disease-related processes, which supports the potential of body composition phenotyping for improved cardiometabolic risk stratification and targeted prevention.

20.
arXiv (CS.AI) 2026-06-16

Why Low-Precision Transformer Training Fails: An Analysis on Flash Attention

arXiv:2510.04212v4 Announce Type: replace-cross Abstract: The pursuit of computational efficiency has driven the adoption of low-precision formats for training transformer models. However, this progress is often hindered by notorious training instabilities. This paper provides the first mechanistic explanation for a long-standing and unresolved failure case where training with flash attention in low-precision settings leads to catastrophic loss explosion. Our in-depth analysis reveals that the failure is not a random artifact but caused by two intertwined phenomena: the emergence of similar low-rank representations within the attention mechanism and the compounding effect of biased rounding errors inherent in low-precision arithmetic. We demonstrate how these factors create a vicious cycle of error accumulation that corrupts weight updates, ultimately derailing the training dynamics. To validate our findings, we introduce a minimal modification to the flash attention that mitigates the bias in rounding errors. This simple change stabilizes the training process, confirming our analysis and offering a practical solution to this persistent problem. Code is available at https://github.com/ucker/why-low-precision-training-fails.

21.
arXiv (CS.AI) 2026-06-19

Reinforcement-aware Knowledge Distillation for LLM Reasoning

arXiv:2602.22495v3 Announce Type: replace-cross Abstract: Reinforcement learning (RL) post-training has recently driven major gains in long chain-of-thought reasoning large language models (LLMs), but the high inference cost of such models motivates distillation into smaller students. Most existing knowledge distillation (KD) methods are designed for supervised fine-tuning (SFT), relying on fixed teacher traces or teacher-student Kullback-Leibler (KL) divergence-based regularization. When combined with RL, these approaches often suffer from distribution mismatch and objective interference: teacher supervision may not align with the student's evolving rollout distribution, and the KL regularizer can compete with reward maximization and require careful loss balancing. To address these issues, we propose RL-aware distillation (RLAD), which performs selective imitation during RL – guiding the student toward the teacher only when it improves the current policy update. Our core component, Trust Region Ratio Distillation (TRRD), replaces the teacher-student KL regularizer with a PPO/GRPO-style likelihood-ratio objective anchored to a teacher–old-policy mixture, yielding advantage-aware, trust-region-bounded distillation on student rollouts and naturally balancing exploration, exploitation, and imitation. Across diverse logic reasoning and math benchmarks, RLAD consistently outperforms offline distillation, standard GRPO, and KL-based on-policy teacher-student knowledge distillation.

22.
arXiv (CS.AI) 2026-06-18

FoMoE: Breaking the Full-Replica Barrier with a Federation of MoEs

arXiv:2606.19025v1 Announce Type: cross Abstract: Pre-training Large Language Models (LLMs) typically demands large-scale infrastructure with tightly coupled hardware accelerators. While increasing model and dataset scale remains the dominant driver of performance, Mixture-of-Experts (MoEs) architectures have recently achieved state-of-the-art results by decoupling parameter count from computational cost. This efficiency enables training massive models on constrained compute budgets, yet it typically requires the high-speed interconnects of a single datacenter. To overcome these physical limits, recent approaches such as DiLoCo and Photon use low-communication data-parallel methods to enable scaling across geographically distributed, weakly connected data centers. However, these methods suffer from a fundamental inefficiency: they require full model replicas at every site, which imposes prohibitive memory constraints and communication overheads. In this work, we introduce FoMoE, a system that breaks the full-replica paradigm by partitioning expert layers across workers. We demonstrate that FoMoE: (I) reduces communication costs by up to 1.42x over efficient baselines and 45.44x over DDP via partial expert replication in the studied regimes; (II) achieves empirical throughput speedups of up to 1.4x through a novel skip-token mechanism; and (III) shows stable routing in the trained proxy regimes and projects the communication/memory benefits to 100B-scale configurations through system modelling.

23.
arXiv (CS.AI) 2026-06-24

Are Safety Guarantees in Neural Networks Safe? How to Compute Trustworthy Robustness Certifications

arXiv:2606.23858v1 Announce Type: cross Abstract: A primary challenge in AI safety is the existence of adversarial examples – slightly distorted inputs that cause a neural network (NN) to misclassify. To mitigate this problem, recent research focuses on the computation of robustness certifications, which, for a given input, determine the largest distortion the input may receive without breaking the network's prediction. Robustness certifications can be interpreted as an axis-aligned hyper-rectangle (multi-dimensional intervals). Most existing approaches focus on maximizing the certification's volume, but recent intractability results prohibit the computation of volume-optimal certifications in reasonable time. We introduce the apothem measure and show how to compute apothem-optimal certifications in a linear number of calls to a NN verifier (oracle) w.r.t. the input domain's diameter. Moreover, we prove that we cannot have a volume-optimal, oracle-based algorithm, even if we discard the oracle costs. Also, we introduce dual certifications – an interval including all instances of a class – thus providing apothem-minimum upper bounds to a robustness certification. Further, we present the ParallelepipedoNN system, which we evaluate on the standard MNIST and Fashion MNIST benchmarks. A preliminary comparison with existing work on the same datasets reveals at least two-fold improvement w.r.t. the minimum edge length.

25.
arXiv (math.PR) 2026-06-24

Conditionally Poissonian random digraphs

arXiv:1705.03801v2 Announce Type: replace Abstract: We define a Poissonian model of directed random graphs which generalises the undirected Poissonian random graph process introduced by Norros and Reittu in Adv. Appl. Probab. 38 (2006), 59–75. Its loopless simple projection is a rank-one independent-arc inhomogeneous digraph of the type studied by Cao and Olvera-Cravioto, Random Struct. Alg. 56 (2020), 722–774. For the Poissonian multigraph itself, we discuss the relation to Norros-Reittu graphs, characterise limiting degree distributions, and record explicit exploration estimates. In particular, we give fixed-depth directed local weak limits, stopped branching-process couplings with weight-mass collision budgets, a comparison with the simple projection, and a rare-event concentration criterion. These estimates are intended as graph-side structural inputs for later dynamics on the graph.