Papers

We introduce EC-EarthFlow, a generative flow matching model that emulates simulations from the physical climate model EC-Earth3. The model is trained on transient simulations from EC-Earth3 (1950-2166, SSP2-4.5) to predict the day ahead temperature field from the previous days temperature as well as annual mean temperature. Predictions are made auto-regressively with rollout periods of between a month and an extended season. Using only this variable of interest, we are able to reproduce the daily variability, spatial patterns, annual cycle and long-term trend from EC-Earth3 at a substantially lower computational cost than the physical model. We demonstrate that EC-EarthFlow is stable for long inference periods, and that it can learn the physical relationships as simulated in EC-Earth3.

We prove an Ax-Lindemann theorem for the rank one Riemann-Hilbert correspondence on a smooth projective complex algebraic curve and recover a result of Simpson.

We study deterministic Rendezvous of two robots with lights under weak geometric and synchronization assumptions. Our focus is on Variable Disorientation (VD), where each robot may arbitrarily rotate, reflect, and rescale its local coordinate system at every \Look, and on VD+Fixed-Unit-Distance (FUD), where each robot's local unit distance remains fixed. We consider the one-sided visibility models FSTA and FCOM, and schedulers ranging from the energy-restricted RSYNCH and R-RSYNCH to fully asynchronous executions. Under VD, we prove color-independent impossibility results for FSTA under SSYNCH and for FCOM under ASYNCH, and establish sharp two-color boundaries under RSYNCH and R-RSYNCH. Under VD+FUD, we introduce periodic distance classes, which turn multiplicative changes of the physical distance into predictable cyclic phase shifts. This technique yields a two-color self-stabilizing non-$L$-Rendezvous algorithm for both FSTA and FCOM under RSYNCH and R-RSYNCH even with Non-Rigid movement. Under Rigid movement, periodic distance classes further yield an optimal three-color non-quasi-self-stabilizing algorithm for FSTA under SSYNCH, a four-color FSTA algorithm under $M$-atomic ASYNCH, and a four-color FCOM algorithm under CM-atomic ASYNCH. By guarding the latter phase mechanism with two additional colors and replacing its initial swap by passive initialization, we obtain a six-color FCOM algorithm under full ASYNCH, improving the previous twelve-color upper bound by a factor of two. These results show that a fixed local metric can serve as persistent cyclic memory, and clarify how metric stability, one-sided visibility, and scheduler atomicity jointly determine Rendezvous solvability and color complexity.

We develop a continuous-time deterministic policy gradient framework for reinforcement learning with reflected state dynamics, where the state process is governed by a controlled reflected stochastic differential equation on a hypercube. Under suitable regularity assumptions, we establish the connection between the value function and the Neumann Bellman equation, introduce an advantage-rate function that yields a deterministic policy gradient formula, and prove the martingale characterization theorem. Motivated by these theoretical results, we propose a continuous-time deep deterministic policy gradient algorithm for reflected stochastic systems, in which the Neumann boundary condition is imposed via either soft penalization or hard architectural constraint. We further quantify the discrepancy between the ideal continuous-time dynamics and the discretely sampled exploratory dynamics executed in practice, showing that the error decays as the time grid is refined and exploration noise vanishes. Our experiments on reservoir control problems illustrate the effectiveness of the RL framework, highlighting that boundary-aware methods substantially reduce Neumann boundary residuals and enhance learning stability.

Let $\mathcal{G}$ be a finite connected simple graph with at least two vertices. We construct a finite regular hypertope $\Gamma$ whose diagram is the graph $\mathcal{G}$ with every edge labelled by $4$, and for which $Cor(\Gamma)/Aut(\Gamma) \cong Aut(\mathcal{G})$. This shows that any possible group of diagram symmetries can be realized by the correlations of a hypertope. The construction uses a group generated by involutions, with nilpotency class two, exponent four and commutator relations dictated by the defining graph.

ANN-to-SNN conversion offers a practical route toward energy-efficient spiking Vision-Language-Action (VLA) models by bypassing the substantial cost of training large-scale SNNs from scratch. However, existing methods often require many timesteps to maintain competitive performance, resulting in substantial inference latency for real-time VLA deployment. To address this challenge, we introduce SpikingVLA, an ANN-to-SNN conversion framework that enables accurate and low-latency spiking VLA inference. Specifically, we propose a Dendritic Integrate-and-Fire (DIF) neuron that alleviates channel-wise activation outliers through dendritic mixing and adaptive somatic firing, enabling accurate ANN-to-SNN conversion with fewer timesteps. Building on DIF neurons, we further introduce an asynchronous execution mechanism that overlaps temporal computation across VLA components, reducing synchronization overhead and latency. Extensive experiments demonstrate that SpikingVLA achieves competitive navigation performance with substantially improved inference efficiency. Compared with existing spiking VLA methods, SpikingVLA improves SR and SPL by 11.9% and 12.6%, respectively, while reducing first-action latency by 11.2$\times$. These results establish SpikingVLA as a practical framework for deploying pretrained VLA models with high-performance and low-latency spiking inference.

Can we determine whether a foundation model will generalize out-of-distribution (OOD) before any target data is available? Existing diagnostics require source or target data, which rules them out before a target domain exists. Those that use the weights alone apply one statistic to every architecture, and do not separate robust models from fragile ones. We show the answer is encoded in the spectral structure of pretrained weights. Two forces shape that structure. Architecture determines how information is stored in weight matrices. Pretraining strategy determines what is rewarded. Together they set a spectral geometry that governs OOD robustness. We prove that the OOD accuracy gap is bounded by how tightly the source representations concentrate. A statistic computed from the pretrained weights alone serves as a proxy for that concentration. The direction of that proxy reverses between architecture families. We operationalize it: the direction is stable within one (architecture X strategy) combination, the finest grouping we test, which we call a cell. Pooled over 116 models spanning 7 modalities, a single statistic ranks OOD robustness weakly, because cells of opposite direction cancel. Within a cell, the statistic selected for it orders 92% of model pairs by OOD robustness in-sample. The selection does not leak the target: for each model family outside the matrix we logged the cell, metric and sign before running its OOD evaluation, and the predicted direction held in every case: EEG, genomic and protein. Acting on spectral concentration narrows the OOD gap by 24% at 87.5% ID retention. The diagnostic operates on released weights alone, so OOD robustness becomes checkable at model-selection time, before data or compute is committed to a target domain.

Vision-Language-Action models (VLAs) are increasingly deployed in safety-critical physical environments, yet their adversarial robustness remains poorly understood. Existing attacks typically assume white-box access or rely on surrogate VLAs, which rarely holds in real-world deployments. Our key insight is that most VLAs are adapted from a publicly released pretrained vision-language model (VLM), inheriting two capabilities essential for action generation: visual perception and instruction-conditioned grounding. Therefore, this paper explores a previously unaddressed question: can an adversary attack deployed VLAs using only their ancestor VLMs? To this end, we propose three adversarial patch attacks that disrupt the inherited capabilities: a vision disruption attack that corrupts the projected visual tokens through relative and absolute terms, an instruction-grounded semantic evidence suppression attack that removes the visual evidence required for instruction-grounded concepts, and a joint attack that unifies both objectives under a two-phase curriculum. Experiments across different VLA families on both simulation and static real-world images show that patches optimized on the ancestor VLM cause substantial degradations in VLA task success rates, demonstrating that VLAs inherit adversarial vulnerabilities alongside their foundational capabilities. This effect is not uniform: it is strongest on tasks that require precise instruction-grounded localization, and nearly vanishes on policies whose adaptation rewrites the shared visual-semantic representation or whose action head iteratively smooths perturbations away. By characterizing the boundary conditions of vulnerability inheritance and providing analysis of why the inheritance effect holds or fails, we advance the understanding of safety for VLA-involved systems.

Assume $V=L$. We prove two uniform theorems producing coanalytic sets of prescribed Hausdorff dimension in Euclidean space. The first concerns countable families of partial operations of finite arity on $\mathbb{R}^n$ whose evaluation is hyperarithmetic and which are tame in each vector variable separately. For such families a single lightface $\Pi^1_1$ relation yields strictly increasing, left-continuous chains of proper sets closed under the operations, one of each Hausdorff dimension in $[0,n]$, uniformly in the family and in $n$; no exchange property of the closure is needed. As an application, there are coanalytic proper subfields of $\mathbb{R}$ of every Hausdorff dimension in $[0,1]$. The second theorem constructs coanalytic bases of every possible Hausdorff dimension for effectively coded pregeometries. As an application, there are coanalytic Hamel bases of $\mathbb{R}^n$ and transcendence bases of $\mathbb{C}$ of every Hausdorff dimension in $[0,n]$ and $[0,2]$, respectively.

Region-based neural style transfer enables fine-grained artistic control by allowing independent stylization of semantic image regions. However, compositing these regions often leads to boundary artifacts, degrading visual quality. We propose Interior-Guided Boundary Repair (IGBR), a lightweight and model-agnostic method that improves boundary handling in multi-region stylization. IGBR repairs boundary pixels using interior-guided propagation and applies inward, distance-based blending restricted to object-background boundaries, preventing inter-object style leakage. The method is derived from a region-wise constrained formulation with a closed-form solution and can be seamlessly integrated into existing stylization pipelines without retraining. To evaluate efficiency of our IGBR, we introduce quantitative metrics that measure boundary consistency, gradient artifacts, inter-object leakage, and interior preservation without requiring annotated stylized images. Our experiments and evaluations demonstrate that the proposed IGBR consistently produces plausible boundaries, outperforming prior blending techniques in boundary consistency, gradient stability, and interior preservation. The code is available at https://github.com/Son-SDT/IGBR.

The organization of interfacial water shapes the electric double layer behavior in aqueous electrolytes. Asymmetries upon charge reversal are commonly attributed to specific ion adsorption or surface chemistry. Yet it is unresolved whether interfacial water is net oriented at zero charge, and whether such orientation alone makes its response charge-asymmetric. Here, using electrochemical heterodyne-detected sum-frequency generation spectroscopy and constant-potential molecular dynamics simulations, we resolve the absolute orientation of water at a weakly interacting graphene electrode. We show that interfacial water retains a net H-down orientation, with hydrogens pointing toward the bulk, at zero net surface charge. Superimposed on this offset, the field-induced reorientation is linear and symmetric upon charge reversal, independent of how the charge is introduced or distributed. The zero-charge offset therefore shifts the potential of zero water orientation negatively relative to the potential of zero charge, making the interfacial water response intrinsically charge-asymmetric. These results establish an electrostatic baseline against which extrinsic contributions at electrified interfaces can be isolated.

We study the $\ell^2$-Betti numbers of graph products of cyclic groups. We relate these invariants to the $\ell^2$-Betti numbers of right-angled Artin groups and obtain explicit formulas in several cases determined by the structure of the finite-order vertices in the defining graph. In particular, we consider the cases where these vertices form a clique or have identical links, and derive a formula for the first $\ell^2$-Betti number when the defining graph has exactly two finite-order vertices. We also compute the BNSR-invariants of graph products of cyclic groups and the first $\ell^2$-Betti number of their automorphism groups when the defining graph has at most two finite-order vertices.

Token-Pruning accelerates Vision-Language Models by removing redundant visual tokens, yet its safety implications remain underexplored. In this work, we present the first comprehensive safety evaluation of Token-Pruning mechanisms and find that: most pruning strategies significantly degrade safety as pruning ratios increase, whereas Query-based Compression shows the opposite, with extreme pruning (up to 99.8%), unexpectedly improves model safety. This sharp contrast prompts a key question: How do different Token-Pruning strategies reshape model safety behavior, and is it possible to enhance safety without sacrificing acceleration? To answer this, we identify an unrecognized mechanism, termed Pruning-Induced Malicious Amplification, where removal of background tokens triggers a side effect: forcing the model's attention to collapse onto a few retained malicious anchors within the foreground, inadvertently amplifying their toxic semantics under jailbreak. To address that, we propose an inference-time and plug-and-play Safety-Aware Pruning (SAP) mechanism that counteracts such dominance via three steps: (1) identifying malicious anchors, (2) restoring pruned benign tokens, and (3) reallocating excessive attention from malicious anchors to benign tokens. Extensive experiments across three safety and four utility benchmarks demonstrate that SAP mitigates pruning-induced vulnerabilities, i.e., reducing ASR by up to 62%, without compromising efficiency or utility.

Ordinary prompt-based editing can cause latent watermark detection to fail without explicitly targeting the watermark. We benchmark eight watermark methods against five editors across four generative backbones, four editing strengths, and five semantic categories, with edit-validity and threshold checks. Separating editing from seven subsequent distortions reveals that editing alone primarily distinguishes Tree-Ring, while added distortions expose a broader spectrum of detection survival. Sequential edits reveal a second hidden difference: score separation can decline while detection rates remain near their ceiling. Across methods, standardized clean score separation ($d'$) organizes composite-survival tiers, whereas spatial overlap adds little to predicting edit-only survival beyond clean detectability. Embedding-strength interventions in two methods link higher clean separation to higher post-edit separation. In HSTR, the margin contrast is positive, while the angular layout contrast at matched clean separation remains unresolved. Together, outcome decomposition and continuous separation expose differences hidden by aggregate TPR. Method tiers are stable under threshold recalibration at the main operating points and alternative composite weights. Clean $d'$ is thus a useful empirical diagnostic within this benchmark, with mixed transfer to unseen methods. Code and supporting artifacts are planned for a separate release.

Spatial audio signal enhancement (SASE) aims to separate a desired soundfield from interference and noise while preserving its spatial cues. However, existing algorithms pay limited attention to preserving the spatial cues of the residual interference, which reduces their effectiveness in practical applications. In this paper, we propose a direction-preserving SASE algorithm based on multi-output beamforming. The proposed algorithm extracts the desired soundfield while preserving the spatial cues of both the desired soundfield and the residual interference soundfield. The simulation study shows that the proposed algorithm achieves performance comparable to the existing SASE algorithm in terms of estimation error, while it better preserves the spatial cues of the residual interference within a region of interest, in a noisy and reverberant environment.

Synthetic hard negatives generated in the representation space have proven effective for unimodal self-supervised learning, but transferring this idea to vision-language pretraining is not straightforward. We analyze six representation-space synthesis strategies and identify two failure modes in their transfer to vision-language pretraining: cross-modal constructions that produce overly easy negatives or pull them toward the query, and intra-modal constructions that incorporate the matched positive. We also observe logit-scale saturation when training with synthetic hard negatives and a learnable temperature, and find that fixing the temperature improves downstream performance. Using this geometric analysis we propose SNAP, which generates intra-modal hard negatives that never involve the positive from either modality, avoiding both failure modes entirely. SNAP is model-agnostic, requires no external generative models, and adds less than 10% training time overhead. Evaluated on top of CLIP and FLIP across multiple architectures and datasets, SNAP delivers consistent improvements on zero-shot retrieval, zero-shot classification, and linear probe evaluation.

Since their introduction, modern evolution strategies, such as the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), have become established as powerful methods for continuous black-box optimization. This success has led to a wide range of proposed modifications, each designed to improve performance or behavior in specific optimization scenarios. However, because these developments have largely been introduced and studied in isolation, their interactions remain comparatively underexplored. In this paper, we present the Modular CMA-ES (ModCMA), a configurable framework that integrates a wide range of mechanisms from modern evolution strategies within a single implementation. By decomposing CMA-ES into modules with interchangeable options for sampling, selection and recombination, step-size adaptation, matrix adaptation, and restarting, ModCMA enables systematic exploration of a large design space of modern evolution strategies and facilitates the construction, comparison, and automated configuration of new algorithm variants. We illustrate the benefits of this modular approach through two example studies. First, we compare several matrix-adaptation mechanisms in terms of their computational cost and optimization performance. Second, we use automated algorithm configuration to specialize ModCMA to individual benchmark problems and analyze the resulting configurations. Together, these examples demonstrate how the framework can be used both to study individual algorithmic design choices and to explore their combinations in a systematic and reproducible manner.

Variational two-particle reduced-density-matrix methods impose hierarchies of semidefinite constraints. We ask whether a fixed level of such a hierarchy can recover even an elementary many-fermion spectrum, using the dispersion test, in which a value is accepted as an eigenvalue when the minimized dispersion vanishes. For an exactly solvable operator that counts fermions in one half of the spin orbitals, whose spectrum consists of integers, we determine the complete set of accepted values at every level of the full hierarchy of particle-hole semidefinite constraints. Besides the integers, this set contains an entire interval of noninteger values unless the hierarchy level grows linearly with the particle number, and we give the exact threshold. The proof constructs explicit pseudo reduced density matrices that are diagonal in the occupation basis, obtained by continuing the occupation probabilities of a fixed-particle-number state to a noninteger particle number. Classical semidefinite constraints on occupation correlations, supplemented by nonnegative occupation-pattern probabilities, suffice to establish fermionic positivity. For this counting model the two relaxations have the same dispersion zero set, not necessarily the same feasible states, and a converse argument shows that no other false eigenvalues occur. The limitation is specific to the hierarchy: two-body inequalities that encode the integer spectrum would remove every false value but are not generated at a fixed level. We also verify that the full semidefinite constraints and their complete projection onto two-particle reduced-density-matrix space are equivalent in finite dimensions, so that the result applies equally to the polar formulation.

Threat analysis and risk assessment (TARA) according to ISO/SAE 21434 treats implementation-level hardware attacks inconsistently. We show that two of the three attack-feasibility approaches exemplified in the standard rate every attack path that requires physical access as very low by construction, independent of the implementation, while the third can assign the same path any of the four ratings, depending on the assumed state of attacker knowledge and on the attacker profile. This paper presents a hardware-aware extension of the TARA that covers side-channel analysis (SCA), fault-injection attacks (FIA), and abuse of debug and test interfaces. It adds an attacker model for physical access over the vehicle lifecycle; a path-dominance relation on the attack-potential factors that identifies when a simpler path makes a sophisticated one irrelevant; a three-level hardware relevance gate (H0 to H2) with an explicit decision rule that fixes whether resistance of a hardware step may be argued, must be documented, or must be demonstrated by test; and a rating rule under which a hardware step receives no credit for implementation-specific resistance until the required evidence exists. The gate is positioned relative to cybersecurity assurance levels and targeted attack feasibility. The method is applied to a reference gateway architecture, and its structural inputs are examined on ten published attacks on automotive components. In seven of the ten studies, the demonstrated attack included a debug, programming, boot, diagnostic, or update mechanism, whereas only four studies included an attack on a cryptographic computation.

Robot manipulation data collection has been shifting from teleoperation toward robot-free demonstrations, through interfaces such as the Universal Manipulation Interface (UMI) or directly from human hands. Vision-Language-Action (VLA) policies trained on such data inherit the demonstrator's timing. Yet human timing does not directly transfer to robots: compliant hands tolerate fast contact, whereas robots may overshoot due to actuator and tracking limitations; conversely, robots can move faster in free space. This motivates a unified approach that reconciles execution speed with contact safety. We present RoboPace, an online retiming layer that preserves the policy's geometric path while adapting its timing, respecting the target robot's kinematic and dynamic constraints. It adapts execution speed based on predicted contact, jointly accounting for contact-dependent speed limits and the robot's motion constraints. The method requires no policy retraining and operates in real time. Across three contact-rich tasks on a dual-arm robot, faster uniform execution and physical-limit-only retiming largely fail. RoboPace instead achieves higher overall success than slow uniform execution while completing four of five commands in approximately half the time, retaining the reliability of slow execution without its time cost.

Visual foundation models (VFMs) are increasingly integrated into end-to-end autonomous driving for their powerful representations, yet it remains unclear when these representations improve driving performance. To investigate this question, we introduce ViRA, a planner-agnostic visual representation alignment framework that keeps the planner architecture and inference cost unchanged. Our study reveals three findings: (1) VFM-guided visual representations consistently improve driving performance across diverse end-to-end planners, with gains extending to zero-shot closed-loop evaluation. (2) The choice of VFM target matters for planning performance, and alignment to a different VFM can further benefit planners with pre-trained VFM encoders. (3) Auxiliary perception supervision reduces sensitivity to VFM target selection, narrowing the EPDMS spread across five targets from 2.7 to 0.5 points and potentially compensating for less effective VFM targets. Guided by these findings, we develop ViRA-Diffusion, a diffusion-based planner trained without auxiliary perception supervision, which achieves 92.3 EPDMS on NAVSIM v2 navtest, outperforming recent methods in our comparison by at least 1.9 points. The results motivate jointly considering target selection and planner supervision when integrating VFMs into end-to-end autonomous driving. The results and demo are available at https://github.com/OpenDriveLab/ViRA.

A combination of recent results of Canning–Larson–Payne–Willwacher and Payne–Willwacher is that the motivic Euler characteristic of the moduli space $M_{g,n}$ of nonsingular curves with $n$ markings is polynomial in the Tate motive if and only if $g=0$ or $3g+2n<25$. In the cases when $g\geq 1$ and $3g+2n<25$, there are $32$ pairs $(g,n)$ where the motivic Euler characteristic has been computed. We compute the motivic Euler characteristic for $9$ more pairs, leaving only $6$ pairs where the motivic Euler characteristic is polynomial but unknown. In genus $4$, we moreover compute the answer when $n=7$, i.e.~the first non-polynomial case.

The wide adoption of LLMs across broad NLG applications heightens the importance of providing users with the means to avert errors and hallucinations. Uncertainty quantification is poised to fill that gap; with low uncertainty (high confidence), as a proxy for correctness, allowing users to be selective (e.g., reject low-confidence, likely incorrect responses). Correlation between confidence and correctness then serves as a useful criterion for evaluation of uncertainty quantifiers (UQs). But in NLG, where diverse responses can be adequate to a prompt, obtaining reliable correctness judgements is not simple, especially without human intervention. Errors in automated judgement are hardly avoidable and known to diminish the reliability of evaluation protocols (Santilli et al., 2025; Ielanskyi et al., 2025). In a meta-analysis of published work, we show that automated judgement is the present norm. Besides, automated judgements are rarely validated against human ones, and the validation of the UQ evaluation they automate is even rarer. With experiments in question answering, using 4 LLMs, human and automated judgements and 7 popular UQs, we find that i) a judge's performance can only coarsely predict the observed impact of its errors on the reliability of UQ evaluation, and that ii) judgement errors tend to misrepresent informative UQs most. We link these observations to patterns of correlation between confidence and categories of judgement error.

Pain perception varies substantially across individuals, making it difficult for population-based classifiers to generalise across all subjects in a dataset. One way to account for subject variability is to train personalised classifiers. In this work, we evaluate Few-Shot Learning, a sub-area of Meta-Learning, as an approach to personalisation in automated pain assessment. We re-interpret the shift from population-level to subject-level evaluation as a task-domain shift, where the observed classes remain fixed but the target subject changes. We evaluate our method on the BioVid Pain Database, the SenseEmotion Database, and the PainMonit Experimental Dataset (PMED), reaching 85.75% and 35.49% accuracy on BioVid and 82.37% and 41.88% on SenseEmotion in the binary and multi-class settings under a Leave-One-Subject-Out CV protocol respectively, and 90.47% on PMED, for which only a binary benchmark exists. Using samples to implement k-shot conditioning, the accuracies can be improved to 86.25%, 40.06%, 83.43%, 44.08%, and 91.25%, respectively. To further evaluate the effects and robustness of our method, we provide additional ablation experiments and investigate the personalisation effects. Our results suggest that support-conditioned few-shot adaptation can improve average performance under inter-subject variability.

We study online learning in market making, where, at each round, a market maker posts bid and ask prices before observing the market price and the private valuation of an incoming trader. In this setting, Maran et al. 2026 introduce a feedback model motivated by limit order books, in which the trader's valuation is revealed only if no transaction occurs. Assuming that trader valuations are drawn i.i.d. from an unknown distribution while market prices are chosen adversarially, they establish an expected regret bound of $\widetilde{\mathcal{O}}(T^{2/3})$. In this work, we improve upon this guarantee by establishing a high-probability regret bound of $\widetilde{\mathcal{O}}(\sqrt{T})$. As a warm-up, we first consider the full-feedback setting. We introduce a discretization of the bid-ask space based on two coupled grids and combine it with Hedge to achieve the desired regret rate. Building on these ideas, we then address the substantially weaker feedback induced by a limit order book and develop an algorithm that achieves the same guarantee. Finally, we investigate the limits of learnability in fully adversarial environments, where the valuations may vary arbitrarily as well. Perhaps surprisingly, we show that when both market prices and trader valuations are chosen adversarially, sublinear regret is impossible even under full feedback, thereby motivating our stochastic assumption on the valuations.

Autonomous underwater vehicles (AUVs) commonly rely on inertial navigation systems (INS) aided by Doppler velocity logs (DVLs) for reliable underwater navigation. Accurate DVL velocity estimation is therefore essential for successful operation. Recent learning-based methods have demonstrated improved DVL velocity estimation, particularly under degraded measurement conditions. However, their training objectives typically rely on conventional regression losses that are highly sensitive to large residuals and corrupted observations. Additionally, they do not explicitly account for the physical consistency and measurement uncertainty associated with the underlying sensing process. To address these limitations, this paper introduces navigation-aware loss (NAViLoss), a robust and uncertainty-aware objective function for learning-based AUV velocity estimation. NAViLoss jointly penalizes the velocity-estimation residual in the navigation-state domain and the beam-consistency residual in the DVL measurement domain. Its bounded formulation limits the influence of large residuals, while an adaptive mechanism regulates the uncertainty in beam geometry. Furthermore, NAViLoss is integrated with a DeepONet architecture to form a novel NAVi-DeepONet model for seamless estimation of an underwater vehicle's velocity. Lastly, our model is evaluated using approximately 10,000m of semi-synthetic AUV experimental data collected during multiple real-world sea trials. Experimental results demonstrate a 44% improvement in velocity-estimation accuracy compared with conventional and learning-based baselines. These results demonstrate the effectiveness of navigation-aware and uncertainty-adaptive loss design for robust learning-based underwater velocity estimation.

We prove an $L^4$ Strichartz estimate for the dispersion $n_1^2+\alpha n_2^2$ on $\mathbb T^2$, for every $\alpha>0$. The loss is controlled by the Shannon entropy of the normalized squared Fourier coefficients and the cardinality of their support $S$. In particular, we obtain a norm loss of $(\log\#S)^{1/2}$ on fixed time intervals, with constants locally uniform in $\alpha$. This provides an irrational-dispersion counterpart of the cardinality estimate of Herr and Kwak [Forum Math. Pi (2024)] and improves the $N^\varepsilon$ loss of Bourgain and Demeter [Ann. of Math. (2015)] to $(\log N)^{1/2}$ on frequency boxes. The proof combines weighted incidence estimates, smooth localization, and an entropy recursion.

We study the stability of optimal transport for the squared Carnot–Carathéodory distance. We prove the sharp $W_1^{1/4}$ stability estimate for optimal transport maps on complete Riemannian and fat sub-Riemannian manifolds, assuming bounded source support and an upper density bound. On John domains with two-sided density bounds, we obtain logarithmic $L^2$ stability of Kantorovich potentials on complete smooth sub-Riemannian manifolds that are equiregular near the source, with arbitrary compact targets. For potentials, we obtain every Hölder exponent $0<\alpha<1/2$ on two-generating structures, and the exponent $1/2$ in the Riemannian and fat settings and on all step-two Carnot groups. Three-atom examples show that the map and potential exponents with respect to $W_1$ cannot exceed $1/4$ and $1/2$, respectively, on any smooth sub-Riemannian manifold of horizontal rank at least two. We also construct smooth transport families whose target variation of order $t$ in $W_1$ produces map variation of order $t^{1/s}$ in an equiregular region of step $s$. The map estimates follow from coercivity of the duality defect, while the potential estimates follow from quantitative concavity of the Kantorovich functional.

Structured recovery phenomena, such as restricted isometry properties in compressed sensing, have shown that high-dimensional objects can often be reconstructed from remarkably low-dimensional linear measurements. This work develops an analogous recovery framework for low-rank signed measures on $\mathbb{R}^2$, defined here as measures that can be expressed as finite sums of product measures with one-dimensional factors. The framework is based on linear operators, termed "silhouette operators," that map a measure to a fixed finite collection of one-dimensional linear pushforwards. The main results show that a suitably chosen collection of $2k$ projected marginals suffices to identify every compactly supported rank-$\le k$ signed measure, that this number is optimal, and that the projection directions cannot be chosen arbitrarily. The framework is also extended to higher-dimensional sums of product measures by establishing sufficient conditions under which collections of pairwise marginals identify the full model. Building on this framework, a computationally efficient estimator, termed "silhouette mixture estimation" (SME), is introduced for constructing a low-rank empirical measure from data by matching its one-dimensional projected marginals to the corresponding empirical marginals in Wasserstein distance. When combined with one-dimensional density estimators, SME yields an efficient nonparametric density estimator that performs strongly relative to a range of parametric, nonparametric, and deep-learning baselines in settings of moderate dimension and sample size.

We study linear Lagrange interpolation on exact geodesic spherical triangles obtained by radial projection of chordal triangles. Factorising the element map into an affine map and a radial correction separates chordal anisotropy from radial distortion. We derive exact identities for the surface measure, Dirichlet energy, and directional derivatives, and use them to transfer planar anisotropic interpolation estimates to the sphere. The resulting local error estimates retain the two chordal directional length scales and keep the geometric factors explicit, with constants independent of the element size, shape, and sphere radius. In the gradient estimate, the maximum-angle factor multiplies only the longer directional scale. Four explicit triangle families quantify the loss caused by replacing these scales with the diameter and establish the sharp maximum-angle dependence. They also show that, within the same two-term directional bound with a constant independent of the element, the maximum-angle factor cannot be moved to the shorter scale or have its exponent reduced below one. The fourth family proves the optimality of the square-root radial-transfer factor while retaining the same transported directional scale. The local estimates extend to compatible spherical triangulations. As an application, they yield an anisotropic energy-error estimate for the conforming approximation of the mean-zero Laplace–Beltrami problem with exact integration.