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arXiv

mmWave Radar Aware Dual-Conditioned GAN for Speech Reconstruction of Signals With Low SNR

Jash Karani, Adithya Chittem, Deepan Roy, Sandeep Joshi

Millimeter-wave (mmWave) radar captures are band-limited and noisy, making for difficult reconstruction of intelligible full-bandwidth speech. In this work, we propose a two-stage speech reconstruction pipeline for mmWave using a Radar-Aware Dual-conditioned Generative Adversarial Network (RAD-GAN), which is capable of performing bandwidth extension on signals with low signal-to-noise ratios (-5 dB to -1 dB), captured through glass walls. We propose an mmWave-tailored Multi-Mel Discriminator (MMD) and a Residual Fusion Gate (RFG) to enhance the generator input to process multiple conditioning channels. The proposed two-stage pipeline involves pretraining the model on synthetically clipped clean speech and finetuning on fused mel spectrograms generated by the RFG. We empirically show that the proposed method, trained on a limited dataset, with no pre-trained modules, and no data augmentations, outperformed state-of-the-art approaches for this specific task. Audio examples of RAD-GAN are available online at https://rad-gan-demo-site.vercel.app/.

arXiv

Adaptive Optimization via Momentum on Variance-Normalized Gradients

Francisco Patitucci, Aryan Mokhtari

We introduce MVN-Grad (Momentum on Variance-Normalized Gradients), an Adam-style optimizer that improves stability and performance by combining two complementary ideas: variance-based normalization and momentum applied after normalization. MVN-Grad scales each coordinate by an exponential moving average of gradient uncertainty and applies momentum to the resulting normalized gradients, removing the cross-time coupling between stale momentum and a stochastic normalizer present in standard Adam-type updates. We prove that this decoupling yields smaller one-step conditional update variance than momentum-then-normalize variance methods, and that MVN-Grad has a uniformly bounded response to isolated gradient spikes. In low-variance regimes, we further show that variance normalization avoids sign-type collapse of second-moment scaling and can yield accelerated convergence. Beyond these comparisons, we prove a general nonconvex convergence guarantee for MVN-Grad under bounded-gradient stochastic assumptions. On CIFAR-100 and GPT-style language modeling, MVN-Grad matches or improves on Adam, AdaBelief, and LaProp, delivering smoother training and better generalization at the cost of one additional state tensor.

arXiv

Last-Iterate Analyses of FTRL with the 1/2-Tsallis Entropy in Stochastic Bandits

Jingxin Zhan, Yuze Han, Zhihua Zhang

The convergence analysis of online learning algorithms is central to machine learning theory, where the last-iterate convergence is particularly important, as it captures the learner's actual decisions and describes the evolution of the learning process over time. However, in multi-armed bandits, most existing algorithmic analyses mainly focus on the order of regret, while the last-iterate (simple regret) convergence rate remains less explored---especially for the widely studied Follow-the-Regularized-Leader (FTRL) algorithms. Recently, FTRL with the $1/2$-Tsallis entropy regularizer $Ψ(p) = -4\sum_{i=1}^d \sqrt{p_i}$ (the $1/2$-Tsallis-INF algorithm, by arXiv:1807.07623) was shown to achieve the desirable Best-of-Both-Worlds (BOBW) guarantees and perform well in both adversarial and stochastic settings. Nevertheless, its last-iterate convergence rate has not yet been fully studied. This paper studies the $1/2$-Tsallis-INF algorithm in stochastic bandits and shows that its sampling simple regret decays at rate $\mathcal{O}(t^{-1})$, without requiring the optimal arm to be unique. Under a unique optimal arm, we further show that the expected Bregman divergence induced by $Ψ$ between the point mass on the optimal arm and the sampling distribution at iteration $t$ decays at rate $\mathcal{O}(t^{-1/2})$. Matching lower bounds under the same uniqueness condition show that both exponents of $t$ are tight.

arXiv

Federated Self-Supervised Modulation Classification under Non-IID and Imbalanced Data

Usman Akram, Yiyue Chen, Haris Vikalo

Automatic modulation classification (AMC) is a core enabler of cognitive wireless systems, providing spectrum awareness and supporting adaptive communication at the network edge. However, training AMC models on centrally aggregated data incurs high communication overhead, raises privacy concerns, and often lacks robustness to real-world conditions. We propose FedSSL-AMC, a federated self-supervised framework for learning AMC models from sparsely labeled, distributed I/Q time-series data. Participating clients collaboratively train a causal, time-dilated CNN encoder using triplet-loss self-supervision on unlabeled signals, followed by lightweight local SVMs trained on limited labeled samples. This enables communication-round-efficient, robust representation learning under class imbalance and channel variability. We establish convergence guarantees for a proximal variant of the encoder-training procedure and derive a separability bound for the downstream classifier under feature noise. Experiments on synthetic and over-the-air datasets demonstrate improvements over supervised FL baselines across all three datasets and nearly all evaluated settings involving heterogeneous SNRs, carrier-frequency offsets, and non-IID label distributions.

arXiv

Enhancing Differentially Private Linear Regression via Public Second-Moment

Zilong Cao, Hai Zhang

Leveraging information from public data has become increasingly crucial in enhancing the utility of differentially private (DP) methods. Traditional DP approaches often require adding noise based solely on private data, which can significantly degrade utility. In this paper, we address this limitation in the context of the ordinary least squares estimator (OLSE) of linear regression based on sufficient statistics perturbation (SSP) under the unbounded data assumption. We propose a novel method that involves transforming private data using the public second-moment matrix to compute a transformed SSP-OLSE, whose second-moment matrix yields a better condition number and improves the OLSE accuracy and robustness. We derive theoretical error bounds about our method and the standard SSP-OLSE to the non-DP OLSE, which reveal the improved robustness and accuracy achieved by our approach. Experiments on synthetic and real-world datasets demonstrate the utility and effectiveness of our method.

arXiv

ProteoKnight: Convolution-based Phage Virion Protein Classification and Uncertainty Analysis

Samiha Afaf Neha, Md. Ishrak Khan, Abir Ahammed Bhuiyan

\textbf{Introduction:} Accurate prediction of Phage Virion Proteins (PVP) is essential for genomic studies due to their crucial role as structural elements in bacteriophages. Computational tools, particularly machine learning, have emerged for annotating phage protein sequences from high-throughput sequencing. However, effective annotation requires specialized sequence encodings. Our paper introduces ProteoKnight, a new image-based encoding method that addresses spatial constraints in existing techniques, yielding competitive performance in PVP classification using pre-trained convolutional neural networks. Additionally, our study evaluates prediction uncertainty in binary PVP classification through Monte Carlo Dropout (MCD). \textbf{Methods:} ProteoKnight adapts the classical DNA-Walk algorithm for protein sequences, incorporating pixel colors and adjusting walk distances to capture intricate protein features. Encoded sequences were classified using multiple pre-trained CNNs. Variance and entropy measures assessed prediction uncertainty across proteins of various classes and lengths. \textbf{Results:} Our experiments achieved 90.8% accuracy in binary classification, comparable to state-of-the-art methods. Multi-class classification accuracy remains suboptimal. Our uncertainty analysis unveils variability in prediction confidence influenced by protein class and sequence length. \textbf{Conclusions:} Our study surpasses frequency chaos game representation (FCGR) by introducing novel image encoding that mitigates spatial information loss limitations. Our classification technique yields accurate and robust PVP predictions while identifying low-confidence predictions.

arXiv

Generative Model Unlearning: A Survey through Target Events, Unlearning Operators, and Evaluation Protocols

XiaoHua Feng, Jiaming Zhang, Fengyuan Yu, Chengye Wang, Li Zhang, Kaixiang Li, Yuyuan Li, Lingjuan Lyu, Chaochao Chen, Jianwei Yin

With the rapid advancement of generative models, privacy, copyright, safety, and reliability risks have attracted growing attention. To mitigate these risks, machine unlearning has been increasingly adapted from traditional classification models to generative settings. Despite notable progress, existing studies remain fragmented in their target definitions, unlearning mechanisms, and evaluation protocols, making objective comparison difficult across models, modalities, and applications. Moreover, modality-specific surveys often overlook the shared structure of Generative Model Unlearning (GenMU). To address this gap, we provide a comprehensive review of GenMU and formulate it as target-constrained distributional projection: given a target event, an unlearning operator transforms the generative distribution to suppress target-related outputs while preserving useful behavior and controlling operator cost. Under this framework, an unlearning request is specified by a target event, implemented through an unlearning operator, and assessed by empirical evidence over target suppression, distribution preservation, and operator cost. We further reorganize existing GenMU studies under this view. From this perspective, we provide the first explicit and unified account of how GenMU connects to mainstream applications, including privacy protection, copyright and style protection, safety alignment, hallucination mitigation, and deployment defense. Finally, we identify key open problems and future directions toward reliable, scalable, robust, and auditable GenMU. We consistently maintain the related open-source materials at https://github.com/caxLee/GenMU-Survey.

arXiv

VirnyFlow: Optimizing ML Pipelines for Accuracy, Fairness, and Stability at Scale

Denys Herasymuk, Anastasiia Mozghova, Nazar Protsiv, Vladyslav Sydorak, Julia Stoyanovich

Developing machine learning (ML) systems for real-world deployment requires navigating context-dependent trade-offs among accuracy, fairness, stability, and other objectives. Existing AutoML frameworks optimize pipelines efficiently, but they fix the optimization objective up front, leave it outside the developer's control during search, and rarely scale beyond a single node. We present VirnyFlow, a system that optimizes ML pipelines jointly for accuracy, fairness, and stability at scale. A user-defined evaluation protocol, with fairness measured over binary and intersectional groups, drives every layer of the optimizer: multi-objective Bayesian optimization of physical pipelines, cost-aware bandit selection of logical pipelines, and multi-criterion pruning. The architecture combines asynchronous execution over Apache Kafka with database-backed experiment management, providing fine-grained parallelism, fault tolerance, and interactive inspection of trade-offs. On six real-world datasets, VirnyFlow achieves competitive or superior performance compared to state-of-the-art AutoML systems (auto-sklearn, Alpine Meadow, FLAML) under identical resource constraints, scales to 128 workers across four nodes on datasets of up to 2.6M records, and achieves up to 7x higher speedup than the best-scaling single-node baseline, while maintaining stable accuracy and fairness as parallelism increases. A clinical case study on distribution shift and an IRB-approved user study demonstrate human-in-the-loop navigation of trade-offs in practice: rather than returning a single "best" model, VirnyFlow lets data scientists define, inspect, and iteratively refine the objectives of the search to fit their deployment context.

arXiv

WATCH: Adaptive Monitoring for AI Deployments via Weighted-Conformal Martingales

Drew Prinster, Xing Han, Anqi Liu, Suchi Saria

Responsibly deploying artificial intelligence (AI) / machine learning (ML) systems in high-stakes settings arguably requires not only proof of system reliability, but also continual, post-deployment monitoring to quickly detect and address any unsafe behavior. Methods for nonparametric sequential testing -- especially conformal test martingales (CTMs) and anytime-valid inference -- offer promising tools for this monitoring task. However, existing approaches are restricted to monitoring limited hypothesis classes or ``alarm criteria'' (e.g., detecting data shifts that violate certain exchangeability or IID assumptions), do not allow for online adaptation in response to shifts, and/or cannot diagnose the cause of degradation or alarm. In this paper, we address these limitations by proposing a weighted generalization of conformal test martingales (WCTMs), which lay a theoretical foundation for online monitoring for any unexpected changepoints in the data distribution while controlling false-alarms. For practical applications, we propose specific WCTM algorithms that adapt online to mild covariate shifts (in the marginal input distribution), quickly detect harmful shifts, and diagnose those harmful shifts as concept shifts (in the conditional label distribution) or extreme (out-of-support) covariate shifts that cannot be easily adapted to. On real-world datasets, we demonstrate improved performance relative to state-of-the-art baselines.

arXiv

Diffusion-model approach to flavor models: A case study for $S_4^\prime$ modular flavor model

Satsuki Nishimura, Hajime Otsuka, Haruki Uchiyama

We propose a numerical method of searching for parameters with experimental constraints in generic flavor models by utilizing diffusion models, which are classified as a type of generative artificial intelligence (generative AI). As a specific example, we consider the $S_4^\prime$ modular flavor model and construct a neural network that reproduces quark masses, the CKM matrix, and the Jarlskog invariant by treating free parameters in the flavor model as generating targets. By generating new parameters with the trained network and local optimization, we find various phenomenologically interesting parameter regions. Additionally, we confirm that the spontaneous CP violation occurs in the $S_4^\prime$ model. The diffusion model enables an inverse problem approach, allowing the machine to provide a series of plausible model parameters from given experimental data.

arXiv

Towards Unified Approaches in Self-Supervised Event Stream Modeling: Progress and Prospects

Levente Zólyomi, Tianze Wang, Sofiane ENNADIR, Oleg Smirnov, Lele Cao

The proliferation of digital interactions across diverse domains, such as healthcare, e-commerce, gaming, and finance, has resulted in the generation of vast volumes of event stream (ES) data. ES data comprises continuous sequences of timestamped events that encapsulate detailed contextual information relevant to each domain. While ES data holds significant potential for extracting actionable insights and enhancing decision-making, its effective utilization is hindered by challenges such as the scarcity of labeled data and the fragmented nature of existing research efforts. Self-Supervised Learning (SSL) has emerged as a promising paradigm to address these challenges by enabling the extraction of meaningful representations from unlabeled ES data. In this survey, we systematically review and synthesize SSL methodologies tailored for ES modeling across multiple domains, bridging the gaps between domain-specific approaches that have traditionally operated in isolation. We present a comprehensive taxonomy of SSL techniques, encompassing both predictive and contrastive paradigms, and analyze their applicability and effectiveness within different application contexts. Furthermore, we identify critical gaps in current research and propose a future research agenda aimed at developing scalable, domain-agnostic SSL frameworks for ES modeling. By unifying disparate research efforts and highlighting cross-domain synergies, this survey aims to accelerate innovation, improve reproducibility, and expand the applicability of SSL to diverse real-world ES challenges.

arXiv

ConfRetro: a 3D-aware template-free method for enhancing retrosynthesis via molecular conformer information

Jiaxi Zhuang, Yu Zhang, Ying Qian, Aimin Zhou

Motivation: Retrosynthesis plays a crucial role in organic synthesis and drug discovery, focusing on identifying a set of reactants capable of synthesizing a target product molecule. Although the existing approaches have shown promising results, they do not fully exploit 3D conformer information and molecular spatial structure, which can hinder stereochemically consistent and chemically plausible predictions. Results: To tackle this problem, we propose ConfRetro, a Transformer-based template-free method that integrates molecular conformer information and spatial structure. We devise an Atom-align Fusion module to combine 3D positional information at the model input stage, ensuring alignment between atom tokens and corresponding 3D representations. Furthermore, we design a Distance-weighted Attention mechanism to guide self-attention, constraining the receptive field of model and emphasizing chemically relevant atom pairs in 3D space. Experiments conducted on the USPTO-50K and USPTO-FULL datasets demonstrate that ConfRetro significantly outperforms existing template-free approaches, achieving a new state-of-the-art performance. Case studies further highlight its capability to predict accurate and chemically plausible reactants, even for target molecules with intricate structures. Moreover, when plugged into a standard retrosynthetic search, ConfRetro recovers feasible synthetic routes for multiple representative drug molecules (e.g. Camptothecin). Availability and implementation: ConfRetro is available at https://github.com/Jesse-zjx/ConfRetro. Archival snapshot is available at https://doi.org/10.5281/zenodo.20785018.

arXiv

Experimentally Extending Quantum Kernel Learning to Quantum Data by NMR

Vivek Sabarad, Vishal Varma, T. S. Mahesh

Quantum kernel learning (QKL) promises efficient machine learning by encoding feature maps onto exponentially large Hilbert spaces inherent in quantum systems. Using the liquid-state nuclear magnetic resonance (NMR) platform, we implement and benchmark QKL for one-dimensional regression and two-dimensional classification tasks. We then classify entangling and non-entangling operators by extending QKL to handle parametrized or non-parameterized operator inputs. We first compute the kernel numerically for a double-layered star system and then experimentally validate it on a 3-qubit NMR register. QKL provides a practical route to compare operators on native quantum hardware without expensive tomography protocols. Our results confirm the superiority of QKL over other classical methods for processing quantum data, thereby highlighting its ability to capture the inherent structure of quantum space and to extend its domain of operation beyond the training domain by exploiting symmetries in the operator space.

arXiv

ODTlearn: A Package for Learning Optimal Decision Trees for Prediction and Prescription

Patrick Vossler, Nathan Justin, Sina Aghaei, Nathanael Jo, Andrés Gómez, Phebe Vayanos

ODTlearn is an open source Python package that provides methods for learning optimal decision trees for high-stakes predictive and prescriptive tasks based on the state-of-the-art mixed-integer optimization (MIO) framework proposed in Aghaei et al. (2025). The current version of the package provides implementations for learning optimal classification trees, optimal fair classification trees, optimal prescriptive trees from observational data, and optimal classification trees robust to distribution shifts. We have designed the package to be easy to maintain and extend as new optimal decision tree problem classes, reformulation strategies, and solution algorithms are introduced. To this end, the package follows object-oriented design principles and supports both commercial (Gurobi) and open source (COIN-OR branch and cut) solvers. The package documentation, user guide, installation instructions, link to source code, and instructions for submitting bug reports and feature requests can all be found at https://d3m-research-group.github.io/odtlearn/.

arXiv

Dynamic Pricing and Advertising with Demand Learning

Shipra Agrawal, Yiding Feng, Wei Tang

We consider a novel pricing and advertising framework in which a seller not only sets the product price but also designs flexible advertising schemes to influence customers' valuations of the product. We impose no structural restriction on the seller's feasible advertising strategies and allow her to advertise the product by disclosing or concealing any information. Following the information design literature, we model this fully flexible advertising as the seller choosing an arbitrary information policy that signals the product quality to customers. Customers observe the advertising signal and form a Bayesian posterior belief over the product quality. We investigate two questions in this work: (1) What is the value of advertising? To what extent can advertising enhance a seller's revenue? (2) Without any a priori knowledge of the customers' demand function, how can a seller adaptively learn and optimize both pricing and advertising strategies using past purchase responses? To study the first question, we quantify the value of advertising by comparing the optimal revenue from jointly designing advertising and a single posted price with the optimal revenue from pricing alone. We show that advertising can increase revenue by at most a factor of two, and this bound is tight. For the second question, we study the seller's dynamic pricing and advertising problem under demand uncertainty. Our main result for this question is a computationally efficient online algorithm that achieves the optimal $O(T^{2/3} (m\log T)^{1/3})$ regret rate when the valuation function is linear in the product quality. Here, $m$ is the cardinality of the discrete product quality domain and $T$ is the time horizon.

arXiv

Score Attack: A Lower Bound Technique for Optimal Differentially Private Learning

T. Tony Cai, Yichen Wang, Linjun Zhang

Achieving optimal statistical performance while ensuring the privacy of personal data is a challenging yet crucial objective in modern data analysis. However, characterizing the optimality, particularly the minimax lower bound, under privacy constraints is technically difficult. To address this issue, we propose a novel approach called the score attack, which provides a lower bound on the differential-privacy-constrained minimax risk of parameter estimation. The score attack method is based on the tracing attack concept in differential privacy and can be applied to any statistical model with a well-defined score statistic. It can optimally lower bound the minimax risk of estimating unknown model parameters, up to a logarithmic factor, while ensuring differential privacy for a range of statistical problems. We demonstrate the effectiveness and optimality of this general method in various examples, such as the generalized linear model in both classical and high-dimensional sparse settings, the Bradley-Terry-Luce model for pairwise comparisons, and nonparametric regression over the Sobolev class.

arXiv

StateSight: Benchmarking Latent Spatial-State Reconstruction in Vision-Language Models

Michelle Lin

Vision-language models are increasingly used for multimodal question answering, yet their ability to reconstruct latent spatial structure from a single image remains difficult to isolate. Broad benchmarks often combine perception, optical character recognition, domain knowledge, linguistic priors, and reasoning in the same evaluation. We introduce StateSight, a procedurally generated benchmark for cube-net opposite-face reasoning, occluded cube-tower counting, and 4-neighbor connected-component counting. Each task family contains 300 single-image prompts with deterministic oracle labels and exact-match scoring. OpenAI GPT-5.5, using the API model identifier gpt-5.5, achieved 59.3%, 33.3%, and 28.3% accuracy across the three tasks, while Claude Sonnet 5 achieved 53.3%, 18.7%, and 7.3%. All final direct runs had zero format errors. A 30-participant human baseline on 60 items exceeded both models on every task, with mean accuracies of 80.8%, 68.8%, and 64.3%. Visible-derivation analysis identified recurring errors in image-state reconstruction and reasoning procedure. We also introduce StateSight-Steps, a companion dataset of 900 interleaved image-text examples and 3,600 deterministic intermediate visual states. The results show that format-valid responses can mask failures to recover the spatial structure required for verifiable visual inference.

arXiv

SUMI: Scalable Unified Model for 3D Point Cloud Inference

Yanlong Li, Kanchana Thilakarathna

Point cloud completion commonly follows a coarse-to-fine paradigm, where a low-density coarse shape is first predicted and then upsampled to the target resolution. Although recent methods have improved global structure recovery, the fine stage often remains limited by simple upsampling and insufficient interaction with coarse structural features, making local detail reconstruction challenging. We propose SUMI, a diffusion-enhanced refinement module for coarse-to-fine point cloud completion. Unlike prior diffusion-based completion methods that use diffusion as a standalone point generator, SUMI injects noisy geometric features into cross-attention with coarse structural features, enabling reverse denoising to refine local geometry while preserving global consistency. SUMI can also be integrated into existing coarse-to-fine models as a flexible refinement module. Experiments on PCN, ShapeNet-55/34, and MVP demonstrate consistent improvements over strong baselines. SUMI achieves the best overall CD and F1-score on PCN, reduces CD by up to 16.1% on ShapeNet-55, and obtains the best CD across all output densities on MVP.

arXiv

MM-BEV: Enhancing Timeliness by Computing Where and When it Matters

Liangkai Liu, Kang G. Shin

Multimodal bird's-eye-view (BEV) perception combines LiDAR depth accuracy with dense camera semantics, but its high computational cost and imperfect sensing conditions make real-time deployment challenging. Existing methods largely compress individual detectors and overlook three opportunities: structured sparsity within camera and LiDAR inputs, timing misalignment between modalities, and the fact that many detected objects do not affect the planner's immediate action. We present MM-BEV, a real-time multimodal BEV system guided by a simple principle: compute where and when it matters. MM-BEV divides perception into mandatory work for safety-critical objects within braking distance of the ego vehicle and with short time-to-collision (TTC), and optional work for less urgent regions. It prioritizes mandatory work and reduces or sheds optional work under tight compute budgets. MM-BEV integrates four mechanisms: (1) a criticality-ranked temporal ROI selector based on motion-extrapolated detections from prior frames; (2) sparse, ROI-aware feature extraction using shared-shape camera crops at context-adaptive resolution and ROI-aware LiDAR voxelization; (3) a latency-aware coordinator that adapts LiDAR sweeps, image resolution, and keyframes according to scene dynamics and TTC; and (4) an asynchronous scheduler that decouples sensing from inference and skips stale frames. On nuScenes, MM-BEV reduces inference latency by 1.96x and end-to-end latency by 2.93x, with no loss in geometry-critical recall and only a 0.2 percentage-point drop in safety-critical recall. On a Clearpath Husky A300 equipped with an Ouster-128 LiDAR, BEV cameras, and a Jetson AGX Orin, MM-BEV further reduces mean latency by 2.11x, demonstrating its potential for real-world autonomous systems.

arXiv

NumerosityVLM: A Cognitively Inspired Benchmark for Interpreting Numerosity Representations in Vision-Language Models

Yiming Fu, Fangjun Li, Xiujin Liu, Ruidong Ma, Hang Yu, Zhichen Lu, Kanwei He, Alessandro Di Nuovo, Angelo Cangelosi, Zhegong Shangguan

Vision-language models (VLMs) achieve strong performance on high-level multimodal tasks, yet numerosity perception, a cognitive ability that emerges in human infants before language acquisition, remains poorly understood in current models, as existing counting benchmarks entangle numerosity with correlated visual factors. We introduce a cognitively inspired diagnostic benchmark, NumerosityVLM, comprising 10,800 synthetic images across six controlled conditions. The benchmark orthogonally manipulates object size, spatial arrangement, and numerosity, while progressively ablating texture, shape, and color. Evaluating seven VLMs in a zero-shot setting, multi-factor analysis reveals that model architecture explains the largest proportion of performance variance (partial $ω^{2}=0.325$), far exceeding visual conditions. Layer-wise probing further shows that linearly separable numerosity signals consistently emerge at early stages of the vision encoder, while performance differences across evaluated models are primarily associated with the language model component. Code and data are publicly available at https://github.com/fuy3/NumerosityVLM-Benchmark, and https://huggingface.co/datasets/fuy3/NumerosityVLM.

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