Counting

Test-time scaling can improve large language model reasoning by generating and combining multiple candidate responses. In sampling-based methods, the inference budget is often described by the number of generated candidates, N. However, N tells us how many candidates are generated, not how they are executed. The same candidate budget can be produced in one batched generation call or split across several sequential calls with smaller batch sizes. We first study the effect of increasing N on reasoning accuracy using Phi-3-mini and Qwen2.5-1.5B on 500 GSM8K prompts. As expected, increasing N from 1 to 8 improves accuracy by 8.4 percentage points for Phi-3-mini and 18.4 points for Qwen2.5-1.5B. However, accuracy alone does not show the systems cost of using a larger candidate budget. We therefore fix N = 8 and compare four generation schedules: 1x8, 2x4, 4x2, and 8x1, where axb denotes a generation calls with b candidates per call. We measure latency, throughput, GPU-hours, and gross GPU-device energy while keeping the total candidate count fixed. On A100 GPUs, eight serial calls use 4.64-4.86x as much gross GPU-device energy and have 5.77-6.12x the P95 latency of one batched call with eight candidates. The same pattern appears across three independently scheduled A100 nodes per model and in short-output SciQ/V100 experiments. These results show that candidate count alone is not enough to describe the systems cost of multi-candidate test-time scaling. When candidates are independent and memory allows it, fewer generation calls with larger batch sizes are more efficient. Evaluations should therefore report not only candidate count and accuracy, but also generation schedule and GPU-level systems metrics.

Leaf counting is an important task in plant phenotyping for monitoring plant growth and estimating crop yield. Most existing methods rely on RGB images, but their performance is often affected by occlusion, lighting variations, and other real-world challenges. Additional modalities, such as depth and thermal images, can provide useful complementary information. However, multimodal leaf counting remains underexplored. Also, many existing methods assume that all training data are available simultaneously, which is impractical in real agricultural settings, where data is collected over time from multiple sources. To address these challenges, we propose MCLC-NET, a multimodal continual learning framework for leaf counting. It learns tasks sequentially using a memory-based strategy with a memory buffer to retain important samples from previous tasks. We also introduce MMLC, a real-world multimodal leaf-counting dataset designed for a domain incremental scenario (DIS) in CL. It contains RGB, depth, and thermal images collected across different crop types under varying environmental conditions, arranged in three orderings: crop-wise, time-wise, and mixed. Experimental results, averaged over three random seeds, demonstrate that MCLC-NET consistently outperforms existing methods across all three task orderings, achieving the lowest AMSE of 0.675$\pm$0.027, 0.542$\pm$0.069, and 0.745$\pm$0.057, respectively.

Visual monitoring of flapping-wing vehicles requires distinguishing individual wingbeats from motion strength and average frequency. This paper presents a controlled MuJoCo evaluation of wingbeat counting from signed optical flow observed by virtual cameras mounted on Crazyflie vehicles. Three flapping-wing models were recorded at optical distances of 1.5 and 3.0 m, producing 1,440 clips from 240 paired scene configurations with a scene-level 3:1 training-test split. A common spatial convolutional encoder was combined with a causal temporal convolutional network, a recurrent leaky integrate-and-fire spiking network, or causal self-attention. Each model predicted phase and activity, followed by the same directed-crossing event counter. The six existing convolutional models were retained, and all twelve new models were frozen before their test predictions were generated. Exact-count accuracies at 1.5 m were 96.67%, 95.00%, and 96.67%, respectively; at 3.0 m they were 94.44%, 92.22%, and 95.00%. All paired scene-bootstrap intervals for differences in exact-count accuracy included zero. Seven far-distance spiking-model clips had correct totals despite event-timing mismatches, demonstrating why total-count and event-level measurements must be reported together. The results support the feasibility of causal optical-flow counting in the tested setting and identify boundary-sensitive errors. They do not establish an architecture ranking across repeated training, real-flight robustness, or hardware efficiency.

Counting equivalent pairs is a common way to report transformation-audit coverage, but it can substantially overstate the constraints imposed by an audit: pairs generated from the same semantic object are correlated, and complete orbit graphs contain algebraically redundant edges. We therefore distinguish four complementary quantities---edge count $m$, effective contrast rank $s$, population support rank $r$, and graph spectral gap $η$---and characterize their roles in audit coverage and deployment reliability. Under a rank-$r$ Gaussian contrast model, a population-invariant calibrated reader exists exactly when the anchor has a component in $\ker T$. When an audit has rank $s < r$, its unobserved risk is $R^\star/U$, with $U \sim \operatorname{Beta}((r-s)/2,s/2)$; when $s \ge r$, exact calibrated interpolation is infeasible. The same distinction appears in orbit topology: a spanning tree imposes the same exact-null constraints as a complete graph, while a sharp graph Poincare inequality propagates edge-level drift to an entire orbit at a cost proportional to $1/η$. Cyclic audits can additionally yield zero pair-level leave-one-out error without holding out any semantic object. To address these failures, we derive exact block-Woodbury leave-one-orbit-out updates and introduce a source-disjoint deployment gate over finitely many candidate readers. The gate retains the original reader unless uncertainty bounds certify lower drift within a prescribed clean-utility budget. etc..

When a language model finds a sentence unusually cheap to predict, it is tempting to conclude that the sentence was in its training data. Almost every published test of that inference has had to guess which sentences were in the training data, the members, and which were not. This paper removes the guessing. Two model families, OLMo-2 and Pythia, publish their pretraining corpora, and a public index over those corpora returns the exact number of times any sentence appeared in each. Those counts make three questions answerable directly. The answers form a pincer, closing from two sides. At the duplication levels ordinary text actually has, five models from 1B to 13B parameters carry at most a faint trace of their own exposure. We measure that trace with a design that reads the same sentence through two models, which cancels fluency and quality by construction, and it comes to a rank correlation near -0.08, where -1 would be a perfect relation and 0 none. Where the trace does become strong, above roughly a thousand copies, the two corpora agree on which sentences those are, because they are the famous ones, so exposure can no longer be told apart from fame. Two further measurements show how apparent membership signal gets manufactured. A common way to build a non-member is to change one word of a member. The model does prefer the original, but the gap is the same whether the original appeared once or a hundred times, so what the model is rewarding is the author's word choice, not memory. Above a thousand copies the gap grows with model size on the twelve sentences we can test there, at the same boundary where the pincer closes. And swapping the controls for sentences that differ from the members in register moves a detector from 0.83 to 0.94 AUC, on a scale where 0.5 is a coin flip and 1.0 is perfect separation. We release the sentence banks, counts, and code.

Excess vocabulary, a word's frequency above its pre-2023 trend, is how the change in scholarly English after 2022 has been measured. We adapt it to Korean with morphological units on 398,296 KCI abstracts (2018-August 2026), with 47,165 Vietnamese abstracts for comparison. Placebo floors are 0.1-2.2 points for the single-word statistic and at most 2.9 for the re-selected split-half set statistic. Korean abstracts show nothing in 2023, onset in late 2024, a rise through 2025 flattening in mid-2026: sisahada "suggest" appears in 21.4% of 2026 abstracts against 5.3% expected; plain verbs like araboda "look into" fall to a quarter of trend. Under stated assumptions the single-word conditional lower bound on LLM-processed abstracts is 3.5%, 10.5% and 16.1% for 2024-2026 and a split-half set bound 7.8%, 20.6% and 33.0%. Holzwarth et al.'s estimator under the same discipline gives 41.9% and 72.1% for 2025-2026. Subject-matter controls reduce but do not remove it: restricting the set to lemmas three language-model annotators all call style leaves 14.7 of the 33.0 points, and pairing each 2026 abstract with its journal's closest base-period abstract leaves 34.1. Tested translation routes do not explain it: the surface marks of translated Korean fall as the markers rise. In the same articles' English abstracts the excess appears a year earlier; where the English side carries none, the Korean shift persists at 30 to 66% of the rate where it does. Control abstracts from three providers reproduce the rising words, with marker turnover consistent with model generations; implied prevalences are scenario-dependent.

Camera traps have become an essential tool for wildlife monitoring, motivating the development of computer vision methods for the automated extraction of information from these data. While most prior work has focused on species identification, many ecological applications also require estimating the number of unique individuals appearing across short image sequences. This task is particularly challenging because camera traps typically acquire bursts of images at approximately one frame per second, creating large temporal discontinuities that may make conventional multi-object tracking methods unreliable, and because manually collecting individual count annotations is prohibitively expensive. In this work, we describe the winning solution to the iWildCam 2021 Challenge, which introduced a benchmark for counting animals at the sequence level under realistic annotation constraints where count annotations are unavailable for training. Our approach, MaxBoxCount, combines a strong species classification pipeline with a simple yet effective counting heuristic based on MegaDetector detections to estimate the number of unique individuals without requiring count annotations. Code is available at https://github.com/alcunha/iwildcam2021ufam.

Flow-based generative models have emerged as powerful image priors for training-free inverse problem solving, capturing coherent semantics and fine-grained structure. Despite these strengths, existing flow-based inverse solvers primarily focus on the design of individual updates, largely overlooking spatio-temporal information allocation under a fixed number of function evaluations (NFEs). Temporally, insufficient early exploration can trap the flow trajectory in an incorrect semantic basin, whereas excessive allocation of NFEs to early stages leaves little budget for late-stage refinement. Spatially, data consistency provides direct constraints only within observed regions, whereas the recovery of missing regions relies mainly on the generative prior. To address these two issues, we introduce two complementary and training-free components, i.e., Spectrum-Adaptive Scheduling (SAS) and Measurement-Prioritized Attention (MPA). For temporal allocation, SAS distributes the available NFEs over flow time according to the degradation spectrum and logSNR geometry, thus better balancing semantic exploration and detail refinement. For spatial propagation, MPA exploits data-prior conflicts to guide information toward weakly constrained regions, thereby enhancing semantic and structural fidelity. Extensive experiments on standard image inverse problems, e.g., super-resolution, motion deblurring, and inpainting, demonstrate that the proposed components can be integrated into existing flow-based inverse solvers in a plug-and-play manner without retraining or additional flow-model evaluations, and can also significantly improve the restoration quality of existing solvers.

How many directions does a neural representation use to encode a concept? A common answer repeatedly erases probe directions and reports the stopping count or cumulative removed rank. We show that both quantities can change under an information-preserving invertible reparameterization, so neither is intrinsically a concept dimension. We distinguish model-defined population quantities (generating dimension, sufficient linear dimension, and minimum guarding rank) from procedure-defined quantities such as stopping count and cumulative edit rank. In a population Gaussian construction, an invertible shear preserves the prediction problem and all three quantities, yet changes the cumulative Euclidean erasure count from one to two. The separation holds for Moore--Penrose ordinary least squares and every finite nonnegative ridge weight. For a two-output full-QR procedure matching our motivating video analysis, cumulative edit rank similarly changes from two to the ambient dimension four. Conversely, the complete cumulative metric-QR trajectory is affine-equivariant when its positive-definite metric, probe, regularizer, and tie-breaking are transported consistently; exact covariance is one corollary, not a canonical semantic metric. In a known-rank finite-sample Adam/QR calibration, identity mixing stops after one accepted update in all 20 large-sample runs, whereas each tested shear $a\in\{.5,.75,1,1.25,2\}$ accepts at least two updates in all 20 runs. Controlled reparameterizations of frozen V-JEPA2 features preserve rank-zero predictions yet alter later Euclidean trajectories under practical optimization. These visual contact experiments are stress tests, not estimates of contact dimension. Iterative erasure therefore returns a procedure-relative estimand jointly determined by representation geometry and the full measurement procedure, not a semantic dimension by itself.

Traffic data collection is dominated today by deep object detectors followed by tracking-by-detection, a pipeline that presupposes what is often missing in practice: a detector already trained on the class one wants to count. We revisit a purely geometric traffic-sensing pipeline for Single Board Computers in which detection is class-agnostic: moving objects come from background subtraction and thresholding, and counting is decided by a geometric rule on an imaginary line across the road, a software inductive loop detector. With no object model, training set or per-object trajectory, it runs faster than real time on Raspberry Pi class hardware. Two counting rules are described: a constant average speed rule, whose expected accuracy is derived analytically as about 86% under a Gaussian speed distribution, and a self-calibrating pre-calibration rule that recovers the lane geometry from blob statistics and counts edges of lane occupancy, additionally yielding per-vehicle average speed at no extra cost. Over four videos the latter counts with 83.3%-100% accuracy; in a field deployment it reaches 91% against 37.5% for a blob-tracking baseline under the same compute budget. We report the observations of that period in detail: the resolution floor below which accuracy collapses, the frame rate floor at which vehicles alias past the counting line, the gap between short curated clips and long uncontrolled footage, and the trade-off between Python (easier to tune, 100% CPU) and C++ (40% CPU, thermally viable). These are properties of the sampling geometry, not of the hardware of the time, and still constrain edge deployments. We close by arguing where motion-based, class-agnostic detection remains the right tool: open-set classes with no annotated data, tight power budgets, privacy-constrained installations, and the cold start of mining training crops to bootstrap a learned detector.

Our primary objective is to advance video object counting in crowded scenes, aiming to robustly count all instances of a target category based on given text or visual prompts. Existing methods rely on RGB information, limiting their discriminative ability in crowded and occluded conditions. To address this, we propose a Depth-Guided Detector (DG-Det) along with a general post-processing pipeline. By integrating depth cues with multi-scale RGB-D cross-attention and explicit occlusion prediction, our method enhances spatial understanding and achieves robust detection in crowded and occluded scenes. Furthermore, we introduce a unified de-duplication framework to eliminate cross-frame redundant counting. To facilitate future research, we also release a new RGB-D Video Object Counting dataset featuring depth information and multiple object categories persequence. Extensive experiments demonstrate that our method achieves a 62.01% reduction in MAE compared to existing baselines, and also produces consistent improvements in RMSE. We provide the source code at https://github.com/streamer-AP/DG-Net and the dataset at https://huggingface.co/datasets/aerospace123/RGBD-VideoCount.

Task B of the 1st DAFx Parameter Estimation Challenge requires estimating the frequencies, decay rates, gains, and number of modes in a dense plate-reverb impulse response. Weak and overlapping modes make sparse peak detection prone to severe undercounting. We train an ExtraTrees regressor on simulator-generated data to predict mode counts in four frequency bands. These counts define dense frequency grids, after which a differentiable all-pole resonator model refines decay and gain while keeping frequency fixed. On two separate synthetic validation sets, the system reduces a local challenge-style error by about 66% relative to the official default peak-picking baseline. The improvement is mainly associated with lower mode-count mismatch, while decay and gain remain the largest error sources. These findings support separating modal-density estimation from continuous parameter fitting.

Multimodal large language models (MLLMs) excel at visual interpretation but fail on spatial reasoning tasks that humans solve reliably. Existing benchmarks evaluate these models as black boxes, limiting their ability to identify the underlying causes of lower performance: when a model fails a spatial reasoning task, it remains difficult to ascertain whether the hurdle is perceptual, such as recognizing object boundaries, or cognitive, such as reasoning about occlusion to infer hidden geometry. We introduce Spatial-IQ, a hierarchical diagnostic framework that decomposes object counting in stacked 3D structures into 9 perceptual and cognitive sub-tasks organized by the developmental stages of human spatial cognition, with mental rotation as an additional target probe. Using NVIDIA Isaac Sim, we procedurally generated a diverse dataset of roughly 80,000 stacked 3D structures with per-task ground truth. We evaluate models across three output formats (free-response text, multiple-choice images, and image editing) alongside a human baseline. The Spatial-IQ framework shows that top-performing models often succeed at the target task (object counting) without succeeding on the lower-level sub-tasks intended to support it, and that models differ in how much of these hierarchical chains they preserve, often revealing shortcut behavior that raw target-task accuracy alone would obscure. Finally, we demonstrate that training models with chain-of-thought (CoT) supervision over our hierarchical sub-tasks, combined with reinforcement learning with verifiable rewards, significantly improves both spatial consistency across sub-tasks and target-task accuracy, supporting the value of the proposed decomposition as both a diagnostic tool and a training signal.

Geometric architectures are often justified by internal mechanisms such as rotations, yet task performance alone cannot show whether those mechanisms drive predictions. Using sheaf neural networks (SNNs) as a testbed, we introduce the first basis-independent measurement of trained triangle-loop products, separating rotation, stalk-space area, and orientation. In a custom high-homophily GraphUniverse regime, Neural Sheaf Propagation (NSP) increases the triangle-weighted mean two-dimensional SO(2) loop rotation from 0.010 to 0.388 radians for triangle counting, while the community-detection comparison ends at 0.029 radians. Across the training-set-size experiment, replacing all learned SO(2) transports with identities sharply increases test error, establishing post-training sensitivity to the complete learned connection. However, a graph-summary ridge predictor is more accurate, diagonal maps also improve, and fixed-degree graphs develop increasing rotation without outperforming the training-mean predictor. This measure-intervene-control study separates geometric change, connection sensitivity, and evidence for triangle-specific computation.

Practitioners make three prompt-design decisions with almost no controlled evidence behind them: how to format instructions and context (markdown, plain text, prose, or tabular), how many simultaneous instructions a system prompt can carry before compliance degrades, and how much context a model can hold before recall and honesty degrade. We report two controlled experiments crossing all three factors on one held, contamination-free synthetic corpus (the "Book of Veyra," 8,780 uniquely-named entities, deterministically regenerable from a fixed seed), evaluated across five models. Experiment 1 (960 calls/model) measures instruction-following decay as rule count N grows from 10 to 160, crossed with four formats and system-prompt vs. user-turn placement. Perfect-response rate collapses to zero by N=80 for every model, format, and placement. Placement produces effects at least as large as format at N=160 in most models, but the direction is model-specific. No model shows a reliable markdown advantage; one 35B model favors plain text instead. Experiment 2 (5,520 calls/model) measures recall accuracy, false-premise sycophancy, and absent-fact fabrication across a 2k-to-512k-token context ladder in the same four formats. Recall stays near ceiling through 64-128k tokens, then degrades sharply and format-dependently: one model's accuracy spread reaches 48 points at 128k tokens. Fabrication never occurs (0/5,760 probes), and sycophancy stays negligible (<=8.3%). What rises sharply near each model's context ceiling is outright refusal to answer (0% to 79-90%), distinct from sycophancy or fabrication. Neither pre-registered format ordering holds, and token overhead (+22% to +37% over plain text) further changes which format is preferable where accuracy spread is genuine. We release the full harness, corpus generator, and raw results (VeyraBench): https://github.com/iNetanel/veyrabench

Generalised object counting aims to estimate the number of instances of an arbitrary object category from a single image, but many recent methods can struggle on structurally complex objects due to limited spatial modelling. We present \textit{UpCount}, a class-agnostic counter designed to better preserve spatial structure. UpCount strengthens the visual representation by extracting multi-layer features from a ViT-B/16 encoder and reassembling them into a refined multi-scale pyramid that is spatially refined using Dense Prediction Transformers and FeatUp, yielding features with improved structural and spatial sensitivity; a proposal--verification counting head then identifies repeated patterns and produces a density map for the final count. On FSC-147, UpCount achieves 12.39 MAE and 100.89 RMSE on the test set, and it transfers effectively to vehicle counting on CARPK (6.27 MAE, 8.79 RMSE). Code: https://github.com/r28112072-rgb/upcount

The ODRL policy language is emerging as the de-facto standard for policy modelling data access and usage preferences, AI governance policies and data workflows in European dataspaces. The current standard has no mathematical formal semantics to describe how a system should implement policy evaluation. This has resulted in a variety of systems and tools that implement their own interpretation of the language, which limits interoperability and cannot guarantee consistent results. Based on an existing semantic model of ODRL, we formalise the problems of ODRL evaluation for the access control and monitoring scenarios, in both static and streaming settings, and we provide a novel, efficient algorithm and implementation. We present the first ODRL Evaluator with transparent formal semantics and supporting all rule types. We experimentally measure its performance, analysing different scalability dimensions related to policy complexity and size of the data on which a policy is evaluated. We compare our system with the state-of-the-art by providing a comparative review of existing ODRL evaluators, which highlights the differences in supported ODRL features and evaluation modes.

Whistleblowers are a key safeguard against organizational wrongdoing, but the threat of retaliation deters reporting. Existing whistleblower-protection proposals lack formal privacy guarantees, and existing differential privacy mechanisms do not directly target the natural threat model -- one in which the audited organization itself observes auditor selection decisions and uses them to identify reporters. We formalize protection against a strong-adversary threat model as per-report $(0, δ)$-differential privacy on the transcript of audit selections. Within this framework we prove that a natural approach -- randomized response applied at the selection step -- can never outperform uniform random auditing by more than $δ$ at any horizon. We then give a generic mechanism that reduces private auditing to private continual counting: any $(0, δ)$-DP continual counter plugs in by post-processing, and the audit transcript inherits the same per-report guarantee. Instantiating the reduction with a recent work in continual counting yields per-report $(0, δ)$-DP with noise scaling as $O(\sqrt{\log T})$ across a horizon of $T$ audit decisions. A utility theorem shows that the selection error vanishes whenever the noisy report gap between the most-reported organization and the runner-up grows faster than $\sqrt{\log T}$. Simulations show a substantial improvement over randomized response.

Turning movement counts are essential for intersection-level traffic management, yet their collection remains predominantly manual due to the cost of per-camera region annotation. This paper presents an unsupervised pipeline that identifies entry and exit regions directly from raw vehicle trajectories extracted via object detection and multi-object tracking, requiring no manual annotation, camera calibration, or prior knowledge of intersection geometry. Unlike trajectory clustering methods that classify individual trajectories using pairwise similarity and must be re-executed on every new batch, the proposed pipeline clusters initial and terminal point locations to produce persistent spatial region polygons that classify future trajectories by point-in-polygon containment at linear cost. The pipeline comprises six sequential steps, five of which introduce configurable parameters evaluated through a systematic statistical analysis spanning 17,100 pipeline executions across 9 surveillance cameras capturing dense heterogeneous traffic in Bengaluru, India, and 10 sequences from the UA-DETRAC benchmark dataset. Both parametric and nonparametric testing frameworks identify three consistently significant parameters and yield an empirically grounded recommended configuration. Under this configuration, the pipeline achieves a median classification error of 3.4% across all 25 Bengaluru cameras, including 16 held-out locations, with a median per-turning-movement GEH of 2.43. Compared with two trajectory clustering baselines, the proposed pipeline exhibits greater stability across camera views and lower computational cost, at the expense of higher median error. Extended evaluation demonstrates that calibration clips of at least 60 minutes and peak-traffic selection further improve region estimation quality.

Despite strong performance on many multimodal tasks, vision-language models (VLMs) still struggle with basic object counting. We investigate whether this reflects missing internal knowledge or a gap between internal representations and verbalized outputs. Training simple probes on activations from four VLMs across five counting datasets reveals that nonlinear probes can reliably detect counting errors, suggesting that VLMs often encode the correct count even when they output the wrong answer. SVCCA analysis shows that probes trained on ground-truth counts and probes trained on model outputs occupy a partially shared activation subspace but read out along misaligned directions. We further validate our findings using a causal steering intervention, proving that strengthening the direction of count-identified probes does improve model counting performance. Motivated by this result, we propose a detector-guided self-correction method that selectively re-prompts the model only when an internal error detector predicts failure. This simple inference-time intervention improves counting accuracy by up to 15.6 absolute percentage points, without any parameter updates. Our results establish activation-based error probing as both a practical tool for improving VLM counting and a mechanistic lens on the gap between internal knowledge and model outputs.

Suppose we observe two sets of $n$ Gaussian vectors in $\mathbb{R}^d$, with the promise that, after applying a permutation of $[n]$ and a rotation of $\mathbb{R}^d$, the two sets are $ρ$-correlated. The Procrustes matching problem asks us to recover the unknown permutation of $[n]$ that aligns the two sets. The problem is well-studied in the low-dimensional regime $d=O(\log n)$, but the high-dimensional regime $d\gg \log n$ has remained largely uncharted: prior matching guarantees require nearly perfect correlation $ρ=1-o(1)$, even for information-theoretic recovery. Our main result is a polynomial-time algorithm for exact recovery at constant correlation. The algorithm works by computing and comparing weighted counts of a specially chosen family of ``wide'' trees. So long as $d\ge \mathrm{polylog}(n)$, the algorithm succeeds with high probability for any $ρ^2>\sqrtα$, where $α\approx 0.338$ is Otter's tree-counting constant. We complement this algorithmic result with an improved information-theoretic guarantee, showing that exact recovery is possible when $ρ^2 \gtrsim \max\{\log n/d,\sqrt{\log n/n}\}$. We also carry out a low-degree advantage calculation, which suggests that the condition $ρ^2 > \sqrtα$ is necessary for any tree-counting algorithm.

Automated crowd counting in Hajj video is difficult not because current models lack capacity, but because the footage violates the assumptions those models were built on: cameras observe the crowd from steep, near-vertical angles, individuals occlude one another extensively, and a single frame can contain well over a thousand people. Benchmarks that test crowd counting in such an environment are either private or not detailed per second. We revisit the HAJJv2 dataset and contribute HAJJv2-CrowdCount: per-second human-annotated crowd counts for its testing videos. Using these annotations, we benchmark three recent zero-shot counting paradigms: an open-vocabulary detector (YOLO-World), a point-based counter (APGCC), and a promptable segmentation-based counter (SAM3Count). SAM3Count attains the lowest overall mean absolute error (MAE 70.4, 95% CI 56.0-86.1), ahead of YOLO-World (92.0) and APGCC (152.9). This ordering reverses, however, in the regime most relevant to deployment: on the densest frames, the detection- and segmentation-based counters both degrade sharply (MAE exceeding 300), while the point-based counter degrades far more gracefully (MAE 114.9). This inversion is decision-relevant for Hajj crowd management, where reliable counts are needed most precisely in the densest and most occluded scenes. The annotations are released to support reproduction and extension of these results.

Visual counting is a fundamental pillar of multimodal intelligence, requiring a seamless integration of fine-grained grounding and spatial reasoning. While Multimodal Large Language Models (MLLMs) have achieved remarkable success in qualitative scene understanding, their quantitative precision remains a significant bottleneck, often characterized by persistent numerical hallucinations. Existing counting benchmarks primarily focus on basic perception in simplified contexts, failing to capture the complex failure modes that emerge under logical constraints or adversarial conditions. To address these limitations, we introduce HoloCount, a holistic and diagnostically rich benchmark structured around a three-level hierarchical taxonomy. HoloCount evaluates MLLMs across: (1) Semantic Counting, focusing on atomic and property-based enumeration; (2) Analytical Counting, assessing logical composition through spatial and set-based reasoning; and (3) Robustness Testing, probing model integrity against adverse scenarios and grounded counter-priors, such as high-density scenes and linguistic biases. Through an exhaustive evaluation of over 20 state-of-the-art MLLMs, we reveal a critical performance gap: even top-tier models degrade significantly as tasks transition from perception to complex analytical reasoning and adverse scenarios. Our findings provide a systematic landscape of current MLLM counting capabilities and offer a roadmap for developing more grounded and reliable multimodal systems. The dataset is available at https://mm-mvr.github.io/HoloCount/.

Diagnosing and monitoring diseases frequently involves the analysis of human biological samples, with blood analysis being pivotal. Specifically, leukocytes, or white blood cells (WBCs), are essential markers for evaluating the body's defense mechanisms against infections. Traditional methods for WBC counting and classification are labor-intensive and prone to inaccuracies, primarily due to human error. The conventional processes for blood cell analysis, especially those concerning WBCs, are beset with difficulties. These include the laborious nature of manual counting and the susceptibility to errors, which can significantly impact the accuracy and reliability of disease diagnosis and monitoring. This study proposes an automated, machine learning-based solution aimed at mitigating the identified challenges. By employing a hybrid model that integrates Yolov5 for the detection of WBCs, coupled with a finely tuned, pre-trained MobileNetV2 model and a Logistic Regression classifier, the study innovates in the accurate identification, counting, and classification of WBCs into four distinct types. The methodology leverages the BCCD dataset for training and validation purposes. The application of the proposed hybrid machine learning model has yielded remarkable results, demonstrating a detection accuracy rate of 98% through the Yolov5 stage, and an unparalleled classification accuracy of 99.04% in subsequent stages utilizing MobileNetV2 and Logistic Regression. Additionally, Our proposed YOLOv5-based RBC detection module achieves an F1 score of 99.73%, which outperforms the baseline. These findings underscore the model's potential in transforming traditional laboratory practices for WBC analysis, offering a path towards more accurate, efficient, and reliable disease diagnostics and monitoring.

Recent work showed that language models represent character counts on curved 1D manifolds, with attention heads performing geometric transformations to enable computation. We test whether this generalizes across four ordinal tasks (bracket depth, indentation, table position, numeric magnitude) in Gemma-2-2B, Gemma-2-9B, and Qwen3-4B. We find that 1D manifolds with place-cell feature tiling emerge for tasks where the ordinal variable is locally computable from token identity, while tasks requiring cross-position integration or semantic extraction produce higher-dimensional or incoherent representations. Geometric computation is architecture-dependent: Qwen3-4B shows substantially stronger twisting than Gemma models for indentation, and its twisters preserve ordinal order, unlike its numeric twisters. Activation patching confirms that the identified manifold subspaces concentrate task-relevant information, with manifold-direction ablation causing dramatically larger probe accuracy drops than random-direction controls.

Satisfiability modulo theory (SMT) solvers have significantly advanced automated reasoning due to their effectiveness in solving problems across various fields. With the advancement in SMT solvers, there is growing interest in exploring capabilities beyond mere satisfiability, similar to the progression observed in Boolean satisfiability solvers that expanded into counting and sampling. In this study, we investigate the following question: Can we rely on modern CNF model counters and CNF samplers to extend modern SMT solvers to handle the problems of counting and sampling over bit-vectors? The main contribution of this work is the development of an efficient and user-friendly tool, csb, that solves a bunch of problems around model counting and sampling on the theory of bit-vectors, namely exact and approximate projected and non-projected model counting, along with the almost-uniform and uniform-like sampling. In the case of exact counting, projected counting, and uniform sampling. Our tool csb converts the bit-vector formula into a CNF formula using bit-blasting techniques before applying CNF model counters or samplers to perform counting or sampling. Our experiments demonstrate significant performance improvements over existing methods.

Zero-shot object counting (ZOC) aims to count instances of arbitrary object categories specified only through textual prompts. Recent training-free approaches leverage foundation models such as SAM to reformulate counting as a prompt-driven segmentation task, eliminating the need for costly counting-specific training data with point-level annotations. More recently, SAM3 introduced promptable concept segmentation, enabling the zero-shot segmentation of all instances corresponding to a text-defined concept. However, SAM3 struggles in densely populated scenes containing numerous small objects, where limited image resolution and insufficient attention to target-relevant regions often lead to missed instances and poor instance separation, hindering accurate object counting. To address this limitation, we propose AdaCount, a training-free framework for ZOC based on similarity-guided spatial and feature adaptation. AdaCount first estimates a prototype-driven similarity map that identifies target-relevant regions. This similarity map subsequently guides two complementary adaptations: (i) similarity-guided spatial warping, which reallocates image resolution toward target instances, and (ii) feature modulation, which amplifies target-relevant encoder representations. Together, these adaptations enable SAM3 to devote greater representational capacity to target-relevant regions while preserving global image context, without requiring any model retraining. Extensive experiments across six diverse counting benchmarks establish AdaCount as a new SOTA among training-free ZOC approaches.

Ensuring model reliability in Explainable AI requires a global assessment of the hypothesis space. We propose a formal framework for the exhaustive analysis of optimal and near-optimal decision trees, called Algebraic Decision Tree Counting (ADTC). Inspired by Algebraic Model Counting (AMC) in knowledge representation, ADTC reformulates diverse analytical tasks, such as optimization, counting, and sampling, into a unified sum-of-products computation over a semiring $R$. While the hypothesis space of decision trees is doubly exponential with respect to the maximum depth $Δ$, our dynamic programming algorithm achieves $O^*(n^{O(Δ)})$ time complexity in the number of features $n$, where $O^*$ suppresses polynomial factors. To handle complex constraints consisting of multiple tree metrics, we introduce model behavior tensors that aggregate semiring values via convolution products over a tensor semiring. This algebraic approach efficiently constructs a model profile that captures the global landscape and trade-offs between criteria such as accuracy, size, and fairness. We demonstrate the utility of our software, emtrees, on real-world datasets, illustrating how ADTC facilitates evidence-based model selection in sensitive domains.

The success of deep learning models in classification and regression is widely attributed to the low-dimensional structure that real-world data tend to exhibit, despite their high-dimensional representation. This work attempts to provide a mathematical framework for binary classification on low-dimensional data, building on Cover's (1965) function-counting theory. With our framework, we aim to address the question of how the low-dimensional structure of the data affects the classification capabilities of learning models. Cover's theory relies on a general position assumption that blinds it to the underlying data structure. We refine this assumption to account for the low-dimensionality of the data and derive dichotomy counts that reflect the data structure. We further extend Cover's separation capacity and problem of generalization to the low-dimensional setting, enabling the impact of the underlying data structure on both to be analyzed.

Private continual counting is a fundamental problem in differential privacy: given a binary stream of length $n$, where each $1$ corresponds to the contribution of one individual, the goal is to release all running counts while protecting the privacy of each individual. The standard algorithm is the binary tree mechanism, whose Gaussian-noise variant achieves expected $\ell_\infty$ error proportional to $\log^{3/2} n$ for approximate differential privacy. Whether this dependence on the stream length is necessary has remained a central open problem. In this work, we resolve the dependence on $n$ by proving that every differentially private mechanism for continual counting must incur expected $\ell_\infty$ error $Ω(\log^{3/2} n)$. This shows that the binary tree mechanism is asymptotically optimal in the approximate-DP setting. As a consequence, we also obtain a largest-possible separation between hereditary discrepancy and private $\ell_\infty$ error for linear queries, showing that the known general upper bound in terms of hereditary discrepancy has the optimal dependence on the number of queries.

In this paper, we attempt to enhance the theoretical understanding of convolutional neural networks (CNNs) as feature extractors in classification tasks by analyzing them through the lens of Cover's function-counting theory. Specifically, our focus lies on the notion of separation capacity, a combinatorial quantity derived from counting the number of realizable dichotomies (i.e., binary label assignments). Our contributions are threefold. First, we extend Cover's framework by establishing a conceptually insightful and practically useful formulation for the separation capacity. Second, leveraging this formulation, we identify the factors governing the separation capacity of feature extractors that employ a specific CNN architecture, so-called scattering networks, in terms of their network building blocks. Third, we provide practical insights for scattering network design.

As a prominent symptom of Parkinson's disease (PD), turning impairment is evaluated through parameters such as turning angle, duration, and particularly, the number of steps required to complete a turn, which directly reflects motor dysfunction. Accurate step counting is challenging due to variability in real-world turning movements and atypical shuffling patterns in parkinsonian gait. Existing methods are predominantly wearable-based, requiring users to wear and manage dedicated devices, which can be inconvenient for continuous daily use. To address this, we propose a passive, video-based framework that estimates step count in a coarse-to-fine manner using diverse motion representations. Specifically, an initial step count is estimated from foot movement signals derived from 3D human mesh recovery, providing high-level motion structures. To incorporate fine-grained motion details, a motion encoder learns complementary gait dynamics from mesh and optical flow to refine the initial estimate. In this process, coarse foot movement signals query the pixel-level motion cues via cross attention to capture subtle parkinsonian gait dynamics. To handle varying video lengths, we partition each video into clips and integrate clip-wise motion embeddings via multiple instance learning (MIL) for step count residual prediction. Extensive experiments show our method consistently outperforms existing step counting methods on real-world PD turning datasets.

Can a statistically significant, large-effect-size finding in computational social science be entirely an artifact of the measurement instrument? We present a case where the answer appears to be yes. Analyzing 85 interviews across four public intellectuals (2016--2026), we find a robust negative-affect/emphatic-certainty lexical co-occurrence pattern under keyword-based scoring ($r = 0.72$--$0.93$, $p < 0.01$ for all four speakers). Replacing keyword counting with LLM-based zero-shot semantic classification on the complete diarized corpus (32,625 sentences) dramatically reduces this correlation: Dalio's $r = 0.851$ drops to $r = 0.206$, with two speakers showing negative $r(\text{neg}, \text{emphatic})$ and one showing null. In contrast, the LLM reveals a strong negative-hedging coupling across speakers -- Rogoff's $r(\text{neg}, \text{hedged}) = 0.875$ ($p = 0.001$) and Zeihan's $r(\text{neg}, \text{hedged}) = 0.722$ ($p = 0.008$) -- consistent with the conventional expectation that pessimistic discourse attracts hedging, not certainty. Sentence-level error analysis traces this discrepancy to three structural failure modes in keyword lexicons -- syntactic blindness, polysemy blindness, and categorical absence -- illustrated through cases where keyword counting inverts semantic meaning (e.g., ''never absolutely totally confident'' scored as high-certainty). We argue that keyword lexicons measure a universal lexical co-occurrence tendency -- negative discourse naturally attracts emphatic vocabulary -- that is orthogonal to, and can systematically invert, rhetorical stance. Treating keyword counts as measurements of epistemic certainty is a category error: a finding that appears to be about a speaker's psychology may be entirely about the counting of words.

We present CombEval, a dynamic benchmark for evaluating combinatorial counting in large language models. CombEval represents each problem as a typed Cofola specification over entities, combinatorial objects, object dependencies, and constraints, enabling controlled generation of natural-language counting problems with exact solver-verified answers. Unlike static collections, CombEval supports systematic variation of object type, entity scale, constraint count, and reasoning depth. We evaluate 11 LLMs under direct and code-augmented settings and find that models remain brittle on ordered objects, indistinguishable elements, relatively positional constraints, and nested object dependencies. Error analysis further identifies failures in constraint interpretation and counting principles. CombEval provides a diagnostic testbed for studying when and why LLMs fail at combinatorial reasoning. The code and generated benchmark suites are publicly available at \url{https://github.com/YuxuZhou-CN/combination-problem-generation}.

Text-guided Open-vocabulary Object Counting (TOOC) aims to estimate the number of objects described by text prompts, which is particularly challenging in dense scenes with large scale variations. Existing TOOC approaches predominantly rely on Transformers, whose quadratic complexity with respect to image resolution limits their scalability. Mamba offers a promising alternative due to its linear complexity. However, previous Mamba-based methods have two main limitations. On the one hand, the inherent causal formulation of Mamba constrains the bidirectional spatial dependency modeling required by non-causal vision tasks. On the other hand, existing Mamba-based vision models often overlook the unconstrained high entropy in the spatial token responses, which can weaken local details and high-frequency cues. To address these limitations, we propose MambaCount, an efficient framework built on the Spatial Sparse State Space Duality (S^4D) block. Specifically, we analyze and reconstruct the decay dynamics of hidden states in Mamba to alleviate the dependency constraints introduced by causal modeling. Moreover, we introduce a Spatial Token Selection (STS) sub-block to reduce the unconstrained high entropy in spatial token responses within Mamba. In addition, we design Multi-Granularity Prototypes (MGP) to identify object-like regions at different semantic levels, improving cross-modal alignment and interpretability. Extensive experiments on FSC-147 demonstrate that MambaCount achieves state-of-the-art performance among methods without secondary querying, obtaining a test MAE of 12.23, while retaining linear complexity.

Text-guided Open-vocabulary Object Counting (TOOC) enables counting arbitrary object categories specified by text prompts, offering substantially greater flexibility than conventional closed-set counting. However, existing TOOC methods are developed and evaluated primarily on ideal images, while real-world scenes often suffer from adverse conditions such as rain, fog, darkness, and sensor noise, which severely degrade visual quality and impair vision-language alignment. To bridge this gap, we introduce Robust-TOOC, the first benchmark for evaluating TOOC under diverse corruption conditions, which covers six representative degradation types: rain, fog, darkness, Gaussian noise, salt-and-pepper noise, and mixed corruption. To improve robustness while preserving the original counting architecture, we propose Dual-TTT, a dual-architecture test-time training framework for TOOC. Specifically, during test-time training, Dual-TTT updates only the Text-guided Lightweight Denoising module (TL-Denoiser), while keeping the original counting network frozen. Inspired by diffusion models, the TL-Denoiser is optimized to remove corruption-aware noise from image representations under degraded conditions. Since only the TL-Denoiser is trained at test time, Dual-TTT is annotation-free and can be seamlessly integrated into existing TOOC models without modifying their original architecture. Extensive experiments on multiple recent TOOC baselines demonstrate the effectiveness of our method.

Text-guided open-vocabulary object counting (TOOC) aims to count objects belonging to the categories specified by natural language descriptions. Although vision-language pre-trained models have been successful applied to TOOC tasks, they still struggle with fine-grained spatial understanding and real-time inference requirements in counting scenarios. To address these limitations, this paper proposes a real-time TOOC framework, called the Real-Time Counter (RT-Counter), that achieves not only good counting accuracy but also high computational efficiency. RT-Counter designs a novel Visual Prototype Textualization (VPT) module that can project learned visual features into a text feature space and then generate features containing the abstract information that is hard to capture with visual prototypes and the detailed prototype information that is difficult to describe in text, enhancing the object-level visual-language model's counting capabilities. Additionally, RT-Counter incorporates our Weaving Transformer (Weaformer) layers, maintaining high descriptive power at a fraction of the computational cost. The Weaformer layer adopts a novel hybrid attention mechanism that can efficiently weave together local and global visual features. Extensive experiments on three public datasets show that RT-Counter successfully breaks the accuracy-speed trade-off in TOOC. While achieving a competitive MAE of 13.30 on FSC147, RT-Counter operates at 112.48 FPS, making it 7.4x faster and over 4$\times$ more parameter-efficient than the existing leading methods in TOOC. Our work aims at balancing high accuracy and real-time performance in TOOC. Code is available at: https://github.com/Jason-Mar1/RT-Counter.

Aggregate accuracy benchmarks conceal a systematic structure in how large language models fail at electronic health record (EHR) question answering: questions requiring more inferential steps produce disproportionately more errors. Motivated by theoretical results on transformer compositionality limits, we introduce a pre-specified hop-count taxonomy -- the number of distinct reasoning steps required to answer a clinical question from an EHR -- as a principled predictor of model failure. We annotate 313 clinician-generated MedAlign EHR question-answer pairs across four hop levels and evaluate 301 questions in a within-model ablation (claude-sonnet-4-6, zero-shot vs. extended thinking) and cross-architecture replications (gpt-4o and gpt-5.4-2026-03-05, zero-shot). All three models, spanning two providers and two OpenAI generations (GPT-4 and GPT-5), show monotone accuracy decline with hop count: Claude Sonnet zero-shot falls from 30.6% (hop=1) to 17.6% (hop=4) (Cochran-Armitage z=-2.30, p=0.011; OR per hop 0.72, 95% CI [0.56,0.92], p=0.008); GPT-4o replicates this (37.8% to 14.7%; OR 0.58 [0.45,0.75], p<0.001); and gpt-5.4-2026-03-05 confirms it (37.8% to 23.5%; OR 0.80 [0.66,0.98], p=0.027). A pre-specified context-sufficiency audit shows higher-hop questions are not differentially disadvantaged by EHR truncation (answerability 93-95% at hops 2-4 vs. 79% at hop=1), so the decline reflects compositional reasoning difficulty. Extended thinking did not significantly flatten the accuracy-depth curve across three reasoning conditions, and thinking-token usage scaled with hop count (r=0.31, p<0.0001), consistent with the predicted O(k) computational requirement. Hop count is thus a theory-motivated, cross-architecture predictor of large-language-model error on EHR question answering, with direct implications for deployment risk stratification of clinical AI.

Ubiquitous perception through RF signals is a pivotal opportunity for future technology: it enables personalized services such as smart living, remote healthcare, automated logistics or interaction through free-space gestures. The ubiquity of Wi-Fi and cellular networks presents a promising platform for the development of innovative sensing tools. Future standards will also introduce dedicated sensing features which, for example, will allow routers to work as frequency modulated continuous wave radios targeting radar applications. Most of the current chip designs support ad-hoc firmware for CSI extraction with MIMO arrangements of the transmitter (TX) and receiver (RX) antennas and OFDM subcarriers. The CSI describes the phase shift and amplitude attenuation of multiple propagation paths on each subcarrier. The latest IEEE 802.11be standard (Wi-Fi 7) offers a wider subcarrier bandwidth of 160MHz (up to 320MHz), providing at least 120 usable pilot subcarriers for CSI or CIR estimation. Additionally, Wi-Fi signals have been recently exploited to track daily human movements and behaviors, while Wi-Fi signal variations have been shown to differ between different people and can consequently be used for their re-identification.

Object counting remains fragmented across domain-specific datasets and task formulations, despite rapid progress in generalist vision models. Existing counting models are often tailored to scenarios such as crowds, vehicles, cells, crops, or remote-sensing objects, and thus struggle to generalize across categories, visual domains, object scales, and density distributions. In this paper, we study text-guided object counting across domains, where a model takes an image and a natural-language query as input and returns an instance-grounded set of target points whose cardinality gives the count. This formulation unifies category-conditioned counting with interpretable spatial localization. To support this setting, we construct CLOC, a Cross-domain Large-scale Object Counting dataset that reorganizes diverse public data sources into a unified benchmark. CLOC covers six visual domains: General Scene, Remote Sensing, Histopathology, Cellular Microscopy, Agriculture, and Microbiology, with about 220K images, 619 categories, and 15M object instances. Based on CLOC, we propose Count Anything, a generalist model for text-guided object counting. Unlike density-map-based methods, which dominate counting models, Count Anything adopts discrete instance points and performs dual-granularity instance enumeration. A Region-level Sparse Counter provides object-level anchors for large and sparse targets, while a Pixel-level Dense Counter handles small, crowded, and weakly bounded targets via dense point prediction. A point-centric supervision strategy enables learning from heterogeneous annotations, and Complementary Count Fusion combines both counters in a parameter-free manner. Extensive experiments show that Count Anything achieves strong accuracy and multi-domain generalization, outperforming existing open-world counting methods. Code is available at: https://github.com/Mengqi-Lei/count-anything.

The parameter counts of the most widely used large language models (LLMs) are often withheld by their developers, leaving model size -- a primary reference point for interpreting capabilities and costs -- largely undisclosed. We propose a black-box method to infer conservative lower bounds on LLM size from generated text outputs alone, requiring nothing beyond the ability to submit text fragments and observe next-token predictions. Our approach is grounded in a key observation: popular, widely-circulated texts -- such as classical literature, religious texts, and foundational documents -- are present in virtually every large-scale pretraining corpus, and how accurately a model predicts the next word across text fragments of varying length is a reliable signal of how much it has memorized them, which in turn is fundamentally limited by its total parameter count. We aggregate this memorization signal across a diverse corpus of texts and fragment lengths into a single accuracy profile vector per model, and build two complementary inference methods on top of it: a pairwise statistical test that determines which of two models is larger, and a scaling-law estimator that extracts a one-dimensional latent index from these vectors via Principal Component Analysis (PCA) to map the aggregated signal to a parameter count. Validated on a broad set of open-weight models, both methods produce accurate and reliable lower bounds. When applied to popular closed-weight models, our framework recovers internal product hierarchies and reveals a clear divergence in industry scaling strategies: while some developers yield significantly higher bounds indicative of large generational parameter growth, others operate under strict parameter ceilings, demonstrating that hidden design choices can be systematically probed even under strict API limitations.

Multilevel Image thresholding is an important preprocessing algorithm in computer vision applications nowadays. Since most common thresholding methods take the desired count of thresholds as input by the user, thresholding methods that automatically determines a suitable count of thresholds from the input image itself are advantageous. In this article, a novel thresholding method based on a dynamic programming algorithm and a modification of Minimum Error Thresholding (MET) criterion is thoroughly presented. An empirical statistical study is performed to pinpoint why this proposed method is superior. Moreover, an extended comparison between this proposed method and other state-of-the-art methods is performed on a comprehensive set of natural, satellite and medical test images. The numerical results show that the proposed MET-DP method takes much less time than traditional dynamic programming thresholding methods when the number of thresholds is high. The proposed method can detect a suitable count of thresholds for most of tested images of different types. However, traditional methods that take the count of thresholds as input produce thresholded images of higher structural similarity index measure (SSIM) and peak signal-to-noise ratio (PSNR) values than MET-DP. Source code can be found on https://w3id.org/met-dp/article1-code

Combinatorial counting problems pervade artificial intelligence, statistics, and discrete mathematics. Whether the task is enumerating subsets, multisets, permutations, partitions, or compositions under structural and arithmetic constraints, solving it remains a stubbornly manual exercise. Closed-form derivations are powerful but brittle, while naive encodings to propositional model counting or constraint satisfaction destroy the exchangeability that makes counting tractable in the first place. We present Cofola (COmbinatorial counting LAnguage with First-Order logic), a typed declarative language whose primitives are the combinatorial objects that recur in everyday counting questions, including sets, bags, tuples, sequences, circles, partitions, and compositions, together with natural relational and arithmetic constraints over them. A denotational semantics maps every Cofola program to a well-defined combinatorial counting problem, and a three-phase compilation pipeline (preprocessing, decomposition, and symmetry-preserving encoding) reduces this problem to a weighted first-order model counting (WFOMC) instance augmented with coefficient-extraction constraints. To stay inside known domain-liftable fragments whenever possible, the encoding groups indistinguishable entities, breaks the symmetry of unordered groupings lexicographically, and encodes sequences and circles via order axioms. On a suite of representative combinatorial counting problems, ranging from textbook math problems to multi-object scenarios that the closest prior framework cannot express, Cofola produces concise specifications and a uniform solving pipeline that is practical end-to-end.

This study tackles the challenge of precise wood log counting, where applications of the proposed methodology can span from automated approaches for materials management, surveillance, and safety science to wood traffic monitoring, wood volume estimation, and others. We introduce an approach leveraging Conditional Generative Adversarial Networks (cGANs) for eucalyptus log segmentation in images, incorporating specialized image processing techniques to handle noise and intersections, coupled with the Connected Components Algorithm for efficient counting. To support this research, we created and made publicly available a comprehensive database of 466 images containing approximately 13,048 eucalyptus logs, which served for both training and validation purposes. Our method demonstrated robust performance, achieving an average Accuracy_pixel of 96.4% and Accuracy_logs of 92.3%, with additional measures such as F1 scores ranging from 0.879 to 0.933 and IoU values between 0.784 and 0.875, further validating its effectiveness. The implementation proves to be efficient with an average processing time of 0.713s per image on an NVIDIA T4 GPU, making it suitable for realtime applications. The practical implications of this method are significant for operational forestry, enabling more accurate inventory management, reducing human errors in manual counting, and optimizing resource allocation. Furthermore, the segmentation capabilities of the model provide a foundation for advanced applications such as eucalyptus stack volume estimation, contributing to a more comprehensive and refined analysis of forestry operations. The methodology's success in handling complex scenarios, including intersecting logs and varying environmental conditions, positions it as a valuable tool for practical applications across related industrial sectors.

As artificial intelligence (AI) systems become more common in our daily lives, it is important to understand how different stakeholders comprehend and envisage the role that these technologies play in shaping social, political, and economic realities. In this paper, we investigate public perceptions of AI based on a corpus of letters submitted during the public consultation for the Trump Administration's US AI Action Plan. To this aim, we release a corpus cleaning pipeline and perform topic modelling and frequency analysis to explore predominant topics discussed by different subgroups (e.g., academia, individuals, private sector) and those appearing in the AI Action Plan. Our results show that individuals voice strong concerns related to the impact of AI on life, while other stakeholders are more concerned with AI development. Our comparison of topics suggests that the AI Action Plan reflects predominantly the concerns of the private sector on security, policies, and development, with individuals' concerns less represented.

Object counting is a foundational vision task with over a decade of dedicated research, yet state-of-the-art models still fail systematically in the mixed-object setting that dominates real-world applications such as industrial inspection and product sorting. We show that this gap is strongly driven by limitations in existing training and evaluation data: real counting datasets are prohibitively expensive to annotate and suffer from labeling noise, while existing synthetic alternatives lack diversity and realism. We address this with MixCount, a dataset and benchmark for mixed-object counting designed to target the failure modes of current counting models. To overcome the high cost of constructing and labeling such data, we develop an automatic generation pipeline that synthesizes images, fine-grained textual descriptions, and pixel-perfect counting annotations at scale, eliminating the labeling ambiguity that plagues prior datasets. Evaluating state-of-the-art counting models on MixCount exposes severe degradation in the mixed-object setting. More importantly, training these models on our synthesized data yields substantial gains on real-world benchmarks, reducing MAE by 20.14% on FSC-147 and by 18.3% on PairTally. These results establish MixCount as both a benchmark and a training dataset for fine-grained counting, and demonstrate that our pipeline, which produces effectively unlimited labeled data, helps address a long-standing bottleneck in counting models.

Counting objects in an image is a task applicable across many domains. For instance, crowd counting, inventory counting, and cell counting have been the focus of recent research. The major challenges in estimating the count of objects include overlapping objects, object scale issues, occlusions, and varying lighting conditions. In this report, we explore the problem of counting machine washer parts. Our technique is an extension of FamNet with an additional loss component, trained on the given dataset. We compare to three baseline methods: a traditional image processing pipeline, instance segmentation, and density map estimation. We evaluate the performance of these algorithms by computing the Mean Absolute Error (MAE) and the Root Mean Squared Error (RMSE) between the true object counts and the model outputs. Our approach achieves a performance of 1.96 MAE.

The expressive limitations of message-passing Graph Neural Networks (GNNs) have motivated a wide range of more powerful graph learning architectures. We advocate Deep Homomorphism Networks (DHNs) as a model particularly well-suited for learning over relational databases, due to their close connection to important fragments of SQL such as conjunctive queries. We study the precise expressive power of DHNs by relating them to various natural fragments and extensions of first-order logic (FO). For DHNs with max, sum, and mean aggregations, we establish connections to the unary negation fragment (UNFO) and to the extensions of UNFO with counting quantifiers and with ratio quantifiers. We further relate sum-aggregation DHNs to the unary quantifier alternation fragment of FO and to an extension of FO with expressive counting. Through the classical correspondence between FO and SQL, these results also illuminate the relation between DHNs and SQL. They also enable us to study the decidability of two fundamental static analysis problems for DHNs, the emptiness problem and the subsumption problem. Finally, we confirm through experiments that the established differences in expressive power are reflected in the performance on suitable prediction tasks.

Vision-Language Models (VLMs) excel at multimodal reasoning, yet it remains unclear whether their answers are grounded in visual evidence or driven by learned language and world priors. Counting provides a precise testbed: when visual evidence conflicts with canonical object knowledge, a model must rely on the image rather than a prototypical count. We introduce CounterCount, a diagnostic framework for counterfactual counting in VLMs, consisting of paired factual and counterfactual images with edited count-relevant attributes, verified answers, and localized evidence annotations. Evaluating recent VLMs, we find strong performance on factual images but consistent degradation under counterfactual attribute changes, indicating reliance on object-level priors even when contradictory visual evidence is present. Using localized annotations, we show that these failures are not solely due to missing or ambiguous visual evidence, but to models underweighting attention to count-relevant visual tokens. We introduce a unified inference-time attention modulation strategy that reweights selected visual tokens, improving counterfactual counting accuracy by up to 8% across multiple VLMs. Overall, CounterCount exposes prior-driven counting failures and provides diagnostic insights for designing future VLMs.

Plaque assays remain the gold standard readout of virus infectivity; however, plaque counting from plate images is labor-intensive and prone to inter-operator variability. We present an end-to-end, computer-aided workflow for cytopathic effect-based virus titration directly from laboratory plaque assay images. The proposed approach combines two models derived from the Segment Anything Model (SAM): a SAM2-based well-segmentation module that localizes assay wells across heterogeneous imaging conditions, and a SAM-based plaque-segmentation model that detects and enumerates plaques within each well. The method was evaluated on a mixed dataset comprising private plaque assay images of Mayaro virus and Coxsackievirus B3, together with public Vaccinia virus images from the VACVPlaque dataset. The pipeline outputs per-well plaque counts, automatically computes plaque-forming units per milliliter (PFU/mL), and is integrated into a web-based platform that allows users to review results and organize experiments. On held-out plates (17 from MAYV/CVB3 and 22 from VACV), the workflow generalized across two plate formats (6-well and 12-well) and showed strong agreement with manual annotations (Pearson correlation coefficients of 0.92 for MAYV/CVB3 and 0.88 for VACV). Automated plaque counts were further compared with annotations from four independent experts, demonstrating high concordance. The proposed system will be open sourced and publicly released upon acceptance of this manuscript to enable reproducible, scalable, and audit-ready plaque assay analysis while substantially reducing manual annotation effort.

Referring expression counting (REC) is an intention-driven task that requires context-aware visual reasoning. While recent vision-language models incorporate language for visual understanding, most existing REC methods rely on rulebased reinforcement learning with rewards focused primarily on final accuracy, overlooking the quality of intermediate reasoning. We propose REC-RL, a reinforcement learning framework that introduces a think-range-answer paradigm to explicitly optimize the visual reasoning process. RECRL employs Group Relative Policy Optimization and two lightweight rewards: an accuracy reward that combines range-based interval supervision with Gaussian-based precision guidance, and a format reward that enforces structured outputs. By modeling intermediate focus prediction as internal decision-making, REC-RL avoids additional annotations and better aligns with human perception. Extensive experiments demonstrate consistent improvements over strong baselines and robust generalization across benchmarks.

Estimating the number of distinct elements in a data stream is well understood when repeated elements are identical. In modern settings, however, observations are high-dimensional and noisy, so repeated instances of the same object are only approximately similar -- for example, different images of the same individual may vary significantly at the pixel level. Classical sketches such as HyperLogLog rely on consistent hash values for identical elements and break down in this regime. Recent work on robust distinct counting in general metric spaces achieves $\widetildeΘ(\sqrt{n})$ memory, which is tight in the worst case. We show that substantially improved memory guarantees are possible under geometric structure common in learned representations. We introduce MaxSketch, a simple max-linear sketch built from random Gaussian projections, and prove that it succeeds in estimating the number of distinct latent objects. Concretely, we show that under this assumption $m = \widetilde{O} (\log n / \varepsilon^2)$ random projections (and hence $\widetilde{O} (\log n/\varepsilon^2)$ memory) suffice to recover the true distinct count within a $(1+\varepsilon)$ factor. Experiments on image streams confirm that MaxSketch accurately estimates distinct counts and generalizes beyond the training regime. Our results bridge classical streaming algorithms and modern representation learning, showing how geometric structure can fundamentally reduce the complexity of distinct counting.

Automated analysis of building facades from street-level imagery has great potential for urban analytics, energy assessment, and emergency planning. However, it requires reasoning over spatially arranged elements rather than solely isolated detections. In this work, we model each facade as a graph over window/door detections with a vertical prior on edges. Additionally, we introduce GATA2Floor, a multi-head Graph Attention v2 (GATv2) based model that predicts the global floor count of a building and, via learnable cross-attention queries, softly assigns elements to latent floor slots, yielding interpretable outputs and robustness to irregular designs. To mitigate the lack of labeled datasets, we demonstrate that the proposed graph-based reasoning can be applied without annotations by leveraging a lightweight label-free proposal mechanism based on self-supervised features and vision-language scoring. Our approach demonstrates the value of graph-attention-based relational reasoning for facade understanding.

Open-world object counting remains brittle: despite rapid advances in vision-language models (VLMs), reliably counting the objects a user intends is far from solved. We argue that a central reason is that counting granularity is left implicit; users may refer to a specific identity, an attribute, an instance type, a category, or an abstract concept, yet most methods treat "what to count" as a single, category-level matching problem. In this work, we redefine open-world counting as multi-grained counting, where visual exemplars specify target appearance and fine-grained text, with optional negative prompts, specifies the intended semantic granularity across five explicit levels. Making granularity explicit, however, exposes a critical data bottleneck: existing counting datasets lack the multi-category scenes, controlled distractors, and instance-level annotations needed to verify fine-grained prompt semantics. To address this, we propose the first fully automatic data-scaling pipeline that integrates controllable 3D synthesis with consistent image editing and VLM-based filtering, and use it to construct KubriCount, the largest and most comprehensively annotated counting dataset to date, supporting both training and multi-grained evaluation. Systematic benchmarking reveals that both multimodal large language models and specialist counting models exhibit severe prompt-following failures under fine-grained distinctions. Motivated by these findings, we train HieraCount, a multi-grained counting model that jointly leverages text and visual exemplars as complementary target specifications. HieraCount substantially improves multi-grained counting accuracy and generalizes robustly to challenging real-world scenarios. The project page is available here: https://verg-avesta.github.io/KubriCount/.

The counting power of Message Passing Neural Networks (MPNN) has been the subject of many recent papers, showing that they can express logic that involves counting up to a threshold or more generally satisfy a linear arithmetic constraint. In this paper, we study the counting capabilities of MPNN beyond linear arithmetic, primarily utilising local and global mean aggregations. In particular, our goal is to tease out conditions required to express extensions of graded modal logic with polynomial counting constraints. We show that global polynomial counting constraints in node-labelled graphs can be checked using mean MPNN under mild assumptions. Checking local constraints is also possible, if we consider formulas with no nested modalities and additionally either (i) permit sum/max aggregations, or (ii) only restrict to regular graphs. We also show how formulas with nested modalities can be captured by mean MPNN over graphs with tree-like structures and similar assumptions.

Large language models fail at counting repeated tokens despite strong performance on broader reasoning benchmarks. These failures are commonly attributed to limitations in internal count tracking. We show this attribution is wrong. Linear probes on the residual stream decode the correct count with near-perfect accuracy at every post-embedding layer, across all model depths. This holds even at the exact layers where the wrong answer crystallizes while the model simultaneously outputs an incorrect count. Attention patterns show no evidence of collapse over repeated tokens and tokenization artifacts account for none of the failure. Instead, a format-triggered multi-layer perceptron (MLP) block overwrites the correctly-encoded count with a fixed wrong answer at roughly 88--93,% network depth. This prior fires for repeated word-tokens in space-separated list format and is absent for repeated digit-tokens. It is suppressed by comma-separated delimiters in larger models but persists in smaller ones. The finding holds across Llama-3.2 (1B and 3B) and Qwen2.5 (1.5B, 3B and 7B) at consistent relative depth. Counting failure is a failure of routing not of representation and the two require different interventions.

High-dimensional count data arise in applications such as single-cell RNA sequencing and neural spike trains, where mappings between distributions across successive batches or time points form critical components of data analysis. The recent success of diffusion- and flow-based deep generative models for images, video, and text motivates extending these ideas to count-valued settings, but many existing methods either treat each count as a categorical state or transform counts into a continuous space, neither of which is natural or efficient when the count range is large. We propose count-FM, a flow-matching framework for count data based on a continuous-time birth-death process with local unit jumps. Count-FM learns marginal transitions efficiently in count space through simulation-free training of conditional transition rates, allowing transport between arbitrary count-distributed source and target populations. In simulation, count-FM variants achieve strong sample quality while using substantially fewer parameters. We further apply count-FM to scRNA-seq and neural spike-train data for unconditional generation, transport, and conditional generation. Across these tasks, count-FM yields improved sample quality, greater modeling efficiency, and interpretable transport paths.

Tool-integrated Text-to-SQL parsing has emerged as a promising paradigm, framing SQL generation as a sequential decision-making process interleaved with tool execution. However, existing reinforcement learning approaches mainly rely on coarse-grained outcome supervision, resulting in a fundamental credit assignment problem: models receive the same reward for any trajectory that yields the correct answer, even when intermediate steps are redundant, inefficient, or erroneous. Consequently, models are encouraged to explore suboptimal reasoning spaces, limiting both efficiency and generalization. To address this problem, we propose FineStep, a novel framework for step-level credit assignment in tool-augmented Text-to-SQL. First, we introduce a reward design with independent process rewards to alleviate the signal sparsity of outcome supervision. Next, we present a step-level credit assignment mechanism to precisely quantify the value of each reasoning step. Finally, we develop a policy optimization method based on step-level advantages for efficient updates. Extensive experiments on BIRD benchmarks show that FineStep achieves state-of-the-art performance and reduces redundant tool interactions, with a 3.25% average EX gain over GRPO at the 4B scale.

Weighted first-order model counting (WFOMC) is a central task in lifted probabilistic inference: It asks for the weighted sum of all models of a first-order sentence over a finite domain. A long line of work has identified domain-liftable fragments of first-order logic, that is, syntactic classes for which WFOMC can be solved in time polynomial in the domain size. Among them, the two-variable fragment with counting quantifiers, $\mathbf{C}^2$, is one of the most expressive known liftable fragments. Existing algorithms for $\mathbf{C}^2$, however, establish tractability through multi-stage reductions that eliminate counting quantifiers via cardinality constraints, which introduces substantial practical overhead as the domain size grows. In this paper, we introduce IncrementalWFOMC3, a lifted algorithm for WFOMC on $\mathbf{C}^2$ and its modulo counting extension, $\mathbf{C}^2_{\text{mod}}$. Instead of relying on reduction techniques, IncrementalWFOMC3 operates directly on a Scott normal form that retains counting quantifiers throughout inference. This direct treatment yields two main results. First, we derive a tighter data-complexity bound for WFOMC in $\mathbf{C}^2$, reducing the degree of the polynomial from quadratic to linear in the counting parameters. Second, we prove that $\mathbf{C}^2_{\text{mod}}$ is domain-liftable, extending tractability from $\mathbf{C}^2$ to a richer fragment with native modulo counting support. Finally, our empirical evaluation shows that IncrementalWFOMC3 delivers orders-of-magnitude runtime improvements and better scalability than both existing WFOMC algorithms and state-of-the-art propositional model counters.

Large language models often fail at simple counting tasks, even when items to count are in the prompt. We investigate whether this failure occurs because transformers do not represent counts internally, or because they cannot convert representations to the correct output tokens. Across three model families: Pythia, Qwen3, and Mistral, ranging from 0.4B to 14B parameters, we find evidence for the second explanation. Linear probes recover the correct count from intermediate layers with $R^2>0.99$, showing that the information is present. However, the internal directions that encode counts are nearly orthogonal to digit-token output-head rows ($|\cos| \leq 0.032$). In other words, the model stores the count in a form that the digit logits do not naturally read out. We localize this failure with two interventions. Updating only the digit rows of the output head (36,864 parameters) substantially improves constrained digit prediction (60.7--100.0% on four tasks), but it does not fix unconstrained generation (0%); we do not claim that digit-row repair fixes open-ended text. By contrast, small LoRA on attention Q/V (7.67M parameters) improves upstream routing and achieves 83.1%$\pm$7.2% in true greedy autoregressive generation (deployable fix). Logit-lens at layer 35 (entity counting; correct-digit rank): (i) median over 3 seeds drops from order-$10^4$ to 1; (ii) seed 42 shows $54{,}332 \to 838$ (median top-1 while one seed stays far below). Norm, logit-lens, and cross-task analyses generalize the bottleneck to counting, addition, and list length; nulls on MMLU and GSM8K and limited DROP transfer. These results identify counting failure as a geometric readout bottleneck, not an internal-representation failure: the model knows the count but the output pathway is misaligned with tokens needed to express it.