Calegari, Marques, and Neves count minimal surfaces in hyperbolic manifolds.
problem Counting minimal surfaces in hyperbolic manifolds.
method Using a laminar measure concept.
result An idea of a proof for counting minimal surfaces.
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48 results for “counting measures”
Study shows central limit theorem for counting measures in non-smooth spaces.
problem Counting measures in non-smooth spaces with coarse negative curvature.
method Established central limit theorems for actions of groups on hyperbolic spaces without properness or smoothness assumptions.
result General framework allows for applications in geometrically finite manifolds and intersection numbers.
Counts arcs in surfaces, proving convergence of geodesic currents.
problem Counting arcs of the same type in compact surfaces and related geometries.
method Derives convergence of geodesic currents to prove arc counts.
result Proves convergence of geodesic currents, leading to arc counting results.
We construct convergent and divergent lattices in negative curvature and give a precise asymptotic description of the behavior of their counting function.
p-SNE embeds Poisson count data into low dimensions preserving structure.
problem Embedding high-dimensional sparse Poisson data into a low-dimensional space.
method p-SNE (Poisson Stochastic Neighbor Embedding) using KL divergence and Hellinger distance.
result p-SNE recovers meaningful structure in real-world count datasets.
Unified framework for comparing clusterings from information-theoretic and pair-counting perspectives.
problem Divergent evaluations of unsupervised models due to different clustering similarity measures.
method Developed an analytical framework that unifies pair-counting and information-theoretic clustering similarity measures.
result Unified framework clarifies when and why the two regimes diverge and provides a principled basis for selecting and interpreting clustering similarity measures.
New method counts boundary pieces in ReLU classifiers for better complexity measure.
problem Current classification complexity measures are misleading and ineffective.
method Developed a novel method using tropical geometry to count exact boundary pieces.
result Boundary piece count is negatively correlated with robustness.
Ergodic Measures in Stratamath.DS
Study counts ergodic measures in surface lamination strata.
problem Counting ergodic measures in surface lamination strata.
method Determined through analysis of geodesic laminations.
result Number of ergodic measures identified in each stratum.
Counting, mixing and equidistribution for GPS systems with applications to relatively Anosov groupsmath.DS
Study equidistribution for flows on geometrically finite convergence group actions.
problem Counting, mixing and equidistribution for flows on geometrically finite convergence group actions.
method Establishing results for finite BMS measures on flow spaces associated to geometrically finite convergence group actions.
result Results apply to flow spaces associated to relatively Anosov groups.
Adjusted for chance measures are widely used to compare partitions/clusterings of the same data set. In particular, the Adjusted Rand Index (ARI) based on pair-counting, and the Adjusted Mutual Information (AMI) based on Shannon information theory are very popular in the clustering community. Nonetheless it is an open …
Counting hyperbolic multi-geodesics with individual component lengths.
problem Counting hyperbolic multi-geodesics with specific component lengths.
method Unified geometric and topological techniques, combining Mirzakhani's results and Margulis's ideas.
result Asymptotic polynomial counts of multi-geodesics in mapping class group orbits, generalizing Wolpert's conjecture.
Counting is a fundamental task in biomedical imaging and count is an important biomarker in a number of conditions. Estimating the uncertainty in the measurement is thus vital to making definite, informed conclusions. In this paper, we first compare a range of existing methods to perform counting in medical imaging and…
We consider an agent's uncertainty about its environment and the problem of generalizing this uncertainty across observations. Specifically, we focus on the problem of exploration in non-tabular reinforcement learning. Drawing inspiration from the intrinsic motivation literature, we use density models to measure uncert…
Counting lattice points in moduli space of Klein surfaces.
problem Count lattice points in moduli space of Klein surfaces.
method Introduced metric Möbius graphs, counted lattice points weighted by non-orientability measure, deduced recursion for volumes.
result Proved refined version of Norbury's recursion and computed refined Euler characteristic.
A new complexity measure for neural networks improves upon classical methods.
problem Lack of a refined complexity measure for comparing different neural network architectures, especially permutation-invariant ones.
method Introduced an equivalence relation among linear functions and counted them relative to this relation.
result The new complexity measure clearly distinguishes between different models and increases exponentially with depth.
Let M be a complete Riemannian manifold with negative curvature, and let C_-, C_+ be two properly immersed closed convex subsets of M. We survey the asymptotic behaviour of the number of common perpendiculars of length at most s from C_- to C_+, giving error terms and counting with weights, starting from the work of Hu…
The paper extends geometric results from negatively-curved spaces to strictly convex Hilbert geometry.
problem Extending geometric results from negatively-curved spaces to strictly convex Hilbert geometry.
method Demonstrates dynamical and counting results for geometrically-finite strictly convex projective structures with Hilbert metric.
result Hilbert geodesic flow is strongly mixing and orbits and primitive closed geodesics equidistribute.
Study counts and equidistributes geodesic orbits on curved spaces.
problem Counting and equidistribution of strongly reversible closed geodesics in negatively curved spaces.
method Generalized techniques from Sarnak and Erlandsson-Souto, thermodynamic formalism, and graphs of groups with 2-torsion.
result Asymptotic counting and equidistribution of geodesic orbits towards the Bowen-Margulis measure.
In this article, associated with each lattice the concept of a harmonic-counting measure on a sphere is introduced and it is applied to determine the asymptotic behavior of the eigenfunctions of the Laplace-Beltrami operator on a lens space. In fact, the asymptotic behavior of …
New insights into neural network complexity reveal better generalization performance.
problem Mysterious generalization in deep models despite high parameter counts.
method Effective dimensionality as a measure of parameter space complexity.
result Double descent behavior in generalization as a function of parameters explained.
Novel Bayesian method for high-dimensional count data prediction.
problem Count data in high-dimensional settings requires feature selection.
method Pseudo-Bayesian framework with scaled Student prior and exponential weights.
result Strong performance compared to Lasso in various settings.
Counting spheres in hyperbolic space with effective methods.
problem Counting spheres in Apollonian and Kleinian packings.
method Spectral methods and orbit counting, extending Kontorovich and Lax-Phillips techniques.
result Best-known effective error rate for sphere packing counting problems.
New framework TVR assesses true DeFi value, revealing substantial double counting.
problem TVL is easily manipulated and inflated in DeFi, leading to unreliable metrics.
method Proposed a new framework TVR to assess true underlying value of DeFi.
result TVR reveals substantial double counting in DeFi, with a gap of $139.87 billion at peak.
Accelerometer measurements are the prime type of sensor information most think of when seeking to measure physical activity. On the market, there are many fitness measuring devices which aim to track calories burned and steps counted through the use of accelerometers. These measurements, though good enough for the aver…
Image reconstruction in low-count PET is particularly challenging because gammas from natural radioactivity in Lu-based crystals cause high random fractions that lower the measurement signal-to-noise-ratio (SNR). In model-based image reconstruction (MBIR), using more iterations of an unregularized method may increase t…
Detecting unusual trading patterns on cryptocurrency exchanges by means of complexity measuresq-fin.TR
Study detects unusual trading patterns on crypto exchanges using complexity measures.
problem Detecting artificial trading activity on cryptocurrency exchanges.
method Complexity and statistical-structure measures derived from high-frequency trade-level data.
result Unusual trading patterns detected on Bitget for BTC and ETH after mid-May 2025.
Anosov groups study matrix coefficients and orbit counting in symmetric spaces.
problem Anosov groups and their matrix coefficients in symmetric spaces.
method Asymptotic analysis of matrix coefficients and higher rank measures.
result Asymptotic behavior of matrix coefficients and orbit counting results.
Classifies invariant measures on specific character varieties.
problem Classifying invariant probability measures on character varieties.
method Measure disintegration along transverse Lagrangian tori fibrations.
result Ergodic measures are either counting measures on finite orbits or Liouville measures.
The seemingly disjoint problems of count and mixture modeling are united under the negative binomial (NB) process. A gamma process is employed to model the rate measure of a Poisson process, whose normalization provides a random probability measure for mixture modeling and whose marginalization leads to an NB process f…
Clustering with variable selection is a challenging yet critical task for modern small-n-large-p data. Existing methods based on sparse Gaussian mixture models or sparse K-means provide solutions to continuous data. With the prevalence of RNA-seq technology and lack of count data modeling for clustering, the current pr…
Gatherings of thousands to millions of people frequently occur for an enormous variety of events, and automated counting of these high-density crowds is useful for safety, management, and measuring significance of an event. In this work, we show that the regularly accepted labeling scheme of crowd density maps for trai…
Counting subgroups of a surface using convex core lengths.
problem Counting conjugacy classes of subgroups of fundamental groups of surfaces.
method Using half the sum of the lengths of the boundaries of the convex core of a subgroup.
result The number of conjugacy classes of subgroups is asymptotic to .
New coherence parameter for GNNs with Fourier measurements improves signal recovery.
problem Characterizing generative compressed sensing with Fourier measurements.
method Subspace counting arguments and high-dimensional probability theory.
result First known restricted isometry guarantee for generative compressed sensing with subsampled isometries.
Semiparametric STAR model improves mental health data analysis.
problem Overdispersed, zero-inflated, bounded count data in self-reported mental health surveys.
method STAR transformation and rounding of latent Gaussian model, nonparametric transformation estimation, EM algorithm for maximum likelihood.
result Substantial improvements in goodness-of-fit compared to existing models.
Bayesian deep learning counts crowds robustly despite occlusions and scale variations.
problem Accurately counting individuals in crowded scenes with occlusions and varying sizes.
method Proposes a Bayesian multi-scale neural network with a ResNet feature extractor, dilated convolutions, and a Perspective-aware Aggregation Module.
result Achieves superior performance on crowd counting benchmarks with uncertainty estimates.
Blind source separation (BSS) aims at recovering signals from mixtures. This problem has been extensively studied in cases where the mixtures are contaminated with additive Gaussian noise. However, it is not well suited to describe data that are corrupted with Poisson measurements such as in low photon count optics or …
Researchers prove a new measure for a financial volatility model.
problem Modeling financial volatility with a Hawkes process.
method Prove existence of equivalent martingale measures for a Heston-Hawkes model.
result Existence of a family of equivalent martingale measures for the model.
Performance of nuclear threat detection systems based on gamma-ray spectrometry often strongly depends on the ability to identify the part of measured signal that can be attributed to background radiation. We have successfully applied a method based on Principal Component Analysis (PCA) to obtain a compact null-space m…
Extrapolation Guarantees for Perturbation Modeling Under the Additive Latent Shift Assumptionstat.ML
The paper tackles extrapolation of gene knockouts effects on RNA counts.
problem Modeling effects of gene knockouts on RNA counts for new perturbations.
method Formulated as a latent variable model with additive perturbation effects, proved identifiability, proposed PDAE for estimation.
result PDAE can accurately predict effects of unseen but identifiable perturbations.
Enhances count process modelling with Markov-modulated non-homogeneous Poisson process.
problem Count data modelling challenges, especially in complex scenarios.
method Introduces a flexible frequency perturbation measure into Markov-modulated Poisson process framework.
result Natural incorporation of observed event arrivals and latent factors.
The study counts geodesics on special manifolds without focusing points.
problem Counting geodesics on specific types of manifolds.
method Margulis-type asymptotic estimates and analysis of geodesic flow.
result The geodesic flow on these manifolds has a unique measure of maximal entropy with the Bernoulli property.
Counting tripods on a flat torus using lattice point counting.
problem Counting finite BPS webs in flat torus geometry.
method Lattice point counting techniques in .
result Asymptotic counting result for tripods on the torus.
The sizes of Markov equivalence classes of directed acyclic graphs play important roles in measuring the uncertainty and complexity in causal learning. A Markov equivalence class can be represented by an essential graph and its undirected subgraphs determine the size of the class. In this paper, we develop a method to …
Fix a translation surface , and consider the measures on coming from averaging the uniform measures on all the saddle connections of length at most . Then as , the weak limit of these measures exists and is equal to the Lebesgue measure on . We also show that any weak limit of a subsequence of …
In unsupervised machine learning, agreement between partitions is commonly assessed with so-called external validity indices. Researchers tend to use and report indices that quantify agreement between two partitions for all clusters simultaneously. Commonly used examples are the Rand index and the adjusted Rand index. …
Flow Matching for Count Datastat.ML
Flow Matching for count data improves sample quality and efficiency.
problem Mapping between count distributions across batches or time points in high-dimensional count data.
method count-FM, a flow-matching framework based on a continuous-time birth-death process with local unit jumps.
result count-FM achieves better sample quality than representative baselines while using fewer parameters.
New findings on geometric flows and equidistribution in Hilbert geometry.
problem Characterizing dynamical and counting results in Hilbert geometry.
method Study of dynamical and counting results in rank-one properly convex projective structures with Hilbert metrics.
result Hilbert geodesic flow is strongly mixing and orbits and primitive closed geodesics equidistribute.
We establish a precise asymptotic formula for the number of homotopy classes of periodic orbits for the geodesic flow on rank one manifolds of nonpositive curvature. This extends a celebrated result of G. A. Margulis to the nonuniformly hyperbolic case and strengthens previous results by G. Knieper. We also establish s…