Paper justifies ideal point forecasts as measurable, clarifying conditions for their existence.
problem Justifying ideal point forecasts as measurable random variables.
method Clarifying and establishing measurability conditions for a wide class of functionals.
result Ideal point forecasts are shown to be measurable, providing theoretical justification.
Fast sampling method for Determinantal Point Processes.
problem Sampling from strongly Rayleigh measures.
method Fast mixing Markov Chain sampler.
result Fast sampling for Determinantal Point Processes.
Probabilistic McShane's identity measures paths through a point.
problem Understanding the probabilistic nature of McShane's identity.
method Interpreting McShane's identity as a measure on path spaces.
result A probabilistic measure on path spaces.
This work tackles community recovery in hypergraphs with measurements of varying sizes.
problem Cluster data points into distinct communities based on measurements with varying sizes.
method Characterizes the fundamental limits on the number of measurements required for community reconstruction in hypergraphs with homogeneity and parity measurements, possibly corrupted by noise.
result Characterizes fundamental limits on the number of measurements required for community reconstruction in hypergraphs.
New Stein Points use Markov chains to select points, reducing computational cost.
problem Approximating a probability measure with an empirical measure.
method Select points based on Markov chain sample paths, solving a non-convex optimisation problem.
result Significantly reduces computational cost and establishes consistency guarantees.
A new PCA method for analyzing point processes.
problem Analyzing variability in replicated point processes.
method Functional Principal Component Analysis (fPCA) on cumulative mass functions.
result Established convergence and introduced principal measures.
A local cut point is by definition a point that disconnectes its sufficiently small neighborhood. We show that there exists an upper bound for the degree of a local cut point in a metric measure space satisfying the generalized Bishop--Gromov inequality. As a corollary, we obtain an upper bound for the number of ends o…
MAS scores cluster size consistency from points, robust to label changes.
problem Desired uniformity in cluster sizes, stability under label perturbations.
method Mass Agreement Score (MAS) measures point-centric cluster size consistency, robust to label changes.
result MAS yields similar scores for partitions with similar bulk structure, sensitive to genuine redistribution of cluster mass.
The paper proves a large deviation principle for Gibbs measures on Polish spaces and applies it to specific cases.
problem Large deviation principles for Gibbs measures on Polish spaces.
method General Laplace principle for non-normalized Gibbs measures, applied to conditional Gibbs measures, Coulomb gases, and Fekete points.
result The Laplace principle is proven and applied to specific cases, providing a deterministic version of Γ-convergence. The study bounds Hausdorff measure of flat singular points in area-minimizing currents.
problem Bounding Hausdorff measure of flat singular points in area-minimizing currents.
method Proving locally finite (m−2)-dimensional Hausdorff measure and Minkowski content bounds. result The set of flat singular points has locally finite (m−2)-dimensional Hausdorff measure. Stein Points improve posterior approximation with minimal points.
problem Approximating posterior distributions with limited points.
method Greedy or conditional gradient method to minimize kernel Stein discrepancy.
result Stein Points enable accurate approximation at low computational cost.
The study proves umbilical points have positive measure in 3D space forms.
problem Characterizing umbilical points in 3D space forms.
method Analyzing mean curvature and proving measure properties.
result Umbilical points have positive measure in 3D space forms.
Optimal quantum change point detection without error.
problem Identifying a change point in a stream of identical quantum particles.
method Sequential local measurements with optimal performance bound.
result Optimal online detection strategy with one bit of memory.
PWGF escapes saddle points in nonconvex optimization.
problem Escaping saddle points in nonconvex optimization.
method PWGF uses noisy perturbations via Gaussian process to escape saddle points.
result PWGF achieves second-order optimality for nonconvex objectives.
We study exponential Levy models with change-point which is a random variable, independent from initial Levy processes. On canonical space with initially enlarged filtration we describe all equivalent martingale measures for change-point model and we give the conditions for the existence of f-divergence minimal equival…
We show that there exists an interval exchange and a point so that the orbit of the point equidistributes for a measure that is not ergodic.
Study equilibrium measures on manifolds without conjugate points with visibility covering.
problem Uniqueness and properties of equilibrium measures on manifolds without conjugate points.
method Analysis of geodesic flows, study of equilibrium measures, ergodic properties, and pressure gap.
result Equilibrium measures satisfy a weak pressure gap under certain conditions.
DDEQs extend DEQs to discrete measure inputs using Wasserstein gradient flows.
problem Applying DEQs to discrete measure inputs like sets or point clouds.
method Wasserstein gradient flows for finding fixed points of discrete measures under permutation-invariance.
result DDEQs can compete with state-of-the-art models in tasks like point cloud classification and completion.
New clustering method using point-set kernel measures similarity.
problem Measuring similarity between objects for clustering.
method Point-set kernel for similarity computation; clustering procedure uses this measure.
result Proposed method is more effective and faster than existing algorithms.
Study shows how point cloud data converges to flat torus measures.
problem Analyzing convergence of point cloud measures to flat torus measures.
method Discrete Wasserstein distance on geometric graphs, Gromov-Hausdorff convergence.
result Wasserstein spaces on point clouds converge to flat torus measures.
The paper measures and limits the extent of non-smooth points in Alexandrov spaces.
problem Understanding the extent of non-smooth points in Alexandrov spaces.
method Defining a non-negative function K(x) to measure the extent of non-smoothness and quantitatively estimating its distribution. result The Hausdorff dimension estimate and quantitative Hausdorff measure estimate for the set of C2-singular points. Study geodesic flows on hyperbolic manifolds without conjugate points, proving unique measure of maximal entropy.
problem Proving uniqueness of measure of maximal entropy for geodesic flows on specific manifolds.
method Analyzing geodesic flows on closed Riemannian manifolds without conjugate points, using properties of Gromov hyperbolic and residually finite groups.
result Proves geodesic flow has a unique measure of maximal entropy under appropriate assumptions.
Algorithm clusters data without specifying cluster number based on similarity.
problem Clustering data without knowing the number of clusters in advance.
method Centroid-based clustering with a similarity measure to decide cluster assignment.
result Algorithm can handle streaming data and clusters based on predefined similarity level.
Unique entropy measure found for geodesic flows on certain surfaces.
problem Finding a unique measure of maximal entropy for geodesic flows.
method Analyzing geodesic flows on surfaces without conjugate points.
result Proved existence of a unique measure of maximal entropy for geodesic flows on certain surfaces.
A method for classifying points with minimal queries using Hermite polynomials.
problem Classifying points from an unknown probability measure with minimal label queries.
method Convex combination of conditional probabilities, Hermite polynomial kernel for hierarchical support estimation.
result The method achieves high F-score for classification in hyper-spectral images and MNIST. The paper proves an inequality for symmetric polynomials under a fixed point measure.
problem An inequality for elementary symmetric polynomials under a fixed point measure of permutations.
method Constructing differential operators to set up a monotone flow.
result The inequality is proven and is sharp.
The paper strengthens a theorem on crossings under linear perturbations with Hausdorff measure estimates.
problem Understanding multiple-point crossings under linear perturbations.
method Establishes a transversality theorem with Hausdorff measure estimates for exceptional parameter sets.
result Explicit upper bounds on the Hausdorff dimension of the exceptional set.
We study the statistical meaning of the minimization of distortion measure and the relation between the equilibrium points of the SOM algorithm and the minima of distortion measure. If we assume that the observations and the map lie in an compact Euclidean space, we prove the strong consistency of the map which almost …
The paper connects geodesic flows and limit sets on visibility manifolds.
problem Understanding dynamics and ergodic properties on non-compact visibility manifolds.
method Analyzing geodesic flows and Patterson-Sullivan measures on visibility manifolds without conjugate points.
result The positivity of the Patterson-Sullivan measure of the Myrberg limit set is equivalent to the conservativity of the geodesic flow.
Characterizes Lebesgue points using nearest neighbor methods.
problem Consistency of classification algorithms based on nearest neighbors.
method Characterization of Lebesgue points via 1-Nearest Neighbor regression.
result Proves convergence of 1-Nearest Neighbor classification algorithms in metric spaces.
This thesis uses Kantorovich-Rubinstein distance for classifying points based on their measures.
problem Classifying points based on their measures in a metric space.
method Using Kantorovich-Rubinstein distance as a metric in the space of measures to capture geometry and topology.
result A large Kantorovich-Rubinstein distance indicates the existence of a 1-Lipschitz classifier that well classifies the points.
New model distinguishes Poisson processes from self-similar ones.
problem Distinguishing Poisson point processes from self-similar processes.
method Machine learning model based on inhomogeneous, compound Poisson point process.
result The model can distinguish Poisson point processes from self-similar processes.
A new method calculates a barycenter for probability measures using Wasserstein distance.
problem Finding a central measure for a set of probability distributions.
method Regularizing the pushforward measure of a set of probability distributions into the Wasserstein space and then finding the barycenter.
result The method yields a uniquely defined barycenter measure supported on the barycentric points of the input measures.
Study sharp convergence rates of empirical UOT for spatio-temporal point processes.
problem Statistical analysis of UOT for spatio-temporal point processes.
method Empirical plug-in estimators for Kantorovich-Rubinstein distance between intensity measures.
result Sharp convergence rates of empirical UOT in terms of intrinsic dimensions of measures.
Transformers can interpolate between arbitrary measures.
problem Understanding the expressive power of Transformers as measure-to-measure maps.
method Provided an explicit choice of parameters for a single Transformer to match N arbitrary input measures to N arbitrary target measures.
result A single Transformer can interpolate between arbitrary measures.
A new measure for traffic safety reduces variability in fatal crash data.
problem Unpredictable clusters in fatal crash data reduce traffic safety measurement accuracy.
method Introduced a fatal point concept and a rounding error method to detect it.
result The proposed measure is not significantly affected by cluster variability.
In this note we show that in metric measure spaces satisfying the reduced curvature-dimension condition CD*(K,N) we always have geodesics in the Wasserstein space of probability measures that satisfy the critical convexity inequality of CD*(K,N) also for intermediate times and in addition the measures along these geode…
A new anomaly detection method using partial identification.
problem Detecting anomalies in large datasets.
method Partial Identification framework and PIDScore geometric anomaly measure.
result PIDForest outperforms other methods in anomaly detection.
Horizontal points of smooth submanifolds in stratified groups play the role of singular points with respect to the Carnot-Carathe'odory distance. When we consider hypersurfaces, they coincide with the well known characteristic points. In two step groups, we obtain pointwise estimates for the Riemannian surface measure …
Study geometric quantization and measured Gromov-Hausdorff convergence of frame bundle metrics.
problem Understanding the convergence of frame bundle metrics and their relation to Gromov-Hausdorff convergence.
method Analyzes the one-parameter family of Riemannian metrics on the frame bundle of a prequantum line bundle, considering the convergence to metric measure spaces with S1-actions. result The frame bundle metrics converge to metric measure spaces with S1-actions, which depend on the choice of base point in Bohr-Sommerfeld fibers. New measures found in 3-uniform geometry.
problem Understanding non-flat uniform measures in geometric measure theory.
method Combining combinatorial methods and distance symmetry properties.
result Infinite family of 3-uniform measures constructed.
New method detects changes online with bounds on delay.
problem Detecting changes in data streams efficiently.
method Maximizes discrepancy between pre-change and post-change distributions.
result Non-asymptotic bounds on average running length and detection delay.
For convex co-compact hyperbolic manifolds Γ\Hn+1 for which the dimension of the limit set satisfies δΓ<n/2, we show that the high-frequency Eisenstein series associated to a point ξ "at infinity" concentrate microlocally on a measure supported by (the closure of) the set of points in the …
New local ID estimators based on data separability.
problem Estimating intrinsic dimensionality locally in multi-dimensional data.
method Local estimators based on concentration of measure.
result Empirical comparison with other ID estimators.
Measuring wave sources uniquely identifies manifold properties.
problem Determining Riemannian manifold structure from wave observations.
method Semilinear wave equation measurements at a single point.
result Topological, differential, and geometric structure can be inferred.
The geodesic flow on certain surfaces is shown to be ergodic.
problem Ergodicity of geodesic flows on surfaces without focal points.
method Analyzing geodesic flows on surfaces with no focal points, proving ergodicity under specific curvature conditions.
result The geodesic flow on the unit tangent bundle of a surface with no focal points is ergodic with respect to the Liouville measure.
Unified interpretation of sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds.
problem Proving a sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds.
method Measure-theoretic perspective, focusing on singular measures and characteristic points.
result Unified interpretation of previous results and natural geometric conditions for the theorem.
Transductive learning considers situations when a learner observes m labelled training points and u unlabelled test points with the final goal of giving correct answers for the test points. This paper introduces a new complexity measure for transductive learning called Permutational Rademacher Complexity (PRC) and …