We prove a Frobenius-type theorem for singular distributions generated by a family of locally Lipschitz continuous vector fields satisfying almost everywhere a quantitative finite type condition.
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48 results for “finite type distributions”
New algorithm for countable bandits with optimal regret.
problem Stochastic bandit problem with countably many arms.
method Fully adaptive online learning algorithm with O(log n) expected cumulative regret.
result Achieves optimal regret of O(log n) after any number of plays n.
Generalizes pseudo-product structures with abnormal extremals.
problem Finiteness of symmetry algebras for non-degenerate pseudo-product structures.
method Modified universal prolongation of graded nilpotent Lie algebras and generalized finiteness criterion.
result Distributions with singularly transitive properties have finite-dimensional symmetries.
Based on the ideas of Optimal Control, we introduce the new basic characteristic of a bracket generating distribution, the Jacobi symbol. In contrast to the classical Tanaka symbol, the set of Jacobi symbols is discrete and classifiable. We give an explicit and unified algebraic procedure for the construction of the ca…
Study tests whether trade-off functions are above or below benchmarks using finite samples.
problem Testing trade-off functions between unknown distributions.
method Identifies a condition for nontrivial testing, constructs a test with error guarantees, and inverts the test for confidence bands.
result Finite-sample testing is possible under specific structural assumptions about rejection regions.
Study non-asymptotic Langevin Monte Carlo for Gibbs distributions.
problem Sampling from Gibbs distributions with dissipative potentials.
method Langevin-type algorithms based on Liptser--Shiryaev theory and Poincaré inequalities.
result Upper bound on 2-Wasserstein distance for accurate approximation.
Unified framework for Arnold-type invariants via dual complexes and finite-difference structures.
problem Study of Arnold-type invariants of immersed curves and surfaces.
method Framework on dual complexes, locally normalized maps, finite-difference structures, and Shumakovitch-type identities.
result Unified evaluation of Arnold-type invariants and on dual skeleta.
PH-VAE models heavy-tailed data with flexible Phase-Type distributions.
problem Standard VAEs fail to capture heavy-tailed behavior in real-world data.
method PH-VAE uses Phase-Type distributions defined by continuous-time Markov chains to adaptively model tail behavior.
result PH-VAE significantly outperforms existing heavy-tail-aware VAEs in approximating diverse heavy-tailed distributions.
In this paper, we study the distribution of the genuses of cluster quivers of finite mutation type. First, we prove that in the exceptional cases, the distribution of genuses is or . Next, we consider the relationship between the genus of an oriented surface and that of cluster quivers from this surface. It…
We analyze the slope gap distribution of Veech surfaces, finding finite non-analytic points and quadratic tail decay.
problem Understanding the slope gap distribution of Veech surfaces.
method Explicit parameterization of a Poincaré section to the horocycle flow, finiteness result for the first return map.
result The limiting gap distribution of slopes of saddle connections on Veech surfaces is piecewise real-analytic with finitely many points of non-analyticity and has quadratic tail decay.
Category theory generalizes finite type invariants using diagrams systems.
problem Generalizing finite type invariants using category theory.
method Relating generating sets for generalized finite type theories with diagrams systems.
result Demonstrates the correspondence between finite type theories and diagrams systems.
New test for conditional independence using kernel embeddings.
problem Testing conditional independence in high-dimensional settings.
method Analytic kernel embeddings, asymptotic distribution.
result New test outperforms existing methods in high-dimensional settings.
Zipf Distributions from Two-Stage Symbolic Processes: Stability Under Stochastic Lexical Filteringstat.ME
Study explains Zipf's law using geometric mechanisms from a finite alphabet.
problem Explains Zipf's law in language without relying on linguistic elements.
method Uses the Full Combinatorial Word Model (FCWM) to generate geometric distributions of word lengths.
result Supports predictions of power-law rank-frequency curves, matching various languages.
We introduce a notion of ternary distributive algebraic structure, give examples, and relate it to the notion of a quandle. Classification is given for low order structures of this type. Constructions of such structures from ternary bialgebras are provided. We also describe ternary distributive algebraic structures com…
Estimates barycenter in geodesic spaces with finite sample bounds.
problem Estimating the barycenter of a distribution in geodesic spaces.
method Finite sample error bounds, Hoeffding- and Bernstein-type concentration inequalities, efficient algorithms.
result Statistical guarantees for efficient barycenter computation.
Develops a framework for distributional Granger causality
problem Identifying predictive dependence in time series beyond Gaussian settings
method Using finite collection of channel-specific restrictions
result Identifies distributional Granger non-causality through testable hypotheses
The one-term distributive homology was introduced by J.H.Przytycki as an atomic replacement of rack and quandle homology, which was first introduced and developed by R.Fenn, C.Rourke and B.Sanderson, and J.S.Carter, S.Kamada and M.Saito. This homology was initially suspected to be torsion-free, but we show in this pape…
We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's triple link homotopy invariant is a finite type invariant, of type 1, in this sense. We also generalize the approach to Milnor's higher order homotopy invariants and show that they are also, in a sense, of finite t…
Submanifolds of coordinate finite-type were introduced in HV1. A submanifold of a Euclidean space is called a coordinate finite-type submanifold if its coordinate functions are eigenfunctions of Δ. In the present study we consider coordinate finite-type surfaces in E^4. We give necessary and sufficient conditions for g…
Empirical Transfer Operators and Finite-Sample Change Detection for Noisy Expanding Interval Mapsstat.ML
Detect changes in noisy dynamical systems using empirical approximations and finite-sample bounds.
problem Change detection in noisy dynamical systems
method Partition-based empirical approximations and finite-state stationary distribution stability
result Finite-sample bound for empirical stationary density
Distributional Random Forests for Complex Survey Designs on Reproducing Kernel Hilbert Spacesstat.ME
SDRF estimates complex survey designs for conditional distributions.
problem Estimating conditional distributions under complex survey designs.
method Survey-calibrated distributional random forest (SDRF) with pseudo-population bootstrap and MMD split criterion.
result Established design consistency and model consistency for survey designs.
The paper gives the complete characterization of all graded nilpotent Lie algebras with infinite-dimensional Tanaka prolongation as extensions of graded nilpotent Lie algebras of lower dimension by means of a commutative ideal. We introduce a notion of weak characteristics of a vector distribution and prove that if a b…
Defines finite type Multivalued Shape using hyperspaces.
problem No specific problem stated; focuses on definition.
method Construction over metric compacta using hyperspaces.
result Defines finite type Multivalued Shape.
Finite-type surfaces have a topological Hopf property.
problem Characterizing surfaces with a topological Hopf property.
method Using topological analogs of the Hopf property.
result Infinite-type surfaces do not have the Hopf property.
New invariant fully describes finite type invariants of knots in homology 3-spheres.
problem Constructing a universal finite type invariant for knots in homology 3-spheres.
method Refined construction of a new invariant that is strictly stronger and universal.
result New invariant fully describes the graded space of finite type invariants of knots in homology 3-spheres.
Compact manifolds with boundary have finite type mapping class groups.
problem Understanding the structure of mapping class groups for manifolds with boundaries.
method Proved finite type for mapping class groups under specific dimension and connectivity assumptions.
result Mapping class groups of compact manifolds with boundaries are of finite type.
We observe that most known results of the form "v is not a finite-type invariant" follow from two basic theorems. Among those invariants which are not of finite type, we discuss examples which are "ft-independent" and examples which are not. We introduce (n,q)-finite invariants, which are generalizations of finite-type…
Homotopy classes of nanowords and nanophrases are combinatorial generalizations of virtual knots and links. Goussarov, Polyak and Viro defined finite type invariants for virtual knots and links via semi-virtual crossings. We extend their definition to nanowords and nanophrases. We study finite type invariants of low de…
Paper shows 3D hyperbolic manifolds are uniquely identified by their finite groups.
problem Identifying hyperbolic 3-manifolds using their finite quotient groups.
method Proves profinite isomorphism of fundamental groups uniquely determines the manifold's isomorphism type.
result Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups.
In this paper, we study ruled surfaces and quadrics in the 3-dimensional Euclidean space which are of finite -type, that is, they are of finite type, in the sense of B.-Y. Chen, with respect to the third fundamental form. We show that helicoids and spheres are the only ruled and quadric surfaces of finite -ty…
Develops robust MDPs for unknown disturbances with performance guarantees.
problem Unknown disturbance distribution in MDPs.
method Empirical distribution, sublevel set of distance function, weak convergence, concentration inequality.
result Robust optimal value function converges to true optimal value function with increasing sample sizes.
We start by showing that the finite-time absolute ruin probability in the classical risk model with constant interest force can be expressed in terms of the transition probability of a positive Ornstein-Uhlenbeck type process, say X. Our methodology applies to the case when the dynamics of the aggregate claims process …
Finite type invariants separate PL links in 3D space.
problem If two PL links are isotopic, are they PL isotopic?
method Proved PL isotopic links are indistinguishable by finite type invariants.
result Finite type invariants separate PL links in 3D space if and only if certain conjectures hold.
The main goal of this article is to understand how the length spectrum of a random surface depends on its genus. Here a random surface means a surface obtained by randomly gluing together an even number of triangles carrying a fixed metric. Given suitable restrictions on the genus of the surface, we consider the number…
Cubic complexes appear in the theory of finite type invariants so often that one can ascribe them to basic notions of the theory. In this paper we begin the exposition of finite type invariants from the `cubic' point of view. Finite type invariants of knots and homology 3-spheres fit perfectly into this conception. In …
Finite Type Enhancementsmath.GT
We enhance the biquandle counting invariant using elements of truncated biquandle-labeled Polyak algebras. These finite type enhancements reduce to the finite type enhancements defined by Goussarov, Polyak and Viro for the trivial biquandle of one element and determine (but are not determined by) the biquandle counting…
A theory of finite type invariants for arbitrary compact oriented 3-manifolds is proposed, and illustrated through many examples arising from both classical and quantum topology. The theory is seen to be highly non-trivial even for manifolds with large first betti number, encompassing much of the complexity of Ohtsuki'…
Robust Independence Tests with Finite Sample Guarantees for Synchronous Stochastic Linear Systemsstat.ML
The paper develops robust tests for detecting independence in synchronous stochastic systems with finite sample guarantees.
problem Detecting independence in synchronous stochastic systems with finite sample guarantees.
method Combines confidence region estimates with permutation tests and dependence measures to detect nonlinear dependence.
result Consistent hypothesis tests for detecting independence under mild assumptions.
Delta finite-type invariants are defined analogously to finite-type invariants, using delta moves instead of crossing changes. We show that they are closely related to the lower central series of the commutator subgroup of the pure braid group.
Characterizes monodromies of projective structures on finite-type surfaces.
problem Understanding monodromies of projective structures on finite-type surfaces.
method Geometrical/topological study of local conical projective structures.
result Any representation can be represented as the holonomy of a branched projective structure.
Finite rigid sets found in complex of curves for surfaces.
problem Finding finite rigid sets in curve complexes of surfaces.
method Exhaustion by finite rigid sets proved for surfaces of finite type and genus ≥3.
result Finite rigid sets exist in the non-separating curve complex of surfaces.
Study on compact and finite-type support in mapping class group homology.
problem Understanding non-trivial classes supported on compact or finite-type subsurfaces.
method Use of shiftable subsurfaces and homological stability for finite-type surfaces.
result Almost-complete answer for surfaces with positive genus, partial answer for zero genus.
Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject had been collected in author's book [Total mean curvature and sub manifolds of finite type, World Scientific, NJ, 1984]. A list of ten open problems and three conjectures on submanifolds of finite type was…
Study finds many Lagrangian fillings for certain Legendrian links.
problem Understanding Lagrangian fillings for Legendrian links of finite type.
method Use of N-graphs and combinatorics of seed patterns.
result Proves existence of at least seeds many exact embedded Lagrangian fillings for Legendrian links of type ADE.
A new measure scales MMD to assess distribution closeness.
problem Testing statistical significance of distribution closeness.
method Norm-adaptive MMD (NAMMD) for distributional discrepancy.
result NAMMD-based DCT has higher test power than MMD-based DCT.
In this paper, we define finite type invariants for cyclic equivalence classes of nanophrases and construct the universal ones. Also, we identify the universal finite type invariant of degree 1 essentially with the linking matrix. It is known that extended Arnold's basic invariants to signed words are finite type invar…
A smooth four manifold is of finite type if its Donaldson invariant satisfies D((x^2-4)^r)=0. We prove that every simply connected manifold is of finite type by using the structure of Donaldson invariants in the presence of immersed spheres. More precisely we prove that if a manifold X contains an immersed sphere w…
Finite Type Invariantsmath.GT
This is an overview article on finite type invariants, written for the Encyclopedia of Mathematical Physics