Characterizes measures preserving compound mixed renewal process properties.
problem Preserving compound mixed renewal process properties under different probability measures.
method Characterization of progressively equivalent probability measures.
result Any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures.
Unified Minkowski problem discussed for (p,q)-mixed quermassintegrals.
problem Unified Minkowski problem for (p,q)-mixed quermassintegrals.
method Introducing (p,q)-mixed quermassintegrals and (p,q)-dual mixed curvature measure to study the Minkowski problem.
result Derivation of important properties and geometric inequalities for (p,q)-mixed quermassintegrals.
Solves Christoffel-Minkowski problem for axially symmetric bodies.
problem Necessary and sufficient conditions for mixed area measures of axially symmetric convex bodies.
method Introduced a new method to transform mixed area measures and mixed volumes of axially symmetric bodies, refining Firey's classification and improving estimates.
result Complete solution to the mixed Christoffel-Minkowski problem for axially symmetric bodies without regularity assumptions.
Characterizes continuity of monotone functionals in mixed topology.
problem Continuity of monotone functionals in mixed topology.
method Characterization through lower semicontinuity and dual representations.
result Continuity in mixed topology is equivalent to dual representation in terms of countably additive measures.
New weighted surface area measures for convex bodies with applications.
problem Generalizing surface area measures to weighted Borel measures.
method Formulating and analyzing weighted surface area measures, proving integral formula and Bézout-type inequality.
result New integral formula for mixed measure of three bodies, generalizing Bézout-type inequality.
We study probability measures induced by set functions with constraints. Such measures arise in a variety of real-world settings, where prior knowledge, resource limitations, or other pragmatic considerations impose constraints. We consider the task of rapidly sampling from such constrained measures, and develop fast M…
This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and f-divergence functionals such as Hellinger and Kullback-Leibl…
The paper proves a unique conformal measure for Anosov groups and shows local mixing.
problem Proving the uniqueness of conformal measures for Anosov groups.
method Analogue of Sullivan's theorem for Anosov subgroups of semisimple groups.
result Uniqueness of conformal measures and local mixing for Anosov groups.
New proof for weak mixing in polygonal billiards.
problem Proving weak mixing in polygonal billiards.
method Using Baire category and eigenvalue analysis.
result Billiard flow is weakly mixing for non-rational polygons.
A new method combines machine learning with mixed-effects models for better repeated measurement analysis.
problem Inference of linear coefficients in partially linear mixed-effects models with complex interactions and high-dimensional variables.
method Double machine learning approach to estimate nonparametrically nonlinear variables, then use standard linear mixed-effects techniques to estimate the linear coefficient.
result The estimated fixed effects coefficient converges at the parametric rate and is semiparametrically efficient.
Rapid mixing of Langevin dynamics on Riemannian manifolds
problem Mixing time of Langevin dynamics on Riemannian manifolds
method Relation between Langevin processes in domain and image
result Achievable polynomial mixing times
New tools for estimating and inferring Wasserstein distance in topic models.
problem Estimating and inferring the Wasserstein distance between mixing measures in topic models.
method New canonical interpretation and tools for inference on Wasserstein distance in topic models.
result First minimax lower bounds and fully data-driven inferential tools for the Wasserstein distance in topic models.
Unique entropy measure found for convex projective manifolds.
problem Entropy measure for convex projective manifolds.
method Developed Patterson--Sullivan densities and mixing theory.
result Unique mixing measure of maximal entropy exists.
Solves a long-standing convex geometry problem about mixed volumes.
problem Characterizing the support of mixed area measures.
method Geometric approach to convex bodies in R^n and R^3.
result Resolved one direction of Schneider's conjecture for arbitrary convex bodies.
New method for summarizing Bayesian mixture models using sliced Wasserstein distances.
problem Estimating the mixing measure in nonparametric Bayesian mixture models.
method Decision-theoretic approach using sliced Wasserstein distances for Gaussian mixtures.
result Effective estimation of the mixing measure and mixture density.
Common perpendiculars equidistribute in negatively curved spaces.
problem Equidistribution of common perpendiculars in negatively curved spaces.
method Analyzing the Bowen-Margulis measure and geodesic flow properties.
result Lebesgue measures of common perpendiculars equidistribute to the Bowen-Margulis measure.
New method for mixed data FI controls type I error and achieves high power.
problem Statistical inadequacy of feature importance measures for mixed data.
method Combining CPI framework with sequential knockoffs for mixed data.
result Our method controls type I error and achieves high power for mixed data.
Paper extends FOFC algorithm to work with mixed data types.
problem Designing causal discovery algorithms for mixed data types.
method Proves tetrad constraint can be entailed for mixed data types and applies FOFC algorithm.
result FOFC algorithm can work on mixed data types.
Solves Christoffel-Minkowski problem and Hessian equations with radial symmetry.
problem Christoffel-Minkowski problem and Hessian equations under rotational symmetries.
method Constructing explicit convex solutions to mixed Monge-Ampère equations on \(\mathbb{R}^n\) under radial symmetry.
result Explicit representation formula for the support function of the resulting convex body.
Frame flows on certain symmetric spaces mix exponentially.
problem Exponential mixing of frame flows in convex cocompact locally symmetric spaces.
method Generalized local non-integrability and non-concentration properties to apply Dolgopyat's method.
result Exponential mixing of frame flows proved for convex cocompact locally symmetric spaces.
The paper proposes a mixed-frequency quantile regression model for VaR and ES forecasting.
problem Forecasting VaR and ES with mixed-frequency data.
method Mixed-frequency quantile regression model to estimate VaR and ES.
result The proposed model outperforms other models in VaR and ES backtesting tests.
Forré introduces a new conditional independence notion for mixed variables.
problem Unified framework for random and non-stochastic variables.
method Unified framework of transitional conditional independence and causal calculus for iDMGs.
result Unified framework connects conditional independencies to graphical separation criteria.
New study on No-U-Turn Sampler for accelerated mixing in Hamiltonian Monte Carlo.
problem Achieving accelerated convergence in Hamiltonian Monte Carlo.
method Combining concentration of measure and coupling analysis for mixing.
result Rigorous mixing guarantees for the No-U-Turn Sampler in certain Gaussian distributions.
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.
We prove a general inequality for mixed Hessian measures by global arguments. Our method also yields a simplification for the case of complex Monge-Ampère equation. Exploiting this and using Kołodziej's mass concentration technique we also prove the uniqueness of the solutions to the complex Hessian equation on compact…
NUTS mixing time scales as d^(1/4) for Gaussian distributions.
problem Improving the efficiency of the No-U-Turn Sampler (NUTS) for Gaussian distributions.
method Coupling argument leveraging geometric structure of Gaussian concentration, uniformity analysis of NUTS transitions.
result The mixing time of NUTS scales as d^(1/4) for Gaussian distributions, up to logarithmic factors.
New method uses dendrograms for better mixture model selection and clustering.
problem Selecting the correct number of components in finite mixture models.
method Hierarchical clustering tree derived from overfitted latent mixing measures.
result Consistently selects the true number of mixing components and optimal convergence rate for parameter estimation.
This work finds mixed equilibria in zero-sum games using interacting particle dynamics.
problem Finding mixed equilibrium points in continuous minmax games.
method A method based on entropic regularisation of two-layer zero-sum games with interacting particle dynamics.
result The sequence of empirical measures of the particle system satisfies a large deviation principle as the number of particles grows to infinity, implying convergence of the empirical measure and the Nikaidô-Isoda error.
Study online learning in RKHS with dependent processes, focusing on \(β\)- and \(φ\)-mixing.
problem Online learning in RKHS with dependent data.
method Online regularized learning algorithm in RKHS, analyzing \(β\)- and \(φ\)-mixing sequences.
result Probabilistic upper bounds and convergence rates for mixing coefficients.
We provide a measure based topology for certain unions of C2 rectifiable submanifolds of mixed dimensions in Rn. In this topology lower dimensional sets remain in the limit as measures when higher dimensional sets collapse down to them. For example a decreasing sequence of spheres may have a limit consisting of just a …
Solves Christoffel problem for disk area measures on spheres.
problem Conditions for a measure to be a disk area measure of convex bodies.
method Integral representation and differential equation reformulation.
result Reconstructs support function from disk area measure.
uHMC achieves fast mixing in high dimensions with gradient evaluations.
problem Quantifying mixing time of uHMC in high dimensions.
method Construction of successful couplings for uHMC.
result uHMC mixes in total variation with logarithmic dependence on dimension.
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.
Bayesian nonparametric (BNP) models provide elegant methods for discovering underlying latent features within a data set, but inference in such models can be slow. We exploit the fact that completely random measures, which commonly used models like the Dirichlet process and the beta-Bernoulli process can be expressed a…
Let X be a proper, geodesically complete CAT(0) space under a proper, non-elementary, isometric action by a group Γ with a rank one element. We construct a generalized Bowen-Margulis measure on the space of unit-speed parametrized geodesics of X modulo the Γ-action. Although the construction of Bowen-Margulis m…
Bayesian test assesses dependence between mixed data types.
problem Assessing dependence between text, image, and sound data.
method Bayesian kernelised correlation test using Dirichlet process model.
result Demonstrated effectiveness compared to other methods.
We consider the problem of recovering a common latent source with independent components from multiple views. This applies to settings in which a variable is measured with multiple experimental modalities, and where the goal is to synthesize the disparate measurements into a single unified representation. We consider t…
This work finds mixed equilibria in machine learning problems using measures and simultaneous gradient ascent-descent.
problem Finding pure equilibria in machine learning problems is computationally hard.
method Entropic regularization, simultaneous gradient ascent-descent, and particle discretization in the Wasserstein metric.
result Global convergence towards the global equilibrium in mixed equilibria problems.
The paper proves exponential mixing for hyperbolic manifolds, with applications to geodesic holonomy.
problem Establishing exponential mixing for frame flows on hyperbolic manifolds.
method Using spectral bounds on transfer operators twisted by holonomy, building on Dolgopyat's method.
result Exponential mixing of frame flows for convex cocompact hyperbolic manifolds.
FAMDAD detects anomalies in mixed data using kurtosis-weighted Factor Analysis.
problem Detecting anomalies in high-dimensional mixed data.
method kurtosis-weighted Factor Analysis of Mixed Data (FAMDAD).
result Anomalies are highly separable in the first and last few dimensions of the FAMDAD embedding.
In this note we consider sampling from (non-homogeneous) strongly Rayleigh probability measures. As an important corollary, we obtain a fast mixing Markov Chain sampler for Determinantal Point Processes.
Abelian covers of hyperbolic 3-manifolds are ubiquitous. We prove the local mixing theorem of the frame flow for abelian covers of closed hyperbolic 3-manifolds. We obtain a classification theorem for measures invariant under the horospherical subgroup. We also describe applications to the prime geodesic theorem as…
New measure of maximal entropy found for a class of geometrically finite groups.
problem Finding a measure of maximal entropy for relatively Anosov groups.
method Constructing reparameterizations and using exponential expansion along unstable foliations.
result The Bowen-Margulis-Sullivan measure is finite and unique for relatively Anosov groups.
New resonance theory for Anosov flows connects spectral properties to mixing measures.
problem Defining and analyzing Ruelle-Taylor resonances for Anosov actions.
method Combining microlocal methods and J. Taylor's cohomological theory, defining Ruelle-Taylor resonances and proving Fredholm theory.
result Ruelle-Taylor resonances form a discrete subset of Cκ with λ=0 being a leading resonance. We propose Gaussian optimal transport for Image style transfer in an Encoder/Decoder framework. Optimal transport for Gaussian measures has closed forms Monge mappings from source to target distributions. Moreover interpolates between a content and a style image can be seen as geodesics in the Wasserstein Geometry. Usi…
The paper solves a generalized Christoffel-Minkowski problem using a curvature flow.
problem Solving the (p,q)-Christoffel-Minkowski problem.
method Investigating the problem via an expanding curvature flow.
result Existence and uniqueness of smooth solutions to the (p,q)-Christoffel-Minkowski problem.
This study deals with the problem of pricing compound options when the underlying asset follows a mixed fractional Brownian motion with jumps. An analytic formula for compound options is derived under the risk neutral measure. Then, these results are applied to value extendible options. Moreover, some special cases of …
We consider the mixed ray transform of tensor fields on a three-dimensional compact simple Riemannian manifold with boundary. We prove the injectivity of the transform, up to natural obstructions, and establish stability estimates for the normal operator on generic three dimensional simple manifold in the case of 1+1 a…