Bayesian optimisation's mean function choice affects convergence speed.
problem The choice of mean function in Bayesian optimisation impacts convergence speed.
method Empirical investigation of 8 mean functions on 10 synthetic and 2 real-world problems.
result Using a constant mean function equal to the worst observed quality value promotes faster convergence.
Median-of-means sampling outperforms mean-of-means for large sample sizes in numerical integration.
problem Improving numerical integration accuracy in high dimensions.
method Median-of-means sampling compared to mean-of-means using RQMC methods.
result Median-of-means sampling is superior for large sample sizes, while mean-of-means is better for smaller sample sizes.
Investigates conditional Chisini means and their application to risk measures.
problem Existence of conditional nonlinear means for bounded random variables.
method Defines a mean as a solution to a functional equation induced by T, and provides conditions for the existence of a unique solution.
result Characterizes the scalarization of conditional Risk Measures.
Paper proves Harnack inequality for f-mean curvature flow.
problem Proving Harnack inequality for f-mean curvature flow. method Gradient flow of the weighed area functional with measure density function e−f. result Proves Li-Yau-Hamilton type Harnack estimate.
In this article we consider means of positive operators on a Hilbert space. We extend the theory of matrix power means to arbitrary operator means in the sense of Kubo-Ando. The basis of the extension is relying on ideas coming from differential geometry. We consider generalized Karcher equations for positive operators…
Unified q-learning for mean-field jump-diffusion models with unobservable population distribution.
problem Continuous-time q-learning in mean-field jump-diffusion models with unobservable population distribution.
method Proposed decoupled Iq-function for unified policy evaluation in MFG and MFC problems; unified q-learning algorithm based on test policies and averaged martingale orthogonality condition.
result Unified policy evaluation rule for MFG and MFC problems based on decoupled Iq-function.
We provide a theoretical foundation for non-parametric estimation of functions of random variables using kernel mean embeddings. We show that for any continuous function f, consistent estimators of the mean embedding of a random variable X lead to consistent estimators of the mean embedding of f(X). For Matérn ke…
The paper constructs surfaces with prescribed mean curvature in a specific space.
problem Finding surfaces with a given mean curvature in a particular geometric space.
method Phase plane analysis to construct entire rotational graphs and catenoid-type surfaces.
result Classification result for surfaces with linearly prescribed mean curvature.
A new formula reveals symmetries between mean excess and ES functions.
problem Optimizing risk measures in financial models.
method Established a reverse ES optimization formula.
result Reveals elegant symmetries and relationships between mean excess and ES functions.
Proves Arnold-Thom conjecture for surfaces' arrival times.
problem Existence of limit tangents for gradient flow lines of surfaces.
method Gradient flow lines of mean curvature flows with neck or cylindrical singularities.
result Proves Arnold's conjecture for all mean convex mean curvature flows of surfaces.
We introduce the functional mean-shift algorithm, an iterative algorithm for estimating the local modes of a surrogate density from functional data. We show that the algorithm can be used for cluster analysis of functional data. We propose a test based on the bootstrap for the significance of the estimated local modes …
Proves optimal regularity for sphere minimizers in 3-sphere.
problem Finding optimal regularity for sphere minimizers.
method Proves C1,1 regularity for minimizers of prescribed mean curvature over isotopy classes. result Proves optimal C1,1 regularity for minimizers. New principles for collapsing law-invariant functionals to means, extending beyond convexity.
problem Conditions for law-invariant functionals to reduce to means.
method Establishing collapse to the mean principles for non-convex functionals.
result General principles apply beyond convexity, including quasiconvex and Choquet integrals.
New neural networks learn mappings between probability measures and functions.
problem Learning mappings between Wasserstein space of probability measures and function spaces.
method Two types of neural networks: bin density and cylindrical approximation, are proposed and supported by universal approximation theorems.
result Accuracy and efficiency of mean-field neural networks in generalization error with various test distributions.
Clustering is a separation of data into groups of similar objects. Every group called cluster consists of objects that are similar to one another and dissimilar to objects of other groups. In this paper, the K-Means algorithm is implemented by three distance functions and to identify the optimal distance function for c…
Optimal fuzzy classification aggregation functions are weighted means.
problem Characterizing optimal fuzzy classification aggregation functions.
method Proving optimality of weighted arithmetic means for fuzzy classification.
result Optimal fuzzy classification aggregation functions are weighted means.
Classification of surfaces with prescribed mean curvature in Heisenberg space and SL2(R).
problem Surfaces with prescribed mean curvature in Heisenberg space and SL2(R).
method Classification result for rotational surfaces with prescribed mean curvature.
result Existence of embedded tori as counterexamples to the Alexandrov problem.
In this paper, we shall prove that space-like surfaces with bounded mean curvature functions in real analytic Lorentzian 3-manifolds can change their causality to time-like surfaces only if the mean curvature functions tend to zero. Moreover, we shall show the existence of such surfaces with non-vanishing mean curvatur…
We use a phase space analysis to give some classification results for rotational hypersurfaces in Rn+1 whose mean curvature is given as a prescribed function of its Gauss map. For the case where the prescribed function is an even function in Sn, we show that a Delaunay-type classification hold…
When fitting Bayesian machine learning models on scarce data, the main challenge is to obtain suitable prior knowledge and encode it into the model. Recent advances in meta-learning offer powerful methods for extracting such prior knowledge from data acquired in related tasks. When it comes to meta-learning in Gaussian…
Paper solves a control problem with robust methods.
problem Monotone mean-variance problems with stochastic coefficients.
method Finding saddle point through BSDEs with unbounded coefficients.
result Optimal control and value match mean-variance problems.
The paper classifies shapes of translating solitons for a specific flow.
problem Understanding the shapes of translating solitons in a specific flow.
method Analyzing functions on a unit sphere and solving an ODE.
result Classification of the shapes of translating solitons.
We prove that, for a generic set of smooth prescription functions h on a closed ambient manifold, there always exists a nontrivial, smooth, closed hypersurface of prescribed mean curvature h. The solution is either an embedded minimal hypersurface with integer multiplicity, or a non-minimal almost embedded hypersur…
Paper characterizes monotonic mean-deviation risk measures.
problem Developing consistent risk measures from mean-deviation models.
method Applying a risk-weighting function to the deviation part of a mean-deviation model.
result Characterizes monotonic mean-deviation measures as consistent risk measures.
In this paper we study mean-field type control problems with risk-sensitive performance functionals. We establish a stochastic maximum principle (SMP) for optimal control of stochastic differential equations (SDEs) of mean-field type, in which the drift and the diffusion coefficients as well as the performance function…
K-Means and RBF networks are shown to be equivalent under certain conditions.
problem Discrete clustering vs. continuous optimization in machine learning.
method Established variational and gradient-based equivalence between K-Means and RBF networks.
result Gradient-based updates of RBF centers recover K-Means centroid update rule.
Study uses actor-critic method for continuous-time mean-field control with entropy regularisation.
problem Continuous-time mean-field control in reinforcement learning.
method Actor-critic approach with entropy regularisation, value function alternation, and Wasserstein space parametrisation.
result Derives exact parametrisation of actor and critic functions in linear-quadratic mean-field framework.
Study shows uniform-time chaos propagation in mean field Langevin dynamics.
problem Understanding the convergence of marginal distributions in mean field dynamics.
method Assumed functional convexity of energy, used Lp-convergence and Wasserstein metrics. result Uniform-in-time propagation of chaos proved in both L2-Wasserstein and relative entropy. New method optimizes portfolio weights as functions, outperforming traditional approaches.
problem Optimizing portfolio weights in mean-variance models.
method Functional optimization approach, treating weights as functions of past values.
result Gradient-ascent algorithms can solve functional optimization problems for mean-variance portfolio management.
Study nondegenerate singularities in mean curvature flow.
problem Understanding the behavior of nondegenerate cylindrical singularities.
method New L2-distance monotonicity formula and discrete almost monotonicity. result Topology change agrees with level sets change near a critical point of a Morse function.
New method optimizes non-linear functionals over probability measures.
problem Optimizing non-linear functionals defined over probability measures.
method N-particle underdamped Langevin algorithm with spacetime discretization.
result Converges globally in total variation distance.
In this paper, we study how the mean shift algorithm can be used to denoise a dataset. We introduce a new framework to analyze the mean shift algorithm as a denoising approach by viewing the algorithm as an operator on a distribution function. We investigate how the mean shift algorithm changes the distribution and sho…
We survey some of the recent advances in mean estimation and regression function estimation. In particular, we describe sub-Gaussian mean estimators for possibly heavy-tailed data both in the univariate and multivariate settings. We focus on estimators based on median-of-means techniques but other methods such as the t…
Study of metrics on positive-definite matrices from power potential, linking to power means.
problem Understanding metrics on positive-definite matrices derived from power potential.
method Explicit expressions for geodesics and distance function derived from Hessian of power potential.
result Geodesics and distance function converge to weighted matrix geometric mean as β tends to zero.
DP-means clustering was obtained as an extension of K-means clustering. While it is implemented with a simple and efficient algorithm, it can estimate the number of clusters simultaneously. However, DP-means is specifically designed for the average distortion measure. Therefore, it is vulnerable to outliers in data, …
The study examines continuous mean curvature functions on manifolds without conjugate points.
problem Understanding properties of manifolds with specific curvature functions.
method Analyzing simply connected Riemannian manifolds with continuous horospherical mean curvature functions.
result Compact rank one manifolds without conjugate points are locally symmetric spaces of negative curvature.
This paper gives a new proof that maximal, globally hyperbolic, flat spacetimes of dimension n≥3 with compact Cauchy hypersurfaces are globally foliated by Cauchy hypersurfaces of constant mean curvature, and that such spacetimes admit a globally defined constant mean curvature time function precisely when they a…
Generative neural network simulates characteristic functions.
problem Simulating from characteristic functions inaccessible in closed form.
method Generative neural network with Maximum-Mean-Discrepancy loss.
result Universal algorithm independent of dimensionality and function properties.
Conditional kernel mean embeddings form an attractive nonparametric framework for representing conditional means of functions, describing the observation processes for many complex models. However, the recovery of the original underlying function of interest whose conditional mean was observed is a challenging inferenc…
Minimal surfaces in harmonic conformally flat space are studied.
problem Minimal surfaces in harmonic conformally flat space.
method Variational geometry approach focusing on mean curvature and Willmore functionals.
result Critical points of mean curvature functional are homeomorphic to the sphere.
The paper explores rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space.
problem Rigidity of hypersurfaces with constant shifted curvature functions in hyperbolic space.
method Characterizations and rigidity investigations for hypersurfaces with constant weighted shifted mean curvatures or ratios.
result Rigidity results for hypersurfaces with constant linear combinations of weighted shifted mean curvatures and radially symmetric shifted mean curvatures.
Study of mean curvature flow in warped products preserving equivariance.
problem Analyzing mean curvature flow in warped products.
method Deriving flow equation and proving existence for infinite time.
result Mean curvature flow exists for infinite time under specific conditions.
Deep Galerkin Method estimates value function for mean-field control problem.
problem Optimal control of agents with average welfare as the objective.
method Apply DGM to estimate value function and distribution evolution.
result Neural network approximations converge to analytical solution.
NP-PROV separates mean and variance spaces to improve function uncertainty.
problem Neural Processes fail on out-of-domain tasks due to shared latent space uncertainty.
method Separates mean and variance into function-value-related and position-related latent spaces.
result NP-PROV achieves state-of-the-art likelihood with bounded variance in drifts.
New KQEs improve probability metrics without mean function constraints.
problem Improving probability metrics without relying on mean function representations.
method Kernel quantile embeddings (KQEs) to construct new distances.
result KQEs offer a competitive alternative to MMD with near-linear cost.
New GP kernels avoid mean reversion without losing smoothness.
problem Pathological behavior in stationary GP regression.
method Improper Gaussian processes with non-positive kernels.
result Stationary, non-reverting covariance functions.
New tests for binary classification regression functions without distribution assumptions.
problem Testing regression functions in binary classification without distributional assumptions.
method Conditional kernel mean embeddings and resampling-based framework.
result Distribution-free hypothesis tests with exact type I error control.
Study expands classical harmonic function results to Riemannian manifolds.
problem Classical harmonic function properties in domains of Riemannian manifolds.
method Generalized classical results to Riemannian manifolds, including pinched negative curvature.
result Generalized results for Riemannian manifolds, including pinched negative curvature.