Geometric Gaussian approximations capture any distribution.
problem Approximating complex probability distributions.
method Geometric Gaussian approximations through diffeomorphisms or exponential maps.
result Geometric Gaussian approximations are universal, capturing any distribution.
Geometric tempering fails for Langevin dynamics, proving convergence limits.
problem Proving convergence and limitations of geometric tempering for Langevin dynamics.
method Theoretical investigation of geometric tempering using Langevin dynamics.
result Geometric tempering can lead to exponential time convergence and poor functional inequalities.
Study explores geometric structure and prior for beta-logistic distribution.
problem Understanding the geometric structure and prior distributions of the beta-logistic distribution.
method Exploring dual geometric structure and uncovering α-parallel prior. result The beta-logistic distribution admits an α-parallel prior for any real number α. Integrates rough geometric forms on manifolds.
problem Integrating rough forms on complex manifolds.
method Combines Whitney's geometric integration and sewing approaches.
result Introduced distributional k-forms for integration.
Geometric structures on surfaces relate to 2-plane distributions in 5D.
problem Understanding geometric properties of vector bundles and distributions.
method Study of horizontal 2-plane distributions on 5-manifolds.
result Established a connection between surface projective differential geometry and 2-plane distribution growth.
Theory explains power-law distributions without complex models.
problem Understanding power-law distributions in geometrically growing systems.
method Developed a theory of geometrically growing systems and applied it to explain various distributions.
result The geometrically growing system's distribution flattens over time, increasing relative size ratios.
We describe the geometric notion of distribution in synthetic terms, utilizing the notion of "first neighbourhood of the diagonal" from algebraic geometry. We characterize involutive distributions in combinatorial terms.
The discrete sum of geometric Brownian motions plays an important role in modeling stochastic annuities in insurance. It also plays a pivotal role in the pricing of Asian options in mathematical finance. In this paper, we study the probability distributions of the infinite sum of geometric Brownian motions, the sum of …
Study curvature and torsion in Gaussian distribution's dual coordinate system.
problem Characterize geometric invariants of Gaussian distribution.
method Investigate Riemannian curvature and torsion in a dual coordinate system of Gaussian distribution.
result Explicitly give Amari formulas in the new coordinate system.
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.
This work introduces a geometric approach to probability representation and option pricing.
problem Representing probability distributions geometrically for better understanding and approximation.
method Introducing a geometric representation of probability using implied volatility and geometric transformations.
result Any probability distribution on positive reals can be represented by a planar curve, facilitating approximation and analysis.
Geometric Variational Inference improves efficiency in complex probability distributions.
problem Efficiently accessing information in non-linear and high-dimensional probability distributions.
method Geometric Variational Inference (geoVI) uses Riemannian geometry and the Fisher information metric to construct a coordinate transformation.
result geoVI provides a more efficient variational approximation by a normal distribution, demonstrated on various problems.
SGLD proves geometric ergodicity via reflection coupling for nonconvex log-concave distributions.
problem Proving geometric ergodicity of SGLD in nonconvex, log-concave settings.
method Reflection coupling technique to handle SGLD's time discretization and minibatch issues.
result SGLD has an invariant distribution and geometric ergodicity in W1 distance. Researchers calculate the Laplace transform of a geometric Brownian motion integral.
problem Calculating the Laplace transform of a specific integral functional of geometric Brownian motion.
method Analytical calculation of the Laplace transform of the cumulative distribution and probability density functions.
result The Laplace transform of the integral functional of geometric Brownian motion is derived.
GDB bridges geometric states with improved accuracy and generality.
problem Challenges in predicting geometric state evolution in complex systems.
method Geometric Diffusion Bridge (GDB) framework using equivariant diffusion bridges.
result GDB surpasses existing methods in accurately bridging geometric states.
Geometric tempering improves sampling from distributions, with exponential convergence rates.
problem Sampling from probability distributions using gradient flow dynamics.
method Geometric tempering of the target distribution in Wasserstein and Fisher-Rao gradient flows.
result Exponential convergence in continuous and discrete time for geometric tempering.
Geometric regularisation improves statistical models by avoiding degeneracy loci.
problem Non-identifiability, singular information, and moment indeterminacy in statistical models.
method Develops the geometric regularisation of distribution-kernel pairs (T,φ) using Whitney, Thom, and Mather theorems. result Finite-dimensional weak transversality theorem for generic kernels, avoiding degeneracy strata of high codimension.
In high dimensions, the mean and geometric median are nearly identical.
problem Understanding the relationship between mean and geometric median in high-dimensional spaces.
method Analytical derivation and simulation of the distance between mean and geometric median.
result The distance between mean and geometric median vanishes with dimensionality in high dimensions.
The aim of this paper is to establish two fundamental measure-metric properties of particular random geometric graphs. We consider ε-neighborhood graphs whose vertices are drawn independently and identically distributed from a common distribution defined on a regular submanifold of RK. We show t…
Paper introduces a geometric approach to model similar probability distributions.
problem Incorporating similar probability distributions into graphical models.
method Information geometric approach to model similar distributions.
result Allows reinterpretation of existing models.
New scheme optimizes BMI through probabilistic and geometric shaping.
problem Optimizing bit-wise mutual information (BMI) for coded modulation.
method Joint optimization of BMI through probabilistic and geometric shaping.
result Joint optimization enables a continuum of constellation geometries and probability distributions.
Entropy corrections improve GBM's predictive accuracy for non-log-normal distributions.
problem Log-normal distribution limitations in GBM predictions.
method Entropy corrections to geometric Brownian motion (GBM).
result Improved predictive accuracy for non-log-normal distributions.
Estimates returns for dollar cost averaging using geometric Brownian motion.
problem Estimating returns for dollar cost averaging investing strategy.
method Uses geometric Brownian motion and log-Normal distribution to construct a lower bound for returns. Computes parameters recursively and in closed form for dollar cost averaging. Compares to lump sum investing for matching wealth distributions.
result Probability of negative returns is less than 2.5% for 40 years of annual dollar cost averaging.
In this paper, we pay our attention to geometric parameters and their applications in economics and finance. We discuss the multiplicative models in which a geometric mean and a geometric standard deviation are more natural than arithmetic ones. We give two examples from Warsaw Stock Exchange in 1995--2009 and from a b…
A recent algorithmic family for distributed optimization, DIGing's, have been shown to have geometric convergence over time-varying undirected/directed graphs. Nevertheless, an identical step-size for all agents is needed. In this paper, we study the convergence rates of the Adapt-Then-Combine (ATC) variation of the DI…
Modified Gibbs-Helmholtz equation geometric models for thermodynamics.
problem Geometric interpretation of Gibbs-Helmholtz equation in thermodynamics.
method Developed new holonomic and non-holonomic geometric models associated to Gibbs-Helmholtz equation.
result Characterized equivalence between Gibbs-Helmholtz entropy and other entropies.
The symmetric product of vector fields on a manifold arises when one studies the controllability of certain classes of mechanical control systems. A geometric description of the symmetric product is provided using parallel transport, along the lines of the flow interpretation of the Lie bracket. This geometric interpre…
Introduces a new canonical connection for Riemannian manifolds and proves Frobenius theorem geometrically.
problem Geometric proof of the Frobenius theorem on Riemannian manifolds.
method Introduces a new canonical connection and applies it to prove the Frobenius theorem.
result Geometric proof of the Frobenius theorem.
A simple geometrical proof shows that any target function can be found in a random network's neighborhood.
problem Finding any target function in a random network's neighborhood.
method Geometrical proof using a simple model of a high-dimensional sphere projected onto a low-dimensional subspace.
result High-dimensional geometry ensures that a uniform distribution over a sphere reduces to a Gaussian distribution with negligible covariances, enabling the presence of any target function in a random network's neighborhood.
In this paper the Buchen's pricing formulae of (higher order) asset and bond binary options are incorporated into the pricing formula of power binary options and a pricing formula of "the normal distribution standard options" with the maturity payoff related to a power function and the density function of normal distri…
The study finds many tight contact structures on hyperbolic 3-spheres.
problem Finding tight contact structures on hyperbolic 3-spheres.
method Constructing hyperbolic homology 3-spheres and analyzing their tight contact structures.
result Produces hyperbolic homology 3-spheres with multiple distinct tight contact structures.
Elliptical slice sampling converges geometrically, providing reliable sampling for Bayesian learning.
problem Sampling from posterior distributions in Bayesian learning.
method Elliptical slice sampling, geometric ergodicity.
result Elliptical slice sampling yields geometric convergence guarantees under weak regularity assumptions.
Geometric interpretation improves VAE performance and robustness.
problem Improving Variational Autoencoder performance and robustness.
method Introducing a geometric perspective on VAEs, sampling from the Riemannian latent space.
result Improved generation and interpolations with competitive or better performance on benchmark datasets.
We develop a comprehensive geometric framework for defining spaces G(M,E) of nonlinear generalized sections of vector bundles E→M containing spaces of distributional sections D′(M,E). Our theory incorporates classical differential geometric operations (like tensor products, covariant deri…
New approach to control diffusion processes with soft constraints.
problem Finding an optimal diffusion process with a target terminal distribution.
method Generalized Schrödinger bridge problem with soft constraints, solving for a geometric mixture of target and other distributions.
result The terminal distribution of the optimally controlled process is a geometric mixture of the target and another distribution.
A field theory is constructed in the context of parameterized absolute parallelism geometry. The theory is shown to be a pure gravity one. It is capable of describing the gravitational field and a material distribution in terms of the geometric structure of the geometry used (the parallelization vector fields). Three t…
New geometric model explains material evolution in morphogenesis.
problem Understanding non-uniform processes of material evolution.
method Groupoid theory and continuous distributions.
result Explicit equation for material distributions.
Introduces q-paths for generalizing geometric annealing paths in machine learning.
problem Limited applicability of existing path methods in machine learning.
method Develops a family of paths derived from a generalized mean, including geometric and arithmetic mixtures.
result Empirical gains in Bayesian inference and generative model evaluation.
The paper geometrically characterizes graded manifolds and proves the Frobenius theorem.
problem Understanding and characterizing graded manifolds.
method Geometric characterization and Frobenius theorem proof.
result Frobenius theorem proven for graded distributions.
The study of record statistics of correlated series is gaining momentum. In this work, we study the records statistics of the time series of select stock market data and the geometric random walk, primarily through simulations. We show that the distribution of the age of records is a power law with the exponent α lyi…
For (2+2)-dimensional nonholonomic distributions, the physical information contained into a spacetime (pseudo) Riemannian metric can be encoded equivalently into new types of geometric structures and linear connections constructed as nonholonomic deformations of the Levi-Civita connection. Such deformations and induced…
Characterizes distribution-free rates in unbalanced classification problems.
problem Minimizing error under two different distributions in unbalanced settings.
method Characterizes minimax rates over all pairs of distributions using a geometric condition.
result Identifies a dichotomy between hard and easy classes based on a three-points-separation condition.
New method learns disentangled representations using Gromov-Monge maps.
problem Learning disentangled representations from unlabelled data.
method Introduces a novel approach based on Gromov-Monge maps to preserve geometric features while aligning data distributions.
result Demonstrates effectiveness on four benchmarks, outperforming other methods.
The 1950's foundational literature on rational mechanics exhibits two somewhat distinct paradigms to the representation of continuous distributions of defects in solids. In one paradigm, the fundamental objects are geometric structures on the body manifold, e.g., an affine connection and a Riemannian metric, which repr…
We present recent results on counting and distribution of circles in a given circle packing invariant under a geometrically finite Kleinian group and discuss how the dynamics of flows on geometrically finite hyperbolic 3 manifolds are related. Our results apply to Apollonian circle packings, Sierpinski curves, Schott…
New HMC method uses asymmetrical momentum distributions and improves performance.
problem Rigorous convergence guarantees for HMC with Gaussian momentum distributions.
method New convergence analysis for HMC with general asymmetrical momentum distributions, proposing AD-HMC.
result AD-HMC exhibits geometric convergence in Wasserstein distance under certain conditions.
Study curvature invariants in sub-Riemannian manifolds.
problem Understand curvature invariants in sub-Riemannian geometry.
method Prove geometrical inequalities for submanifolds with orthogonal distributions.
result Inequalities for submanifolds with orthogonal distributions are derived.
Building upon recent advances in entropy-regularized optimal transport, and upon Fenchel duality between measures and continuous functions , we propose a generalization of the logistic loss that incorporates a metric or cost between classes. Unlike previous attempts to use optimal transport distances for learning, our …