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.
The paper explores geometric calculations on probability manifolds derived from master equations.
problem Understanding geometric properties of probability manifolds from master equations.
method Deriving geometric quantities like Levi-Civita connection, gradient, Hessian, parallel transport, and curvatures on probability manifolds.
result Calculation of geometric quantities in probability manifolds, including curvatures and connections.
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.
We study the problem of supervised learning for both binary and multiclass classification from a unified geometric perspective. In particular, we propose a geometric regularization technique to find the submanifold corresponding to a robust estimator of the class probability P(y∣x). The regularization term meas…
The space of all probability measures having positive density function on a connected compact smooth manifold M, denoted by P(M), carries the Fisher information metric G. We define the geometric mean of probability measures by the aid of which we investigate information geometry of P(M), equ…
Reflected geometric Brownian motion models are not arbitrage-free.
problem No-arbitrage condition violation in financial markets.
method Analysis of reflected geometric Brownian motion models.
result Models violate even the weakest no-arbitrage condition.
Develops geometric framework for uncertainty-aware multi-class classification.
problem Silent failure of AI models when uncertain, especially in multi-class settings.
method Geometric framework treating probability vectors as points on the (c−1)-dimensional probability simplex, using Fisher--Rao metric for calibration and uncertainty quantification. result Empirical validation shows 72.5% of errors captured while deferring 34.5% of ambiguous predictions, reducing automated decision error rates from 16.8% to 6.9%.
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.
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.
Paper offers a fast method to assess DeFi liquidation risk.
problem Assessing liquidation risk in DeFi stablecoin lending.
method Modeling collateral exchange rate as zero-drift geometric Brownian motion.
result Derives an exact formula for liquidation probability.
This paper proposes a new geometric framework for asset pricing.
problem The asymmetry between risk-neutral and physical measures in asset pricing.
method Information geometry, focusing on the relativity of probabilistic reference frames.
result Unified explanation for price fluctuations, event-driven behavior, and risk premia.
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.
Study investigates ruin probability with random premiums and risky investments.
problem Ruin probability with random premiums and risky investments.
method Laplace transform applied to a model with geometric Brownian motion.
result Asymptotic behavior of ruin probability for large initial capital values.
We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We then prove this fact.
Study classifies submanifolds in probability simplex.
problem Classifying submanifolds in the probability simplex.
method Complete classification through geometric analysis.
result Doubly totally-umbilical submanifolds identified and classified.
This letter introduces an abstract learning problem called the "set embedding": The objective is to map sets into probability distributions so as to lose less information. We relate set union and intersection operations with corresponding interpolations of probability distributions. We also demonstrate a preliminary so…
This paper studies geometrical structure of the manifold of escort probability distributions and shows its new applicability to information science. In order to realize escort probabilities we use a conformal transformation that flattens so-called alpha-geometry of the space of discrete probability distributions, which…
Random hyperbolic surfaces are mostly tangle-free, with geometric implications.
problem Understanding the structure of random hyperbolic surfaces.
method Introduced and analyzed L-tangle-free compact hyperbolic surfaces.
result Random surfaces are (a log g)-tangle-free for any a < 1, almost optimal.
On the probability simplex, we can consider the standard information geometric structure with the e- and m-affine connections mutually dual with respect to the Fisher metric. The geometry naturally defines submanifolds simultaneously autoparallel for the both affine connections, which we call {\em doubly autoparallel s…
Formula found for probability of random triangles on flat tori being homotopically trivial.
problem Calculating the probability of random triangles on flat tori being homotopically trivial.
method Reduced problem to new invariant of measurable sets in the plane unchanged by area-preserving affine transformations.
result Probability is minimized on rectangular tori and maximized on regular hexagonal tori.
A novel method compares 3D point clouds using information geometry.
problem Comparing 3D point clouds in machine learning applications.
method Interprets point clouds as probability density functions on a statistical manifold, using GMM and Modified Symmetric KL divergence.
result Demonstrates effectiveness through various case studies.
If M is a smooth compact connected Riemannian manifold, let P(M) denote the Wasserstein space of probability measures on M. We describe a geometric construction of parallel transport of some tangent cones along geodesics in P(M). We show that when everything is smooth, the geometric parallel transport agrees with earli…
The geometric approach to optimal transport and information theory has triggered the interpretation of probability densities as an infinite-dimensional Riemannian manifold. The most studied Riemannian structures are Otto's metric, yielding the L2-Wasserstein distance of optimal mass transport, and the Fisher--Rao me…
A new method for averaging probability distributions based on optimal weak mass transport.
problem Averaging probability distributions in a geometric way.
method Weak barycenters based on optimal weak mass transport.
result Extracts common geometric information shared by all input distributions.
We propose a novel algebraic framework for treating probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of finding the subspace on which several probability distributions agree. Instead of minimizing an objective f…
In this paper we revisit the integral functional of geometric Brownian motion It=∫0te−(μs+σWs)ds, where μ∈R, σ>0, and (Ws)s>0 is a standard Brownian motion. Specifically, we calculate the Laplace transform in t of the cumulative distribution function and of the probability density …
Explains how geometry and statistics intertwine, focusing on information geometry.
problem Understanding the interplay between geometry and statistics.
method Introduces differential topology, geometry, probability, and (pre-)Frobenius manifolds.
result Discovers connections between geometry and statistics, particularly in information geometry.
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 …
The volume of a credal set correlates with epistemic uncertainty in binary classification but not in multi-class.
problem Representing and quantifying epistemic uncertainty in machine learning.
method Examined the geometric representation of credal sets as d-dimensional polytopes and their volume as a measure of uncertainty. result The volume of a credal set is a meaningful measure of epistemic uncertainty in binary classification but not in multi-class.
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.
For undirected graphs, the Ricci curvature introduced by Lin-Lu-Yau has been widely studied from various perspectives, especially geometric analysis. In the present paper, we discuss generalization problem of their Ricci curvature for directed graphs. We introduce a new generalization by using the mean transition proba…
Study on geometric Jensen-Shannon divergence for Gaussian measures in Hilbert space.
problem Computing divergence between Gaussian measures in infinite-dimensional Hilbert space.
method Closed form expression and regularization for divergence calculation.
result Closed form expression and regularization for Geometric Jensen-Shannon divergence.
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.
This study presents a long-term alternative formula for stock price variation described by a geometric Brownian motion on the basis of median instead of mean or expected values. The proposed method is motivated by the observation made in remote fields, where optimality of bet-hedging or diversification strategies is ex…
New discrepancy function compares discrete probability measures considering space geometry.
problem Comparing discrete probability measures in a geometrically meaningful way.
method Proposes the Fourier Discrepancy Function, proving convexity, differentiability, and providing gradient formula.
result Proves the Fourier Discrepancy is convex, twice differentiable, and provides an explicit gradient formula.
Introduces a new geometric method for optimal experimental design.
problem Restrictive invariance properties of traditional OED approaches based on probability densities.
method Mutual transport dependence (MTD) using optimal transport theory.
result Demonstrates high-quality designs and flexibility compared to standard methods.
Random covers of hyperbolic surfaces follow a specific probability measure.
problem Understanding the distribution of random covers of hyperbolic surfaces.
method Analyzing random covers subject to specific group isomorphism conditions.
result Asymptotic distribution of random covers according to a probability measure on moduli space of metric graphs.
By studying the group of rigid motions, PSH(1), in the 3D-Heisenberg group H1, we define the density and the measure for the sets of horizontal lines. We show that the volume of a convex domain D⊂H1 is equal to the integral of length of chord over all horizontal lines intersecting D. As the classical r…
New game approximates mean curvature flow evolution.
problem Approximating geometric mean curvature flow evolution.
method Two-player zero-sum game with probabilistic elements.
result Value function approximates mean curvature flow.
Introduces geometric formulation of EM algorithm for robust inference and various applications.
problem Statistical inference with missing data or unobservables.
method Information geometric formulation of EM algorithm and its extensions.
result Outlier-robust inference algorithm and various applications in deep learning.
The long-term dependence of Bitcoin (BTC), manifesting itself through a Hurst exponent H>0.5, is exploited in order to predict future BTC/USD price. A Monte Carlo simulation with 104 geometric fractional Brownian motion realisations is performed as extensions of historical data. The accuracy of statistical inferen…
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…
New method recovers manifold distances from noisy data.
problem Reconstructing manifold geometry from noisy distance measurements.
method Develops new framework to estimate L2-norms of expectation-functions, uses geometric clusters to recover distances.
result Recovery of true distances up to an additive error of O(ε log ε⁻¹) under mild geometric assumptions.
We formulate the Riemannian calculus of the probability set embedded with L2-Wasserstein metric. This is an initial work of transport information geometry. Our investigation starts with the probability simplex (probability manifold) supported on vertices of a finite graph. The main idea is to embed the probability m…
We obtain a lower asymptotic bound on the decay rate of the probability of a portfolio's underperformance against a benchmark over a large time horizon. It is assumed that the prices of the securities are governed by geometric Brownian motions with the coefficients depending on an economic factor, possibly nonlinearly.…
Develops efficient projections for multivariate probability measures.
problem Estimating causal effects and optimal weights in multivariate data.
method Tangent Wasserstein projections using generalized geodesics.
result Provides a unique solution for causal inference and optimal weights.
Estimating dimension from sparse random geometric graphs.
problem Estimating the dimension of the underlying space from a random geometric graph.
method An estimator of dimension is derived using the adjacency matrix of the graph, under specific conditions on the density and threshold.
result An estimator converges to the true dimension with high probability under certain conditions.