New geometric structures defined on SPD matrices for better understanding.
problem Understanding SPD matrices and their geometric properties.
method Introducing Finslerian and dual information-geometric structures on James' bicone domain.
result Geodesics correspond to straight lines in coordinate systems, and new dissimilarities generalize existing ones.
Paper finds local normal forms for wavefronts in flat coordinates.
problem Understanding local diffeomorphic types of wavefronts.
method Using connections and the metric, criteria for wavefront types are derived in affine flat coordinates.
result Local normal forms of e/m-wavefronts in affine flat coordinates are derived. New accelerators for EM improve convergence speed in complex mixture models.
problem Improving the convergence speed of the EM algorithm for complex mixture models.
method Derive a new operator connecting global descent and local convergence, and use it to develop two acceleration strategies.
result Two new acceleration strategies (G-Accelerator and Geo-Adaptive) significantly improve EM algorithm performance.
We examine the Banados-Teitelboim-Zanelli (BTZ) black hole in terms of the information geometry and consider what kind of quantum information produces the black hole metric in close connection with the anti-de Sitter space/conformal field theory (AdS/CFT) correspondence. We find a Hessian potential that exactly produce…
This research smooths out fluid equations to avoid sudden shocks.
problem Formation of shock singularities in compressible fluid equations.
method Information geometric regularization of unidimensional pressureless Euler equations.
result Smooth global solutions without artificial viscosity.
This paper analyzes Barlow Twins' representation efficiency using information-geometric methods.
problem Understanding and comparing the efficiency of self-supervised learning methods.
method Introduces an information-geometric framework to quantify representation efficiency and applies it to Barlow Twins.
result Proves that Barlow Twins achieves optimal representation efficiency (η=1).
Information Geometric Causal Inference (IGCI) is a new approach to distinguish between cause and effect for two variables. It is based on an independence assumption between input distribution and causal mechanism that can be phrased in terms of orthogonality in information space. We describe two intuitive reinterpretat…
This report concerns the problem of dimensionality reduction through information geometric methods on statistical manifolds. While there has been considerable work recently presented regarding dimensionality reduction for the purposes of learning tasks such as classification, clustering, and visualization, these method…
Introduces new info-geometric structure for dynamics on graphs and hypergraphs.
problem Modeling dynamics on discrete structures like graphs and hypergraphs.
method Introduces two dually flat structures: one on vertex space and another on edge space.
result Extends gradient flows to include nonequilibrium dynamics.
FRESH combines patient-level and aggregate-level data for better clinical decision making.
problem Combining patient-level and aggregate-level data for clinical decision making.
method FRESH method that re-calibrates a patient-level model to match specified aggregate statistics.
result Unified data-efficient model for clinical decision making.
The Fisher information metric is an important foundation of information geometry, wherein it allows us to approximate the local geometry of a probability distribution. Recurrent neural networks such as the Sequence-to-Sequence (Seq2Seq) networks that have lately been used to yield state-of-the-art performance on speech…
A family of replicator-like dynamics, called the escort replicator equation, is constructed using information-geometric concepts and generalized information entropies and diverenges from statistical thermodynamics. Lyapunov functions and escort generalizations of basic concepts and constructions in evolutionary game th…
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…
Upper bound on CRN reaction rates derived using information geometry.
problem Challenging task of deriving an upper bound on reaction rates of nonlinear, discrete CRNs.
method Information geometric approach using natural gradient.
result Validated through numerical simulations, demonstrating faster convergence in specific CRNs.
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.
The study proves that certain noncompact Hessian manifolds are diffeomorphic to R^n.
problem Characterizing complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature.
method Using a geometric flow on noncompact affine Riemannian manifolds, constructing Hessian metrics, and proving diffeomorphism.
result Complete noncompact Hessian manifolds with nonnegative Hessian sectional curvature are diffeomorphic to R^n if their tangent bundle has maximal volume growth.
New metric for probability measures connects physics and geometry.
problem Developing a new metric for probability measures.
method Transport Hessian metric, formulated dynamical systems.
result Connections to physics equations and mathematical models.
Investigates regularity of solutions to complex Hessian equation.
problem Regularity of solutions to complex Hessian equation.
method Analyzes solutions to Dirichlet problem with specific density condition.
result Establishes conditions for regularity of solutions.
Abstract: Survey on quadratic Hessian equations, their properties, and open problems.
problem Understanding quadratic Hessian equations and their solutions.
method Survey and review of existing research.
result Survey of entire solutions, viscosity solutions, and Hessian estimates.
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…
Paper solves Hessian equations on Kähler manifolds.
problem Solving Hessian equations on Kähler manifolds.
method Combines elementary symmetric functions; provides sufficient and necessary condition.
result Generalizes results for Hessian and Hessian quotient equations.
New Hessian estimates for heat equations on manifolds.
problem Estimating Hessian matrices for heat-type equations on Riemannian manifolds.
method Using Bismut-Stroock Hessian formula, with explicit coefficients and delay/growth rate functions.
result Novel backward weak Harnack inequality and precise pointwise Hessian estimates for eigenfunctions.
Adaptive sampling improves graph diffusion models by maintaining uniform information speed.
problem Standard diffusion models overlook non-homogeneous dynamics on complex manifolds.
method Information-geometric framework using Fisher-Rao metric and Drift Variation Score (DVS).
result DVS solver ensures uniform rate of distributional change, improving structural fidelity and efficiency.
Study calculates Hessian of Busemann function on Damek-Ricci spaces.
problem Calculating Hessian of Busemann function on Damek-Ricci spaces.
method Calculates eigenvalues of Hessian and proves positive definiteness.
result Hessian of Busemann function is positive definite.
The paper shows infinitely many components in Floer Hessians space.
problem Understanding the structure of Floer Hessians.
method Proving the existence of infinitely many connected components.
result Proves infinitely many connected components in Floer Hessians space.
The paper describes flat Hessian metrics on surfaces and their potentials.
problem Understanding Hessian metrics on surfaces.
method Theoretical description and explicit construction using integrable systems.
result Explicit construction of potentials for flat Hessian metrics on surfaces.
The paper quantizes Hessian structures on R^2 using KV-algebras.
problem Quantizing Hessian structures on a 2D space.
method Deformation quantization within Koszul-Vinberg algebras.
result Established links between deformation theory and Hessian geometry.
Information-Geometric Optimization (IGO) is a unified framework of stochastic algorithms for optimization problems. Given a family of probability distributions, IGO turns the original optimization problem into a new maximization problem on the parameter space of the probability distributions. IGO updates the parameter …
We prove that, in dimensions greater than 2, the generic metric is not a Hessian metric and find a curvature condition on Hessian metrics in dimensions greater than 3. In particular we prove that the forms used to define the Pontryagin classes in terms of the curvature vanish on a Hessian manifold. By contrast all anal…
Superintegrable systems on curved manifolds found to have Hessian structures.
problem Characterizing superintegrable systems on curved manifolds.
method Identifying and computing Hessian coordinates for superintegrable systems.
result Examples of superintegrable systems in 2D and 3D have natural Hessian coordinates.
Curved Frobenius manifolds link to Hessian metrics in geometry.
problem Understanding curved Frobenius manifolds and their relation to Hessian metrics.
method Analyzing the relationship between curved Frobenius structures and Hessian metrics on spaces with non-vanishing curvature.
result Consistent curved Frobenius structures on constant curvature spaces are linked to Hessian metrics.
New rigidity results for a generalized Ricci-Hessian equation on manifolds.
problem Understanding rigidity in generalized Ricci-Hessian equations on manifolds.
method Proving new rigidity results related to a generalized Ricci-Hessian equation.
result New rigidity results for the generalized Ricci-Hessian equation on Riemannian manifolds.
Optimal transport and information geometry both study geometric structures on spaces of probability distributions. Optimal transport characterizes the cost-minimizing movement from one distribution to another, while information geometry originates from coordinate-invariant properties of statistical inference. Their con…
Study classifies 3D Hessian manifolds, proving their topology.
problem Global topology of 3D Hessian manifolds.
method Proved structure and analyzed Betti numbers.
result Complete topological classification of 3D Hessian manifolds.
Criterion for solvability of complex 2-Hessian equation on compact Kähler manifolds.
problem Solvability of complex 2-Hessian equation on compact Kähler manifolds.
method Nakai--Moishezon-type criterion associated with the complex 2-Hessian equation.
result Criterion equivalent to existence of a smooth 2-admissible representative in complex dimension three.
Establishes a concavity property for positive Hessian quotient operators.
problem Analyzing positive Hessian quotient operators on Riemannian manifolds.
method Proves a special concavity property and a Jacobi inequality.
result Proves a Jacobi inequality for symmetric tensors.
Estimates for complex Hessian equations on Hermitian manifolds.
problem Establishing estimates for solutions to complex Hessian equations.
method Using concavity inequality for complex sum-of-Hessian operators.
result Second-order estimates for admissible solutions on Hermitian manifolds.
This report provides an in-depth overview over the implications and novelty Generalized Variational Inference (GVI) (Knoblauch et al., 2019) brings to Deep Gaussian Processes (DGPs) (Damianou & Lawrence, 2013). Specifically, robustness to model misspecification as well as principled alternatives for uncertainty quantif…
Derives Hessian estimates for Lagrangian mean curvature equation.
problem Lagrangian mean curvature equation with supercritical phase and bounded second derivatives.
method Derives a priori interior Hessian estimates.
result Hessian estimates for Lagrangian mean curvature equation.
A new unbiased Hessian estimator for expectation-based objectives.
problem Estimating Hessian for objectives with non-reparameterizable nodes.
method GO Hessian estimator for expectation-based objectives.
result GO Hessian provides unbiased and low-variance estimation of Hessian.
Constructs homogeneous Kähler structures on tangent bundles of Hessian manifolds.
problem Creating Kähler structures on tangent bundles of Hessian manifolds.
method Endowing Hessian manifolds with Kähler structures using group actions and homothetic vector fields.
result Homogeneous conformally Kähler structures on tangent bundles of selfsimilar Hessian manifolds.
Paper proves inequalities on Hermitian manifolds with applications to bounded solutions.
problem Establishing mixed Hessian inequalities on Hermitian manifolds.
method Weak convergence theorem of complex Hessian operators and general mixed Hessian inequality.
result Existence of bounded solutions of complex Hessian equations.
A selfsimiar manifold is a Riemannian manifold (M,g) endowed with a homothetic vector field ξ. We characterize global selfsimilar manifolds and describe the structure of local selfsimilar manifolds. We prove that any selfsimilar manifold with a potential homothetic vector field is a conical Riemannian ma…
Study of contravariant pseudo-Hessian manifolds and their Poisson structures.
problem Understanding properties of contravariant pseudo-Hessian manifolds.
method Investigation of flat connections and symmetric bivector fields satisfying a contravariant Codazzi equation.
result Association of a Poisson tensor to contravariant pseudo-Hessian manifolds.
This paper uncovers the low-rank structure of neural network Hessians.
problem Understanding the structure of Hessians in neural networks.
method Proposes a decoupling conjecture to decompose layer-wise Hessians into Kronecker products of smaller matrices.
result Proves the structure of top eigenspaces in 2-layer networks and shows high overlap in top eigenvectors across different models.
The paper studies efficient Hessian fitting methods for stochastic optimization.
problem Efficient Hessian fitting for stochastic optimization.
method Preconditioned Stochastic Gradient Descent (PSGD) method and Lie groups.
result Hessian fitting problem is strongly convex in certain Lie groups.
Study solves complex Hessian equation on Hermitian manifolds.
problem Solving Hessian equations on Hermitian manifolds with mixed structure.
method Derive a priori estimates and solve Dirichlet problem under conditions.
result Solvability of the Dirichlet problem for mixed Hessian equations.
New divergence measures improve KL approximation.
problem Improving KL divergence approximation without AC condition.
method Introduced α-geodesical skew divergence. result Properties of α-geodesical skew divergence studied.