Geometric methods solve sampling, optimisation, inference, and adaptive decision-making.
problem Efficient solutions for sampling, optimisation, inference, and adaptive decision-making.
method Derive algorithms exploiting geometric structures of Hamiltonian systems, Hilbertian subspaces, and information geometry.
result Wide range of geometric theories emerge in these fields, enabling efficient solutions.
GeoPhy uses geometric gradients to efficiently infer phylogenetic trees from molecular data.
problem Challenges in accurately inferring species relationships from molecular data due to combinatorially vast tree topologies.
method Introduces a novel, fully differentiable formulation of phylogenetic inference using geometric spaces and variational Bayesian methods.
result Significantly outperforms other approximate Bayesian methods in inferring phylogenetic trees.
Geometric analysis improves convergence of variational inference.
problem Challenges in analyzing convergence of variational inference due to non-convexity and non-smoothness.
method Exploits exponential family structure and Bregman divergences to geometrically analyze the optimization landscape.
result Establishes non-asymptotic convergence rates for gradient descent algorithms.
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.
This paper presents a unified geometric framework for the statistical analysis of a general ill-posed linear inverse model which includes as special cases noisy compressed sensing, sign vector recovery, trace regression, orthogonal matrix estimation, and noisy matrix completion. We propose computationally feasible conv…
This work investigates a mixture of LMC and RMHMC with MMALA for geometric ergodicity.
problem Lack of geometric ergodicity study in Riemannian manifold and Lagrangian Monte Carlo methods.
method Investigates a mixture of LMC and RMHMC with MMALA to achieve geometric ergodicity.
result Demonstrates geometric ergodicity in the mixture of LMC and RMHMC with MMALA.
We propose a geometric algorithm for topic learning and inference that is built on the convex geometry of topics arising from the Latent Dirichlet Allocation (LDA) model and its nonparametric extensions. To this end we study the optimization of a geometric loss function, which is a surrogate to the LDA's likelihood. Ou…
Develops geometric causal models for causal inference from dependent data.
problem Causal inference from structured, dependent data (e.g., spatial, network, molecular).
method Geometric causal models (GCMs) exploiting symmetries of data generating process, combining group theory, ergodic theory, and Bayesian inference.
result Establishes identification and estimation of causal effects from dependent data.
NeRF-VAE generates 3D scenes with geometric structure from few images.
problem Generating 3D scenes from few images with geometric consistency.
method Combines NeRF and VAE, incorporating shared geometric structure.
result NeRF-VAE can infer and render geometrically-consistent scenes from unseen environments.
This paper develops geometric tools for causal inference using information flow concepts.
problem Developing a geometric interpretation of causal inference from probabilistic measures.
method Introducing a new measure, GeoC_{y
ightarrow x}, based on fractal correlation dimension.
result Avoids boundedness issues in transfer entropy, providing a more robust measure of causal inference.
This paper proposes a new method for learning covers of geometric datasets to improve topological inference and visualization.
problem Improving topological inference and visualization of large-scale geometric datasets.
method Proposes a method for learning topologically-faithful covers of geometric datasets using optimization.
result Simplicial complexes obtained from learned covers outperform standard methods in terms of size and representation of large-scale topology.
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. We develop new models and algorithms for learning the temporal dynamics of the topic polytopes and related geometric objects that arise in topic model based inference. Our model is nonparametric Bayesian and the corresponding inference algorithm is able to discover new topics as the time progresses. By exploiting the c…
Study shows TAP free energy minimization provides better posterior inference in high-dimensional linear models.
problem Deviation from true posterior mean and underestimation of posterior uncertainty in variational inference.
method Minimization of TAP free energy in a high-dimensional asymptotic framework, showing geometric and statistical properties.
result Local minimizer of TAP free energy provides consistent estimate of posterior marginals and correctly calibrated posterior inference.
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.
In this study, we present and analyze a framework for geometric and topological estimation for mapping of unknown environments. We consider agents mimicking motion behaviors of cyborg insects, known as biobots, and exploit coordinate-free local interactions among them to infer geometric and topological information abou…
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.
Geometric framework analyzes bias in variational inference for posterior functionals.
problem Analyzing the bias of posterior functionals under variational approximations.
method Developed a geometric framework to evaluate the bias of posterior functionals using the variational tangent space.
result The leading-order bias of a posterior functional is determined by its component orthogonal to the variational tangent space.
Bayesian models for networks are often misspecified, leading to overconfident inference.
problem Real-world networks violate assumptions of geometry and link function in latent space models.
method Proposes a generalized posterior framework for random geometric graphs, using Link-Sequential R-SafeBayes to adaptively tune posterior regularization.
result Improved calibration and better link prediction performance demonstrated on synthetic and real-world networks.
Deep learning models complex multivariate extremes using geometric shapes.
problem Modeling complex extremal dependencies in high-dimensional data.
method Geometric representation and deep learning for flexible semi-parametric models.
result First approach to modeling limit sets using deep learning for high-dimensional data.
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…
BBVI with STL converges geometrically under perfect specification, with quadratic variance bound.
problem Convergence rate of BBVI with STL estimator.
method Proved geometric convergence rate with quadratic variance bound for BBVI with STL estimator.
result BBVI with STL converges geometrically under perfect variational family specification.
Develops methods to analyze feature-outcome associations in subpopulations.
problem Challenges in understanding feature-outcome associations in high-dimensional data.
method Geometric decomposition framework using gradient flow and co-monotonicity decomposition.
result Identifies context-dependent patterns and improves statistical power and interpretability.
New method learns dynamics from sparse data using geometric constraints.
problem Learning dynamics from sparse, undersampled data.
method Reformulates inference as a stochastic control problem, using geometry-driven path augmentation.
result Accurately recovers stochastic dynamics from extremely undersampled data.
Paper introduces geometry-aware normalizing flows for improved causal inference.
problem Disparity between sample and population distributions in causal inference.
method Integrates continuous normalizing flows with parametric submodels, employing Wasserstein gradient flows and optimal transport.
result Significantly reduces parameter estimation bias and variance in finite-sample settings.
Cryo-em images are found to be low-dimensional.
problem Understanding the geometric structure of cryo-em data.
method Applied manifold learning techniques to CryoSBI representations.
result Cryo-em data inherently populate low-dimensional manifolds.
Extends geometric approach to model non-stationary extremal dependence.
problem Capturing evolving extremal dependence in multivariate data.
method Geometric framework for non-stationary multivariate extreme value modelling.
result Framework can capture various dependence forms and is robust to different model formulations.
A novel circuit motif uses sister cells for inference with correlated priors.
problem Structured priors in neural systems pose architectural challenges.
method Proposes a novel circuit motif using sister cells to implement correlated priors without direct interactions.
result Demonstrates the efficacy of correlated priors for inference in noisy environments.
This paper develops embeddings that preserve likelihood-based statistical inference.
problem Modern machine learning embeddings destroy the geometric structure required for likelihood-based inference.
method Developed a rigorous theory of likelihood-preserving embeddings and introduced the Likelihood-Ratio Distortion metric.
result Controlling the distortion Δn is necessary and sufficient for preserving inference. In this letter we borrow from the inference techniques developed for unbounded state-cardinality (nonparametric) variants of the HMM and use them to develop a tuning-parameter free, black-box inference procedure for Explicit-state-duration hidden Markov models (EDHMM). EDHMMs are HMMs that have latent states consisting…
We review the information geometry of linear systems and its application to Bayesian inference, and the simplification available in the Kähler manifold case. We find conditions for the information geometry of linear systems to be Kähler, and the relation of the Kähler potential to information geometric quantities such …
We propose Dirichlet Simplex Nest, a class of probabilistic models suitable for a variety of data types, and develop fast and provably accurate inference algorithms by accounting for the model's convex geometry and low dimensional simplicial structure. By exploiting the connection to Voronoi tessellation and properties…
Cooperation information sharing is important to theories of human learning and has potential implications for machine learning. Prior work derived conditions for achieving optimal Cooperative Inference given strong, relatively restrictive assumptions. We relax these assumptions by demonstrating convergence for any disc…
Algorithm reconstructs vertex positions in random geometric graphs with improved accuracy.
problem Reconstructing vertex positions in random geometric graphs with high accuracy.
method Hybrid of graph distances and short-range estimates based on common neighbors.
result Algorithm reconstructs vertex positions with error of O(nβ), improving over previous results. 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…
Transformers mimic Bayesian reasoning in controlled settings, revealing geometric mechanisms.
problem Verifying if transformers perform Bayesian reasoning rigorously in natural data.
method Constructing Bayesian wind tunnels with known posteriors and proving memorization impossibility.
result Transformers achieve 10−3-10−4 bit accuracy in Bayesian posteriors, while MLPs fail. The problem of completing high-dimensional matrices from a limited set of observations arises in many big data applications, especially, recommender systems. Existing matrix completion models generally follow either a memory- or a model-based approach, whereas, geometric matrix completion models combine the best from b…
In this paper, we examine a geometrical projection algorithm for statistical inference. The algorithm is based on Pythagorean relation and it is derivative-free as well as representation-free that is useful in nonparametric cases. We derive a bound of learning rate to guarantee local convergence. In special cases of m-…
New method uses hyperbolic space for faster phylogenetic tree inference.
problem Inefficient Euclidean-based phylogenetic inference in high dimensions.
method Developed novel hyperbolic extensions of sequential search algorithms and variational inference methods.
result Improved speed, scalability and performance in phylogenetic inference.
Paper bounds tensor decomposition's RLCT, aiding Bayesian inference.
problem Unclear mathematical property of tensor decomposition.
method Algebraic geometrical method for upper bound derivation.
result Upper bound of real log canonical threshold (RLCT) derived.
We present a general method for privacy-preserving Bayesian inference in Poisson factorization, a broad class of models that includes some of the most widely used models in the social sciences. Our method satisfies limited precision local privacy, a generalization of local differential privacy, which we introduce to fo…
Paper estimates manifold reach using convexity defect function.
problem Estimating the reach of submanifolds from point clouds.
method Relates reach to convexity defect function, uses stability properties, and combines with recent estimators.
result Uniform expected loss bound and minimax rate lower bounds for reach estimation are provided.
New risk models use chaotic attractors to predict extreme events.
problem Predicting Black Swan events in financial markets.
method Combining heavy-tailed priors with chaotic dynamics (Lorenz and Rossler systems).
result Models generate volatility clustering, fat tails, and extreme events.
Bayesian design improves accuracy without extra cost.
problem Nested inference in complex systems limits BED accuracy and efficiency.
method Grouped geometric pooled posterior with EKI formulation.
result Improved accuracy and stable estimators at comparable cost.
Efficient unsupervised training and inference in deep generative models remains a challenging problem. One basic approach, called Helmholtz machine, involves training a top-down directed generative model together with a bottom-up auxiliary model used for approximate inference. Recent results indicate that better genera…
GNPE improves inference for astrophysical systems.
problem Efficiently incorporating geometric properties like equivariances in neural density estimation.
method GNPE integrates equivariances into neural posterior estimation, standardizing data pose while estimating parameters.
result GNPE achieves state-of-the-art accuracy in astrophysical binary black hole inference, reducing inference times by 3 orders of magnitude.
The paper connects neural network ensembles to Bayesian inference using variational methods.
problem Explaining the behavior of ensemble methods in neural networks.
method Deriving conditions for ensemble optimization to reduce divergence to the posterior distribution.
result Ensemble methods can be a valid alternative to approximate Bayesian inference.
Stein variational gradient descent (SVGD) is a particle-based inference algorithm that leverages gradient information for efficient approximate inference. In this work, we enhance SVGD by leveraging preconditioning matrices, such as the Hessian and Fisher information matrix, to incorporate geometric information into SV…