Non-trivialization probability of arc system in 3D space
problem Defining and generalizing the knotting probability of an arc diagram in 3D space
method Transforming polygonal arcs in 3D space into unique arc diagrams
result Introducing and generalizing the Non-Trivialization probability (NT probability) for arc systems in 3D space
NT probability measures knotting in 3D arc systems.
problem Measuring knotting in 3D arc systems.
method Transforming polygonal arcs into unique diagrams, generalizing NT probability.
result Properties of NT probability for 3D arc systems are shown.
NNLMs optimize poorly for word probabilities due to embedding space structure.
problem NNLMs assign suboptimal probabilities to some words.
method Analyzed the inductive bias of NNLMs and the structure of word embeddings.
result Words on the convex hull have bounded probability, affecting others.
This study redefines probability for finite outcomes using axioms and examples.
problem Defining probability for finite outcomes and preserving information.
method Developed three axioms for relative probability functions and provided examples and a system for their composition.
result Proved the topological closure of the relative probability space, preserving information under limits.
A new method for adapting to label shifts using class probability matching.
problem Adapting to label shifts where class probabilities differ between source and target domains.
method Class Probability Matching using Kernel Methods (CPMKM) framework.
result CPMKM outperforms existing methods on real datasets.
Quantum probability metrics improve distribution comparison in high dimensions.
problem Challenges in comparing probability distributions, especially in high-dimensional and non-compact domains.
method Quantum probability metrics (QPMs) derived from quantum state spaces, overcoming limitations of MMD.
result QPMs offer enhanced sensitivity to subtle distributional differences in high dimensions and improve performance in generative modeling.
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…
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.
The paper explores geometry of probability measures and barycenter maps.
problem Understanding the space of probability measures and their barycenter.
method Information geometry, Fisher metric, dualistic structures, divergences, geodesics.
result Recent developments in the geometry of probability measures and barycenter.
This paper develops GPCA for probability distributions using Otto-Wasserstein geometry.
problem Analyzing modes of variation in datasets of probability measures.
method Geodesic Principal Component Analysis (GPCA) on Wasserstein space with neural networks.
result Identification of geodesic curves that capture modes of variation in probability distributions.
We give a complete characterization of both comonotone and not comonotone coherent risk measures in the discrete finite probability space, where each outcome is equally likely. To the best of our knowledge, this is the first work that characterizes \textit{and} distinguishes comonotone and not comonotone coherent risk …
Permutation invariant network learns Wasserstein metrics.
problem Understanding the space of probability measures and comparing distributions.
method Permutation invariant network mapping samples to a low-dimensional space.
result Network can generalize to compute distances between unseen densities and learn moments.
New method calibrates photometric redshift PDFs more accurately.
problem Inaccurate photometric redshift uncertainties lead to systematic errors.
method Local re-calibration using feature-space regression of Probability Integral Transform (PIT) distributions.
result Calibrated PDFs are more accurate at all locations in feature space.
Wavelet-based online learning adapts to noisy Besov spaces with high probability.
problem Minimizing integrated squared error in Besov spaces with noisy observations.
method Adaptive wavelet-based online learning algorithm that dynamically adjusts to gradient noise.
result Achieves minimax-optimal integrated squared error with high probability.
A new metric for comparing probability measures on graphs, scalable and negative definite.
problem Optimal transport's high complexity and indefiniteness for kernel machines.
method Sobolev transport metric for graph metrics, closed-form formula, negative definiteness.
result Sobolev transport yields a scalable and negative definite metric.
We describe a Groebner basis of relations among conditional probabilities in a discrete probability space, with any set of conditioned-upon events. They may be specialized to the partially-observed random variable case, the purely conditional case, and other special cases. We also investigate the connection to generali…
Develops methods to find most probable paths on complex manifolds.
problem Identifying optimal paths for manifold-valued processes, especially those with non-trivial structures.
method Constructs a general approach to defining and identifying most probable paths by measuring the Onsager-Machlup function on the anti-development of such processes.
result Derives explicit equations for development most probable paths that encompass various manifold-valued processes.
We generalize the notion of monetary value measures developed with category theory in [Adachi, 2014] by extending their base category from the category \c{hi} to the category of probability spaces Prob introduced in [Adachi and Ryu, 2016].
A pivotal problem in Bayesian nonparametrics is the construction of prior distributions on the space M(V) of probability measures on a given domain V. In principle, such distributions on the infinite-dimensional space M(V) can be constructed from their finite-dimensional marginals---the most prominent example being the…
New dimension concept for groups based on percolation probability.
problem Defining a new dimension for groups using percolation probability.
method Introducing percolation dimension pdim(G) for groups G using symmetric probability measures. result The percolation dimension pdim(G) has natural properties like monotonicity and coincides with growth rate exponents for various groups. 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.
We build a new probability measure on closed space and plane polygons. The key construction is a map, given by Knutson and Hausmann using the Hopf map on quaternions, from the complex Stiefel manifold of 2-frames in n-space to the space of closed n-gons in 3-space of total length 2. Our probability measure on polygon s…
Aggregates probability models using Wasserstein space and variational approach.
problem Model aggregation in the Wasserstein space of distributions.
method Data-driven calibration framework based on Γ-convergence. result Empirical minimizers converge to the minimizers of the actual problem.
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.
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.
Gradient flows on distributions of distributions for machine learning tasks.
problem Designing gradient flows for datasets of probability distributions.
method Representing classes as conditional distributions, modeling datasets as mixture distributions, using Wasserstein over Wasserstein (WoW) distance and gradients.
result Demonstrated gradient flows for dataset transfer and distillation tasks.
Study shows zero probability of cut locus for Fréchet mean on Riemannian manifolds.
problem Understanding the cut locus of Fréchet mean on Riemannian manifolds.
method Analytical proof and examples.
result Cut locus of Fréchet mean has zero probability.
New method speeds up optimization over probability measures.
problem High computational overhead in optimizing probability measures.
method Randomized coordinate descent on Wasserstein space.
result Significant speedups over full-gradient methods.
Optimizes functionals on probability space using ICNNs.
problem Optimizing functionals on the space of probabilities with high-dimensional convex functions.
method Proposes an approach using input-convex neural networks (ICNNs) to approximate the JKO scheme.
result Demonstrates feasibility and validity in approximating solutions of PDEs and molecular discovery.
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 study rays and co-rays in the Wasserstein space Pp(X) (p>1) whose ambient space X is a complete, separable, non-compact, locally compact length space. We show that rays in the Wasserstein space can be represented as probability measures concentrated on the set of rays in the ambient spac…
The Wasserstein metric is an important measure of distance between probability distributions, with applications in machine learning, statistics, probability theory, and data analysis. This paper provides upper and lower bounds on statistical minimax rates for the problem of estimating a probability distribution under W…
Sharp bounds for max-sliced Wasserstein distances derived for empirical distributions.
problem Estimating the expected max-sliced Wasserstein distance between a probability measure and its empirical distribution.
method Banach space version and operator norm approach for upper bounds.
result Upper bounds for max-sliced Wasserstein distances are essentially matching and sharp up to a log factor.
New method estimates and samples high-dimensional probability distributions avoiding optimization and approximation curse.
problem Estimating high-dimensional probability distributions from data samples.
method Hierarchic probability flow from coarse to fine scales, defined by conditional probabilities across scales.
result Sampling hierarchic models avoids critical slowing down at phase transitions and generates turbulence and dark matter images.
HLoOP detects outliers in hyperbolic 2-space.
problem Detecting local outliers in hyperbolic 2-space.
method Combines nearest neighbor finding and probabilistic scoring in hyperbolic space.
result Promising results on WordNet dataset.
A new kernel for probability measures based on optimal transport.
problem Efficiently comparing and modeling distributions.
method Kernel over probability measures using regularized optimal transport and Hilbertian embedding.
result The proposed kernel enables Gaussian process modeling on distributions with theoretical and computational advantages.
We investigate Fano schemes of conditionally generic intersections, i.e. of hypersurfaces in projective space chosen generically up to additional conditions. Via a correspondence between generic properties of algebraic varieties and events in probability spaces that occur with probability one, we use the obtained resul…
Formulates mechanics for probability distributions on statistical manifold.
problem Formulating mechanics for probability distributions on statistical manifold.
method Information-geometric formulation of Classical Mechanics on statistical manifold, using dually-flat connection and Hilbert bundle structure.
result Provides coherent formalism for Lagrangian and Hamiltonian mechanics on statistical bundle.
New framework transforms labeled datasets for various machine learning tasks.
problem Lack of principled methods to transform labeled datasets.
method Wasserstein gradient flows in probability space for optimization of data-generating distributions.
result Framework can impose constraints, adapt for transfer learning, or re-purpose models.
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.
Obtaining accurate and well calibrated probability estimates from classifiers is useful in many applications, for example, when minimising the expected cost of classifications. Existing methods of calibrating probability estimates are applied globally, ignoring the potential for improvements by applying a more fine-gra…
Method generates i.i.d. samples from GT data using space-time mixing.
problem Generating synthetic i.i.d. samples from high-dimensional real-valued distributions.
method Space-time mixing strategies, diffusion bridges, and score-matching.
result Optimal transport from initial to target distribution.
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.
Let S be a non-exceptional oriented surface of finite type. We discuss the action of subgroups of the mapping class group of S on the CAT(0)-boundary of the completion of Teichmueller space with respect to the Weil-Petersson metric. We show that the set of invariant Borel probability measures for the Weil-Petersson flo…
New MCMC method for complex models with large variables.
problem Inference on posterior model probabilities in large model spaces.
method Reversible genetically modified mode jumping Markov chain Monte Carlo (GMJMCMC).
result Introduced a proper MCMC with correct limiting distribution.
Extends Gaussian process theory to Banach spaces.
problem Extending Gaussian process theory to Banach spaces.
method Investigates the connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces.
result Characterizes positive definite functions that arise from covariance operators in Banach space setting.
The paper studies the geometry of probability measures on the unit circle.
problem Understanding the Riemannian geometry of probability measures on the unit circle.
method Developed an intrinsic framework using the Peter-Weyl Theorem to study the differential geometry of Wasserstein spaces of compact Lie groups.
result Explicitly demonstrated that the Wasserstein space of the unit circle is flat with vanishing curvature.
The notion of utility maximising entropy (u-entropy) of a probability density, which was introduced and studied by Slomczynski and Zastawniak (Ann. Prob 32 (2004) 2261-2285, arXiv:math.PR/0410115 v1), is extended in two directions. First, the relative u-entropy of two probability measures in arbitrary probability space…