Revisits information metric as pseudo metric on observables, with applications to conditional independence.
problem Clarifying the nature of information metric on manifolds of observables.
method Characterizes geodesics and applies Pythagorean theorem to conditional independence.
result Illustrates computation of information metric on Diabetes dataset.
Proves a conjecture for Calabi-Yau manifolds.
problem Maximal degeneration of Calabi-Yau manifolds.
method Valuative independence condition for section ring.
result Metric SYZ conjecture proven.
Paper develops metrics for random dynamical systems using vector-valued RKHSs.
problem Creating metrics for random nonlinear dynamical systems.
method Develops metrics on random dynamical systems using Perron-Frobenius operators in vector-valued reproducing kernel Hilbert spaces (vvRKHSs). Uses operator-valued kernels and time-wise independence criteria.
result Extends existing metrics for deterministic systems and introduces kernel maximal mean discrepancy for random processes.
The number of functionally independent scalar invariants of arbitrary order of a generic pseudo--Riemannian metric on an n--dimensional manifold is determined.
Paper solves a metric-independent problem on almost Kähler 4-manifolds.
problem Find a metric-independent generalization of Bott-Chern and Aeppli numbers.
method Introduced a new approach to generalize Bott-Chern and Aeppli numbers.
result Found a solution valid on almost Kähler 4-manifolds.
New metrics for high-dimensional data improve on energy distance.
problem Testing equality of distributions and independence in high dimensions.
method Proposed new metrics inheriting properties of energy distance and others.
result Improved metrics detect homogeneity and independence in high dimensions.
This research introduces a new strategy in cluster ensemble selection by using Independency and Diversity metrics. In recent years, Diversity and Quality, which are two metrics in evaluation procedure, have been used for selecting basic clustering results in the cluster ensemble selection. Although quality can improve …
Derives derivatives and geometric framework for functions with non-independent variables.
problem Characterizing functions with non-independent variables in probabilistic models.
method Derives actual and dependent partial derivatives, dependent Jacobian matrix, and tensor metric.
result Derives gradient, Hessian, and Taylor expansion for functions with non-independent variables.
In this paper we investigate the relationship between the existence of parallel semi-Riemannian metrics of a connection and the reducibility of the associated holonomy group. The question as to whether the holonomy group necessarily reduces in the presence of a specified number of independent parallel semi-Riemannian m…
New spectral invariant generalizes analytic torsion for manifolds with geometric product structure.
problem Generalizing analytic torsion for manifolds with specific geometric product structures.
method Defined multi-torsion as a spectral invariant for compact manifolds with a local geometric product structure, proving metric-independence using Stokes' theorem.
result Proved multi-torsion is metric-independent under suitable conditions.
This work explores the connection between distances and kernels for conditional independence.
problem Measuring conditional independence in various fields like causal discovery and feature selection.
method Investigates the relationship between conditional independence measures induced by distances and reproducing kernels.
result Some kernel-based conditional independence measures are not equivalent to distance-based measures.
Classifies connections on Galilei manifolds, generalizing known results.
problem Classifying general affine connections on Galilei manifolds.
method Classification through tensor fields, extending known Galilei connections.
result Additional freedom in connections not metric-compatible, linked to clock form and space metric.
This research tackles group fairness in predictive process monitoring by ensuring predictions are independent of sensitive group membership.
problem Predictive models using biased historical data can perpetuate unfair behavior in new cases.
method Investigates independence through metrics like ΔDP and a composite loss function balancing predictive performance and fairness.
result Proposes and validates a composite loss function for training models that balance fairness and performance.
Survey on warped products and their curvature properties.
problem Understanding warped products and their curvature bounds.
method Construction and analysis of warped products between manifolds and metric spaces.
result Warp products have nice curvature properties, especially sectional and Ricci bounds.
Paper investigates conditions for independence of weak gradients on metric spaces.
problem Dependence of weak gradients on p in arbitrary metric measure spaces. method Investigates the Bounded Interpolation Property to ensure independence of weak gradients.
result Bounded Interpolation Property guarantees independence of weak gradients.
Kähler-Ricci flow singularity type is independent of initial metric.
problem Independence of singularity type for Kähler-Ricci flows.
method Analyzing solutions to the Kähler-Ricci flow on numerically effective manifolds.
result The singularity type of solutions is independent of the initial metric.
We define and address the problem of unsupervised learning of disentangled representations on data generated from independent factors of variation. We propose FactorVAE, a method that disentangles by encouraging the distribution of representations to be factorial and hence independent across the dimensions. We show tha…
Model estimates lung well-aerated volume from CT images, independent of patient and imaging parameters.
problem Lack of clear connection between quantitative metrics in lung CT images and physiology.
method Patient-independent model using Gaussian fit to lower CT histogram data points.
result Model estimates well-aerated volume (WAVE) independent of CT reconstruction parameters and respiratory cycle.
New definition of disentanglement for non-independent factors of variation.
problem Current disentanglement definitions assume independent factors of variation, limiting their applicability.
method Definition based on information theory, related to Information Bottleneck Method, proposed measurement method.
result Proposed method correctly measures disentanglement with non-independent factors of variation.
Decomposes ultrametric spaces into scaled simplices.
problem Understanding the structure of ultrametric spaces.
method Introducing metric resolutions and coarse disjoint union.
result Constructs universal spaces for asymptotic dimension 0.
Efficiently compares independence structures in log-linear models.
problem Limited direct measures for comparing log-linear model independence structures.
method Direct comparison method based on independence structure, efficient computation.
result First metric for direct comparison of log-linear model independence structures.
New metric reduces estimation error in survival model evaluation.
problem Dependent censoring complicates survival model evaluation.
method Dependent Brier score based on Archimedean copula and Copula-Graphic estimator.
result Reduces estimation error by 12-16% on average.
We analyze disentangled representations under a causal generative process, proposing new metrics and datasets.
problem Addressing fairness and interpretability through disentangled representations with a causal perspective.
method Work under a causal generative process, proposing new metrics and datasets to study disentanglement.
result Proposed metrics capture the desiderata of disentangled causal process.
We define analytic torsion of Z_2-graded elliptic complexes as an element in the graded determinant line of the cohomology of the complex, generalizing most of the variants of Ray-Singer analytic torsion in the literature. It applies to a myriad of new examples, including flat superconnection complexes, twisted analyti…
In this paper, we are interested in short homologically and homotopically independent loops based at the same point on Riemannian surfaces and metric graphs. First, we show that for every closed Riemannian surface of genus g≥2 and area normalized to g, there are at least $\ceil{\log(2g)+1}$ homotopically indep…
We consider three-dimensional Lorentzian metrics that locally admit four independent Killing vectors. Their classification is summarized, and conditions for characterizing them are found. These consist of algebraic classification of the traceless Ricci tensor, and other conditions satisfied by the curvature and its der…
The norm of Cartan torsion plays an important role for studying of immersion theory in Finsler geometry. Indeed, Finsler manifold with unbounded Cartan torsion can not be isometrically imbedded into any Minkowski space. In this paper, we find two subclasses of (?, ?)-metrics which have bounded Cartan torsion. Then, we …
Study null conformal Killing vector fields on complex surfaces.
problem Characterize pseudo-Hermitian surfaces with null vector fields.
method Analyze topological types and use vector fields to define para-hyperhermitian structures.
result Classify compact four-manifolds with orthogonal null Killing vector fields.
The study defines Finsler metrics on special surfaces and constructs geodesic currents.
problem Defining and studying Finsler metrics on specific geometric structures.
method Defined compatible Finsler distances, studied geodesics, and constructed Liouville currents.
result Constructs a Liouville current for each metric, encoding curve lengths.
The paper finds new metrics for geodesic flows with rational integrals.
problem Finding Riemannian metrics with rational integrals for geodesic flows.
method Explicit construction of metrics and integrals.
result New examples of metrics with rational integrals are provided.
For elliptic principal bundles $π:X\ra B$ over Kähler manifolds it was shown by Blanchard that X has a Kähler metric if and only both Chern classes (with real coefficients) of π vanish. For some elliptic principal bundles, when the span of these Chern classes is 1-dimensional, it was shown by Vaisman that X carry…
LCIT tests conditional independence using latent representations.
problem Detecting conditional independencies in statistical and machine learning tasks.
method Generative framework for learning latent representations of target variables X and Y, then testing for remaining dependencies.
result LCIT outperforms state-of-the-art baselines consistently under different metrics and settings.
We use an index-theoretic technique of Hitchin to show that the space of complete Riemannian metrics of nonnegative sectional curvature on certain open spin manifolds has nontrivial homotopy groups in infinitely many degrees. A new ingredient of independent interest is homotopy density of the subspace of metrics with c…
Study classifies metrics on a twice-punctured sphere, proving Delaunay metrics are complete.
problem Classifying metrics on a twice-punctured sphere.
method Analyzes Delaunay metrics and proves a sharp conformal factor bound.
result Proves that most conformal flat metrics on a twice-punctured sphere are Delaunay metrics.
AutoPC optimizes hyperparameters for the PC algorithm to improve its performance.
problem The unsupervised nature of the PC algorithm makes it difficult to tune the Type I α level. method AutoPC optimizes α directly for a chosen metric and ensures stability through a second run. result AutoPC consistently outperforms state-of-the-art methods across multiple metrics.
Let M be a pseudo-Riemannian spin manifold of dimension n and signature s and denote by N the rank of the real spinor bundle. We prove that M is locally homogeneous if it admits more than 3/4N independent Killing spinors with the same Killing number, unless n≡1(mod4) and s≡3(mod4). We …
Geodesic flows with diagonalisable integrals are orthogonal.
problem Understanding geodesic flows with specific integrals.
method Analyzing quadratic integrals for geodesic flows.
result Diagonalisable integrals imply orthogonal separation of variables.
New method for reducing dimensions of distributional data.
problem Nonlinear sufficient dimension reduction for distribution-on-distribution regression.
method Building universal kernels on metric spaces to characterize conditional independence.
result Method outperforms competing methods in synthetic and real data applications.
In this paper, we derive apriori estimates for constant scalar curvature Kähler metrics on a compact Kähler manifold. We show that higher order derivatives can be estimated in terms of a C0 bound for the Kähler potential. We also discuss some local versions of these estimates which can be of independent interest.
We show that for any solvable Lie group of real type, any homogeneous Ricci flow solution converges in Cheeger-Gromov topology to a unique non-flat solvsoliton, which is independent of the initial left-invariant metric. As an application, we obtain results on the isometry groups of non-flat solvsoliton metrics and Eins…
It is often stated in papers tackling the task of inferring Bayesian network structures from data that there are these two distinct approaches: (i) Apply conditional independence tests when testing for the presence or otherwise of edges; (ii) Search the model space using a scoring metric. Here I argue that for complete…
We describe a family of locally conformal Kaehler metrics on class 1 Hopf surfaces H containing some recent metrics constructed by P. Gauduchon and L. ornea. We study some canonical foliations associated to these metrics, in particular a 2-dimensional foliation E that is shown to be independent of the metric. We elemen…
The paper studies contact pseudo-metric manifolds with a specific curvature condition.
problem Investigating properties of contact pseudo-metric manifolds under a nullity condition.
method Introducing and analyzing (κ,μ)-contact pseudo-metric manifolds and generalized (κ,μ)-contact pseudo-metric manifolds. result The curvature of these manifolds is constant if the φ-sectional curvature is independent of the φ-section. The Gauss-Bonnet Theorem is studied for edge metrics as a renormalized index theorem. These metrics include the Poincaré-Einstein metrics of the AdS/CFT correspondence. Renormalization is used to make sense of the curvature integral and the dimensions of the L2-cohomology spaces as well as to carry out the heat equa…
We prove that two homogeneous ultra-metric spaces X,Y are coarsely equivalent if and only if Ent♯(X)=Ent♯(Y) where Ent♯(X) is the so-called sharp entropy of X. This classification implies that each homogeneous proper ultra-metric space is coarsely equivalent to the a…
New metric measures dynamical richness without relying on accuracy.
problem Lack of a reliable metric for measuring dynamical richness.
method Developed a computationally efficient, performance-independent metric based on low-rank bias.
result Metric recovers neural collapse as a special case and captures known transitions without accuracy.
Solves Yamabe problem for 3D metrics of Sobolev class W2,q.
problem Yamabe problem on closed 3-manifolds for Sobolev metrics.
method Developed elliptic theory for conformal Laplacian on rough metrics.
result Existence, regularity, and blow-up analysis for Green function.
The Freund family of distributions becomes a Riemannian 4-manifold with Fisher information as metric; we derive the induced α-geometry, i.e., the α-curvature, α-Ricci curvature with its eigenvales and eigenvectors, the α-scalar curvature etc. We show that the Freund manifold has a positive constant 0-scalar cur…