A tensor field generates separation of variables for certain metrics.
problem Finding metrics with specific tensor field properties.
method Constructing differential invariants for a (1,1)-tensor field. result Explicit system of invariants for metrics generating separation of variables.
A new metric assesses latent variable models using data and model moments.
problem Difficulty in assessing the quality of unsupervised learning models.
method A moment-matching metric using matrix norms to compare data and model moments.
result The proposed metric is faster and has less variance than alternative methods.
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.
According to [8] if the stationary Schroedinger equation on n-dim. Riemann space admits R-separation of variables (i.e. separation of variables with a factor R), then the underlying metric is necessarily isothermic. An important sub-class of isothermic metrics are the so called binary metrics. In this paper we study co…
Proposes a new metric selection for VM-PG with improved convergence.
problem Improves convergence of VM-PG methods for ill-conditioned problems.
method Diagonal Barzilai-Borwein stepsize for adaptive metric selection.
result Improved convergence results for ill-conditioned problems.
Model-based clustering defines population level clusters relative to a model that embeds notions of similarity. Algorithms tailored to such models yield estimated clusters with a clear statistical interpretation. We take this view here and introduce the class of G-block covariance models as a background model for varia…
Study of dHYM connections on ruled surfaces with variable background metrics.
problem Finding new dHYM connections on ruled surfaces with variable metrics.
method Using momentum construction and moment map partial differential equations, coupled to scalar curvature of the background.
result Provide many new examples of dHYM connections coupled to a variable background Kähler metric.
In this paper, we address the problem of hidden common variables discovery from multimodal data sets of nonlinear high-dimensional observations. We present a metric based on local applications of canonical correlation analysis (CCA) and incorporate it in a kernel-based manifold learning technique.We show that this metr…
This research proposes a new distance metric using Isolation Forests.
problem Approximating spatial distance between data points.
method Isolation Forests for outlier detection, transforming separation depth into a distance metric.
result The method produces a distance metric invariant to variable scales and capable of handling non-linear relationships.
Defines a new metric to measure importance of predictors in complex machine learning models.
problem Measuring importance of predictors in black box machine learning models.
method Introduces a new metric, GVIM, based on true conditional expectation functions and causal interpretation.
result The GVIM can be represented as a function of Conditional Average Treatment Effect (CATE), providing a causal interpretation.
We describe algorithms for learning Bayesian networks from a combination of user knowledge and statistical data. The algorithms have two components: a scoring metric and a search procedure. The scoring metric takes a network structure, statistical data, and a user's prior knowledge, and returns a score proportional to …
New method identifies latent relationships in deep models without additional constraints.
problem Latent representations in deep latent variable models are not statistically identifiable.
method Identifies relationships between latent variables (distances, angles, volumes) under mild model conditions.
result Empirically demonstrates more reliable latent distances without additional labeled data.
Proposes a new method for variable importance using targeted learning.
problem Uncertainty quantification in variable importance metrics.
method Employing the targeted learning (TL) framework for conditional permutation variable importance.
result Improved accuracy in finite sample contexts compared to traditional methods.
A method to derive Lagrangians from field equations in metric-affine theories of gravity.
problem Deriving Lagrangians from field equations in metric-affine theories of gravity.
method Variational completion method to transform field equations into Euler-Lagrange equations and find a Lagrangian.
result Starting from metric equations, full metric equations and Lagrangian can be derived up to metric-independent terms.
Paper defines Fisher co-metric on cotangent bundle and clarifies its relation to variance.
problem Defines Fisher co-metric on cotangent bundle and clarifies its relation to variance.
method Defines Fisher co-metric directly from Fisher metric without going through tangent bundle, using a natural correspondence between cotangent vectors and random variables.
result Clarifies the relation between Fisher co-metric and variance/covariance, trivializing the Cramér-Rao inequality.
The paper connects calculus, gauge theory, and noncommutative worlds.
problem Exploring how gauge theoretic structures emerge in non-commutative calculus.
method Develops a non-commutative calculus framework to study gauge theory, Hamiltonian mechanics, and quantum mechanics.
result A covariant Levi-Civita connection is derived in this non-commutative calculus, satisfying specific properties.
We classify invariant Lagrangians of the form L(gij,gij,k,gij,kl,DI,DI,j) depending at most quadratically on the variables gij,k,gij,kl and DI,DI,j, where g is a Lorentz metric and D is a tensor field of arbitrary rank on a smooth manifold. As a corollary, we prove a conjecture of Bray'…
We model pseudo-Finsler geometries, with pseudo-Euclidean signatures of metrics, for two classes of four dimensional nonholonomic manifolds: a) tangent bundles with two dimensional base manifolds and b) pseudo-Riemannian/ Einstein manifolds. Such spacetimes are enabled with nonholonomic distributions and associated non…
A new metric tensor improves Riemann manifold Monte Carlo for Bayesian models.
problem Improving sampling efficiency in Bayesian hierarchical models.
method Metric tensor derived from log-density gradient covariance matrices.
result Metric tensors enhance sampling for complex Bayesian models.
Reproducibility in reinforcement learning is challenging: uncontrolled stochasticity from many sources, such as the learning algorithm, the learned policy, and the environment itself have led researchers to report the performance of learned agents using aggregate metrics of performance over multiple random seeds for a …
New metric measures variability in bandit algorithms, linking regret and variability.
problem Variability in multi-armed bandit allocations harms modern applications.
method Introduces allocation variability as a new metric and establishes a trade-off with regret.
result Any minimax regret-optimal algorithm must incur worst-case allocation variability Θ(T).
We construct a family of split signature Einstein metrics in four dimensions, corresponding to particular classes of third order ODEs considered modulo fiber preserving transformations of variables.
Investigates parallel spinors on Eguchi-Hanson metrics.
problem Analyzing parallel spinors on specific metrics.
method Investigated parallel spinors on Eguchi-Hanson metrics with harmonic spinors.
result Found complex 2-dimensional space of complex parallel spinors and solutions for metrics with zero scalar curvature.
Proposes ICE-based metric for better understanding interactions in black-box models.
problem Misleading global sensitivity metrics in black-box models due to interaction effects.
method Individual Conditional Expectation (ICE) curves to compute feature importance and interactions.
result ICE-based metric provides richer insights into feature importance and interactions.
This paper improves probabilistic latent models on hyperbolic spaces.
problem Uncertainty in predictions due to geodesics crossing low-data regions.
method Augmenting hyperbolic manifold with a pullback metric for probabilistic pullback metrics.
result Geodesics on pullback metric respect both geometry and data distribution, reducing uncertainty.
Two proteins are homologous if they have a common evolutionary origin, and the binary classification problem is to identify proteins in a candidate set that are homologous to a particular native protein. The feature (explanatory) variables available for classification are various measures of similarity of proteins. The…
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.
Geodesic flows on Kähler manifolds are quantum integrable when metrics are c-projectively equivalent.
problem Quantum integrability of geodesic flows on Kähler manifolds under c-projective equivalence.
method Construction of Poisson-commuting integrals of motion and their quantum counterparts.
result The geodesic flow's integrals of motion commute as quantum operators, leading to separation of variables in Schrödinger's equation.
Motivated by a classical comparison result of J. C. F. Sturm we introduce a curvature-dimension condition CD(k,N) for general metric measure spaces and variable lower curvature bound k. In the case of non-zero constant lower curvature our approach coincides with the celebrated condition that was proposed by K.-T. Sturm…
Unified Bayesian Optimisation for mixed variables improves performance.
problem Efficient optimisation of problems with both categorical and continuous variables.
method Derive value proposals from the Expected Improvement criterion to optimise both categorical and continuous variables under a single acquisition metric.
result Unified approach significantly outperforms existing methods across mixed-variable tasks.
We determine the Christoffel's symbols for the Siegel-Jacobi ball endowed with the balanced metric. We study the equations of geodesics on the Siegel-Jacobi ball. We calculate the covariant derivative of one-forms in the variables in which is expressed the balanced metric on the Siegel-Jacobi ball.
A new fairness metric for decision-making algorithms, conditioning on known fair variables.
problem Fairness issues in decision-making systems.
method Conditional fairness metric, Derivable Conditional Fairness Regularizer (DCFR), adversarial representation.
result Traditional fairness notations are special cases of the new conditional fairness notation.
Algorithms learn and test variable partitions in various groups and error metrics.
problem Learning and testing variable partitions in different groups and error metrics.
method Algorithms for agnostically learning and testing k-partitionability over various groups and error metrics. result Learning algorithms for k-partitionability with polynomial time complexity and testing with adaptive queries. We outline a new approach for solving optimization problems which enforce triangle inequalities on output variables. We refer to this as metric-constrained optimization, and give several examples where problems of this form arise in machine learning applications and theoretical approximation algorithms for graph cluste…
This paper uses diffusion models for lossy image compression, improving perceptual metrics and practicality.
problem Lossy image compression with improved perceptual metrics and practicality.
method End-to-end optimized lossy image compression using conditional diffusion models.
result The model yields stronger FID scores and competitive performance in distortion metrics.
Latent variable models (LVMs) learn probabilistic models of data manifolds lying in an \emph{ambient} Euclidean space. In a number of applications, a priori known spatial constraints can shrink the ambient space into a considerably smaller manifold. Additionally, in these applications the Euclidean geometry might induc…
New metric for disentangling multivariate representations, accounting for more complex entanglements.
problem Current disentanglement metrics fail to detect entanglements involving more than two variables.
method Partial Information Decomposition framework to analyze information sharing and propose a new disentanglement metric.
result The proposed metric correctly identifies entanglements in high-dimensional spaces.
The anholonomic frame method is generalized for non--Riemannian gravity models defined by string corrections to the general relativity and metric-affine gravity (MAG) theories. Such spacetime configurations are modeled as metric-affine spaces provided with generic off-diagonal metrics (which can not be diagonalized by …
New metric improves latent dynamics inference from neural data.
problem Limitations of co-smoothing in predicting latent dynamics.
method Few-shot co-smoothing to assess latent dynamics.
result High co-smoothing models often have extraneous dynamics, which few-shot co-smoothing detects.
Note shows equivalence of recent NEC reformulation to classical NEC for C2-metrics.
problem Consistency of null energy condition in Lorentzian length spaces.
method Shows equivalence of recent reformulation of null energy condition to classical formulation for C2-metrics. result Equivalence of recent reformulation of null energy condition to classical formulation for C2-metrics. The sequence of moments of a vector-valued random variable can characterize its law. We study the analogous problem for path-valued random variables, that is stochastic processes, by using so-called robust signature moments. This allows us to derive a metric of maximum mean discrepancy type for laws of stochastic proce…
A method for constructing explicit Calabi-Yau metrics in six dimensions in terms of an initial hyperkahler structure is presented. The equations to solve are non linear in general, but become linear when the objects describing the metric depend on only one complex coordinate of the hyperkahler 4-dimensional space and i…
This article describes the R package varrank. It has a flexible implementation of heuristic approaches which perform variable ranking based on mutual information. The package is particularly suitable for exploring multivariate datasets requiring a holistic analysis. The core functionality is a general implementation of…
We construct the Einstein equation for an invariant Riemannian metric on the exceptional full flag manifold M=G2/T. By computing a Gröbner basis for a system of polynomials of multi-variables we prove that this manifold admits exactly two non-Kähler invariant Einstein metrics. Thus G2/T turns out to be the first …
S2MAM improves semi-supervised learning by selecting relevant variables and updating similarity metrics.
problem Joint learning from labeled and unlabeled data with geometric structure.
method Bilevel optimization scheme for automatic variable selection and similarity matrix update.
result The proposed S2MAM achieves robust and interpretable predictions.
Study of deformed Hermitian Yang-Mills equations with variable Kähler metrics.
problem Solving special Lagrangian type equations with variable metrics.
method Introducing extended gauge group to couple moment maps and scalar curvature.
result Solutions satisfy a mixture of K-stability and Bridgeland-type stability.
The paper introduces toric separable geometries and finds new extremal metrics.
problem Finding explicit extremal Kähler metrics on toric manifolds.
method Introducing toric separable geometries and analyzing their moduli space.
result Explicit computation of scalar curvature and derivation of necessary conditions for extremality.
SIC measures dependency between variables, promoting feature selection.
problem Measuring and selecting features in high-dimensional data.
method Gradient regularized Integral Probability Metric (IPM) with sparsity inducing penalties.
result SIC can be used for reliable and interpretable feature selection.