Synthetic Ricci flows defined for metric measure spaces.
problem Characterizing Ricci flows for non-smooth spaces.
method Heat flow, optimal transport, volume asymptotics.
result Equivalent characterizations of weighted Ricci flow.
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48 results for “synthetic notions”
Develops new synthetic Ricci flow concepts for metric measure spaces.
problem No specific problem stated; focuses on new mathematical concepts.
method Formulated in terms of dynamic convexity and local concavity of entropy, and global/short-time asymptotic transport cost estimates.
result Shows these properties characterise smooth (weighted) Ricci flows.
We present an axiomatic/synthetic account of the Huygens Principle of wave fronts. The primitive notions are "touching", and (a weak notion of ) metric. The paper simplifies some of the exposition of the author's "Metric spaces and SDG", Theory and Appl. of Categories 32 (2017), 803-822
We describe the geometric notion of distribution in synthetic terms, utilizing the notion of "first neighbourhood of the diagonal" from algebraic geometry. We characterize involutive distributions in combinatorial terms.
Proves a synthetic Lorentzian Cartan-Hadamard theorem.
problem Formulates and proves a theorem for Lorentzian geometry.
method Uses an appropriate notion of local concavity for Lorentzian (pre-)length spaces.
result Establishes existence and uniqueness of timelike geodesics.
We prove that in two dimensions the synthetic notions of lower bounds on sectional and on Ricci curvature coincide.
Survey talk on certain aspects of the subject, stressing the neighbor relation as a basic notion in differential geometry.
Develops optimal transport in Lorentzian spaces with synthetic curvature bounds.
problem Synthetic curvature bounds for Lorentzian spaces.
method Optimal transport, convexity analysis of entropy functionals.
result Synthetic notion of timelike Ricci curvature lower bounds.
Equivalence found between smooth and synthetic timelike curvature bounds.
problem Understanding timelike sectional curvature bounds in spacetime geometry.
method Established equivalence between sectional curvature bounds on timelike planes and synthetic timelike bounds.
result Equivalence proved for sectional curvature bounds on timelike planes and synthetic timelike bounds on strongly causal spacetimes.
We study the low-regularity (in-)extendibility of spacetimes within the synthetic-geometric framework of Lorentzian length spaces developed in [KS:17]. To this end, we introduce appropriate notions of geodesics and timelike geodesic completeness and prove a general inextendibility result. Our results shed new light on …
Develops a private synthetic graph generator using Gromov-Wasserstein distance.
problem Creating private synthetic networks for complex data.
method Random connection model, fused Gromov-Wasserstein distance, differential privacy.
result Effective algorithm for generating private synthetic graphs with theoretical guarantees.
New relation between curvature bounds and spacetime inextendibility.
problem Inextendibility of spacetimes under low regularity conditions.
method Synthetic curvature and causal character analysis.
result Low-regularity spacetimes with unbounded curvature.
Timelike curves in Lorentzian length spaces have a total curvature notion that agrees with smooth curves.
problem Estimating the length of timelike curves in Lorentzian length spaces.
method Introducing a synthetic timelike total curvature notion.
result Proving timelike curves of finite total curvature are rectifiable.
New relation between curvature bounds and spacetime inextendibility.
problem Inextendibility of spacetimes under low regularity conditions.
method Synthetic curvature and causal character maximizers.
result Low-regularity inextendibility linked to unbounded curvature.
The paper explores conjugate points in Lorentzian spaces, comparing different definitions and proving related theorems.
problem Understanding conjugate points in Lorentzian geometry.
method Introducing and comparing different definitions of conjugate points in synthetic Lorentzian length spaces.
result All defined notions of conjugate points are compatible with the smooth spacetime setting.
Unified framework for measuring causality-based fairness in machine learning.
problem Challenges of measuring causality-based fairness from observational data.
method Unified definition of PC fairness and constrained optimization method.
result Correctness and effectiveness of the proposed method demonstrated on synthetic and real-world datasets.
Generates synthetic data for benchmarking unsupervised outlier detection.
problem Difficulty in benchmarking unsupervised outlier detection due to rare and varied outliers in real data.
method Proposes a generic process to generate synthetic data with insightful characteristics.
result Demonstrates practicality of the generic process through a benchmark with state-of-the-art detection methods.
Extends online learning to metric spaces using exponential weights.
problem Online learning in metric spaces.
method Exponentially weighted average forecaster, barycenters, Jensen's inequality, measure contraction property.
result Results in a statistical learning framework.
While harms of allocation have been increasingly studied as part of the subfield of algorithmic fairness, harms of representation have received considerably less attention. In this paper, we formalize two notions of stereotyping and show how they manifest in later allocative harms within the machine learning pipeline. …
Synthetic approach to conformal transformations in metric and Lorentzian spaces.
problem Defining consistent conformal transformations in spaces of low regularity.
method Introducing conformal transformations in metric and Lorentzian spaces, focusing on Lorentzian pre-length spaces.
result Established a consistent notion of conformal length and proved its properties.
Synthetic scalar curvature defined via Gaussian integrals, applied to manifolds and flows.
problem Defining scalar curvature for non-smooth spaces and flows.
method Gaussian integral approach to scalar curvature, applied to manifolds and flows.
result Characterizes Ricci flows as minimal super-Ricci flows.
The adoption of automated, data-driven decision making in an ever expanding range of applications has raised concerns about its potential unfairness towards certain social groups. In this context, a number of recent studies have focused on defining, detecting, and removing unfairness from data-driven decision systems. …
Ranking Differential Privacystat.ML
This paper protects rankings from differential privacy breaches.
problem Leakage of personal information in rankings.
method Develops ε-ranking differential privacy and a multistage ranking algorithm.
result Establishes the connection between Mallows model and ε-ranking differential privacy.
Equal experience fairness in recommender systems reduces bias.
problem Bias in data leads to unfair item recommendations.
method Introduces equal experience fairness notion and optimization framework.
result Mitigates unfairness in recommendations with minimal accuracy loss.
We present a new method for online prediction and learning of tensors (-way arrays, ) from sequential measurements. We focus on the specific case of 3-D tensors and exploit a recently developed framework of structured tensor decompositions proposed in [1]. In this framework it is possible to treat 3-D tensors …
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.
Given the widespread popularity of spectral clustering (SC) for partitioning graph data, we study a version of constrained SC in which we try to incorporate the fairness notion proposed by Chierichetti et al. (2017). According to this notion, a clustering is fair if every demographic group is approximately proportional…
A new method compares synthetic power networks to actual ones using multiscale flat norm.
problem Comparing synthetic power networks to actual ones due to lack of correspondence.
method Proposes a multiscale flat norm approach to compute distance between networks.
result The flat norm distance captures variations more accurately than Hausdorff distance.
Paper proves Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.
problem Proving equivalence between Brunn-Minkowski inequality and curvature dimension condition.
method Analyzes weighted Riemannian manifolds, proving equivalence without optimal transport or differential structure.
result Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.
Almost-Riemannian manifolds fail to meet a synthetic curvature condition.
problem Proving almost-Riemannian manifolds do not satisfy the condition.
method Developed a new strategy to contradict the 1-dimensional condition.
result 2D and strongly regular almost-Riemannian manifolds do not satisfy for any and .
We extend to a scheme-theoretic context the notion of a combinatorial differential form, due to A.Kock in the framework of synthetic differential geometry. We show that group-valued combinatorial forms on a scheme may be identified, under very general hypotheses, with traditional Lie algebra-valued differential forms, …
The paper prices long-term options with a reflecting barrier model.
problem Pricing long-term options with asset price limits.
method Model asset price as geometric Brownian motion with a lower reflecting barrier, pricing options using compound options.
result Option prices can be determined using standard risk-neutral arguments, and hedging strategies are available.
Synthetic theory defines orbifolds as microlinear types with finite identifications.
problem Defining orbifolds in traditional set-level foundations with internal symmetries.
method Synthetic differential cohesive homotopy type theory, microlinearity, finite identifications.
result Proper étale groupoids are orbifolds in synthetic theory.
Sparse NMF with archetypal regularization aims to robustly represent data points.
problem Representing data points as sparse linear combinations of archetypes.
method Sparse NMF with archetypal regularization, introducing strong and weak robustness.
result Theoretical robustness guarantees hold under minimal assumptions.
Synthetic splitting theorem for Lorentzian spaces with non-negative curvature.
problem Proving a splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature.
method Synthetic approach using triangle comparison and parallelity of timelike lines.
result Establishes a splitting of a neighborhood of a complete timelike line, leading to global inextendibility.
New bounds for low-regularity Riemannian metrics defined via distributional curvature.
problem Establishing curvature bounds for Riemannian metrics of low regularity.
method Introducing a distributional version of sectional curvature for and metrics.
result New bounds for low-regularity metrics recover classical bounds in Alexandrov spaces.
Refines d'Alembertian for signed Lorentz distance functions in metric measure spacetimes.
problem Exact representation and bounds of d'Alembertian for signed Lorentz distance functions.
method Metric geometry techniques, localization, Sobolev calculus.
result Distributional d'Alembertian is a signed measure with integration by parts formula.
Defines a geometric dimension for Lorentzian spaces, distinguishing spacelike and null subspaces.
problem No specific problem stated; focuses on defining a new geometric dimension.
method Introduces a one-parameter family of volume measures and a doubling condition for causal diamonds.
result Defines a geometric dimension for synthetic spacetimes, distinguishing between spacelike and null subspaces.
FaiREE provides fair classification with guarantees for small datasets.
problem Fairness in classification often requires large sample sizes and distributional assumptions.
method FaiREE offers finite-sample and distribution-free fairness guarantees.
result FaiREE achieves optimal accuracy and satisfies various fairness notions.
We present a novel class of convolutional neural networks (CNNs) for set functions, i.e., data indexed with the powerset of a finite set. The convolutions are derived as linear, shift-equivariant functions for various notions of shifts on set functions. The framework is fundamentally different from graph convolutions b…
A novel notion of barycenter for probability distributions based on optimal weak mass transportstat.ML
A new method for averaging probability distributions based on optimal weak mass transport.
problem Averaging probability distributions in a geometric way.
method Weak barycenters based on optimal weak mass transport.
result Extracts common geometric information shared by all input distributions.
The paper shows averaging gradients leads to memorization, proposing an alternative algorithm to focus on invariances.
problem The principle that 'good explanations are hard to vary' in deep learning is investigated.
method Formalizing consistency for loss surface minima, proposing an alternative algorithm based on logical AND.
result The alternative algorithm prevents memorization and focuses on invariances.
New method for multiway clustering of 3rd order tensors.
problem Clustering of 3rd order tensors.
method MCAM method based on affinity matrix and clustering.
result Competitive results on synthetic and real datasets.
Frolicher spaces and smooth mappings form a cartesian closed category. It was shown in our previous paper [Far East Journal of Mathematical Sciences, 35 (2009), 211-233] that its full subcategory of Weil exponentiable Frolicher spaces is cartesian closed. By emancipating microlinearity from within a well-adapted model …
We introduce the notions of `super-Ricci flows' and `Ricci flows' for time-dependent families of metric measure spaces . The former property is proven to be stable under suitable space-time versions of mGH-convergence. Uniformly bounded families of super-Ricci flows are compact. In the spirit of t…
Boosted Control Functions improve prediction under distributional shifts.
problem Prediction under distributional shifts in the presence of hidden confounding.
method Boosted Control Function (BCF) and ControlTwicing algorithm.
result BCF allows for distribution generalization and invariance under nonlinear, non-identifiable structural functions.
Study new Ricci bounds for metric measure spaces, preserving properties under time changes.
problem Extend Ricci bounds to non-synthetic spaces and understand their behavior under time changes.
method Introduce distribution-valued lower Ricci bounds BE, prove equivalence with gradient estimates, and show preservation under time changes.
result Distribution-valued Ricci bounds BE are preserved under arbitrary time changes and imply sharp gradient estimates.
The paper formalizes feature attribution to address inconsistent definitions and evaluate methods.
problem Inconsistent definitions of feature relevance in feature attribution.
method Formalization based on relaxed functional dependence, extended to instance-wise setting.
result State-of-the-art methods often fail to verify necessary properties for candidate selection.