The paper introduces new geometric methods to analyze radar electromagnetic wave statistics.
problem Analyzing spatio-temporal and polarimetric fluctuations of radar electromagnetic waves.
method Using statistical mechanics and Information Geometry, the paper defines a Fréchet barycentre and maximum entropy density for radar measurements.
result New tools for describing radar electromagnetic wave fluctuations, including a distance on covariance matrices.
The asymptotic concentration of the Fr{é}chet mean of IID random variables on a Rieman-nian manifold was established with a central limit theorem by Bhattacharya \& Patrangenaru (BP-CLT) [6]. This asymptotic result shows that the Fr{é}chet mean behaves almost as the usual Euclidean case for sufficiently concentrated di…
In [16], a new family of vector-valued risk measures called multivariate expectiles is introduced. In this paper, we focus on the asymptotic behavior of these measures in a multivariate regular variations context. For models with equivalent tails, we propose an estimator of these multivariate asymptotic expectiles, in …
Study on geometry of Dirichlet distributions using Fisher-Rao metric.
problem Understanding the geometry of Dirichlet distributions.
method Analysis of Fisher-Rao metric on Dirichlet distribution parameter space.
result Geodesic completeness and negative sectional curvature of the space.
New metrics defined for full-rank correlation matrices, ensuring unique operations.
problem No suitable problem statement as the abstract does not describe a problem to be solved.
method New Riemannian metrics defined on full-rank correlation matrices, providing unique operations.
result Unique Riemannian logarithm and Fréchet mean defined for full-rank correlation matrices.
Constructs supermartingale couplings with full marginals constraints.
problem Optimal transport for supermartingale couplings with multiple marginals.
method Markovian iteration of one-period optimal supermartingale couplings.
result Explicit construction of supermartingale processes solving optimal transport problem.
New formulas for geodesics on Stiefel and flag manifolds using trust-region method.
problem Computing geodesics and logarithms on Stiefel and flag manifolds.
method Closed-form geodesic formulas, trust-region solver, Fréchet derivatives.
result Efficient computation of geodesic distance and logarithm map.
Let G be a compact connected simple Lie group and let $M=G^{\bb{C}}/P=G/K$ be a generalized flag manifold. In this article we focus on an important invariant of G/K, the so called $\fr{t}$-root system $R_{\fr{t}}$, and we introduce the notion of symmetric $\fr{t}$-triples, that is triples of $\fr{t}$-roots $ξ, ζ, η…
A new method for spectral barycentre of graph datasets.
problem Creating a summary graph from a set of graphs with community structure.
method Using multiscale spectral distance based on normalized graph Laplacian eigenvalues.
result The barycentre inherits the topological structure of the graphs in the sample dataset.
We consider in this paper the FRS-deformations of a family of space curves with codimension ≤3. Some geometric aspects of a space curve such as flattenings, vertices and twistings points has been studied.
A new method for estimating large-scale linear models with improved precision.
problem Estimating large-scale linear statistical models efficiently.
method Sequential Least-Squares Estimators with Fast Randomized Sketching (SLSE-FRS), integrating Sketch-and-Solve and Iterative-Sketching methods.
result SLSE-FRS produces high-precision estimators, outperforming state-of-the-art methods.
Study shows splitting schemes can approximate WFR flows faster than the exact flow.
problem Improving sampling efficiency in Wasserstein-Fisher-Rao gradient flows.
method Investigates operator splitting techniques to numerically approximate WFR flows.
result A judicious choice of step size and operator ordering can lead to faster convergence of split schemes to the target distribution.
Phishing as one of the most well-known cybercrime activities is a deception of online users to steal their personal or confidential information by impersonating a legitimate website. Several machine learning-based strategies have been proposed to detect phishing websites. These techniques are dependent on the features …
Extends optimal transport to dynamic and martingale settings.
problem Dynamic and martingale relaxation of optimal transport problems.
method Extends Benamou-Brenier formula to weak optimal transport and introduces barycentric optimal transport.
result Relates barycentric optimal transport to martingale Benamou-Brenier formula.
We quantify conditions that ensure that a signed measure on a Riemannian manifold has a well defined centre of mass. We then use this result to quantify the extent of a neighbourhood on which the Riemannian barycentric coordinates of a set of n+1 points on an n-manifold provide a true coordinate chart, i.e., the ba…
FR-Train improves fair and robust AI training by detecting and reducing poisoned data.
problem Training AI models that are fair and robust in the presence of data bias and poisoning.
method Mutual information-based adversarial training with an additional discriminator.
result FR-Train maintains fairness and accuracy even in the presence of poisoned data.
Proves FR-NGD optimally approximates evolutionary dynamics and continuous Bayesian inference.
problem Optimizing continuous time replicator equations and continuous Bayesian inference.
method Fisher-Rao natural gradient descent (FR-NGD) and its correspondence with evolutionary dynamics.
result FR-NGD optimally approximates continuous time replicator equations and continuous Bayesian inference.
FR-LUX optimizes portfolio management by learning cost-aware policies robust to market conditions.
problem Transaction costs and regime shifts cause failure in live trading portfolios.
method Integrates three ingredients: microstructure-consistent execution model, trade-space trust region, and explicit regime conditioning.
result Achieves top average Sharpe ratio, maintains flat cost-performance slope, and superior risk-return efficiency.
Consider an anchored bundle (E,ρ), i.e. a vector bundle E→M equipped with a bundle map ρ:E→TM covering the identity. M.~Kapranov showed in the context of Lie-Rinehard algebras that there exists an extension of this anchored bundle to an infinite rank universal free Lie algebroid FR(E)⊃E. We …
Paper tackles measure estimation in barycentric coding model.
problem Estimating an unknown measure in the barycentric coding model.
method Geometric, statistical, and computational insights; quadratic optimization problem; empirical i.i.d. samples algorithm.
result Proves precise rates of convergence for algorithm, ensuring statistical consistency.
Introduces a new geometric framework for field theories.
problem Developing a rigorous mathematical framework for field theories.
method Introduces supergeometric homotopy theory to physics.
result Classical bosonic field theories fit naturally into smooth sets.
The Riemannian barycentre is one of the most widely used statistical descriptors for probability distributions on Riemannian manifolds. At present, existing algorithms are able to compute the Riemannian barycentre of a probability distribution, only if i.i.d. samples of this distribution are readily available. However,…
This paper investigates the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. We first propose a new and general type of family of subspaces in manifolds that we call barycentric subspaces. They are implicitly defined as the locus of points which are weighted means of k+1 reference points.…
We recall an extension of Kirby's Calculus on non-simply connected 3-manifolds given in [FR], and the surgery calculus of bridged links from [Ke], which involves only local moves. We give a short combinatorial proof that the two calculi are equivalent, and thus describe the same classes of 3-manifolds. This makes the p…
This study provides an explicit expansion of KL divergence's gradient flow in Fisher-Rao geometry.
problem Sampling techniques struggle to traverse between modes in non-convex potential functions.
method Explicit expansion of KL divergence's gradient flow in Fisher-Rao geometry.
result The convergence rate to π is independent of the potential function.
Discrete Morse functions induce shellings with critical tiles corresponding to function's critical faces.
problem Mapping discrete Morse functions to shellings for topological analysis.
method Inducing Morse shellings on the second barycentric subdivision of a simplicial complex.
result Critical tiles of induced shellings correspond to critical faces of the discrete Morse function.
Develop a framework for barycentric projections of optimal transport plans on Riemannian manifolds.
problem Optimal transport couplings are probabilistic objects, while many learning pipelines require deterministic maps.
method Develop a framework for barycentric projections of transport couplings on Riemannian manifolds.
result The intrinsic projection maps each source point to the conditional Fréchet mean of its destination law and is shown to be the best deterministic representative under squared geodesic loss.
We study geodesics of the form γ(t)=π(exp(tX)exp(tY)), $X,Y\in \fr{g}=\operatorname{Lie}(G)$, in homogeneous spaces G/K, where π:G→G/K is the natural projection. These curves naturally generalise homogeneous geodesics, that is orbits of one-parameter subgroups of G (i.e. γ(t)=π(exp(tX)), $X\in …
We study the barycentric straightening of simplices in irreducible symmetric spaces of non-compact type. We show that, for an n-dimensional symmetric space of rank r>1, the p-Jacobian has uniformly bounded norm, as soon as p is at least n-r+2. As a consequence, for a non-compact, connected, semisimple real Lie group G,…
Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a contractible cubical complex Sigma_L (the Davis complex) on which W_L acts properly and cocompactly, and such that the link of each vertex is L. It follows that if L is a generalized homology sphere, then Sigma_L is a contractible h…
New theorem proves convergence of various discrete conformal structures to conformal maps.
problem Proving convergence of discrete conformal structures to conformal maps.
method General theorem using piecewise linear discrete conformal mappings and Riemannian barycentric coordinates.
result Discrete conformal mappings converge to conformal maps under certain conditions.
Combines expert models using Kullback-Leibler divergence to create a combined model.
problem Combining expert views on stochastic processes.
method Minimizes weighted Kullback-Leibler divergence to create a barycentre model.
result Existence and uniqueness of the barycentre model with explicit representation.
We study geodesics in generalized Wallach spaces which are expressed as orbits of products of three exponential terms. These are homogeneous spaces M=G/K whose isotropy representation decomposes into a direct sum of three submodules m=m1⊕m2⊕m3, satisfying the relations $[\fr…
A new embedding method for high-dimensional data.
problem Handling large sample sizes in high-dimensional spaces.
method Partitioning space into simplices and embedding into barycentric coordinates.
result Linear classifier in rich feature space yields highly non-linear decision boundaries.
This article follow the article {http://hal.archives-ouvertes.fr/hal-00361030/fr/} in which the author characterize the fact of being of finite volume for a convex projective surface. We show here that the moduli space βf(Σg,p) of the convex projective structure on the surface Σg,p of genius g with p pun…
New method solves tree-structured Schrödinger Bridge problems.
problem Computing Schrödinger Bridge between tree-structured distributions.
method Iterative Markovian Fitting (IMF) procedure for tree-structured costs.
result Extends IMF to tree-structured Schrödinger Bridge problems.
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.
Minimal displacement set in weakly systolic complexes is systolic and embeds isometrically.
problem Structure of minimal displacement set in weakly systolic complexes.
method Investigation of minimal displacement set properties and embeddings.
result Minimal displacement set is systolic and embeds isometrically into the complex.
A new category generates 1D tangle invariants.
problem Developing a new category for 1D tangles.
method Proving a new (∞,1)-category has universal mapping property. result The new category generates link invariants.
We introduce canonical measures on a locally finite simplicial complex K and study their asymptotic behavior under infinitely many barycentric subdivisions. We also compute the face polynomial of the asymptotic link and dual block of a simplex in the dth barycentric subdivision Sdd(K) of K, d≫0. It is a…
New method improves fairness of facial recognition systems.
problem Facial recognition systems exhibit bias across different demographic groups.
method Optimizes centroid-based scores to reduce bias in pre-trained models.
result Demonstrates significant improvement in fairness with minimal loss in accuracy.
The paper analyzes rates of convergence for optimal transport map estimators using barycentric projections.
problem Estimating optimal transport maps from data sampled according to two distributions.
method Comprehensive analysis of rates of convergence for plug-in estimators defined via barycentric projections.
result New stability estimate for barycentric projections under minimal smoothness assumptions.
A method models nonlinear dynamics from data using barycentric coordinates and memory.
problem Modeling complex dynamical systems from data.
method SPA for data projection, barycentric coordinates, delay-embedding theorem for memory.
result Stable models of chaotic dynamics and attractors are reproduced.
Tree complex linked to polyhedral shapes like associahedra and cyclohedra.
problem Understanding the structure of mapping class groups and complex dynamics.
method Characterizing associahedra and cyclohedra using planar tree embeddings and barycentric subdivision.
result Tree complex is a barycentric subdivision of a polyhedral cell complex made of associahedra and cyclohedra.
Proposes a fair pricing framework insensitive to protected covariates.
problem Ensuring fair prices for financial products without using discriminatory covariates.
method Develops a discrimination-insensitive pricing framework using optimization and KL divergence.
result Proves existence and uniqueness of discrimination-insensitive pricing measures.
This paper introduces a new Urban Point Cloud Dataset for Automatic Segmentation and Classification acquired by Mobile Laser Scanning (MLS). We describe how the dataset is obtained from acquisition to post-processing and labeling. This dataset can be used to learn classification algorithm, however, given that a great a…
Simpler algorithms for morphing planar and toroidal graphs.
problem Constructing smooth transitions between isomorphic drawings of planar and toroidal graphs.
method Barycentric interpolation and scaling strategy.
result Simplified and more natural morphs with improved computational efficiency.
We introduce \texttt{pycobra}, a Python library devoted to ensemble learning (regression and classification) and visualisation. Its main assets are the implementation of several ensemble learning algorithms, a flexible and generic interface to compare and blend any existing machine learning algorithm available in Pytho…