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Statistical initialization of intrinsic K-means clustering on homogeneous manifolds
The K -means algorithm is widely applied for clustering, and its clustering effect is influenced by its initialization. However, most existing works...
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Rethinking the Riemannian Logarithm on Flag Manifolds as an Orthogonal Alignment Problem
Flags are sequences of nested linear subspaces of increasing dimension. They belong to smooth manifolds generalizing Grassmannians and bring a richer...
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An Optimization Method for Accurate Nonparametric Regressions on Stiefel Manifolds
We consider the problem of regularized nonlinear regression on Riemannian Stiefel manifolds when only few observations are available. In this paper,...
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Dimension Estimates on Manifolds
In this chapter generalizations of the Douady-Oesterlé theorem (Theorem 5.1 , Chap. 5) are obtained for maps and vector fields on Riemannian...
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Federated Learning Under Statistical Heterogeneity on Riemannian Manifolds
Federated learning (FL) is a collaborative machine learning paradigm in which clients with limited data collaborate to train a single “best” global...
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Nonlinear Spectral Processing of Shapes via Zero-Homogeneous Flows
In this work we extend the spectral total-variation framework, and use it to analyze and process 2D manifolds embedded in 3D. Analysis is performed...
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Introducing Poset-Based Connected n-Manifolds and \(\mathcal {P}\)-well-composedness in Partially Ordered Sets
In discrete topology, discrete surfaces are well-known for their strong topological and regularity properties. Their definition is recursive, and...
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Online Learning of Riemannian Hidden Markov Models in Homogeneous Hadamard Spaces
Hidden Markov models with observations in a Euclidean space play an important role in signal and image processing. Previous work extending to models...
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Music genre profiling based on Fisher manifolds and Probabilistic Quantum Clustering
Probabilistic classifiers induce a similarity metric at each location in the space of the data. This is measured by the Fisher Information Matrix....
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On First Integrals and Invariant Manifolds in the Generalized Problem of the Motion of a Rigid Body in a Magnetic Field
Differential equations describing the motion of a rigid body with a fixed point under the influence of both a magnetic field generated by the...
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Nested Grassmanns for Dimensionality Reduction with Applications to Shape Analysis
Grassmann manifolds have been widely used to represent the geometry of feature spaces in a variety of problems in medical imaging and computer vision...
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Coupling matrix manifolds assisted optimization for optimal transport problems
Optimal transport (OT) is a powerful tool for measuring the distance between two probability distributions. In this paper, we introduce a new...
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Submanifolds of Fixed Degree in Graded Manifolds for Perceptual Completion
We extend to a Engel type structure a cortically inspired model of perceptual completion initially proposed in the Lie group of positions and...
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Quasi-arithmetic Centers, Quasi-arithmetic Mixtures, and the Jensen-Shannon \(\nabla \) -Divergences
We first explain how the information geometry of Bregman manifolds brings a natural generalization of scalar quasi-arithmetic means that we term...
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Geometry-Preserving Lie Group Integrators for Differential Equations on the Manifold of Symmetric Positive Definite Matrices
In many applications, one encounters time series that lie on manifolds rather than a Euclidean space. In particular, covariance matrices are...
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Geometric deep learning and equivariant neural networks
We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop...
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Equivalence of Invariant Star-Products: The “Retract” Method
In this article, we present a general method for enlarging the group of symmetries (symplectomorphisms) of a given star-product (or deformation...
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Algebraic Geometry
In this chapter, we will study algebraic geometry and its surroundings. First, we will learn about algebraic sets and manifolds. A manifold is like a...
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Nonlinear Regression on Manifolds for Shape Analysis using Intrinsic Bézier Splines
Intrinsic and parametric regression models are of high interest for the statistical analysis of manifold-valued data such as images and shapes. The...
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Unsupervised Assignment Flow: Label Learning on Feature Manifolds by Spatially Regularized Geometric Assignment
This paper introduces the unsupervised assignment flow that couples the assignment flow for supervised image labeling (Åström et al. in J Math...
