Karcher Mean Research Articles

Estimations of Karcher mean by Hadamard product

In this paper, we estimate the difference between the Hadamard product and the Karcher mean of n positive invertible operators on the Hilbert space in terms of the Specht ratio and the Kantorovich constant. Also, we improve the obtained inequalities in the case of n=2. Moreover, we give ratio inequalities of the operator power means by the Hadamard product.

Stochastic gradient descent for barycenters in Wasserstein space

Abstract We present and study a novel algorithm for the computation of 2-Wasserstein population barycenters of absolutely continuous probability measures on Euclidean space. The proposed method can be seen as a stochastic gradient descent procedure in the 2-Wasserstein space, as well as a manifestation of a law of large numbers therein. The algorithm aims to find a Karcher mean or critical point in this setting, and can be implemented ‘online’, sequentially using independent and identically distributed random measures sampled from the population law. We provide natural sufficient conditions for this algorithm to almost surely converge in the Wasserstein space towards the population barycenter, and we introduce a novel, general condition which ensures uniqueness of Karcher means and, moreover, allows us to obtain explicit, parametric convergence rates for the expected optimality gap. We also study the mini-batch version of this algorithm, and discuss examples of families of population laws to which our method and results can be applied. This work expands and deepens ideas and results introduced in an early version of Backhoff-Veraguas et al. (2022), in which a statistical application (and numerical implementation) of this method is developed in the context of Bayesian learning.

Testing Outlier Detection Algorithms for Identifying Early Stage Solute Clusters in Atom Probe Tomography.

Atom probe tomography (APT) is commonly used to study solute clustering and precipitation in materials. However, standard techniques used to identify and characterize clusters within atom probe data, such as the density-based spatial clustering applications with noise (DBSCAN), often underperform with respect to small clusters. This is a limitation of density-based cluster identification algorithms, due to their dependence on the parameter Nmin, an arbitrary lower limit placed on detectable cluster sizes. Therefore, this article attempts to consider the characterization of clustering in atom probe data as an outlier detection problem of which k-nearest neighbors local outlier factor and learnable unified neighborhood-based anomaly ranking algorithms were tested against a simulated dataset and compared to the standard method. The decision score output of the algorithms was then auto thresholded by the Karcher mean to remove human bias. Each of the major models tested outperforms DBSCAN for cluster sizes of <25 atoms but underperforms for sizes >30 atoms using simulated data. However, the new combined k-nearest neighbors (k-NN) and DBSCAN method presented was able to perform well at all cluster sizes. The combined k-NN and seven methods are presented as a new approach to identifying clusters in APT.

Differentiable Geodesic Distance for Intrinsic Minimization on Triangle Meshes

Computing intrinsic distances on discrete surfaces is at the heart of many minimization problems in geometry processing and beyond. Solving these problems is extremely challenging as it demands the computation of on-surface distances along with their derivatives. We present a novel approach for intrinsic minimization of distance-based objectives defined on triangle meshes. Using a variational formulation of shortest-path geodesics, we compute first and second-order distance derivatives based on the implicit function theorem, thus opening the door to efficient Newton-type minimization solvers. We demonstrate our differentiable geodesic distance framework on a wide range of examples, including geodesic networks and membranes on surfaces of arbitrary genus, two-way coupling between hosting surface and embedded system, differentiable geodesic Voronoi diagrams, and efficient computation of Karcher means on complex shapes. Our analysis shows that second-order descent methods based on our differentiable geodesics outperform existing first-order and quasi-Newton methods by large margins.

Weighted Karcher means on unipotent Lie groups

Weighted Karcher means on unipotent Lie groups

Computing the Riemannian center of mass on meshes

The Riemannian center of mass (a.k.a. Karcher mean or Fréchet mean) provides the equivalent to the Euclidean affine average on manifolds. In spite of its many potential applications in computer graphics and geometric modeling, there exist surprisingly few algorithms to compute it. We present a direct method for computing the Riemannian center of mass on a triangle mesh. Our method works in the polyhedral metric and uses a piecewise-linear interpolation of gradients of the distance fields from a set of control points. We present applications for tracing splines on a surface, comparing to other methods at the state of the art, and showing that we produce quality results while supporting user interaction.

Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics: A Comprehensive Numerical Framework

This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing parallel transport along paths of surfaces. Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence when computing geodesics between unparametrized surfaces, while also yielding versatile algorithms that allow us to compare surfaces with varying sampling or mesh structures. Importantly, we demonstrate how our relaxed variational framework can be extended to tackle partially observed data. The different benefits of our numerical pipeline are illustrated over various examples, synthetic and real.

The Karcher mean of equilateral triangles

In this paper we present a class of Riemannian equilateral triangles ΔABC of the Cartan-Hadamard Riemannian manifold of N×N positive definite matrices of determinant 1 whose Karcher mean of vertices is of the formΛ(A,B,C)=A+B+Cdet⁡(A+B+C)N. The obtained class of triples (A,B,C) forms an analytic manifold with an analytic parameterization over SL(N,C)±×R. The determinantal identity det⁡(A+B+C)=det⁡(A−1+B−1+C−1) and an extended arithmetic-Karcher-harmonic mean inequalities is derived for such triples.

Continuation Methods for Riemannian Optimization

Numerical continuation in the context of optimization can be used to mitigate convergence issues due to a poor initial guess. In this work, we extend this idea to Riemannian optimization problems, that is, the minimization of a target function on a Riemannian manifold. For this purpose, a suitable homotopy is constructed between the original problem and a problem that admits an easy solution. We develop and analyze a path-following numerical continuation algorithm on manifolds for solving the resulting parameter-dependent equation. To illustrate our developments, we consider two typical classical applications of Riemannian optimization: the computation of the Karcher mean and low-rank matrix completion. We demonstrate that numerical continuation can yield improvements for challenging instances of both problems.

Non-Euclidean estimation of acoustic cross-spectral density matrices: Evaluation by matched-field localization in an ocean waveguide.

A set of non-Euclidean cross-spectral density matrix (CSDM) estimators based on the Frechet mean and median is applied to spatially localize an acoustic source in a stochastic shallow water environment driven by internal gravity waves. The resulting geometric-based matrix estimates are incorporated into matched-field processors defined through steering matrices using intrinsic distance measures between pairs of CSDMs on a Riemannian manifold for each replica source location. Their performance is evaluated using the probability of correct localization to determine any improvement in source localization over the conventional (maximum likelihood) statistical approach for CSDM estimation.

Tracking and Regret Bounds for Online Zeroth-Order Euclidean and Riemannian Optimization

We study numerical optimization algorithms that use zeroth-order information to minimize time-varying geodesically convex cost functions on Riemannian manifolds. In the Euclidean setting, zeroth-order algorithms have received a lot of attention in both the time-varying and time-invariant cases. However, the extension to Riemannian manifolds is much less developed. We focus on Hadamard manifolds, which are a special class of Riemannian manifolds with global nonpositive curvature that offer convenient grounds for the generalization of convexity notions. Specifically, we derive bounds on the expected instantaneous tracking error, and we provide algorithm parameter values that minimize the algorithm's performance. Our results illustrate how the manifold geometry in terms of the sectional curvature affects these bounds. Additionally, we provide dynamic regret bounds for this online optimization setting. To the best of our knowledge, these are the first regret bounds even for the Euclidean version of the problem. Lastly, via numerical simulations, we demonstrate the applicability of our algorithm on an online Karcher mean problem.

Hyperbolicity of the Karcher mean

The main concern of this paper is the Karcher mean of linearly independent triples (A,B,C) on the hyperbolic manifold of 2×2 positive definite matrices of determinant 1. We show that the Karcher mean is of the formΛ(A,B,C)=xA+y(B+C),0<x,yandx+2y<1 under the trace condition tr(AB−1)=tr(AC−1). We further find an invertible hyperbolic matrix M depending only on the trace values tr(AB−1) and tr(BC−1) such that [xy]=M[cosh⁡θsinh⁡θ] for some (unique) θ∈R.

A Multistage Framework With Mean Subspace Computation and Recursive Feedback for Online Unsupervised Domain Adaptation.

In this paper, we address the Online Unsupervised Domain Adaptation (OUDA) problem and propose a novel multi-stage framework to solve real-world situations when the target data are unlabeled and arriving online sequentially in batches. Most of the traditional manifold-based methods on the OUDA problem focus on transforming each arriving target data to the source domain without sufficiently considering the temporal coherency and accumulative statistics among the arriving target data. In order to project the data from the source and the target domains to a common subspace and manipulate the projected data in real-time, our proposed framework institutes a novel method, called an Incremental Computation of Mean-Subspace (ICMS) technique, which computes an approximation of mean-target subspace on a Grassmann manifold and is proven to be a close approximate to the Karcher mean. Furthermore, the transformation matrix computed from the mean-target subspace is applied to the next target data in the recursive-feedback stage, aligning the target data closer to the source domain. The computation of transformation matrix and the prediction of next-target subspace leverage the performance of the recursive-feedback stage by considering the cumulative temporal dependency among the flow of the target subspace on the Grassmann manifold. The labels of the transformed target data are predicted by the pre-trained source classifier, then the classifier is updated by the transformed data and predicted labels. Extensive experiments on six datasets were conducted to investigate in depth the effect and contribution of each stage in our proposed framework and its performance over previous approaches in terms of classification accuracy and computational speed. In addition, the experiments on traditional manifold-based learning models and neural-network-based learning models demonstrated the applicability of our proposed framework for various types of learning models.

Nonlinear distributed state estimation on the Stiefel manifold using diffusion particle filters

Many relevant problems in engineering demand the estimation of dynamic system states that are constrained to non-Euclidean spaces. Furthermore, it is generally advantageous to implement estimation methods in a network of cooperating nodes instead of relying on a single data fusion center. With this in mind, we propose in this paper two new distributed particle filtering methods to track the states of a dynamic system that evolves on the Stiefel manifold, which arises naturally when the states are subject to nonlinear orthogonal constraints. The proposed algorithms are based on the Random Exchange and Adapt-then-Combine diffusion techniques, and perform parametric approximations using the matrix von Mises-Fisher distribution to compress information exchanged between network nodes. Estimates of the state are then determined via an empirical averaging method that approximates the centroid (Karcher mean) of the particle set. As we verify via numerical simulations, the proposed methods show improved performance compared to previous Particle and Extended Kalman filters designed for Euclidean state variables, and compared to a non-cooperative particle filtering algorithm.

Projection-free nonconvex stochastic optimization on Riemannian manifolds

Abstract We study stochastic projection-free methods for constrained optimization of smooth functions on Riemannian manifolds, i.e., with additional constraints beyond the parameter domain being a manifold. Specifically, we introduce stochastic Riemannian Frank–Wolfe (Fw) methods for nonconvex and geodesically convex problems. We present algorithms for both purely stochastic optimization and finite-sum problems. For the latter, we develop variance-reduced methods, including a Riemannian adaptation of the recently proposed Spider technique. For all settings, we recover convergence rates that are comparable to the best-known rates for their Euclidean counterparts. Finally, we discuss applications to two classic tasks: the computation of the Karcher mean of positive definite matrices and Wasserstein barycenters for multivariate normal distributions. For both tasks, stochastic Fw methods yield state-of-the-art empirical performance.

Analyticity of the Karcher mean coefficient maps

The main concern of this paper is the three-variable Karcher (alternatively, Riemannian or Cartan) matrix mean for 2×2 linearly independent triples of positive definite matrices. We show that such triples form an analytic manifold and obtain three analytic functions from (1,∞)2×(0,π/2) to (0,1) realizing the Karcher mean as a linear combination of variables.

A Mean Ergodic Theorem for Nonexpansive Mappings in Hadamard Spaces

In this paper, we prove a mean ergodic theorem for nonexpansive mappings in Hadamard (nonpositive curvature metric) spaces, which extends the Baillon nonlinear ergodic theorem. The main result shows that the sequence given by the Karcher means of iterations of a nonexpansive mapping with a nonempty fixed point set converges weakly to a fixed point of the mapping. This result also remains true for a 1-parameter continuous semigroup of contractions.

Global Sensitivity Analysis and Wasserstein Spaces

Sensitivity indices are commonly used to quantity the relative inuence of any specic group of input variables on the output of a computer code. In this paper, we focus both on computer codes the output of which is a cumulative distribution function and on stochastic computer codes. We propose a way to perform a global sensitivity analysis for these kinds of computer codes. In the rst setting, we dene two indices: the rst one is based on Wasserstein Frechet means while the second one is based on the Hoeding decomposition of the indicators of Wasserstein balls. Further, when dealing with the stochastic computer codes, we dene an ideal version of the stochastic computer code thats ts into the frame of the rst setting. Finally, we deduce a procedure to realize a second level global sensitivity analysis, namely when one is interested in the sensitivity related to the input distributions rather than in the sensitivity related to the inputs themselves. Several numerical studies are proposed as illustrations in the dierent settings.

LogDet Metric-Based Domain Adaptation.

Domain adaptation has proven to be successful in dealing with the case where training and test samples are drawn from two kinds of distributions, respectively. Recently, the second-order statistics alignment has gained significant attention in the field of domain adaptation due to its superior simplicity and effectiveness. However, researchers have encountered major difficulties with optimization, as it is difficult to find an explicit expression for the gradient. Moreover, the used transformation employed here does not perform dimensionality reduction. Accordingly, in this article, we prove that there exits some scaled LogDet metric that is more effective for the second-order statistics alignment than the Frobenius norm, and hence, we consider it for second-order statistics alignment. First, we introduce the two homologous transformations, which can help to reduce dimensionality and excavate transferable knowledge from the relevant domain. Second, we provide an explicit gradient expression, which is an important ingredient for optimization. We further extend the LogDet model from single-source domain setting to multisource domain setting by applying the weighted Karcher mean to the LogDet metric. Experiments on both synthetic and realistic domain adaptation tasks demonstrate that the proposed approaches are effective when compared with state-of-the-art ones.

Understanding the topology and the geometry of the space of persistence diagrams via optimal partial transport

Despite the obvious similarities between the metrics used in topological data analysis and those of optimal transport, an optimal-transport based formalism to study persistence diagrams and similar topological descriptors has yet to come. In this article, by considering the space of persistence diagrams as a space of discrete measures, and by observing that its metrics can be expressed as optimal partial transport problems, we introduce a generalization of persistence diagrams, namely Radon measures supported on the upper half plane. Such measures naturally appear in topological data analysis when considering continuous representations of persistence diagrams (e.g. persistence surfaces) but also as limits for laws of large numbers on persistence diagrams or as expectations of probability distributions on the space of persistence diagrams. We explore topological properties of this new space, which will also hold for the closed subspace of persistence diagrams. New results include a characterization of convergence with respect to Wasserstein metrics, a geometric description of barycenters (Frechet means) for any distribution of diagrams, and an exhaustive description of continuous linear representations of persistence diagrams. We also showcase the strength of this framework to study random persistence diagrams by providing several statistical results made meaningful thanks to this new formalism.