Karcher Mean Research Articles

Matrix geometric means based on shuffled inductive sequences

We propose a new deterministic sequence converging to the least squares mean of N symmetric positive definite (SPD) matrices. By “least squares mean”, we refer to the Riemannian barycenter with respect to the natural metric (also known as the trace metric or affine-invariant metric) on the set Pn of n×n SPD matrices. In some papers, this mean is also referred to as the Karcher mean. The sequence we propose belongs to the family of inductive sequences, obtained by letting a point take successive steps towards the data points, with step lengths progressively diminishing to zero. We use the word “inductive” since each point of those sequences can be seen as the inductive mean of a well-chosen set of points containing multiple replications of the data points. We show that visiting the data points in a cyclic order usually results in a slow convergence, and remedy this weakness by reordering repeatedly the data points using a shuffling algorithm. We prove the convergence of the resulting inductive sequence to the least squares mean of the data in the general framework of NPC spaces (complete metric spaces with non-positive curvature), which includes the set Pn. We also illustrate numerically on Pn that some points of this sequence are fairly accurate estimates of the least squares mean, while being considerably cheaper to compute, and perform well as initializers for state-of-the-art algorithms. To reduce further the computation time, we finally exploit inductive sequences to produce a family of parallelizable algorithms for estimating the least squares mean.

The Hadamard Product and the Karcher Mean of Positive Invertible Operators

In this paper, we show several operator inequalities involving the Hadamard product and the Karcher mean of n (≥3) positive invertible operators on a separable Hilbert space, which are regarded as an n -variable operator version of results due to Ando and Aujla–Vasudeva. As applications, we show estimates from above for an n -variable version of the Fiedler-type theorem due to Fujii. Moreover, we show an n -variable version of the majorization relation due to Ando for the Hadamard product via the Karcher mean of n (≥ 3) positive-definite matrices.

Template Estimation in Computational Anatomy: Fréchet Means Top and Quotient Spaces Are Not Consistent

In this article, we study the consistency of the template estimation with the Frechet mean in quotient spaces. The Frechet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases.

Further generalizations of some operator inequalities involving positive linear map

We obtain a generalized conclusion based on an ?-geometric mean inequality. The conclusion is presented as follows: If m1,M1,m2,M2 are positive real numbers, 0 < m1 ? A ? M1 and 0 < m2 ? B ? M2 for m1 < M1 and m2 < M2, then for every unital positive linear map ? and ? ? (0,1], the operator inequality below holds: (?(?)#??(B))p ? 1/16 {(M1+m1)2((M1+m1)-1(M2+m2))2?)/(m2M2)?(m1M1)1- ?}p ?p(A#?B), p ? 2. Likewise, we give a second powering of the Diaz-Metcalf type inequality. Finally, we present p-th powering of some reversed inequalities for n operators related to Karcher mean and power mean involving positive linear maps.

A Numerical Framework for Sobolev Metrics on the Space of Curves

Statistical shape analysis can be done in a Riemannian framework by endowing the set of shapes with a Riemannian metric. Sobolev metrics of order two and higher on shape spaces of parametrized or unparametrized curves have several desirable properties not present in lower order metrics, but their discretization is still largely missing. In this paper, we present algorithms to numerically solve the geodesic initial and boundary value problems for these metrics. The combination of these algorithms enables one to compute Karcher means in a Riemannian gradient-based optimization scheme and perform principal component analysis and clustering. Our framework is sufficiently general to be applicable to a wide class of metrics. We demonstrate the effectiveness of our approach by analyzing a collection of shapes representing HeLa cell nuclei.

Endpoint Geodesics on the Stiefel Manifold Embedded in Euclidean Space

The Stiefel manifold has no known analytical formula for endpoint geodesics, i.e., locally shortest length curves between two given points. In this work, we consider the Stiefel manifold as a submanifold of Euclidean space, and its geometry and metric are inherited from this ambient Hilbert space. With this geometric interpretation, we derive two numerical schemes to compute endpoint geodesics on the Stiefel manifold: the shooting method and path-straightening. We perform a variety of numerical experiments to showcase and compare the performance of these two algorithms under various conditions. Additionally, we show an application of the Stiefel geodesic and Karcher mean calculation in the context of affine-invariant shape analysis.

A Majorization-Minimization Algorithm for Computing the Karcher Mean of Positive Definite Matrices

An algorithm for computing the Karcher mean of $n$ positive definite matrices is proposed, based on the majorization-minimization (MM) principle. The proposed MM algorithm is parameter-free, does not need to choose step sizes, and has a theoretical guarantee of asymptotic linear convergence.

The relative operator entropy and the Karcher mean

The Karcher mean for positive invertible operators on a Hilbert space, which is the unique solution of the Karcher operator equation, was established by Lawson–Lim based on the geometric considerations for positive invertible operators. In this note, we extend both this mean and the equation to the non-invertible case. The key concept is the relative operator entropy. For operators whose ranges are closed, we show that the Karcher mean for the non-invertible case is the unique solution of the extended Karcher equation. As a byproduct, we can show the uniqueness of the Karcher solution for the invertible case based only on that of the power means.

Limiting behaviour of Fréchet means in the space of phylogenetic trees

As demonstrated in our previous work on T4, the space of phylogenetic trees with four leaves, the topological structure of the space plays an important role in the non-classical limiting behaviour of the sample Frechet means in T4. Nevertheless, the techniques used in that paper cannot be adapted to analyse Frechet means in the space Tm of phylogenetic trees with m(⩾5)m(⩾5) leaves. To investigate the latter, this paper first studies the log map of Tm. Then, in terms of a modified version of this map, we characterise Frechet means in Tm that lie in top-dimensional or co-dimension one strata. We derive the limiting distributions for the corresponding sample Frechet means, generalising our previous results. In particular, the results show that, although they are related to the Gaussian distribution, the forms taken by the limiting distributions depend on the co-dimensions of the strata in which the Frechet means lie.

Incremental Gradient Method for Karcher Mean on Symmetric Cones

In this paper, we deal with the minimization problem for computing Karcher mean on a symmetric cone. The objective of this minimization problem consists of the sum of squares of the Riemannian distances with many given points in a symmetric cone. Moreover, the problem can be reduced to a bound-constrained minimization problem. These motivate us to adapt an incremental gradient method. So we propose an incremental gradient method and establish its global convergence properties exploiting the Lipschitz continuity of the gradient of the Riemannian distance function.

LBO-Shape densities: A unified framework for 2D and 3D shape classification on the hypersphere of wavelet densities

Recent years have seen a sharp rise in shape classification applications, and following suit, several frameworks have been proposed for efficient indexing of shape models. Here we propose a state-of-the-art shape matching framework which concomitantly provides transformation invariance and computationally efficient querying. Shapes are represented as probability density functions estimated on the eigenspace of the shape’s Laplace–Beltrami operator (LBO) and the ensuing manifold geometry is leveraged to classify query shapes. Specifically, we estimate a nonparametric, square-root wavelet density on the low-order eigenvectors of the LBO, capturing the rich geometric structure of 2D and 3D shapes with very minimal pre-processing requirements. By estimating 3D, square-root wavelet densities on each shape’s eigenspace (LBO-Shape densities), both 2D and 3D shapes become identifiable with the unit hypersphere. Leveraging the hypersphere’s simple geometry, our avant-garde model-to-mean indexing scheme computes the intrinsic Karcher mean for each shape category, and then uses the closed-form distance between a query shape and the means to assign labels. In 2D, the need for burdensome preprocessing like extracting closed curves along with topological and equal point set cardinality requirements are eliminated. Similarly, in 3D we gain isometry invariance and rid ourselves of the need for superfluous feature extraction schemes. The extensive experimental results demonstrate that our approach is competitive with the state-of-the-art in 2D/3D shape matching algorithms.

Scattered manifold-valued data approximation

We consider the problem of approximating a function f from an Euclidean domain to a manifold M by scattered samples (f(xi _i))_{iin mathcal {I}}, where the data sites (xi _i)_{iin mathcal {I}} are assumed to be locally close but can otherwise be far apart points scattered throughout the domain. We introduce a natural approximant based on combining the moving least square method and the Karcher mean. We prove that the proposed approximant inherits the accuracy order and the smoothness from its linear counterpart. The analysis also tells us that the use of Karcher’s mean (dependent on a Riemannian metric and the associated exponential map) is inessential and one can replace it by a more general notion of ‘center of mass’ based on a general retraction on the manifold. Consequently, we can substitute the Karcher mean by a more computationally efficient mean. We illustrate our work with numerical results which confirm our theoretical findings.

Compute Karcher means on SO(n) by the geometric conjugate gradient method

In this paper, numerical methods to compute the Karcher means on the n-order rotation group SO(n) are considered. First, after recalling Karcher means as solutions of a kind of minimization problems on SO(n), a super-linearly convergent numerical method, namely conjugate gradient method, has been used to deal with them. By the geometric structure of SO(n), the proposed algorithm is structure preserving. Then, a variety of numerical experiments are presented to demonstrate the performance and efficiency of the proposed algorithm by comparing with a recent structure preserving method.

The Hadamard product for the weighted Karcher means

The weighted Karcher mean of positive definite matrices is defined as the unique minimizer of the weighted sum of squares of the Riemannian distances to each of given points. Using the well-known connection between the tensor product and the Hadamard product, we show that the Hadamard product of weighted Karcher means for permuted tuples with fixed weight is bounded by the Hadamard product of given positive definite matrices. It generalizes the results for the case of two-variable geometric means and Sagae–Tanabe inductive means.

A fixed point mean approximation to the Cartan barycenter of positive definite matrices

In this paper we define a new weighted inductive geometric mean Hn on the Riemannian manifold of positive definite matrices of fixed size and present its fixed point mean approximation to the Cartan barycenter (Karcher mean). We show that for t∈(0,1], the weighted geometric mean equation X=Hn+1(ωt;A1,…,An,X), where ωt=t1+t(w1,…,wn,1/t), has a unique positive definite solution that approaches as t→0+ to the ω-weighted Cartan mean of A1,…,An. Numerical computations for the stochastic convergence in terms of Hn to derive a new contractive barycenter other than the Cartan barycenter are presented.

An Approach to the Log-Euclidean Mean via the Karcher Mean on Symmetric Cones

In a general symmetric cone, we show that certain sequence of the Karcher means converges to the Log-Euclidean mean by using the fact that the Karcher mean is the limit of inductive means. One can see this as a generalization of the Lie-Trotter formula of positive definite matrices into a symmetric cone setting via the least squares mean.

A Riemannian Limited-Memory BFGS Algorithm for Computing the Matrix Geometric Mean

Various optimization algorithms have been proposed to compute the Karcher mean (namely the Riemannian center of mass in the sense of the affine-invariant metric) of a collection of symmetric positive-definite matrices. Here we propose to handle this computational task with a recently developed limited-memory Riemannian BFGS method using an implementation tailored to the symmetric positive-definite Karcher mean problem. We also demonstrate empirically that the method is best suited for large-scale problems in terms of computation time and robustness when comparing to the existing state-of-the-art algorithms.

A characterization of matrix inequality A≥B≥C via Karcher mean

A characterization of matrix inequality A≥B≥C via Karcher mean

Operator means of probability measures and generalized Karcher equations

In this article we consider means of positive bounded linear operators on a Hilbert space. We present a complete theory that provides a framework which extends the theory of the Karcher mean, its approximating matrix power means, and a large part of Kubo–Ando theory to arbitrary many variables, in fact, to the case of probability measures with bounded support on the cone of positive definite operators. This framework characterizes each operator mean extrinsically as unique solutions of generalized Karcher equations which are obtained by exchanging the matrix logarithm function in the Karcher equation to arbitrary operator monotone functions over the positive real half-line. If the underlying Hilbert space is finite dimensional, then these generalized Karcher equations are Riemannian gradients of convex combinations of strictly geodesically convex log-determinant divergence functions, hence these new means are their global minimizers, in analogy to the case of the Karcher mean as pointed out. Our framework is based on fundamental contraction results with respect to the Thompson metric, which provides us nonlinear contraction semigroups in the cone of positive definite operators that form a decreasing net approximating these operator means in the strong topology from above.

Fruit Flies and Moduli: Interactions between Biology and Mathematics

Possibilities for using geometry and topology to analyze statistical problems in biology raise a host of novel questions in geometry, probability, algebra, and combinatorics that demonstrate the power of biology to influence the future of pure mathematics. This expository article is a tour through some biological explorations and their mathematical ramifications. The article starts with evolution of novel topological features in wing veins of fruit flies, which are quantified using the algebraic structure of multiparameter persistent homology. The statistical issues involved highlight mathematical implications of sampling from moduli spaces. These lead to geometric probability on stratified spaces, including the sticky phenomenon for Frechet means and the origin of this mathematical area in the reconstruction of phylogenetic trees.