Newton Algorithms for Riemannian Distance Related Problems on Connected Locally Symmetric Manifolds

IEEE Trans. Inform. Theory

Bombrun

Berthoumieu

et al. 2017

111

181

Data which lie in the space Pm , of m × m symmetric positive definite matrices, (sometimes called tensor data), play a fundamental role in applications including medical imaging, computer vision, and radar signal processing. An open challenge, for these applications, is to find a class of probability distributions, which is able to capture the statistical properties of data in Pm , as they arise in real-world situations. The present paper meets this challenge by introducing Riemannian Gaussian distributions on Pm . Distributions of this kind were first considered by Pennec in 2006. However, the present paper gives an exact expression of their probability density function for the first time in existing literature. This leads to two original contributions. First, a detailed study of statistical inference for Riemannian Gaussian distributions, uncovering the connection between maximum likelihood estimation and the concept of Riemannian centre of mass, widely used in applications. Second, the derivation and implementation of an expectation-maximisation algorithm, for the estimation of mixtures of Riemannian Gaussian distributions. The paper applies this new algorithm, to the classification of data in Pm , (concretely, to the problem of texture classification, in computer vision), showing that it yields significantly better performance, in comparison to recent approaches. Index TermsSymmetric positive definite matrices, tensor, Riemannian metric, Gaussian distribution, expectation-maximisation, texture where, again, d : P m × P m → R + is Rao's Riemannian distance. SinceŶ N minimises the sum of squares of distances to the points Y 1 , . . . , Y N , it is widely viewed as a representative, average, or mode of these points.Distributions of the form (1) were considered by Pennec, who defined them on general Riemannian manifolds [20]. However, in existing literature, their treatment remains incomplete, as it is based on asymptotic formulae, valid only in the limit where the parameter σ is small, see [20]- [22]. In addition to being inexact, such formulae are quite difficult, both to evaluate and to apply. These issues, (lack of an exact expression and difficulty of application), are overcome in the following.

Section: B Statistical Inference Problemsmentioning

confidence: 99%

“…where Log Y denotes the Riemannian logarithm mapping, (whose expression is (40), given below). To prove (72), note that for all Y ∈ P m , Pm p(Z| Y, σ)dv(Z) = 1 (74) since p(Z| Y, σ), as defined by (20), is a probability density.…”

mentioning

confidence: 99%

Riemannian Gaussian Distributions on the Space of Symmetric Positive Definite Matrices

IEEE Trans. Inform. Theory

Bombrun

Berthoumieu

et al. 2017

111

181

“…This is here only briefly indicated. Expression (19c) is a slight improvement of the one in [15] (see Theorem IV.1, Page 636), where it is enough to note that if R is the curvature tensor of M , then the operator R v (u) = R(v, u)v has the eigenvalues 0 and (λ(a)) 2 for each λ ∈ ∆ + , whenever v, u ∈ T x M p with v = Ad(s) a [7] [12]. It is well-known, by properties of the Jacobi equation [6], that H x (φ(s, a)) has the same eigenspace decomposition as R v , in this case.…”

Section: Proof Of Corollarymentioning

confidence: 94%

The Riemannian Barycentre as a Proxy for Global Optimisation

Lecture Notes in Computer Science

Manton

2019

Let M be a simply-connected compact Riemannian symmetric space, and U a twice-differentiable function on M , with unique global minimum at x * ∈ M . The idea of the present work is to replace the problem of searching for the global minimum of U , by the problem of finding the Riemannian barycentre of the Gibbs distribution PT ∝ exp(−U/T ). In other words, instead of minimising the function U itself, to minimiseThe following original result is proved : if U is invariant by geodesic symmetry about x * , then for each δ < 1 2 rcx (rcx the convexity radius of M ), there exists Tδ such that T ≤ Tδ implies ET is strongly convex on the geodesic ball B(x * , δ) , and x * is the unique global minimum of ET . Moreover, this Tδ can be computed explicitly. This result gives rise to a general algorithm for black-box optimisation, which is briefly described, and will be further explored in future work.

“…In conclusion, the EM algorithm (45a)-(45c) provides an approach to the problem of density estimation in M, which can be expected to offer a suitable rate of convergence and which is not greedy in terms of memory. The main computational requirement of this algorithm is the ability to find Riemannian barycentres, a task for which there exists an increasing number of high-performance routines [42][44][45] [11][31] [32]. The fact that the EM algorithm reduces the problem of probability density estimation in M to one of repeated computation of Riemannian barycentres is due to the unique connection which exists between Gaussian distributions in M and the concept of Riemannian barycentre.…”

Section: Comparison To Kernel Density Estimationmentioning

confidence: 99%

Gaussian Distributions on Riemannian Symmetric Spaces: Statistical Learning With Structured Covariance Matrices

IEEE Trans. Inform. Theory

Hajri

Bombrun

et al. 2018

108

The Riemannian geometry of covariance matrices has been essential to several successful applications, in computer vision, biomedical signal and image processing, and radar data processing. For these applications, an important ongoing challenge is to develop Riemannian-geometric tools which are adapted to structured covariance matrices. The present paper proposes to meet this challenge by introducing a new class of probability distributions, Gaussian distributions of structured covariance matrices. These are Riemannian analogs of Gaussian distributions, which only sample from covariance matrices having a preassigned structure, such as complex, Toeplitz, or block-Toeplitz. The usefulness of these distributions stems from three features : (1) they are completely tractable, analytically or numerically, when dealing with large covariance matrices, (2) they provide a statistical foundation to the concept of structured Riemannian barycentre (i.e. Fréchet or geometric mean), (3) they lead to efficient statistical learning algorithms, which realise, among others, density estimation and classification of structured covariance matrices. The paper starts from the observation that several spaces of structured covariance matrices, considered from a geometric point of view, are Riemannian symmetric spaces. Accordingly, it develops an original theory of Gaussian distributions on Riemannian symmetric spaces, of their statistical inference, and of their relationship to the concept of Riemannian barycentre. Then, it uses this original theory to give a detailed description of Gaussian distributions of three kinds of structured covariance matrices, complex, Toeplitz, and block-Toeplitz. Finally, it describes algorithms for density estimation and classification of structured covariance matrices, based on Gaussian distribution mixture models.