In this paper, the issue of averaging data on a manifold is addressed. While the Fréchet mean resulting from Riemannian geometry appears ideal, it is unfortunately not always available and often computationally very expensive. To overcome this,$R$-barycenters have been proposed and successfully applied to Stiefel and Grassmann manifolds. However,$R$-barycenters still suffer severe limitations as they rely on iterative algorithms and complicated operators. We propose simpler, yet efficient, barycenters that we call$RL$-barycenters. We show that, in the setting relevant to most applications, our framework yields astonishingly simple barycenters: arithmetic means projected onto the manifold. We apply this approach to the Stiefel and Grassmann manifolds. On simulated data, our approach is competitive with respect to existing averaging methods, while computationally cheaper.

Kernel methods are powerful tools in machine learning. Classical kernel methods are based on positive definite kernels, which enable learning in reproducing kernel Hilbert spaces (RKHS). For non-Euclidean data spaces, positive definite kernels are difficult to come by. In this case, we propose the use of reproducing kernel Krein space (RKKS) based methods, which require only kernels that admit a positive decomposition. We show that one does not need to access this decomposition to learn in RKKS. We then investigate the conditions under which a kernel is positively decomposable. We show that invariant kernels admit a positive decomposition on homogeneous spaces under tractable regularity assumptions. This makes them much easier to construct than positive definite kernels, providing a route for learning with kernels for non-Euclidean data. By the same token, this provides theoretical foundations for RKKS-based methods in general.

Riemannian Gaussian distributions were initially introduced as basic building blocks for learning models which aim to capture the intrinsic structure of statistical populations of positive-definite matrices (here called covariance matrices). While the potential applications of such models have attracted significant attention, a major obstacle still stands in the way of these applications: there seems to exist no practical method of computing the normalising factors associated with Riemannian Gaussian distributions on spaces of high-dimensional covariance matrices. The present paper shows that this missing method comes from an unexpected new connection with random matrix theory. Its main contribution is to prove that Riemannian Gaussian distributions of real, complex, or quaternion covariance matrices are equivalent to orthogonal, unitary, or symplectic log-normal matrix ensembles. This equivalence yields a highly efficient approximation of the normalising factors, in terms of a rather simple analytic expression. The error due to this approximation decreases like the inverse square of dimension. Numerical experiments are conducted which demonstrate how this new approximation can unlock the difficulties which have impeded applications to real-world datasets of high-dimensional covariance matrices. The paper then turns to Riemannian Gaussian distributions of block-Toeplitz covariance matrices. These are equivalent to yet another kind of random matrix ensembles, here called “acosh-normal” ensembles. Orthogonal and unitary “acosh-normal” ensembles correspond to the cases of block-Toeplitz with Toeplitz blocks, and block-Toeplitz (with general blocks) covariance matrices, respectively.

This paper studies fixed step-size stochastic approximation (SA) schemes, including stochastic gradient schemes, in a Riemannian framework. It is motivated by several applications, where geodesics can be computed explicitly, and their use accelerates crude Euclidean methods. A fixed step-size scheme defines a family of time-homogeneous Markov chains, parametrized by the step-size. Here, using this formulation, non-asymptotic performance bounds are derived, under Lyapunov conditions. Then, for any step-size, the corresponding Markov chain is proved to admit a unique stationary distribution, and to be geometrically ergodic. This result gives rise to a family of stationary distributions indexed by the step-size, which is further shown to converge to a Dirac measure, concentrated at the solution of the problem at hand, as the step-size goes to $0$. Finally, the asymptotic rate of this convergence is established, through an asymptotic expansion of the bias, and a central limit theorem.

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. This 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); and 3) they lead to efficient statistical learning algorithms, which realise, among others, density estimation and classification of structured covariance matrices. This 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.

Many signal and image processing applications, including SAR polarimetry and texture analysis, require the classification of complex covariance matrices. The present paper introduces a geometric learning approach on the space of complex covariance matrices based on a new distribution called Riemannian Gaussian distribution. The proposed distribution has two parameters, the centre of mass Y̅ and the dispersion parameter σ. After having derived its maximum likelihood estimator and its extension to mixture models, we propose an application to texture recognition on the VisTex database.

This paper presents a novel framework for visual content classification using jointly local mean vectors and covariance matrices of pixel level input features. We consider local mean and covariance as realizations of a bivariate Riemannian Gaussian density lying on a product of submanifolds. We first introduce the generalized Mahalanobis distance and then we propose a formal definition of our product-spaces Gaussian distribution on Rm× SPD(m). This definition enables us to provide a mixture model from a mixture of a finite number of Riemannian Gaussian distributions to obtain a tractable descriptor. Mixture parameters are estimated from training data by exploiting an iterative Expectation-Maximization (EM) algorithm. Experiments in a texture classification task are conducted to evaluate this extended modeling on several color texture databases, namely popular Vistex, 167-Vistex and CUReT. These experiments show that our new mixture model competes with state-of-the-art on the experimented datasets.

This paper deals with parametric techniques for the description of texture on very high resolution (VHR) remote sensing images. These techniques focus on the property of anisotropy as described by the local structure tensor (LST). The novelty of this paper consists in proposing several comprehensive statistical frameworks to handle LST fields for rotation-invariant texture discrimination tasks. These frameworks are all based on probability models defined on the Riemannian manifold of positive definite matrices: a recent Riemannian Gaussian model on the affine-invariant metric space and a multivariate Gaussian distribution on the Log-Euclidean space. A thorough comparison of the proposed methods is performed with respect to some state-of-the-art texture analysis methods. Three experimental protocols are considered based on VHR remote sensing data. The first one consists of a content-based image retrieval (CBIR) protocol for browsing oyster field patches. The second one concerns a supervised classification protocol for grouping maritime pine forest stands in different age classes. The third one is, again, a CBIR protocol performed on the UC Merced land use/land cover patch collection. Tensor-based approaches show similar or even better results than the state-of-the-art texture analysis methods considered for comparison in all the experimental contexts.

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 the maximum likelihood estimation and the concept of Riemannian centre of mass, widely used in applications. Second, the derivation and the 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.

This paper describes the application of particle filtering to the solution of the problem of filtering with observations in a manifold. Mathematically, this is based on an original use of so-called connector maps. It is shown that well-chosen connector maps can be used to transform successive samples from a continuous time observation process, evolving on a manifold, into a discrete sequence of random vectors, which are asymptotically independent and normally distributed, in the limit where the sampling interval goes to zero. Roughly speaking, this “innovation sequence” can be used as the input of a sequential Monte Carlo algorithm. As a concrete application, numerical simulation results are presented, for the problem of estimating the angular velocity of a rigid body from noisy observations of its attitude.

This paper presents a new Bayesian approach to texture classification, yielding enhanced performance in the presence of intraclass diversity. From a mathematical point of view, it specifies an original EM algorithm for mixture estimation on Riemannian manifolds, generalising existing, non probabilistic, clustering analysis methods. For texture classification, the chosen feature space is the Riemannian manifold known as the Poincaré half plane, here denoted H, (this is the set of univariate normal distributions, equipped with Rao's distance). Classes are modelled as finite mixtures of Riemannian priors, (Riemannian priors are probability distributions, recently introduced by the authors, which represent clusters of points in H). During the training phase of classification, the EM algorithm, proposed in this paper, computes maximum likelihood estimates of the parameters of these mixtures. The algorithm combines the structure of an EM algorithm for mixture estimation, with a Riemannian gradient descent, for computing weighted Riemannian centres of mass.

Multivariate generalized Gaussian distribution (MGGD) has been an attractive solution to many signal processing problems due to its simple yet flexible parametric form, which requires the estimation of only a few parameters, i.e., the scatter matrix and the shape parameter. Existing fixed-point (FP) algorithms provide an easy to implement method for estimating the scatter matrix, but are known to fail, giving highly inaccurate results, when the value of the shape parameter increases. Since many applications require flexible estimation of the shape parameter, we propose a new FP algorithm, Riemannian averaged FP (RA-FP), which can effectively estimate the scatter matrix for any value of the shape parameter. We provide the mathematical justification of the convergence of the RA-FP algorithm based on the Riemannian geometry of the space of symmetric positive definite matrices. We also show using numerical simulations that the RA-FP algorithm is invariant to the initialization of the scatter matrix and provides significantly improved performance over existing FP and method-of-moments (MoM) algorithms for the estimation of the scatter matrix.

Partial observation of stochastic processes can occur for various reasons, ranging from faulty sensors to occultation issues. In this paper, we consider the problem of estimating the angular velocity of a rotating system from partial observation corrupted by noise. The system is assumed to evolve on the rotation group SO(n), and only k noisy measurements with k <; n are available. We propose an optimal filter to track the angular velocity. We show that, under some conditions, it is possible to recover the angular velocity of the rotating system and we propose a solution based on a Monte-Carlo method (particle filter). In particular, we show that if the angular velocity is stepwise constant, our algorithm succeed in estimating it. Simulations illustrate the proposed approach.

Wide sense stationary processes are a mainstay of classical signal processing. It is well known that they can be obtained by solving ordinary differential equations with constant coefficients whose right-hand side is a white noise. This paper addresses the extension of this construction to random fields defined on compact Lie groups. On an underlying compact Lie group, the paper studies left invariant second-order elliptic partial differential equations whose right-hand side is a spatial white noise. Quite often, the solution of a partial differential equation is not defined as a function but as a distribution. To adapt to this situation, the paper introduces a definition of wide sense stationary distributions on a compact Lie group. This is shown to be consistent with the more restricted definition of wide sense stationary fields given in a classic paper by Yaglom. It is proved that the solution of a partial differential equation, of the kind being studied, is a wide sense stationary distribution whose covariance structure is determined by the fundamental solution of the equation. As a concrete example, this paper describes the fundamental solution of the Helmholtz equation on the rotation group and the resulting covariance structure.

This paper studies Brownian distributions on compact Lie groups. These are defined as the marginal distributions of Brownian processes and are intended as a natural extension of the well-known normal distributions to compact Lie groups. It is shown that this definition preserves key properties of normal distributions. In particular, Brownian distributions transform in a nice way under group operations and satisfy an extension of the central limit theorem. Brownian distributions on a compact Lie groupGbelong to one of two parametric familiesNL(g,C) andNR(g,C)-g∈GandCa positive-definite symmetric matrix. In particular, the parametergappears as a location parameter. An approach based on the extrinsic mean for estimation of the parametersgandCis studied in detail. It is shown thatgis the unique extrinsic mean for a Brownian distributionNL(g,C) orNR(g,C). Resulting estimates are proved to be consistent and asymptotically normal. While they may also be used to simultaneously estimategandC, it is seen this requires thatGbe embedded into a higher dimensional matrix Lie group. Going beyond Brownian distributions, it is shown the extrinsic mean can be used to recover the location parameter for a wider class of distributions arising more generally from Lévy processes. The compact Lie group structure places limitations on the analogy between normal distributions and Brownian distributions. This is illustrated by the study of multivariate Brownian distributions. These are introduced as Brownian distributions on some product group-e.g.,G×G. This paper describes their covariance structure and considers its transformation under group operations.

Noncommutative harmonic analysis is used to solve a nonparametric estimation problem stated in terms of compound Poisson processes on compact Lie groups. This problem of decompounding is a generalization of a similar classical problem. The proposed solution is based on a characteristic function method. The treated problem is important to recent models of the physical inverse problem of multiple scattering.

Motivated by applications in multiple scattering, we study the problem of decompounding on compact Lie groups. Employing tools from harmonic analysis, we give a nonparametric approach to this problem. The case of the special orthogonal group SO(3) is discussed in detail.