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We consider the problem of computing tractable approximations of time-dependent dxd large positive semidefinite (PSD) matrices defined as solutions of a matrix differential equation. We propose to use "low-rank plus diagonal" PSD matrices as approximations that can be stored with a memory cost being linear in the high dimension d. To constrain the solution of the differential equation to remain in that subset, we project the derivative at all times onto the tangent space to the subset, following the methodology of dynamical low-rank approximation. We derive a closedform formula for the projection and show that after some manipulations, it can be computed with a numerical cost being linear in d, allowing for tractable implementation. Contrary to previous approaches based on pure low-rank approximations, the addition of the diagonal term allows for our approximations to be invertible matrices that can moreover be inverted with linear cost in d. We apply the technique to Riccati-like equations, then to two particular problems: first, a lowrank approximation to our recent Wasserstein gradient flow for Gaussian approximation of posterior distributions in approximate Bayesian inference and, second, a novel low-rank approximation of the Kalman filter for high-dimensional systems. Numerical simulations illustrate the results. [ABSTRACT FROM AUTHOR]

We propose a unifying and versatile framework for a class of discrete time systems whose state is an element of a general group G , that we call linear observed systems on groups. Those systems strictly mimic linear systems in the sense that ＋ is replaced with group multiplication, and linear maps with automorphisms. We argue they are the true generalization of linear systems of the form X n + 1 = F n X n + B n u n in the context of state estimation, since 1—when G = R N the latter systems are recovered, 2—they are proved to possess the "preintegration" property, a characteristic property of linear systems that relates continuous time to discrete time, and has recently proved extremely useful in robotics applications, and 3—we can build observers that ensure the evolution between the true state and estimated state does not depend on the followed trajectory, a characteristic feature of Luenberger (and invariant) observers. The theory is applied to a 3D inertial navigation example. Interestingly, this example cannot be put in the form of an invariant system and the proposed generalization is required. [ABSTRACT FROM AUTHOR]

Trajectories of aerial and marine vehicles are typically made of a succession of smooth trajectories, linked by abrupt changes, i.e. maneuvers. Notably, modern highly maneuvering targets are capable of very brutal changes in the heading, with accelerations of up to 15 g. As a result, we model the target behavior using piecewise deterministic Markov models, driven by parameters that jump at unknown times. Over the past years, real-time (or incremental) optimization-based smoothing methods have become a popular alternative to nonlinear filters, such as the Extended Kálmán Filter (EKF), owing to the successive relinearizations that mitigate the linearization errors that inherently affect the EKF estimates. In the present paper, we propose to combine such methods for tracking the target during non-jumping phases with a probabilistic approach to detect jumps. Our algorithm is shown to compare favorably to the state-of-the-art Interacting Multiple Model (IMM) algorithm, especially in terms of target's velocity estimation, on a set of meaningful and challenging trajectories. [ABSTRACT FROM AUTHOR]

Statistical analysis of diffusion tensor imaging (DTI) data requires a computational framework that is both numerically tractable (to account for the high dimensional nature of the data) and geometric (to account for the nonlinear nature of diffusion tensors). Building upon earlier studies exploiting a Riemannian framework to address these challenges, the present paper proposes a novel metric and an accompanying computational framework for DTI data processing. The proposed approach grounds the signal processing operations in interpolating curves. Well-chosen interpolating curves are shown to provide a computational framework that is at the same time tractable and information relevant for DTI processing. In addition, and in contrast to earlier methods, it provides an interpolation method which preserves anisotropy, a central information carried by diffusion tensor data. [ABSTRACT FROM AUTHOR]

Abstract: The generalization of the geometric mean of positive scalars to positive definite matrices has attracted considerable attention since the seminal work of Ando. The paper generalizes this framework of matrix means by proposing the definition of a rank-preserving mean for two or an arbitrary number of positive semi-definite matrices of fixed rank. The proposed mean is shown to be geometric in that it satisfies all the expected properties of a rank-preserving geometric mean. The work is motivated by operations on low-rank approximations of positive definite matrices in high-dimensional spaces. [Copyright &y& Elsevier]

Abstract: This work presents a low-cost vehicle localization system, using measurements from one gyroscope, two wheel speed sensors and a GPS, to estimate the heading, velocity and position of a vehicle. Taking advantage of the nonholonomic constraints, the design of the observer (or “filter”) takes into account imperfections of the embedded sensor measurements, such as a slowly time-varying gyroscope bias or some uncertainty in the wheel diameter value. Thanks to a well-chosen nonlinear structure, the estimator is easy to tune, easy to implement, and well-behaved even at very low speed. Moreover, the proposed filter has guaranteed convergence properties when GPS is available and keeps providing good estimations during GPS losses. Simulation and experiment results in urban area illustrate the good performance of the algorithm. [Copyright &y& Elsevier]

The paper addresses the problem of learning a regression model parameterized by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear nature of the search space and on scalability to high-dimensional problems. The mathematical developments rely on the theory of gradient descent algorithms adapted to the Riemannian geometry that underlies the set of fixed-rank positive semidefinite matrices. In contrast with previous contributions in the literature, no restrictions are imposed on the range space of the learned matrix. The resulting algorithms maintain a linear complexity in the problem size and enjoy important invariance properties. We apply the proposed algorithms to the problem of learning a distance function parameterized by a positive semide?nite matrix. Good performance is observed on classical benchmarks. [ABSTRACT FROM AUTHOR]

Abstract We present a method of exploiting symmetries of discrete-time optimal control problems to reduce the dimensionality of dynamic programming iterations. The results are derived for systems with continuous state variables, and can be applied to systems with continuous or discrete symmetry groups. We prove that symmetries of the state update equation and stage costs induce corresponding symmetries of the optimal cost function and the optimal policies. We then provide a general framework for computing the optimal cost function based on gridding a space of lower dimension than the original state space. This method does not require algebraic manipulation of the state update equations; it only requires knowledge of the symmetries that the state update equations possess. Since the method can be performed without any knowledge of the state update map beyond being able to evaluate it and verify its symmetries, this enables the method to be applied in a wide range of application problems. We illustrate these results on two six-dimensional optimal control problems that are computationally difficult to solve by dynamic programming without symmetry reduction. [ABSTRACT FROM AUTHOR]

Motivated by the problem of learning a linear regression model whose parameter is a large fixed-rank non-symmetric matrix, we consider the optimization of a smooth cost function defined on the set of fixed-rank matrices. We adopt the geometric framework of optimization on Riemannian quotient manifolds. We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian quotient geometry of the search space in the design of a class of gradient descent and trust-region algorithms. The proposed algorithms generalize our previous results on fixed-rank symmetric positive semidefinite matrices, apply to a broad range of applications, scale to high-dimensional problems, and confer a geometric basis to recent contributions on the learning of fixed-rank non-symmetric matrices. We make connections with existing algorithms in the context of low-rank matrix completion and discuss the usefulness of the proposed framework. Numerical experiments suggest that the proposed algorithms compete with state-of-the-art algorithms and that manifold optimization offers an effective and versatile framework for the design of machine learning algorithms that learn a fixed-rank matrix. [ABSTRACT FROM AUTHOR]

• Isotopic niche reflects stable carbon and nitrogen isotopes in animals' tissues. • Celebrated indicators for niche description often rely on Gaussian assumptions. • We advocate minimum convex polygons (MCP) for isotopic niche assessment. • The method allows for outlier rejection and relaxes Gaussian assumption. • A R code allows for illustration on real data. Ecological (isotopic) niche refers to a surface in a two-dimensional space, where the axes correspond to environmental variables that reflect values of stable isotopes incorporated in an animal's tissues. Carbon and nitrogen stable isotope ratios (δ13C-δ15N) notably provide precious information about trophic ecology, resource and habitat use, and population dynamics. Various metrics allow for isotopic niche size and overlap assessment. In this paper, we advocate α-minimum convex polygons (MCP) - that have long been used for home range estimation – as a relevant tool for isotopic niche size, overlap, and characteristics. The method allows for outlier rejection while being suited to data that are not Gaussian in the bivariate isotopic (δ13C-δ15N) space. The proposed indicators are compared to other existing approaches and are shown to be complementary. Notably an indicator of divergence within the niche is introduced, and allows for comparisons at low (n > 6) and different sample sizes. The R code is made publicly available and will enable ecologists to perform isotopic niche comparison, contraction and expansion assessment, and overlap, based on various methods. [ABSTRACT FROM AUTHOR]