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Anisotropy preserving DTI processing Collard, Anne ; ; Phillips, Christophe et al in International Journal of Computer Vision (2014), 107 Detailed reference viewed: 117 (23 ULg)Rank-preserving geometric means of positive semi-definite matrices ; Collard, Anne ; Sepulchre, Rodolphe in Linear Algebra & its Applications (2013), 438(8), 3202-3216 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 ... [more ▼] 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. [less ▲] Detailed reference viewed: 44 (9 ULg)Anisotropy preserving interpolation of diffusion tensors Collard, Anne ; ; Phillips, Christophe et al Poster (2012, June) Detailed reference viewed: 16 (10 ULg)Anisotropy preserving interpolation of diffusion tensors Collard, Anne ; ; Phillips, Christophe et al Conference (2012, March) Detailed reference viewed: 25 (5 ULg)Fixed-rank matrix factorizations and Riemannian low-rank optimization Mishra, Bamdev ; Meyer, Gilles ; et al E-print/Working paper (2012) 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 ... [more ▼] 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 optimization framework of optimization on Riemannian matrix manifolds. We study the underlying geometries of several well-known fixed-rank matrix factorizations and then exploit the Riemannian 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 relative usefulness of the proposed framework. Numerical experiments suggest that the proposed algorithms compete with the state-of-the-art and that manifold optimization offers an effective and versatile framework for the design of machine learning algorithms that learn a fixed-rank matrix. [less ▲] Detailed reference viewed: 41 (5 ULg)Regression on fixed-rank positive semidefinite matrices: a Riemannian approach Meyer, Gilles ; ; Sepulchre, Rodolphe in Journal of Machine Learning Research (2011), 12(Feb), 593625 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 ... [more ▼] 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 semidefinite matrix. Good performance is observed on classical benchmarks. [less ▲] Detailed reference viewed: 109 (18 ULg)Linear regression under fixed-rank constraints: a Riemannian approach Meyer, Gilles ; ; Sepulchre, Rodolphe in Proceedings of the 28th International Conference on Machine Learning (2011) In this paper, we tackle the problem of learning a linear regression model whose parameter is a fixed-rank matrix. We study the Riemannian manifold geometry of the set of fixed-rank matrices and develop ... [more ▼] In this paper, we tackle the problem of learning a linear regression model whose parameter is a fixed-rank matrix. We study the Riemannian manifold geometry of the set of fixed-rank matrices and develop efficient line-search algorithms. The proposed algorithms have many applications, scale to high-dimensional problems, enjoy local convergence properties and confer a geometric basis to recent contributions on learning fixed-rank matrices. Numerical experiments on benchmarks suggest that the proposed algorithms compete with the state-of-the-art, and that manifold optimization offers a versatile framework for the design of rank-constrained machine learning algorithms. [less ▲] Detailed reference viewed: 158 (4 ULg)Rank-constrained linear regression: a Riemannian approach Meyer, Gilles ; ; Sepulchre, Rodolphe Poster (2010, December) Detailed reference viewed: 18 (0 ULg)Adaptive filtering for estimation of a low-rank positive semidefinite matrix ; Meyer, Gilles ; Sepulchre, Rodolphe in Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems (2010, July) In this paper, we adopt a geometric viewpoint to tackle the problem of estimating a linear model whose parameter is a fixed-rank positive semidefinite matrix. We consider two gradient descent flows ... [more ▼] In this paper, we adopt a geometric viewpoint to tackle the problem of estimating a linear model whose parameter is a fixed-rank positive semidefinite matrix. We consider two gradient descent flows associated to two distinct Riemannian quotient geometries that underlie this set of matri- ces. The resulting algorithms are non-linear and can be viewed as a generalization of Least Mean Squares that instrically constrain the parameter within the manifold search space. Such algorithms designed for low-rank matrices find applications in high-dimensional distance learning problems for classification or clustering. [less ▲] Detailed reference viewed: 52 (16 ULg)Géométrie des matrices positives semi-définies de rang fixé : un peu de théorie et beaucoup d’applications Sepulchre, Rodolphe ; ; in Proceedings of Sixième Conférence Internationale Francophone d'Automatique (CIFA 2010) (2010, June) Cet article est une introduction au calcul et `a l’optimisation sur les matrices sym´etriques positives semid ´efinies de rang (faible) fix´e. L’approche propos´ee est bas´ee sur deux g´eom´etries ... [more ▼] Cet article est une introduction au calcul et `a l’optimisation sur les matrices sym´etriques positives semid ´efinies de rang (faible) fix´e. L’approche propos´ee est bas´ee sur deux g´eom´etries riemanniennes quotient, qui permettent de calculer efficacement tout en pr´eservant le rang et le caract` ere positif des matrices consid´er´ees. Le champ d’applications est vaste, et l’article survole quelques d´eveloppements r´ecents qui illustrent l’int´erˆet de l’approche consid´er´ee dans les probl`emes de tr`es grande taille rencontr´es en contrˆole, statistiques, et apprentissage. [less ▲] Detailed reference viewed: 77 (7 ULg)Coordinated motion design on Lie groups Sarlette, Alain ; ; Sepulchre, Rodolphe in IEEE Transactions on Automatic Control (2010), 55(5), 1047-1058 The present paper proposes a unified geometric framework for coordinated motion on Lie groups. It first gives a general problem formulation and analyzes ensuing conditions for coordinated motion. Then, it ... [more ▼] The present paper proposes a unified geometric framework for coordinated motion on Lie groups. It first gives a general problem formulation and analyzes ensuing conditions for coordinated motion. Then, it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integrator dynamics. It thereby shows that coordination can be studied in a systematic way once the Lie group geometry of the configuration space is well characterized. Applying the proposed general methodology to particular examples allows to retrieve control laws that have been proposed in the literature on intuitive grounds. A link with Brockett's double bracket flows is also made. The concepts are illustrated on SO(3) , SE(2) and SE(3). [less ▲] Detailed reference viewed: 70 (19 ULg)From subspace learning to distance learning: a geometrical optimization approach Meyer, Gilles ; ; et al in Proceedings of the 2009 IEEE Workshop on Statistical Signal Processing (SSP2009) (2009) In this paper, we adopt a differential-geometry viewpoint to tackle the problem of learning a distance online. As this prob- lem can be cast into the estimation of a fixed-rank positive semidefinite (PSD ... [more ▼] In this paper, we adopt a differential-geometry viewpoint to tackle the problem of learning a distance online. As this prob- lem can be cast into the estimation of a fixed-rank positive semidefinite (PSD) matrix, we develop algorithms that ex- ploits the rich geometry structure of the set of fixed-rank PSD matrices. We propose a method which separately updates the subspace of the matrix and its projection onto that subspace. A proper weighting of the two iterations enables to continu- ously interpolate between the problem of learning a subspace and learning a distance when the subspace is fixed. [less ▲] Detailed reference viewed: 77 (27 ULg)Riemannian Metric and Geometric Mean for Positive Semidefinite Matrices of Fixed Rank ; Sepulchre, Rodolphe in SIAM Journal on Matrix Analysis & Applications (2009), 31(3), 1055--1070 Detailed reference viewed: 36 (11 ULg)Coordination on Lie groups Sarlette, Alain ; ; Sepulchre, Rodolphe in Proceedings of the 47th IEEE Conference on Decision and Control (2008, December) This paper studies the coordinated motion of a group of agents evolving on a Lie group. Left- or right-invariance with respect to the absolute position on the group lead to two different characterizations ... [more ▼] This paper studies the coordinated motion of a group of agents evolving on a Lie group. Left- or right-invariance with respect to the absolute position on the group lead to two different characterizations of relative positions and two associated definitions of coordination (fixed relative positions). Conditions for each type of coordination are derived in the associated Lie algebra. This allows to formulate the coordination problem on Lie groups as consensus in a vector space. Total coordination occurs when both types of coordination hold simultaneously. The discussion in this paper provides a common geometric framework for previously published coordination control laws on SO(3), SE(2) and SE(3). The theory is illustrated on the group of planar rigid motion SE(2). [less ▲] Detailed reference viewed: 42 (16 ULg) |
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