43 research outputs found
Regression-Based Elastic Metric Learning on Shape Spaces of Elastic Curves
We propose a metric learning paradigm, Regression-based Elastic Metric
Learning (REML), which optimizes the elastic metric for geodesic regression on
the manifold of discrete curves. Geodesic regression is most accurate when the
chosen metric models the data trajectory close to a geodesic on the discrete
curve manifold. When tested on cell shape trajectories, regression with REML's
learned metric has better predictive power than with the conventionally used
square-root-velocity (SRV) metric.Comment: 4 pages, 2 figures, derivations in appendi
A General Framework for Robust G-Invariance in G-Equivariant Networks
We introduce a general method for achieving robust group-invariance in
group-equivariant convolutional neural networks (-CNNs), which we call the
-triple-correlation (-TC) layer. The approach leverages the theory of the
triple-correlation on groups, which is the unique, lowest-degree polynomial
invariant map that is also complete. Many commonly used invariant maps - such
as the max - are incomplete: they remove both group and signal structure. A
complete invariant, by contrast, removes only the variation due to the actions
of the group, while preserving all information about the structure of the
signal. The completeness of the triple correlation endows the -TC layer with
strong robustness, which can be observed in its resistance to invariance-based
adversarial attacks. In addition, we observe that it yields measurable
improvements in classification accuracy over standard Max -Pooling in
-CNN architectures. We provide a general and efficient implementation of the
method for any discretized group, which requires only a table defining the
group's product structure. We demonstrate the benefits of this method for
-CNNs defined on both commutative and non-commutative groups - ,
, , and (discretized as the cyclic , dihedral ,
chiral octahedral and full octahedral groups) - acting on
and on both -MNIST and -ModelNet10
datasets
Defining a mean on Lie group
National audienceThis master thesis explores the properties of three different definitions of the mean on a Lie group : the Riemannian Center of Mass, the Riemannian exponential barycenter and the group exponential barycenter.Cette thèse de master étudie trois différentes définitions de la moyenne sur un groupe de Lie : le centre de masse riemannien, le barycentre exponentiel riemannien et le barycentre exponentiel de groupe
Statistics on Lie groups : a need to go beyond the pseudo-Riemannian framework
Abstract. Lie groups appear in many fields from Medical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ's shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform statistics on Lie groups. A Lie group G is a manifold that carries an additional group structure. Statistics on Riemannian manifolds have been well studied with the pioneer work of Fréchet, Karcher and Kendall [1, 2, 3, 4] followed by other
Architectures of Topological Deep Learning: A Survey on Topological Neural Networks
The natural world is full of complex systems characterized by intricate
relations between their components: from social interactions between
individuals in a social network to electrostatic interactions between atoms in
a protein. Topological Deep Learning (TDL) provides a comprehensive framework
to process and extract knowledge from data associated with these systems, such
as predicting the social community to which an individual belongs or predicting
whether a protein can be a reasonable target for drug development. TDL has
demonstrated theoretical and practical advantages that hold the promise of
breaking ground in the applied sciences and beyond. However, the rapid growth
of the TDL literature has also led to a lack of unification in notation and
language across Topological Neural Network (TNN) architectures. This presents a
real obstacle for building upon existing works and for deploying TNNs to new
real-world problems. To address this issue, we provide an accessible
introduction to TDL, and compare the recently published TNNs using a unified
mathematical and graphical notation. Through an intuitive and critical review
of the emerging field of TDL, we extract valuable insights into current
challenges and exciting opportunities for future development
A survey of mathematical structures for extending 2D neurogeometry to 3D image processing
International audienceIn the era of big data, one may apply generic learning algorithms for medical computer vision. But such algorithms are often "black-boxes" and as such, hard to interpret. We still need new constructive models, which could eventually feed the big data framework. Where can one find inspiration for new models in medical computer vision? The emerging field of Neurogeometry provides innovative ideas.Neurogeometry models the visual cortex through modern Differential Geometry: the neuronal architecture is represented as a sub-Riemannianmanifold R2 x S1. On the one hand, Neurogeometry explains visual phenomena like human perceptual completion. On the other hand, it provides efficient algorithms for computer vision. Examples of applications are image completion (in-painting) and crossing-preserving smoothing. In medical image computer vision, Neurogeometry is less known although some algorithms exist. One reason is that one often deals with 3D images, whereas Neurogeometry is essentially 2D (our retina is 2D). Moreover, the generalization of (2D)-Neurogeometry to 3D is not straight-forward from the mathematical point of view. This article presents the theoretical framework of a 3D-Neurogeometry inspired by the 2D case. We survey the mathematical structures and a standard frame for algorithms in 3D- Neurogeometry. The aim of the paper is to provide a "theoretical toolbox" and inspiration for new algorithms in 3D medical computer vision
Statistics on Lie groups : a need to go beyond the pseudo-Riemannian framework
International audienceLie groups appear in many fields from Medical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ's shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform statistics on Lie groups. A Lie group G is a manifold that carries an additional group structure. Statistics on Riemannian manifolds have been well studied with the pioneer work of Fréchet, Karcher and Kendall [1, 2, 3, 4] followed by others [5, 6, 7, 8, 9]. In order to use such a Riemannian structure for statistics on Lie groups, one needs to define a Riemannian metric that is compatible with the group structure, i.e a bi-invariant metric. However, it is well known that general Lie groups which cannot be decomposed into the direct product of compact and abelian groups do not admit a bi-invariant metric. One may wonder if removing the positivity of the metric, thus asking only for a bi-invariant pseudo-Riemannian metric, would be sufficient for most of the groups used in Computational Anatomy. In this paper, we provide an algorithmic procedure that constructs bi-invariant pseudo-metrics on a given Lie group G . The procedure relies on a classification theorem of Medina and Revoy. However in doing so, we prove that most Lie groups do not admit any bi-invariant (pseudo-) metric. We conclude that the (pseudo-) Riemannian setting is not the richest setting if one wants to perform statistics on Lie groups. One may have to rely on another framework, such as affine connection space