133 research outputs found
DIMAL: Deep Isometric Manifold Learning Using Sparse Geodesic Sampling
This paper explores a fully unsupervised deep learning approach for computing
distance-preserving maps that generate low-dimensional embeddings for a certain
class of manifolds. We use the Siamese configuration to train a neural network
to solve the problem of least squares multidimensional scaling for generating
maps that approximately preserve geodesic distances. By training with only a
few landmarks, we show a significantly improved local and nonlocal
generalization of the isometric mapping as compared to analogous non-parametric
counterparts. Importantly, the combination of a deep-learning framework with a
multidimensional scaling objective enables a numerical analysis of network
architectures to aid in understanding their representation power. This provides
a geometric perspective to the generalizability of deep learning.Comment: 10 pages, 11 Figure
Analysis of (sub-)Riemannian PDE-G-CNNs
Group equivariant convolutional neural networks (G-CNNs) have been successfully applied in geometric deep learning. Typically, G-CNNs have the advantage over CNNs that they do not waste network capacity on training symmetries that should have been hard-coded in the network. The recently introduced framework of PDE-based G-CNNs (PDE-G-CNNs) generalizes G-CNNs. PDE-G-CNNs have the core advantages that they simultaneously (1) reduce network complexity, (2) increase classification performance, and (3) provide geometric interpretability. Their implementations primarily consist of linear and morphological convolutions with kernels. In this paper, we show that the previously suggested approximative morphological kernels do not always accurately approximate the exact kernels accurately. More specifically, depending on the spatial anisotropy of the Riemannian metric, we argue that one must resort to sub-Riemannian approximations. We solve this problem by providing a new approximative kernel that works regardless of the anisotropy. We provide new theorems with better error estimates of the approximative kernels, and prove that they all carry the same reflectional symmetries as the exact ones. We test the effectiveness of multiple approximative kernels within the PDE-G-CNN framework on two datasets, and observe an improvement with the new approximative kernels. We report that the PDE-G-CNNs again allow for a considerable reduction of network complexity while having comparable or better performance than G-CNNs and CNNs on the two datasets. Moreover, PDE-G-CNNs have the advantage of better geometric interpretability over G-CNNs, as the morphological kernels are related to association fields from neurogeometry
Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking
We introduce a data-driven version of the plus Cartan connection on the
homogeneous space of 2D positions and orientations. We formulate
a theorem that describes all shortest and straight curves (parallel velocity
and parallel momentum, respectively) with respect to this new data-driven
connection and corresponding Riemannian manifold. Then we use these shortest
curves for geodesic tracking of complex vasculature in multi-orientation image
representations defined on . The data-driven Cartan connection
characterizes the Hamiltonian flow of all geodesics. It also allows for
improved adaptation to curvature and misalignment of the (lifted) vessel
structure that we track via globally optimal geodesics. We compute these
geodesics numerically via steepest descent on distance maps on
that we compute by a new modified anisotropic fast-marching method.
Our experiments range from tracking single blood vessels with fixed endpoints
to tracking complete vascular trees in retinal images. Single vessel tracking
is performed in a single run in the multi-orientation image representation,
where we project the resulting geodesics back onto the underlying image. The
complete vascular tree tracking requires only two runs and avoids prior
segmentation, placement of extra anchor points, and dynamic switching between
geodesic models.
Altogether we provide a geodesic tracking method using a single, flexible,
transparent, data-driven geodesic model providing globally optimal curves which
correctly follow highly complex vascular structures in retinal images.
All experiments in this article can be reproduced via documented Mathematica
notebooks available at GitHub
(https://github.com/NickyvdBerg/DataDrivenTracking)
Analysis of (sub-)Riemannian PDE-G-CNNs
Group equivariant convolutional neural networks (G-CNNs) have been
successfully applied in geometric deep learning. Typically, G-CNNs have the
advantage over CNNs that they do not waste network capacity on training
symmetries that should have been hard-coded in the network. The recently
introduced framework of PDE-based G-CNNs (PDE-G-CNNs) generalises G-CNNs.
PDE-G-CNNs have the core advantages that they simultaneously 1) reduce network
complexity, 2) increase classification performance, and 3) provide geometric
interpretability. Their implementations primarily consist of linear and
morphological convolutions with kernels.
In this paper we show that the previously suggested approximative
morphological kernels do not always accurately approximate the exact kernels
accurately. More specifically, depending on the spatial anisotropy of the
Riemannian metric, we argue that one must resort to sub-Riemannian
approximations. We solve this problem by providing a new approximative kernel
that works regardless of the anisotropy. We provide new theorems with better
error estimates of the approximative kernels, and prove that they all carry the
same reflectional symmetries as the exact ones.
We test the effectiveness of multiple approximative kernels within the
PDE-G-CNN framework on two datasets, and observe an improvement with the new
approximative kernels. We report that the PDE-G-CNNs again allow for a
considerable reduction of network complexity while having comparable or better
performance than G-CNNs and CNNs on the two datasets. Moreover, PDE-G-CNNs have
the advantage of better geometric interpretability over G-CNNs, as the
morphological kernels are related to association fields from neurogeometry.Comment: 29 pages, 21 figure
Recommended from our members
Immunization expands B cells specific to HIV-1 V3 glycan in mice and macaques.
Broadly neutralizing monoclonal antibodies protect against infection with HIV-1 in animal models, suggesting that a vaccine that elicits these antibodies would be protective in humans. However, it has not yet been possible to induce adequate serological responses by vaccination. Here, to activate B cells that express precursors of broadly neutralizing antibodies within polyclonal repertoires, we developed an immunogen, RC1, that facilitates the recognition of the variable loop 3 (V3)-glycan patch on the envelope protein of HIV-1. RC1 conceals non-conserved immunodominant regions by the addition of glycans and/or multimerization on virus-like particles. Immunization of mice, rabbits and rhesus macaques with RC1 elicited serological responses that targeted the V3-glycan patch. Antibody cloning and cryo-electron microscopy structures of antibody-envelope complexes confirmed that immunization with RC1 expands clones of B cells that carry the anti-V3-glycan patch antibodies, which resemble precursors of human broadly neutralizing antibodies. Thus, RC1 may be a suitable priming immunogen for sequential vaccination strategies in the context of polyclonal repertoires
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