562 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
Project Elements: A computational entity-component-system in a scene-graph pythonic framework, for a neural, geometric computer graphics curriculum
We present the Elements project, a computational science and computer
graphics (CG) framework, that offers for the first time the advantages of an
Entity-Component-System (ECS) along with the rapid prototyping convenience of a
Scenegraph-based pythonic framework. This novelty allows advances in the
teaching of CG: from heterogeneous directed acyclic graphs and depth-first
traversals, to animation, skinning, geometric algebra and shader-based
components rendered via unique systems all the way to their representation as
graph neural networks for 3D scientific visualization. Taking advantage of the
unique ECS in a a Scenegraph underlying system, this project aims to bridge CG
curricula and modern game engines, that are based on the same approach but
often present these notions in a black-box approach. It is designed to actively
utilize software design patterns, under an extensible open-source approach.
Although Elements provides a modern, simple to program pythonic approach with
Jupyter notebooks and unit-tests, its CG pipeline is not black-box, exposing
for teaching for the first time unique challenging scientific, visual and
neural computing concepts.Comment: 8 pages, 8 figures, 2 listings, submitted to EuroGraphics 2023
education trac
Geometric Clifford Algebra Networks
We propose Geometric Clifford Algebra Networks (GCANs) for modeling dynamical
systems. GCANs are based on symmetry group transformations using geometric
(Clifford) algebras. We first review the quintessence of modern (plane-based)
geometric algebra, which builds on isometries encoded as elements of the
group. We then propose the concept of group action
layers, which linearly combine object transformations using pre-specified group
actions. Together with a new activation and normalization scheme, these layers
serve as adjustable that can be refined via
gradient descent. Theoretical advantages are strongly reflected in the modeling
of three-dimensional rigid body transformations as well as large-scale fluid
dynamics simulations, showing significantly improved performance over
traditional methods
Three-dimensional quadrics in extended conformal geometric algebras of higher dimensions from control points, implicit equations and axis alignment
International audienceWe introduce the quadric conformal geometric algebra (QCGA) inside the algebra of R 9,6. In particular, this paper presents how three-dimensional quadratic surfaces can be defined by the outer product of conformal geometric algebra points in higher dimensions, or alternatively by a linear combination of basis vectors with coefficients straight from the implicit quadratic equation. These multivector expressions code all types of quadratic surfaces in arbitrary scale, location, and orientation. Furthermore, we investigate two types of definitions of axis aligned quadric surfaces, from contact points and dually from linear combinations of R 9,6 basis vectors
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