This library implements manifolds and associated metrics on Keras. MNIST on hyperspheres manifold example code here). I would advise reading the geomstats associated paper, which does a great job at showing what it is and how it can be used, along with example codes (e.g. Regarding deep neural networks on manifolds This paper also describes a typical case for which manifolds are interesting: where graphs (the other example of GDL/DL on non euclidean data) are better at handling data from social or sensor networks, manifolds are good at modeling 3D objects endowed with properties like color texture in computer vision. This analogy reminds us of the spherical surface planet model from Bronstein's paper already quoted above. But if you instead move within the local earth (say spherical) coordinates you will stay on the surface and get to see all the cool stuff. If you don't understand the structure of earth you'll quickly find yourself in space or inside the earth. Now say you're on the surface of the earth which is a 2-manifold and you start moving in a random direction (let's assume gravity doesn't exist and you can go through solid objects). Perhaps a good analogy here is that of a solar system: the surface of our planets are the manifolds we're interested in, one for each digit. One could say you're looking for the best information geometry for your task, the one that best captures the desirable data distribution properties.To develop this intuition consider the solar system analogy for manifold learning of this Kaggle kernel: I see it as a part of the effort of deep learning formalization. Why bother learning on manifolds ?ĭefining a clearer/better adapted set (understand that it's a sort of constraint!) on which to learn parameters and features can make it simpler to formally understand what your model is doing, and can lead to better results. For example, you can choose to work on images and videos by representing the samples using Symmetric Positive Definite (SPD) matrices (see this paper), the space of SPD matrices being a manifold itself. Your data can also be represented thanks to a practical manifold. training with parameters constrained on a hypersphere, among the geomstats paper examples). By work, you can typically understand that you constrain the neural net parameters to the manifold you chose (e.g. Very shortly put, it's an interesting mathematical set on which to work (different kinds exist, see papers at the end of this answer for DL related manifolds uses). Other Wikipedia examples to develop not too abstract understanding Good other not-so-technical explanation on stats.stackexchange (homeomorphic) to a d-dimensional Euclidean space, called the tangent (differentiable) d-dimensional manifold X is a topological space whereĮach point x has a neighborhood that is topologically equivalent Point, it seems to be planar, which has led generations of people toīelieve in the flatness of the Earth. Simplest examples is a spherical surface modeling our planet: around a Roughly, a manifold is a space that is locally Euclidean. In case you don't know what a manifold is ![]() Note that the description of GDL through the explanation of what are DL on graphs and manifolds, in opposition to DL on euclidean domains, comes from the 2017 paper Geometric deep learning: going beyond Euclidean data (this paper is excellent at clarifying both the intuition and the mathematics of what I'm writing). ![]() Such a system would significantly accelerate the process of scientific discovery and engineering innovation.To complete the first answer that is rather graph oriented, I will write a little about deep learning on manifolds, which is quite general in terms of GDL thanks to the nature of manifolds. ![]() ![]() The goal is to create an AI system that can help scientists and engineers gain a deeper understanding of complex systems by automating the process of deriving equations and building test prototypes. However, this process can be labor-intensive and time-consuming, leading to the question of whether automation can assist in this endeavor.ĭumancic and his team want to explore the possibility of using artificial intelligence (AI) to help uncover explanations for the natural phenomena that surround us. Scientists and engineers follow this principle when they distil natural phenomena into concise mathematic equations and build test prototypes of complex machines. Scientists and engineers adopt this principle when they simplify natural phenomena and create mathematical equations to explain them. This statement emphasizes the importance of gaining a comprehensive understanding of a concept by starting from scratch, using fundamental principles and building blocks. Richard Feynman famously said "What I cannot create, I do not understand".
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