相关论文: Modelling curvature of a bent paper leaf

We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised…

Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this…

In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations,…

The purpose of this paper is to employ the language of Cartan moving frames to study the geometry of the data manifolds and its Riemannian structure, via the data information metric and its curvature at data points. Using this framework and…

Geometric graphs are a special kind of graph with geometric features, which are vital to model many scientific problems. Unlike generic graphs, geometric graphs often exhibit physical symmetries of translations, rotations, and reflections,…

The clear understanding of the non-convex landscape of neural network is a complex incomplete problem. This paper studies the landscape of linear (residual) network, the simplified version of the nonlinear network. By treating the gradient…

This paper presents a transformative framework for artificial neural networks over graded vector spaces, tailored to model hierarchical and structured data in fields like algebraic geometry and physics. By exploiting the algebraic…

The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to…

On logarithmic paper some real algebraic curves look like smoothed broken lines. Moreover, the broken lines can be obtained as limits of those curves. The corresponding deformation can be viewed as a quantization, in which the broken line…

Document structure analysis, such as zone segmentation and table recognition, is a complex problem in document processing and is an active area of research. The recent success of deep learning in solving various computer vision and machine…

This paper presents a mathematical framework for analyzing machine learning models through the geometry of their induced partitions. By representing partitions as Riemannian simplicial complexes, we capture not only adjacency relationships…

Neural representations of 3D data have been widely adopted across various applications, particularly in recent work leveraging coordinate-based networks to model scalar or vector fields. However, these approaches face inherent challenges,…

What is the best representation for doing euclidean geometry on computers? These notes from a SIGGRAPH 2019 short course entitled "Geometric algebra for computer graphics" introduce projective geometric algebra (PGA) as a modern framework…

The development of computational techniques in the last decade has made possible to attack some classical problems of algebraic geometry. In this survey, we briefly describe some open problems related to algebraic curves which can be…

Network models with latent geometry have been used successfully in many applications in network science and other disciplines, yet it is usually impossible to tell if a given real network is geometric, meaning if it is a typical element in…

There is strong interest in developing mathematical methods that can be used to understand complex neural networks used in image analysis. In this paper, we introduce techniques from Linear Algebra to model neural network layers as maps…

This work proposes an algorithm for explicitly constructing a pair of neural networks that linearize and reconstruct an embedded submanifold, from finite samples of this manifold. Our such-generated neural networks, called Flattening…

Past approaches for statistical shape analysis of objects have focused mainly on objects within the same topological classes, e.g., scalar functions, Euclidean curves, or surfaces, etc. For objects that differ in more complex ways, the…

Deep learning models are often considered black boxes due to their complex hierarchical transformations. Identifying suitable architectures is crucial for maximizing predictive performance with limited data. Understanding the geometric…

In this survey I should like to introduce some concepts of algebraic geometry and try to demonstrate the fruitful interaction between algebraic geometry and computer algebra and, more generally, between mathematics and computer science. One…