相关论文: The algebra and the geometry aspect of Deep learni…

In this paper, we overview one promising avenue of progress at the mathematical foundation of deep learning: the connection between deep networks and function approximation by affine splines (continuous piecewise linear functions in…

As datasets used in scientific applications become more complex, studying the geometry and topology of data has become an increasingly prevalent part of the data analysis process. This can be seen for example with the growing interest in…

Although statistical learning theory provides a robust framework to understand supervised learning, many theoretical aspects of deep learning remain unclear, in particular how different architectures may lead to inductive bias when trained…

Deep neural networks have reshaped modern machine learning by learning powerful latent representations that often align with the manifold hypothesis: high-dimensional data lie on lower-dimensional manifolds. In this paper, we establish a…

Multilayered artificial neural networks are becoming a pervasive tool in a host of application fields. At the heart of this deep learning revolution are familiar concepts from applied and computational mathematics; notably, in calculus,…

There have been several attempts to mathematically understand neural networks and many more from biological and computational perspectives. The field has exploded in the last decade, yet neural networks are still treated much like a black…

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…

While deep learning has enabled significant advances in many areas of science, its black-box nature hinders architecture design for future artificial intelligence applications and interpretation for high-stakes decision makings. We…

Thanks to the availability of large scale digital datasets and massive amounts of computational power, deep learning algorithms can learn representations of data by exploiting multiple levels of abstraction. These machine learning methods…

This article provides an expository account of training dynamics in the Deep Linear Network (DLN) from the perspective of the geometric theory of dynamical systems. Rigorous results by several authors are unified into a thermodynamic…

Advances in experimental neuroscience have transformed our ability to explore the structure and function of neural circuits. At the same time, advances in machine learning have unleashed the remarkable computational power of artificial…

This work starts with the intention of using mathematics to understand the intriguing vulnerability observed by ~\citet{szegedy2013} within artificial neural networks. Along the way, we will develop some novel tools with applications far…

Deep neural networks are widely used for classification. These deep models often suffer from a lack of interpretability -- they are particularly difficult to understand because of their non-linear nature. As a result, neural networks are…

Deep neural networks have been demonstrated to achieve phenomenal success in many domains, and yet their inner mechanisms are not well understood. In this paper, we investigate the curvature of image manifolds, i.e., the manifold deviation…

In this expository paper we want to give a brief introduction, with few key references for further reading, to the inner functioning of the new and successfull algorithms of Deep Learning and Geometric Deep Learning with a focus on Graph…

For classification tasks, the performance of a deep neural network is determined by the structure of its decision boundary, whose geometry directly affects essential properties of the model, including accuracy and robustness. Motivated by a…

Deep learning networks have been trained to recognize speech, caption photographs and translate text between languages at high levels of performance. Although applications of deep learning networks to real world problems have become…

Deep neural networks have shown great potential in image reconstruction problems in Euclidean space. However, many reconstruction problems involve imaging physics that are dependent on the underlying non-Euclidean geometry. In this paper,…

Artificial networks have been studied through the prism of statistical mechanics as disordered systems since the 80s, starting from the simple models of Hopfield's associative memory and the single-neuron perceptron classifier. Assuming…

The prediction of molecular properties is a crucial task in the field of material and drug discovery. The potential benefits of using deep learning techniques are reflected in the wealth of recent literature. Still, these techniques are…