Related papers: A Geometric Framework for Convolutional Neural Net…

In this paper, we propose a geometric framework to analyze the convergence properties of gradient descent trajectories in the context of linear neural networks. We translate a well-known empirical observation of linear neural nets into a…

Neural computation in biological and artificial networks relies on the nonlinear summation of many inputs. The structural connectivity matrix of synaptic weights between neurons is a critical determinant of overall network function, but…

In this paper, we study the dynamics of gradient descent in learning neural networks for classification problems. Unlike in existing works, we consider the linearly non-separable case where the training data of different classes lie in…

Deep Neural Networks (DNNs) have become very popular for prediction in many areas. Their strength is in representation with a high number of parameters that are commonly learned via gradient descent or similar optimization methods. However,…

In contrast to the somewhat abstract, group theoretical approach adopted by many papers, our work provides a new and more intuitive derivation of steerable convolutional neural networks in $d$ dimensions. This derivation is based on…

We address the problem of determining correspondences between two images in agreement with a geometric model such as an affine or thin-plate spline transformation, and estimating its parameters. The contributions of this work are…

Feature descriptors play a crucial role in a wide range of geometry analysis and processing applications, including shape correspondence, retrieval, and segmentation. In this paper, we introduce Geodesic Convolutional Neural Networks…

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,…

Machine learning problems have an intrinsic geometric structure as central objects including a neural network's weight space and the loss function associated with a particular task can be viewed as encoding the intrinsic geometry of a given…

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 compact, matrix-based representation of neural networks in a self-contained tutorial fashion. Specifically, we develop neural networks as a composition of several vector-valued functions. Although neural networks are…

The Convolutional Neural Network (CNN) has been successfully applied in many fields during recent decades; however it lacks the ability to utilize prior domain knowledge when dealing with many realistic problems. We present a framework…

This paper introduces the Neural Differential Manifold (NDM), a novel neural network architecture that explicitly incorporates geometric structure into its fundamental design. Departing from conventional Euclidean parameter spaces, the NDM…

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…

We propose a principled convolutional neural pyramid (CNP) framework for general low-level vision and image processing tasks. It is based on the essential finding that many applications require large receptive fields for structure…

This paper concerns models and convergence principles for dealing with stochasticity in a wide range of algorithms arising in nonlinear analysis and optimization in Hilbert spaces. It proposes a flexible geometric framework within which…

In this paper, we investigate the geometric structure of activation spaces of fully connected layers in neural networks and then show applications of this study. We propose an efficient approximation algorithm to characterize the convex…

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…

Many supervised learning tasks have intrinsic symmetries, such as translational and rotational symmetry in image classifications. These symmetries can be exploited to enhance performance. We formulate the symmetry constraints into a concise…

Designing composite materials as per the application requirements is fundamentally a challenging and time consuming task. Here we report the development of a deep neural network based computational framework capable of solving the forward…