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We introduce a method for learning provably stable deep neural network based dynamic models from observed data. Specifically, we consider discrete-time stochastic dynamic models, as they are of particular interest in practical applications…

Machine Learning · Computer Science 2021-03-30 Nathan P. Lawrence , Philip D. Loewen , Michael G. Forbes , Johan U. Backström , R. Bhushan Gopaluni

While deep learning is successful in a number of applications, it is not yet well understood theoretically. A satisfactory theoretical characterization of deep learning however, is beginning to emerge. It covers the following questions: 1)…

Machine Learning · Computer Science 2019-08-27 Tomaso Poggio , Andrzej Banburski , Qianli Liao

In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…

Dynamical Systems · Mathematics 2025-07-08 Dennis Chemnitz , Maximilian Engel , Christian Kuehn , Sara-Viola Kuntz

We consider the problem of learning a graph from a finite set of noisy graph signal observations, the goal of which is to find a smooth representation of the graph signal. Such a problem is motivated by the desire to infer relational…

Machine Learning · Computer Science 2023-02-08 Xiaolu Wang , Yuen-Man Pun , Anthony Man-Cho So

The existence of instabilities, for example in the form of adversarial examples, has given rise to a highly active area of research concerning itself with understanding and enhancing the stability of neural networks. We focus on a popular…

Numerical Analysis · Mathematics 2025-10-28 Matthias J. Ehrhardt , Davide Murari , Ferdia Sherry

Feature propagation in Deep Neural Networks (DNNs) can be associated to nonlinear discrete dynamical systems. The novelty, in this paper, lies in letting the discretization parameter (time step-size) vary from layer to layer, which needs to…

Optimization and Control · Mathematics 2022-04-20 Harbir Antil , Hugo Díaz , Evelyn Herberg

Intermediate features at different layers of a deep neural network are known to be discriminative for visual patterns of different complexities. However, most existing works ignore such cross-layer heterogeneities when classifying samples…

Computer Vision and Pattern Recognition · Computer Science 2016-07-20 Xiaojie Jin , Yunpeng Chen , Jian Dong , Jiashi Feng , Shuicheng Yan

Deep Learning's recent successes have mostly relied on Convolutional Networks, which exploit fundamental statistical properties of images, sounds and video data: the local stationarity and multi-scale compositional structure, that allows…

Machine Learning · Computer Science 2015-06-18 Mikael Henaff , Joan Bruna , Yann LeCun

In the context of classification problems, Deep Learning (DL) approaches represent state of art. Many DL approaches are based on variations of standard multi-layer feed-forward neural networks. These are also referred to as deep networks.…

Machine Learning · Computer Science 2023-11-21 Andrea Apicella , Francesco Isgrò , Roberto Prevete

Robustness against adversarial attack in neural networks is an important research topic in the machine learning community. We observe one major source of vulnerability of neural nets is from overparameterized fully-connected layers. In this…

Machine Learning · Computer Science 2021-02-01 Bingyuan Liu , Christopher Malon , Lingzhou Xue , Erik Kruus

Existing methods for estimating uncertainty in deep learning tend to require multiple forward passes, making them unsuitable for applications where computational resources are limited. To solve this, we perform probabilistic reasoning over…

Machine Learning · Statistics 2020-12-08 Javier Antorán , James Urquhart Allingham , José Miguel Hernández-Lobato

Scaling deep reinforcement learning networks is challenging and often results in degraded performance, yet the root causes of this failure mode remain poorly understood. Several recent works have proposed mechanisms to address this, but…

Deep neural networks have become widely used, obtaining remarkable results in domains such as computer vision, speech recognition, natural language processing, audio recognition, social network filtering, machine translation, and…

Neural and Evolutionary Computing · Computer Science 2020-02-03 Divya Gopinath , Guy Katz , Corina S. Pasareanu , Clark Barrett

Even though probabilistic treatments of neural networks have a long history, they have not found widespread use in practice. Sampling approaches are often too slow already for simple networks. The size of the inputs and the depth of typical…

Computer Vision and Pattern Recognition · Computer Science 2018-05-30 Jochen Gast , Stefan Roth

Even though deep neural networks have shown tremendous success in countless applications, explaining model behaviour or predictions is an open research problem. In this paper, we address this issue by employing a simple yet effective method…

Computer Vision and Pattern Recognition · Computer Science 2023-01-12 Ryan Benkert , Oluwaseun Joseph Aribido , Ghassan AlRegib

A neural network is locally specialized to the extent that parts of its computational graph (i.e. structure) can be abstractly represented as performing some comprehensible sub-task relevant to the overall task (i.e. functionality). Are…

Machine Learning · Computer Science 2022-02-09 Shlomi Hod , Daniel Filan , Stephen Casper , Andrew Critch , Stuart Russell

Learning deep representations to solve complex machine learning tasks has become the prominent trend in the past few years. Indeed, Deep Neural Networks are now the golden standard in domains as various as computer vision, natural language…

Machine Learning · Computer Science 2020-12-04 Vincent Gripon , Carlos Lassance , Ghouthi Boukli Hacene

Deep learning is a powerful tool for solving nonlinear differential equations, but usually, only the solution corresponding to the flattest local minimizer can be found due to the implicit regularization of stochastic gradient descent. This…

Numerical Analysis · Mathematics 2021-03-17 Yiqi Gu , Chunmei Wang , Haizhao Yang

Deep learning is recognized to be capable of discovering deep features for representation learning and pattern recognition without requiring elegant feature engineering techniques by taking advantage of human ingenuity and prior knowledge.…

Machine Learning · Computer Science 2020-04-02 Zhi Han , Siquan Yu , Shao-Bo Lin , Ding-Xuan Zhou

In current deep network architectures, deeper layers in networks tend to contain hundreds of independent neurons which makes it hard for humans to understand how they interact with each other. By organizing the neurons by correlation,…

Computer Vision and Pattern Recognition · Computer Science 2020-07-14 Randy Tan , Naimul Khan , Ling Guan