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Machine learning methods are commonly used to solve inverse problems, wherein an unknown signal must be estimated from few indirect measurements generated via a known acquisition procedure. In particular, neural networks perform well…

Deep learning has become a pivotal technology in fields such as computer vision, scientific computing, and dynamical systems, significantly advancing these disciplines. However, neural Networks persistently face challenges related to…

We study the generalization of over-parameterized deep networks (for image classification) in relation to the convex hull of their training sets. Despite their great success, generalization of deep networks is considered a mystery. These…

How to understand deep learning systems remains an open problem. In this paper we propose that the answer may lie in the geometrization of deep networks. Geometrization is a bridge to connect physics, geometry, deep network and quantum…

Generalization, the ability to perform well beyond the training context, is a hallmark of biological and artificial intelligence, yet anticipating unseen failures remains a central challenge. Conventional approaches often take a…

Deep neural networks have achieved impressive results in many image classification tasks. However, since their performance is usually measured in controlled settings, it is important to ensure that their decisions remain correct when…

This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the…

Machine learning (ML) models are typically optimized for their accuracy on a given dataset. However, this predictive criterion rarely captures all desirable properties of a model, in particular how well it matches a domain expert's…

Model reparametrization, which follows the change-of-variable rule of calculus, is a popular way to improve the training of neural nets. But it can also be problematic since it can induce inconsistencies in, e.g., Hessian-based flatness…

Deep learning approaches have provided state-of-the-art performance in many applications by relying on large and overparameterized neural networks. However, such networks have been shown to be very brittle and are difficult to deploy on…

Successful training of convolutional neural networks is often associated with sufficiently deep architectures composed of high amounts of features. These networks typically rely on a variety of regularization and pruning techniques to…

Deep neural networks have recently drawn considerable attention to build and evaluate artificial learning models for perceptual tasks. Here, we present a study on the performance of the deep learning models to deal with global optimization…

One of the most prominent attributes of Neural Networks (NNs) constitutes their capability of learning to extract robust and descriptive features from high dimensional data, like images. Hence, such an ability renders their exploitation as…

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…

With the proliferation of Deep Machine Learning into real-life applications, a particular property of this technology has been brought to attention: robustness Neural Networks notoriously present low robustness and can be highly sensitive…

Two major uncertainties, dataset bias and adversarial examples, prevail in state-of-the-art AI algorithms with deep neural networks. In this paper, we present an intuitive explanation for these issues as well as an interpretation of the…

We consider the problem of data classification where the training set consists of just a few data points. We explore this phenomenon mathematically and reveal key relationships between the geometry of an AI model's feature space, the…

Embedding of large but redundant data, such as images or text, in a hierarchy of lower-dimensional spaces is one of the key features of representation learning approaches, which nowadays provide state-of-the-art solutions to problems once…

In this paper, we propose the $\textit{geometric invariance hypothesis (GIH)}$, which argues that the input space curvature of a neural network remains invariant under transformation in certain architecture-dependent directions during…

Considering smooth mappings from input vectors to continuous targets, our goal is to characterise subspaces of the input domain, which are invariant under such mappings. Thus, we want to characterise manifolds implicitly defined by level…