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Deep Learning (DL) has attracted a lot of attention for its ability to reach state-of-the-art performance in many machine learning tasks. The core principle of DL methods consists of training composite architectures in an end-to-end fashion, where inputs are associated with outputs trained to optimize an objective function. Because of their compositional nature, DL architectures naturally exhibit several intermediate representations of the inputs, which belong to so-called latent spaces. When treated individually, these intermediate representations are most of the time unconstrained during the learning process, as it is unclear which properties should be favored. However, when processing a batch of inputs concurrently, the corresponding set of intermediate representations exhibit relations (what we call a geometry) on which desired properties can be sought. In this work, we show that it is possible to introduce constraints on these latent geometries to address various problems. In more detail, we propose to represent geometries by constructing similarity graphs from the intermediate representations obtained when processing a batch of inputs. By constraining these Latent Geometry Graphs (LGGs), we address the three following problems: (i) reproducing the behavior of a teacher architecture is achieved by mimicking its geometry, (ii) designing efficient embeddings for classification is achieved by targeting specific geometries, and (iii) robustness to deviations on inputs is achieved via enforcing smooth variation of geometry between consecutive latent spaces. Using standard vision benchmarks, we demonstrate the ability of the proposed geometry-based methods in solving the considered problems. 

For the past few years, deep learning (DL) robustness (i.e. the ability to maintain the same decision when inputs are subject to perturbations) has become a question of paramount importance, in particular in settings where misclassification can have dramatic consequences. To address this question, authors have proposed different approaches, such as adding regularizers or training using noisy examples. In this paper we introduce a regularizer based on the Laplacian of similarity graphs obtained from the representation of training data at each layer of the DL architecture. This regularizer penalizes large changes (across consecutive layers in the architecture) in the distance between examples of different classes, and as such enforces smooth variations of the class boundaries. We provide theoretical justification for this regularizer and demonstrate its effectiveness to improve robustness on classical supervised learning vision datasets for various types of perturbations. We also show it can be combined with existing methods to increase overall robustness. 

Graphs are nowadays ubiquitous in the fields of signal processing and machine learning. As a tool used to express relationships between objects, graphs can be deployed to various ends: (i) clustering of vertices, (ii) semi-supervised classification of vertices, (iii) supervised classification of graph signals, and (iv) denoising of graph signals. However, in many practical cases graphs are not explicitly available and must therefore be inferred from data. Validation is a challenging endeavor that naturally depends on the downstream task for which the graph is learnt. Accordingly, it has often been difficult to compare the efficacy of different algorithms. In this work, we introduce several ease-to-use and publicly released benchmarks specifically designed to reveal the relative merits and limitations of graph inference methods. We also contrast some of the most prominent techniques in the literature.