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 in 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 details, 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.
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