A singular Riemannian Geometry Approach to Deep Neural Networks III. Piecewise Differentiable Layers and Random Walks on $n$-dimensional Classes

Neural networks are playing a crucial role in everyday life, with the most modern generative models able to achieve impressive results. Nonetheless, their functioning is still not very clear, and several strategies have been adopted to study how and why these model reach their outputs. A common approach is to consider the data in an Euclidean settings: recent years has witnessed instead a shift from this paradigm, moving thus to more general framework, namely Riemannian Geometry. Two recent works introduced a geometric framework to study neural networks making use of singular Riemannian metrics. In this paper we extend these results to convolutional, residual and recursive neural networks, studying also the case of non-differentiable activation functions, such as ReLU. We illustrate our findings with some numerical experiments on classification of images and thermodynamic problems.

A Modular Deep Learning-based Approach for Diffuse Optical Tomography Reconstruction

Medical imaging is nowadays a pillar in diagnostics and therapeutic follow-up. Current research tries to integrate established - but ionizing - tomographic techniques with technologies offering reduced radiation exposure. Diffuse Optical Tomography (DOT) uses non-ionizing light in the Near-Infrared (NIR) window to reconstruct optical coefficients in living beings, providing functional indications about the composition of the investigated organ/tissue. Due to predominant light scattering at NIR wavelengths, DOT reconstruction is, however, a severely ill-conditioned inverse problem. Conventional reconstruction approaches show severe weaknesses when dealing also with mildly complex cases and/or are computationally very intensive. In this work we explore deep learning techniques for DOT inversion. Namely, we propose a fully data-driven approach based on a modularity concept: first data and originating signal are separately processed via autoencoders, then the corresponding low-dimensional latent spaces are connected via a bridging network which acts at the same time as a learned regularizer.

Majorization-Minimization for sparse SVMs

Several decades ago, Support Vector Machines (SVMs) were introduced for performing binary classification tasks, under a supervised framework. Nowadays, they often outperform other supervised methods and remain one of the most popular approaches in the machine learning arena. In this work, we investigate the training of SVMs through a smooth sparse-promoting-regularized squared hinge loss minimization. This choice paves the way to the application of quick training methods built on majorization-minimization approaches, benefiting from the Lipschitz differentiabililty of the loss function. Moreover, the proposed approach allows us to handle sparsity-preserving regularizers promoting the selection of the most significant features, so enhancing the performance. Numerical tests and comparisons conducted on three different datasets demonstrate the good performance of the proposed methodology in terms of qualitative metrics (accuracy, precision, recall, and F 1 score) as well as computational cost.

Unsupervised deep learning techniques for powdery mildew recognition based on multispectral imaging

Objectives. Sustainable management of plant diseases is an open challenge which has relevant economic and environmental impact. Optimal strategies rely on human expertise for field scouting under favourable conditions to assess the current presence and extent of disease symptoms. This labor-intensive task is complicated by the large field area to be scouted, combined with the millimeter-scale size of the early symptoms to be detected. In view of this, image-based detection of early disease symptoms is an attractive approach to automate this process, enabling a potential high throughput monitoring at sustainable costs. Methods. Deep learning has been successfully applied in various domains to obtain an automatic selection of the relevant image features by learning filters via a training procedure. Deep learning has recently entered also the domain of plant disease detection: following this idea, in this work we present a deep learning approach to automatically recognize powdery mildew on cucumber leaves. We focus on unsupervised deep learning techniques applied to multispectral imaging data and we propose the use of autoencoder architectures to investigate two strategies for disease detection: i) clusterization of features in a compressed space; ii) anomaly detection. Results. The two proposed approaches have been assessed by quantitative indices. The clusterization approach is not fully capable by itself to provide accurate predictions but it does cater relevant information. Anomaly detection has instead a significant potential of resolution which could be further exploited as a prior for supervised architectures with a very limited number of labeled samples.

A singular Riemannian geometry approach to Deep Neural Networks II. Reconstruction of 1-D equivalence classes

In a previous work, we proposed a geometric framework to study a deep neural network, seen as sequence of maps between manifolds, employing singular Riemannian geometry. In this paper, we present an application of this framework, proposing a way to build the class of equivalence of an input point: such class is defined as the set of the points on the input manifold mapped to the same output by the neural network. In other words, we build the preimage of a point in the output manifold in the input space. In particular. we focus for simplicity on the case of neural networks maps from n-dimensional real spaces to (n - 1)-dimensional real spaces, we propose an algorithm allowing to build the set of points lying on the same class of equivalence. This approach leads to two main applications: the generation of new synthetic data and it may provides some insights on how a classifier can be confused by small perturbation on the input data (e.g. a penguin image classified as an image containing a chihuahua). In addition, for neural networks from 2D to 1D real spaces, we also discuss how to find the preimages of closed intervals of the real line. We also present some numerical experiments with several neural networks trained to perform non-linear regression tasks, including the case of a binary classifier.

A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations

Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. The strategies employed to investigate their theoretical properties mainly rely on Euclidean geometry, but in the last years new approaches based on Riemannian geometry have been developed. Motivated by some open problems, we study a particular sequence of maps between manifolds, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the structures induced trough pullbacks on the other manifolds of the sequence and on some related quotients. In particular, we show that the pullbacks of the final Riemannian metric to any manifolds of the sequence is a degenerate Riemannian metric inducing a structure of pseudometric space, we show that the Kolmogorov quotient of this pseudometric space yields a smooth manifold, which is the base space of a particular vertical bundle. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between manifolds implementing neural networks of practical interest and we present some applications of the geometric framework we introduced in the first part of the paper.

A Deep Learning Generative Model Approach for Image Synthesis of Plant Leaves

Objectives. We generate via advanced Deep Learning (DL) techniques artificial leaf images in an automatized way. We aim to dispose of a source of training samples for AI applications for modern crop management. Such applications require large amounts of data and, while leaf images are not truly scarce, image collection and annotation remains a very time--consuming process. Data scarcity can be addressed by augmentation techniques consisting in simple transformations of samples belonging to a small dataset, but the richness of the augmented data is limited: this motivates the search for alternative approaches. Methods. Pursuing an approach based on DL generative models, we propose a Leaf-to-Leaf Translation (L2L) procedure structured in two steps: first, a residual variational autoencoder architecture generates synthetic leaf skeletons (leaf profile and veins) starting from companions binarized skeletons of real images. In a second step, we perform translation via a Pix2pix framework, which uses conditional generator adversarial networks to reproduce the colorization of leaf blades, preserving the shape and the venation pattern. Results. The L2L procedure generates synthetic images of leaves with a realistic appearance. We address the performance measurement both in a qualitative and a quantitative way; for this latter evaluation, we employ a DL anomaly detection strategy which quantifies the degree of anomaly of synthetic leaves with respect to real samples. Conclusions. Generative DL approaches have the potential to be a new paradigm to provide low-cost meaningful synthetic samples for computer-aided applications. The present L2L approach represents a step towards this goal, being able to generate synthetic samples with a relevant qualitative and quantitative resemblance to real leaves.