Graph theory and geometric deep learning

La seguente tesi propone un’introduzione al geometric deep learning. Nella prima parte vengono presentati i concetti principali di teoria dei grafi ed introdotta una dinamica di diffusione su grafo, in analogia con l’equazione del calore. A seguire, iniziando dal linear classifier verranno introdotte le architetture che hanno portato all’ideazione delle graph convolutional networks. In conclusione, si analizzano esempi di alcuni algoritmi utilizzati nel geometric deep learning e si mostra una loro implementazione sul Cora dataset, un insieme di dati con struttura a grafo.

Geometric Deep Learning per il denoising di mesh 3D

La crescente disponibilità di scanner 3D ha reso più semplice l’acquisizione di modelli 3D dall’ambiente. A causa delle inevitabili imperfezioni ed errori che possono avvenire durante la fase di scansione, i modelli acquisiti possono risultare a volte inutilizzabili ed affetti da rumore. Le tecniche di denoising hanno come obiettivo quello di rimuovere dalla superficie della mesh 3D scannerizzata i disturbi provocati dal rumore, ristabilendo le caratteristiche originali della superficie senza introdurre false informazioni. Per risolvere questo problema, un approccio innovativo è quello di utilizzare il Geometric Deep Learning per addestrare una Rete Neurale in maniera da renderla in grado di eseguire efficacemente il denoising di mesh. L’obiettivo di questa tesi è descrivere il Geometric Deep Learning nell’ambito del problema sotto esame.

Geometric deep learning for polygonal mesh denoising

Il seguente lavoro si propone come analisi degli operatori convoluzionali che caratterizzano le graph neural networks. ln particolare, la trattazione si divide in due parti, una teorica e una sperimentale. Nella parte teorica vengono innanzitutto introdotte le nozioni preliminari di mesh e convoluzione su mesh. In seguito vengono riportati i concetti base del geometric deep learning, quali le definizioni degli operatori convoluzionali e di pooling e unpooling. Un'attenzione particolare è stata data all'architettura Graph U-Net. La parte sperimentare riguarda l'applicazione delle reti neurali e l'analisi degli operatori convoluzionali applicati al denoising di superfici perturbate a causa di misurazioni imperfette effettuate da scanner 3D.

Exact Combinatorial Optimization with Graph Convolutional Neural Networks

Combinatorial optimization problems are typically tackled by the branch-and-bound paradigm. We propose to learn a variable selection policy for branch-and-bound in mixed-integer linear programming, by imitation learning on a diversified variant of the strong branching expert rule. We encode states as bipartite graphs and parameterize the policy as a graph convolutional neural network. Experiments on a series of synthetic problems demonstrate that our approach produces policies that can improve upon expert-designed branching rules on large problems, and generalize to instances significantly larger than seen during training.

Spectral methods in geometric deep learning

Questa tesi propone una panoramica sul funzionamento interno delle architetture alla base del deep learning e in particolare del geometric deep learning. Iniziando a discutere dalla storia degli algoritmi di intelligenza artificiale, vengono introdotti i principali costituenti di questi. In seguito vengono approfonditi alcuni elementi della teoria dei grafi, in particolare il concetto di laplaciano discreto e il suo ruolo nello studio del fenomeno di diffusione sui grafi. Infine vengono presentati alcuni algoritmi utilizzati nell'ambito del geometric deep learning su grafi per la classificazione di nodi. I concetti discussi vengono poi applicati nella realizzazione di un'architettura in grado di classficiare i nodi del dataset Zachary Karate Club.

Morphometric shape analysis using learning vector quantization neural networks - an example distinguishing two microtine vole species

Closely related species may be very difficult to distinguish morphologically, yet sometimes morphology is the only reasonable possibility for taxonomic classification. Here we present learning-vector-quantization artificial neural networks as a powerful tool to classify specimens on the basis of geometric morphometric shape measurements. As an example, we trained a neural network to distinguish between field and root voles from Procrustes transformed landmark coordinates on the dorsal side of the skull, which is so similar in these two species that the human eye cannot make this distinction. Properly trained neural networks misclassified only 3% of specimens. Therefore, we conclude that the capacity of learning vector quantization neural networks to analyse spatial coordinates is a powerful tool among the range of pattern recognition procedures that is available to employ the information content of geometric morphometrics.

Implicit sequence learning in recurrent neural networks

Magdeburg, Univ., Fak. für Elektrotechnik, Diss., 2013

Optimal Unsupervised Learning in Feedforward Neural Networks

We investigate the properties of feedforward neural networks trained with Hebbian learning algorithms. A new unsupervised algorithm is proposed which produces statistically uncorrelated outputs. The algorithm causes the weights of the network to converge to the eigenvectors of the input correlation with largest eigenvalues. The algorithm is closely related to the technique of Self-supervised Backpropagation, as well as other algorithms for unsupervised learning. Applications of the algorithm to texture processing, image coding, and stereo depth edge detection are given. We show that the algorithm can lead to the development of filters qualitatively similar to those found in primate visual cortex.

Matching Recall and Storage in Sequence Learning with Spiking Neural Networks

Storing and recalling spiking sequences is a general problem the brain needs to solve. It is, however, unclear what type of biologically plausible learning rule is suited to learn a wide class of spatiotemporal activity patterns in a robust way. Here we consider a recurrent network of stochastic spiking neurons composed of both visible and hidden neurons. We derive a generic learning rule that is matched to the neural dynamics by minimizing an upper bound on the Kullback–Leibler divergence from the target distribution to the model distribution. The derived learning rule is consistent with spike-timing dependent plasticity in that a presynaptic spike preceding a postsynaptic spike elicits potentiation while otherwise depression emerges. Furthermore, the learning rule for synapses that target visible neurons can be matched to the recently proposed voltage-triplet rule. The learning rule for synapses that target hidden neurons is modulated by a global factor, which shares properties with astrocytes and gives rise to testable predictions.

Dynamics of on-line gradient descent learning for multilayer neural networks

We consider the problem of on-line gradient descent learning for general two-layer neural networks. An analytic solution is presented and used to investigate the role of the learning rate in controlling the evolution and convergence of the learning process.

On-line learning in multilayer neural networks

We present an analytic solution to the problem of on-line gradient-descent learning for two-layer neural networks with an arbitrary number of hidden units in both teacher and student networks. The technique, demonstrated here for the case of adaptive input-to-hidden weights, becomes exact as the dimensionality of the input space increases.

Finite-size effects in on-line learning of multilayer neural networks

We complement recent advances in thermodynamic limit analyses of mean on-line gradient descent learning dynamics in multi-layer networks by calculating fluctuations possessed by finite dimensional systems. Fluctuations from the mean dynamics are largest at the onset of specialisation as student hidden unit weight vectors begin to imitate specific teacher vectors, increasing with the degree of symmetry of the initial conditions. In light of this, we include a term to stimulate asymmetry in the learning process, which typically also leads to a significant decrease in training time.

Globally optimal parameters for on-line learning in multilayer neural networks

We present a framework for calculating globally optimal parameters, within a given time frame, for on-line learning in multilayer neural networks. We demonstrate the capability of this method by computing optimal learning rates in typical learning scenarios. A similar treatment allows one to determine the relevance of related training algorithms based on modifications to the basic gradient descent rule as well as to compare different training methods.

Forming of Learning Set for Neural Networks in Problems of Losless Data Compression

questions of forming of learning sets for artificial neural networks in problems of lossless data compression are considered. Methods of construction and use of learning sets are studied. The way of forming of learning set during training an artificial neural network on the data stream is offered.