FlowPool: Pooling Graph Representations with Wasserstein Gradient Flows

In several machine learning tasks for graph structured data, the graphs under consideration may be composed of a varying number of nodes. Therefore, it is necessary to design pooling methods that aggregate the graph representations of varying size to representations of fixed size which can be used in downstream tasks, such as graph classification. Existing graph pooling methods offer no guarantee with regards to the similarity of a graph representation and its pooled version. In this work we address this limitation by proposing FlowPool, a pooling method that optimally preserves the statistics of a graph representation to its pooled counterpart by minimizing their Wasserstein distance. This is achieved by performing a Wasserstein gradient flow with respect to the pooled graph representation. We propose a versatile implementation of our method which can take into account the geometry of the representation space through any ground cost. This implementation relies on the computation of the gradient of the Wasserstein distance with recently proposed implicit differentiation schemes. Our pooling method is amenable to automatic differentiation and can be integrated in end-to-end deep learning architectures. Further, FlowPool is invariant to permutations and can therefore be combined with permutation equivariant feature extraction layers in GNNs in order to obtain predictions that are independent of the ordering of the nodes. Experimental results demonstrate that our method leads to an increase in performance compared to existing pooling methods when evaluated in graph classification tasks.

Graph Representation Learning with Optimal Transport: Analysis and Applications

Effrosyni Simou

In several machine learning settings, the data of interest are well described by graphs. Examples include data pertaining to transportation networks or social networks. Further, biological data, such as proteins or molecules, lend themselves well to graph-structured descriptions. In such settings, it is desirable to design representation learning algorithms that can take into account the information captured by the underlying structure. This thesis focuses on providing new methods to ingrain geometrical information in end-to-end deep learning architectures for graphs through the use of elements from Optimal Transport theory.First, we consider the setting where the data can be described by a fixed graph. In this setting, each datapoint corresponds to a node of the graph and the edges among nodes signify their pairwise relations. We propose an autoencoder architecture that consists of a linear layer in the encoder and a novel Wasserstein barycentric layer at the decoder. Our proposed barycentric layer takes into account the underlying geometry through the diffusion distance of the graph and provides a way to obtain structure-aware, non-linear interpolations, which lead to directly interpretable node embeddings that are stable to perturbations of the graph structure. Further, the embeddings obtained with our proposed autoencoder achieve competitive or superior results compared to state-of-the-art methods in node classification tasks.Second, we consider the setting where the data consist of multiple graphs. In that setting, the graphs can be of varying size and, therefore, a global pooling operation is needed in order to obtain representations of fixed size and enable end-to-end learning with deep networks. We propose a global pooling operation that optimally preserves the statistical properties of the representations. In order to do so, we take into account the geometry of the representation space and introduce a global pooling layer that minimizes the Wasserstein distance between a graph representation and its pooled counterpart, of fixed size. This is achieved by performing a Wasserstein gradient flow with respect to the pooled representation. Our proposed method demonstrates promising results in the task of graph classification.Overall, in this thesis we provide new methods for incorporating geometrical information in end-to-end deep learning architectures for graph structured data. We believe that our proposals will contribute to the development of algorithms that learn meaningful representations and that take fully into account the geometry of the data under consideration.

EPFL2022

Theory of representation learning in cortical neural networks

Carlos Stein Naves de Brito

Our brain continuously self-organizes to construct and maintain an internal representation of the world based on the information arriving through sensory stimuli. Remarkably, cortical areas related to different sensory modalities appear to share the same functional unit, the neuron, and develop through the same learning mechanism, synaptic plasticity. It motivates the conjecture of a unifying theory to explain cortical representational learning across sensory modalities. In this thesis we present theories and computational models of learning and optimization in neural networks, postulating functional properties of synaptic plasticity that support the apparent universal learning capacity of cortical networks. In the past decades, a variety of theories and models have been proposed to describe receptive field formation in sensory areas. They include normative models such as sparse coding, and bottom-up models such as spike-timing dependent plasticity. We bring together candidate explanations by demonstrating that in fact a single principle is sufficient to explain receptive field development. First, we show that many representative models of sensory development are in fact implementing variations of a common principle: nonlinear Hebbian learning. Second, we reveal that nonlinear Hebbian learning is sufficient for receptive field formation through sensory inputs. A surprising result is that our findings are independent of specific details, and allow for robust predictions of the learned receptive fields. Thus nonlinear Hebbian learning and natural statistics can account for many aspects of receptive field formation across models and sensory modalities. The Hebbian learning theory substantiates that synaptic plasticity can be interpreted as an optimization procedure, implementing stochastic gradient descent. In stochastic gradient descent inputs arrive sequentially, as in sensory streams. However, individual data samples have very little information about the correct learning signal, and it becomes a fundamental problem to know how many samples are required for reliable synaptic changes. Through estimation theory, we develop a novel adaptive learning rate model, that adapts the magnitude of synaptic changes based on the statistics of the learning signal, enabling an optimal use of data samples. Our model has a simple implementation and demonstrates improved learning speed, making this a promising candidate for large artificial neural network applications. The model also makes predictions on how cortical plasticity may modulate synaptic plasticity for optimal learning. The optimal sampling size for reliable learning allows us to estimate optimal learning times for a given model. We apply this theory to derive analytical bounds on times for the optimization of synaptic connections. First, we show this optimization problem to have exponentially many saddle-nodes, which lead to small gradients and slow learning. Second, we show that the number of input synapses to a neuron modulates the magnitude of the initial gradient, determining the duration of learning. Our final result reveals that the learning duration increases supra-linearly with the number of synapses, suggesting an effective limit on synaptic connections and receptive field sizes in developing neural networks.

EPFL2016

Deep Generative Models and Applications

Tatjana Chavdarova

Over the past few years, there have been fundamental breakthroughs in core problems in machine learning, largely driven by advances in deep neural networks. The amount of annotated data drastically increased and supervised deep discriminative models exceeded human-level performances in certain object detection tasks. The increasing availability in quantity and complexity of unlabelled data also opens up exciting possibilities for the development of unsupervised learning methods. Among the family of unsupervised methods, deep generative models find numerous applications. Moreover, as real-world applications include high dimensional data, the ability of generative models to automatically learn semantically meaningful subspaces makes their advancement an essential step toward developing more efficient algorithms. Generative Adversarial Networks (GANs) are a family of unsupervised generative algorithms that have demonstrated impressive performance for data synthesis and are now used in a wide range of computer vision tasks. Despite this success, they gained a reputation for being difficult to train, which results in a time-consuming and human-involved development process to use them. In the first part of this thesis, we focus on improving the stability and the performances of GANs. Foremost, we consider an alternative training process to the standard one, named SGAN, in which several adversarial âlocalâ pairs of networks are trained independently so that a âglobalâ supervising pair of networks can be trained against them. Experimental results on both toy and real-world problems demonstrate that this approach outperforms standard training in terms of better mitigating mode collapse, stability while converging and that it surprisingly, increases the convergence speed as well. To further reduce the computational footprint while maintaining the stability and performance advantages of SGAN, we focus on training a single pair of adversarial networks using variance reduced gradient. More precisely, we study the effect of the stochastic gradient noise on the training of generative adversarial networks (GANs) and show that it can prevent the convergence of standard game optimization methods, while the batch version converges. We address this issue with two stochastic variance-reduced gradient and extragradient optimization algorithms for GANs, named SVRG-GAN and SVRE, respectively. We observe empirically that SVRE performs similarly to a batch method on the MNIST dataset, while being computationally cheaper, and that SVRE yields more stable GAN training on standard datasets. In the second part of the thesis we present our work on people detection. People detection methods are highly sensitive to occlusions between pedestrians, and using joint visual information from multiple synchronized cameras gives the opportunity to improve detection performance. We address the problem of multi-view people occupancy map estimation using an endâtoâend deep learning algorithm called DeepMCD that jointly utilizes the correlated streams of visual information. DeepMCD empirically outperformed the classical approaches by a large margin. Finally, we present a new large-scale and high-resolution dataset, named WILDTRACK. We provide an accurate joint calibration, as well as a series of benchmark results using baseline algorithms published over the recent months for multi-view detection with deep neural networks, and trajectory estimation using a non-Markovian model.

EPFL2020