Statistic | EPFL Graph Search

A statistic (singular) or sample statistic is any quantity computed from values in a sample which is considered for a statistical purpose. Statistical purposes include estimating a population parameter, describing a sample, or evaluating a hypothesis. The average (or mean) of sample values is a statistic. The term statistic is used both for the function and for the value of the function on a given sample. When a statistic is being used for a specific purpose, it may be referred to by a name indicating its purpose. When a statistic is used for estimating a population parameter, the statistic is called an estimator. A population parameter is any characteristic of a population under study, but when it is not feasible to directly measure the value of a population parameter, statistical methods are used to infer the likely value of the parameter on the basis of a statistic computed from a sample taken from the population. For example, the sample mean is an unbiased estimator of the population mean. This means that the expected value of the sample mean equals the true population mean. A descriptive statistic is used to summarize the sample data. A test statistic is used in statistical hypothesis testing. A single statistic can be used for multiple purposes - for example, the sample mean can be used to estimate the population mean, to describe a sample data set, or to test a hypothesis. Some examples of statistics are: "In a recent survey of Americans, 52% of Republicans say global warming is happening." In this case, "52%" is a statistic, namely the percentage of Republicans in the survey sample who believe in global warming. The population is the set of all Republicans in the United States, and the population parameter being estimated is the percentage of all Republicans in the United States, not just those surveyed, who believe in global warming. "The manager of a large hotel located near Disney World indicated that 20 selected guests had a mean length of stay equal to 5.6 days." In this example, "5.

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

Domain adaptation using manifold alignment

Maxime Pierre Trolliet

Domain adaptation is a major challenge for future remote sensing applications. Both financial and temporal constraints of data acquisition lead to the developing of new techniques able to use knowledge from alternative sources. Different approaches have been developed by considering the statistical properties of images or by modifying already existing classifiers. We propose a intermediary approach of these two kinds of methods by using a manifold alignment technique constrained by similarity between two images. The two images are mapped in a high dimensional latent space which maximizes the proximity of similar elements, thus allowing classification of the images suffering from label scarcity by using the knowledge of the other image. Such a classification offers improvement compared to various used processes.

2012

Graph Representation Learning with Optimal Transport: Analysis and Applications

Effrosyni Simou

EPFL2022