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Franca Hoffmann, RLM 10.176: From PDE analysis to graph-based semi-supervised learning
Friday, February 21, 2020, 01:00pm - 02:00pm
In this talk, we focus on two different directions, combining model-driven approaches and data-driven approaches. Certain diffusive PDEs can be viewed as infinite-dimensional gradient flows. This fact has led to the development of new tools in PDE Analysis. In the first part of the talk we use gradient flows for analyzing non-linear and non-local aggregation-diffusion equations when the corresponding energy functionals are not necessarily convex. Moreover, the gradient flow structure enables us to make connections to well-known functional inequalities, revealing possible links between the optimizers of these inequalities and the equilibria of certain aggregation-diffusion PDEs. In the second part, we discuss consistency of semi-supervised learning algorithms on graphs. Semi-supervised learning is the task of propagating observed labels from a small subset to the full data set. Combining labels with the geometric information contained in the graph, we show that our method results in unique classifiers that predict the correct labels of vertices under some conditions and in appropriate limits. Taking the big data limit, one can develop an analogous theory in the continuum setting, making use of weighted elliptic operators and thus linking back to PDE theory.
Location: RLM 10.176

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