In my talk I will consider the following question: how many random hyperplanes are needed to uniformly tessellate a given subset of Rn with high probability? In my talk I will present an optimal answer to this question for selected distributions for the random hyperplanes and sketch three applications of these results in the mathematical foundations of data science. First, I will show how to create a fast encoding of any given dataset into a minimal number of bits. Second, I will consider performance guarantees for one-bit compressed sensing methods, which aim to reconstruct a signal from a small number of measurements that are each quantized to a single bit using an efficient analog-to-digital converter. Third, I will discuss implications for the robustness of ReLU neural networks. The talk will be a survey-style presentation for a general mathematical audience. Based on joint works with Shahar Mendelson (ANU Canberra), Alexander Stollenwerk (Louvain), Patrick Finke, Nigel Strachan (Utrecht), Paul Geuchen, Dominik Stöger, Felix Voigtlaender (Eichstätt-Ingolstadt) ___________________________ Invited by Prof. Johannes Maly

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Consider an unknown random vector X, taking values in R^d. Is it possible to "guess" its mean accurately if the only information one is given consists of N independent copies of X? More accurately, given an arbitrary norm on R^d, the goal is to find a mean estimation procedure: upon receiving a wanted confidence parameter \delta and N independent copies X_1,...,X_N of an unknown random vector X - that has a finite mean and covariance -, the procedure returns \hat{\mu} for which the error \| \hat{\mu} - E X\| is as small as possible with probability at least 1-\delta (with respect to the product measure).

The mean estimation problem has been studied extensively over the years and I will present some of the ideas that have led to its solution (and to the solution of other questions of a similar flavour that I will outline). Two surprising facts are that in all these problems the obvious choices fail miserably (for mean estimation, that choice is N^{-1}\sum_{i=1}^N X_i); and, that the solution behaves as if the (arbitrary) random vector X were gaussian. ________________________________________ Invited by Prof. Holger Rauhut

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