École Normale Supérieure, Paris
The mathematical curse of big data
The world is flooded by digital information, which records complex phenomena involving a large number of variables. Analyzing data amounts to approximate functions from sample values in high dimension. It faces a curse of dimensionality, because the space volume grows exponentially with the dimension. Avoiding this curse will lead us into a mathematical trip, which begins with statistics and probability and goes through functional approximation, harmonic analysis, high dimensional geometry, and group theory.
Despite this mathematical curse, learning algorithms have been considerably improved over the last 10 years, sometimes reaching a remarkable precision. These results are raising challenging mathematical questions which are fully open. Image and audio applications will illustrate the mathematics.