QuRe-ViMaL · Quantitative Rectifiability: from Vitushkin's conjecture to Manifold Learning

Quantitative Rectifiability: from Vitushkin's conjecture to Manifold Learning

The project QuRe-ViMal links two a priori very different area of quantitative research: quantitative geometric measure theory (Q-GMT) on one hand, and statistical learning theory (SLT) on the other. The former is a branch of pure mathematics. It grew out of purely theoretical questions in complex analysis and it deals with understanding the geometry of sets and measures in Euclidean and more general spaces. Imagine for example a two-dimensional surface in the three dimensional Euclidean space. Its degree of smoothness is one of its key properties: for example, it determines whether this surface represents an amenable domain where to solve PDEs, compute derivatives or integrals. Creating holes and sharp corners in a smooth surface turn it rough: these are features which, if too abundantly present, prevent doing analysis and PDEs. The main focus of Q-GMT is then to study the geometry of possibly very high dimensional objects (sets and measures) via quantifying the presence of holes and corners. In the last twenty years Q-GMT proved a very powerful tool in extending analysis and PDEs to a much larger class of `surfaces' (of arbitrary dimension), which are known as Quantitatively Rectifiable sets (or measures). In other words: smoothness is not needed. What is necessary, instead, is a precise quantification and control of holes and corners. The latter research area, STL, is tasked to formalise when a model constructed out of observations, that is, data sets, has predictive capacities with respect to the phenomenon observed. For example, it gives criteria to check whether an interpolating function is overfitting or underfitting. STL and machine learning are afflicted by the so-called curse of dimensionality: the fact that the computational costs scale exponentially with the dimension of the dataset. What saves the day is that more often than not data points coagulate near geometric objects, such as smooth submanifolds, whose intrinsic dimension is vastly smaller than the ambient dimension. This is known as the `latent (smooth) manifold', and the hypothesis of its existence is known as `manifold hypothesis'. There is a vast literature on how to detect and describe the latent manifold in a dataset. However, there is plenty of empirical evidence that the manifold hypothesis is too restrictive: data sets do tend to group around lower dimensional objects, but these may be not `nice' as smooth manifold. The main achievement of QuRe-ViMal is to develop techniques to use the Q-GMT tools to detect the presence of latent Quantitative Rectifiable sets in datasets, with consequences both in unsupervised and supervised learning. These tools, however, had to be developed and deepened. This is the other main achievement of QuRe-ViMal.

Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз