Mircea Petrache

Scientific themes I care about (storyline):

1. Topological singularities and large point configurations (my mathematical upbringing):

• In geometry and physics, vortices or topological defects are formed and studied via regularity theory. 

• Groups of defects form (in a setup including, but not restricted to "points = topological singularities") interacting point configurations. 

• These configurations often organize themselves in relation with the geometry/shape of the ambient space.

• How to quantify "regularity" for discrete structures?  We may look at distance/Fourier/spectral statistics of the points, and quantify noise levels: we then go from random, to hyperuniform, to regular/crystalline configurations. Persistence/recovery questions of the above statistics, are closely related questions.

• Algorithmic complexity viewpoint: a limiting factor in working with general point configurations is their exponential complexity (=the space of configurations grows in complexity exponentially in the number of points). And high regularity = low complexity (to describe points forming a regular lattice we need to fix a basis and an origin, while to describe a random point cloud we need to enumerate all the points one by one!)

The nicest models of large point configurations we use, combine elegance & generality, with built-in exponential complexity.

But if we stick with these models, we might miss that the data itself is actually much better behaved !

2. How to improve our way of thinking about (our classical models of) large groups of points in space, to make better sense real data points?

A powerful principle is to quantify the group behavior of the points, through problem-specific structures, that in turn instruct low-complexity approximations. Examples of what maths to use:

• Statistical mechanics measures group behavior and correlations within the theory of crystallization and in the study of other phases of matter.

• In Material science in macroscopic/continuum limits the properties of large numbers of atoms are summarized by (fewer) continuum/multiscale variables.  

• In probability theory, one can start with large deviation principles, concentration of measure bounds.

• In computer science, we have multi-scale analysis, metric dimension reduction are studied (also cf. compressed sensing). 

3. Similarly to the (green part) above,  Deep Neural Networks are experimentally shown to "learn surprisingly well" real-life datasets, but we have no good theory.

What are we missing in order to understand this success? Some threads I'm following at the moment:

• Quantifying the information flow through Neural Networks, especially models which are easily testable in "real data", in order to understand the Lottery Ticket Hypothesis.

• How Neural Networks process geometric/topological features of the input datasets (topics including, but not restricted to, Topological Data Analysis and Persistence Theory).

• Geometry Informed Neural Networks : why keep building neural networks based on imagining data in R^d or in L^2 norm? I'm thinking especially about ways to use Optimal Transport for DNN theory, and how to use Gromov Hyperbolicity to implement better learning algorithms.

• Memory mechanisms in Neural Networks and in the brain: what can Neuromorphic Neural Networks teach us about memory? What's their mathematical theory going to look like? This is related to understanding in a principled way the role of recurrence and of time-dependencies, in deep learning.