New applications of compression in numerical simulation in high dimension
given at strobl11 (18.06.11 14:00)
In our recent research activity (http://hdsparse.ricam.oeaw.ac.at/) we
formulated and explored new models and methods for
- compressive sensing and algorithms;
- learning functions in high-dimension;
- particle, kinetic, and hydrodynamic models of social interaction.
We are interested to address the combination of these ingredients for
learning, simulation, and control of particle systems, kinetic equations, and fluid dynamics models of interacting agents in high-dimension.
In particular, in this talk we explore how concepts of high-dimensional
data compression via random projections onto lower-dimensional spaces can be applied for tractable simulation of certain dynamical systems modeling complex interactions. In such systems, one has to deal with a large number of agents
(typically millions) in spaces of parameters describing each agent of high-dimension (thousands or more).
Even with todays powerful computers, numerical simulations for such
systems are prohibitively expensive. We propose an approach for the simulation of dynamical systems governed by functions of adjacency matrices in high-dimension, by random projections via Johnson-Lindenstrauss embeddings, and recovery
by compressed sensing techniques. We call this technique the Projection
Method. We further show how these concepts can be generalized to work for associated kinetic equations, by addressing the phenomenon of the delayed curse of dimensionality, known in information based complexity for numerical integration
– Joint work with Jan Haskovec and Jan Vybıral.