Stochastic interpolants
a generative method

The investigation of data in any context leads to the question of their statistics and ultimately their distribution. Images, timeseries,  and data in general can be interpreted to be originating from some distribution. There is a long hisdtory of mathematical echniques which attempt to directly characterize this distributin by their density function or by their moments. The modern perspective is to devise techniques which generates random samples which have the same distribution as the target. They are called generative methods [6].

An important class of generative methods are flow-based - they start with the same simple distribution μ_0 such as the standard normal distribution, and try to find a transformation F that transports it to the target distribution μ_1. There has been experiments with various choices for F.  A major breakthrough was the idea of realizing F as the flow-map of an ODEs [4]. Subsequent improvements were the use if efficient trace-estimators [2] and the addition of diffusion [3]. The theoretical challenges that still remained was the proof of convergence, non-uniqueness of the ODE transport, and a determination of the precise role played by diffusion.

A sweeping generalization to all flow-based techniques was the idea of stochastic interpolants [1]. It is a generic framework rooted in basic measure theory and Analysis. It provides many avenues for algorithmic improvement, real-world applications, and theoretical investigations.

Future directions.

1) Implementing codes in Julia language.

2) Investigating the range of optimal values for parameters such as noise level, time-divisions, and integration steps.

3) Applying generative techniques to discovering paterns in data generated by dynamical systems.

4) Implementing kernel integral methods [5] to determine the drift and score functions in stochastic interpolation.

Prerequisites. 

1) Programming : familiarity with Matlab / Julia; familiarity with programming techniques in Linear Algebra

2) Linear Algebra : matrix theory, SVD, eigen-decomposition, least squares methods

3) Analysis : continuity and differntiability, limits and convergence

4) Functional analysis : Banach spaces, Hilbert spaces, kernel integrals