Project Description

A cognitive system can be defined as any organized complex that can recieve inputs (usually a high number of them), synthesize and fuse collected data, and then export it in outputs (of a similarly complex manner). One common problem that arises when dealing with these systems is the extremely high volumes of data that is collected-- one must be able to interpret and understand such datasets if we are to hope to imitate cognitive systems.

In this project I will study the geometry of these high-dimensional data sets, and focus on the mathematics of the diffusion geometry process. On the way I will look into studying the heat kernel in relation to the probabilitistic random walk on the data set.

Progress Log

Note: All such intervals of time will be denoted by semi-closed intervals [a,b), as to form a proper sigma-algebra.

Week 1: [June 1, June 6)

• Started DIMACS REU program, attended orientation and started working on product formula decomposition.
• Read chapters on linear functionals in Bachman's "Functional Analysis".
• Proved the product formula decomposition in the case of dyadic step functions.

• Read chapters 1-4 in Do Carmo's "Differential Geometry of Curves and Surfaces".
• Set a goal to understand heat kernel constructions on Riemannian manifolds.
• Presented a powerpoint on the project-- link will be up soon.
• Finally got some intuition as to the diffusion geometry process!

• Slow crawl through formal differential forms and Riemannian geometry.
• As most high-dimensional datasets are fractal by nature, looked into analysis on fractals.
• Looked into RR Coifman's papers on diffusion geometry for focus.