AMMCS-2013 Plenary Talk

Information Integration/Organization and Numerical Harmonic Analysis.

Ronald R. Coifman

Abstract: We provide an overview of recent developments in methodologies for empirical organization of data. We present a geometric/analytic mathematical framework for learning, which revolves around building a network or a graph whose nodes are observations. In our framework, connections between observations are constantly reconfigured in order to achieve learning for specific tasks. In particular we will provide a synthesis of a range of ideas from mathematics and machine learning, which address the transition from a local similarity model to a global configuration. This is analogous to Newtonian Calculus, which from of a local linear model of variability, calculates a global solution to a differential, or partial differential equation. We apply these fundamentals to jointly organize the rows and columns of a matrix, viewed either as the matrix of a linear operator, or as a Database. Here the rows are viewed as functions on the columns and the columns as functions of the rows, a dual geometry is built to optimize prediction and processing .

We relate these methods to ideas from classical Harmonic Analysis and indicate tools to measure success of information extraction. In particular we introduce methodologies that resemble "signal processing" on data matrices, enabling functional regression, prediction, denoising, compression fast numerics, and so on. We illustrate these ideas to organize and map out in an automatic and purely data driven fashion on music databases of audio segments, text documents, psychological questionnaires, medical profiles, physical sensor data, financial data.

Ronald R. Coifman is Phillips professor of mathematics at Yale University. He received his Ph.D. from the University of Geneva in 1965. Prior to coming to Yale in 1980, he was a professor at Washington University in St Louis. Prof. Coifman's recent publications have been in the areas of nonlinear Harmonic Analysis , Fourier Analysis, wavelet theory, numerical analysis and scattering theory. Professor Coifman is currently leading a research program to develop new mathematical tools for efficient transcription and organization of data, with applications to feature extraction, learning classification and denoising. He was chairman of the Yale mathematics department 1986-89. He is a member of the National Academy of Sciences, American Academy of Arts and Sciences, and the Connecticut Academy of Sciences and Engineering. He received the DARPA Sustained Excellence Award in 1996, and the 1996 Connecticut Science Medal. The 1999 Pioneer award from the International Society for Industrial and applied Mathematics , the National Science Medal 1999, And the Wavelet Pioneer award 2009.