Harmonic Analysis on Data Bases
Our goal is to illustrate how methods of classical Harmonic Analysis can be applied to organize and “process“ matrices. Here we view a matrix either as an array of data, say a response to a questionnaire, or as the matrix of a linear transformation (say an eigenvector expansion) including discretizations of integral operators.
We claim that given such a matrix we can build two coupled geometries and corresponding analysis, one on the rows, the other on the columns. These geometries are such that the matrix viewed as a function on the tensor product is decomposable as bi smooth function + an outlier supported on a small fraction of the entries. ( Calderon Zygmund decomposition).
We will illustrate this approach as it applies to a psychological questionnaire, to a Science News data base, to a music data base and to various mathematical transforms viewed as matrices.