Parameter identification using intrinsic dimensionality

Let be an -dimensional vector space and be a - dimensional topological hypersurface in . The intrinsic dimensionality problem can be stated as follows. Given randomly selected points , estimate , which is the dimensionality of and is called the intrinsic dimensionality of the points . This problem has been attacked by several authors. If signals are represented by vectors in an abstract signal space , rather than as time or frequency functions, the locus of signals that are generated by a hypothetical signal generator possessing free parameters is a -dimensional hypersurface in the signal space. The purpose of this paper is to identify some of the free parameters of the signals. Let be a one-parameter group that acts on . To say that a parameter associated with a group is a free parameter of means that is closed under , i.e., . To decide whether , the estimated dimensionalities of and are compared. The method is illustrated by presenting several examples from the field of signal analysis. Finally, a slight modification is suggested so that parameters can he identified even when the vectors of interest do not represent time or frequency functions.