Parameter identification using intrinsic dimensionality

Let
W
be an
N
-dimensional vector space and
V
be a
K
- dimensional topological hypersurface in
W
. The intrinsic dimensionality problem can be stated as follows. Given
M
randomly selected points
\nu_i, \nu_i \in V
, estimate
K
, which is the dimensionality of
V
and is called the intrinsic dimensionality of the points
\nu_i
. This problem has been attacked by several authors. If signals are represented by vectors in an abstract signal space
W
, rather than as time or frequency functions, the locus of signals that are generated by a hypothetical signal generator possessing
K
free parameters is a
K
-dimensional hypersurface
V
in the signal space. The purpose of this paper is to identify some of the free parameters of the signals. Let
H
be a one-parameter group that acts on
W, HW = W
. To say that a parameter associated with a group
H
is a free parameter of
V
means that
V
is closed under
H
, i.e.,
HV = V
. To decide whether
HV = V
, the estimated dimensionalities of
HV
and
V
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.