US 2004/0107079 A1 · Determining a tangent space and filtering data onto a manifold

Abstract

A technique for determining the number of constraints on a set of input data, or equivalently the topological dimension, especially when such data are produced by a nonlinear system, such as a pathological vocal system or econometric data and the like. The technique characterizes the tangent space about a predetermined base point by identifying a maximal set of non-redundant nonlinear fits to the data. It needs only a modest number of data points and does not assume prior knowledge of the functional form of the true constraints, other than smoothness. Each fit is equivalent to a set of contours (including curves, surfaces, and other manifolds), with the data themselves all lying along the zero-value contour of the fit. For each fit, the gradient of the fit at the base point in the uphill direction across the contours identifies the constraint direction. Considering all fits simultaneously, the number of constraint directions that are linearly independent provides the number of constraints in the neighborhood of the base point. The remaining unconstrained directions define the tangent space, and its dimensionality, which is precisely the number of linearly independent unconstrained directions, is precisely the inferred topological dimensionality of the original data.

Claims17

Dependent claims

A method for estimating the topological dimension of a set of data points representing a nonlinear system response, each data point having the same number of coordinates, the method comprising the steps of:

identifying a maximal set of non-redundant, nonlinear single-constraint fits to data points which are in the neighborhood of a predetermined base point, in which the gradient of each fit in the neighborhood of the base point identifies a constrained direction;
estimating the number of constraints in the neighborhood of the base point to be the same as the number of such constrained directions that are linearly independent;
estimating the topological dimension of the set of data points to be the original number of coordinates of the data minus the estimated number of constraints;
wherein the step of identifying a set of fits further comprises the step of:
using a matrix decomposition technique to find singular values and singular vectors, or eigenvalues and eigenvectors, of a design matrix formed from basis functions constructed from the data values;
and
wherein each fit is a linear combination of a set of basis functions for which the zero-contours of the fit, that is, the curves at which the fit has the value zero, pass near the data points and a set of individual coefficients multiplying the individual basis functions, wherein the coefficients of the basis functions are the components of singular vectors obtained from a decomposition of the design matrix.

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Dependencies are inferred from the claim text.

Determining a tangent space and filtering data onto a manifold

Life of this patent

Determining a tangent space and filtering data onto a manifold

Life of this patent

Abstract

Claims17