Manifold unwrapping using critical surfaces

Natural high dimensional data distributions often exhibit clear low-dimensional underlying structures, sometimes referred to as the underlying manifold. In general such underlying low-dimensional surfaces may have complicated shapes and may have to be defined locally at best. Under the premise that d-dimensional critical surfaces of a probability density function (pdf) over Rⁿ provide a natural skeleton for the data, we propose that local nonlinear coordinate transformations to unwrap the so-called manifold are based on curvilinear coordinate systems defined in a manner that is consistent with the critical surfaces. Specifically, we provide a convenient characterization of all critical surfaces in the form of the zero level set of the determinant of a matrix we introduce. This allows the characterization of the underlying manifold using only the gradient and Hessian of the pdf. Especially for the family of distributions in the form of exponential of polynomials, we show that this determinant expression is a polynomial, hence the underlying critical surfaces are zero level sets of a polynomial, and that the factorization of this polynomial will lead to a suitable curvilinear coordinate system. We demonstrate the use of these concepts in globally and locally unwrapping distributions from the family of exponential of polynomials.