Local conditions for critical and principal manifolds

Principal manifolds are essential underlying structures that manifest canonical solutions for significant problems such as data de- noising and dimensionality reduction. The traditional definition of self-consistent manifolds rely on a least-squares construction error approach that utilizes semi-global expectations across hyperplanes orthogonal to the solution. This definition creates various practical difficulties for algorithmic solutions to identify such manifolds, besides the theoretical shortcoming that self-intersecting or nonsmooth manifolds are not acceptable in this framework. We present local conditions for critical and principal manifolds by introducing the concept of subspace local maxima. The conditions generalize the two conditions that characterize stationary points of a function to stationary surfaces. The proposed framework yields a unique set of principal points which can be partitioned into principal curves and manifolds of any intrinsic dimensionality. A subspace-constrained fixed-point algorithm is proposed to determine the principal graph.