Geometry of curves in [formula omitted] from the local singular value decomposition.

Abstract We establish a connection between the local singular value decomposition and the geometry of n -dimensional curves. In particular, we link the left singular vectors to the Frenet-Serret frame, and the generalized curvatures to the singular values. Specifically, let γ : I → R n be a parametric curve of class C n + 1 , regular of order n. The Frenet-Serret apparatus of γ at γ (t) consists of a frame e 1 (t) , ... , e n (t) and generalized curvature values κ 1 (t) , ... , κ n − 1 (t). Associated with each point of γ there are also local singular vectors u 1 (t) , ... , u n (t) and local singular values σ 1 (t) , ... , σ n (t). This local information is obtained by considering a limit, as ϵ goes to zero, of covariance matrices defined along γ within an ϵ -ball centered at γ (t). We prove that for each t ∈ I , the Frenet-Serret frame and the local singular vectors agree at γ (t) and that the values of the curvature functions at t can be expressed as a fixed multiple of a ratio of local singular values at t. To establish this result we prove a general formula for the recursion relation of a certain class of sequences of Hankel determinants using the theory of monic orthogonal polynomials and moment sequences. [ABSTRACT FROM AUTHOR]