Intrinsic Polynomials for Regression on Riemannian Manifolds

Abstract

We develop a framework for polynomial regression on Riemannian manifolds. Unlike recently developed spline models on Riemannian manifolds, Riemannian polynomials offer the ability to model parametric polynomials of all integer orders, odd and even. An intrinsic adjoint method is employed to compute variations of the matching functional, and polynomial regression is accomplished using a gradient-based optimization scheme. We apply our polynomial regression framework in the context of shape analysis in Kendall shape space as well as in diffeomorphic landmark space. Our algorithm is shown to be particularly convenient in Riemannian manifolds with additional symmetry, such as Lie groups and homogeneous spaces with right or left invariant metrics. As a particularly important example, we also apply polynomial regression to time-series imaging data using a right invariant Sobolev metric on the diffeomorphism group. The results show that Riemannian polynomials provide a practical model for parametric curve regression, while offering increased flexibility over geodesics.

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“…12 In some textbooks it is called the basic Euler-Poincaré equation to emphasize that it only includes the kinetic energy of a free-body. 13 In [121,83] it is also found that…”

Highly-Expressive Spaces of Well-Behaved Transformations: Keeping it Simple

“…[3,42,11,29,13,20], we deal with the nonlinearity of the space of interest by utilizing the linear tangent space at the identity. Though not explored here, other types of statistics on manifolds [32,12,34,38,36,21,19,41,28,44,10,23,14] may also be applicable.…”

Geodesic Regression on the Grassmannian

“…A variety of methods extending concepts of regression in Euclidean spaces to nonflat manifolds have been proposed. Rentmeesters [24], Fletcher [11] and Hinkle et al [15] address the problem of geodesic fitting on Riemannian manifolds, mostly focusing on symmetric spaces. Niethammer et al [22] generalized linear regression to the manifold of diffeomorphisms to model image time-series data, followed by works extending this concept [16,25,26].…”

“…Unlike Jacobi field approaches, solutions using adjoint methods do not require computation of the curvature explicitly and easily extend to higher-order models, e.g., polynomials [15], splines [26], or piecewise regression models. Our approach is a representative of the first category which ensures extensibility to more advanced models.…”