Manifold Kernel Rank Reduced Regression

Abstract

The Kernel Rank Reduced Regression (KRRR) technique works well on highly dependent dataset with a latent variable structure. When we extended the KRRR to the Reproducing Kernel Hilbert Space (RKHS), the powerful kernel presentation and reproducing ability can enhance the regression ability. But previous research always work on Euclidean space with vector data presentation, which omit the intrinsic geometric shape of the data distribution. If the whole dataset can be thought as a manifold, the regression result will only rely on the intrinsic data distribution instead of the extrinsic frame. So we present the manifold kernel rank reduced regression model (MKRRR). We fist give the definition of the MKRRR model. Then with leveraging Kendall shape space for representing sample manifold data, we derive the closed-form solution of the regression model and prediction result. Moreover, we discuss the convergent and robust ability of the model, with presenting the robustness proof. At last, the we present a skull repair application by the MKRRR model for 3D mandibular reconstruction. The experiment result validate effective of our model even on the data with high-level noise.