Group Orbit Optimization: A Unified Approach to Data Normalization

TL;DR

This paper introduces Group Orbit Optimization (GOO), a unified framework for matrix and tensor decompositions that can also be applied to normalize point cloud data, demonstrating robustness against various distortions.

Contribution

The paper presents GOO as a novel unified approach to matrix and tensor decompositions, bridging existing methods and enabling new applications like data normalization.

Findings

GOA can induce classical matrix decompositions such as SVD, LU, QR, Schur, and Cholesky.

The generalized GOO framework extends to tensor decompositions.

The normalization method effectively recovers data distorted by shearing, rotation, and squeezing.

Abstract

In this paper we propose and study an optimization problem over a matrix group orbit that we call \emph{Group Orbit Optimization} (GOO). We prove that GOO can be used to induce matrix decomposition techniques such as singular value decomposition (SVD), LU decomposition, QR decomposition, Schur decomposition and Cholesky decomposition, etc. This gives rise to a unified framework for matrix decomposition and allows us to bridge these matrix decomposition methods. Moreover, we generalize GOO for tensor decomposition. As a concrete application of GOO, we devise a new data decomposition method over a special linear group to normalize point cloud data. Experiment results show that our normalization method is able to obtain recovery well from distortions like shearing, rotation and squeezing.