Grassmann Manifold Flows for Stable Shape Generation

TL;DR

This paper introduces a novel method using continuous normalization flows on Grassmann manifolds to generate stable shapes, effectively handling transformations and outperforming existing methods in quality and likelihood.

Contribution

It establishes the theoretical foundations for learning distributions on Grassmann manifolds with flows, enabling robust shape generation by removing extraneous transformations.

Findings

Generated high-quality shape samples.

Outperformed state-of-the-art methods in likelihood metrics.

Enhanced stability in shape analysis and generation.

Abstract

Recently, studies on machine learning have focused on methods that use symmetry implicit in a specific manifold as an inductive bias. Grassmann manifolds provide the ability to handle fundamental shapes represented as shape spaces, enabling stable shape analysis. In this paper, we present a novel approach in which we establish the theoretical foundations for learning distributions on the Grassmann manifold via continuous normalization flows, with the explicit goal of generating stable shapes. Our approach facilitates more robust generation by effectively eliminating the influence of extraneous transformations, such as rotations and inversions, through learning and generating within a Grassmann manifold designed to accommodate the essential shape information of the object. The experimental results indicated that the proposed method could generate high-quality samples by capturing the data structure. Furthermore, the proposed method significantly outperformed state-of-the-art methods in terms of the log-likelihood or evidence lower bound. The results obtained are expected to stimulate further research in this field, leading to advances for stable shape generation and analysis.