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 manifolds designed to accommodate the essential shape information of the object. The experimental results indicated that the proposed method can 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.

翻译：近期，机器学习研究聚焦于利用特定流形隐含的对称性作为归纳偏置的方法。格拉斯曼流形具备处理以形状空间表示的基本形状的能力，从而实现稳定的形状分析。本文提出了一种新颖方法，通过连续归一化流为格拉斯曼流形上的分布学习奠定理论基础，明确目标为生成稳定形状。我们的方法通过在学习与生成过程中采用旨在容纳物体本质形状信息的格拉斯曼流形，有效消除旋转、反转等无关变换的影响，从而实现更稳健的生成。实验结果表明，所提方法能够通过捕捉数据结构生成高质量样本。此外，该方法在似然对数或证据下界指标上显著优于现有最优方法。所得结果有望推动该领域的进一步研究，促进稳定形状生成与分析领域的进展。