Riemannian representation learning typically relies on an encoder to estimate densities on chosen manifolds. This involves optimizing numerically brittle objectives, potentially harming model training and quality. To completely circumvent this issue, we introduce the Riemannian generative decoder, a unifying approach for finding manifold-valued latents on any Riemannian manifold. Latents are learned with a Riemannian optimizer while jointly training a decoder network. By discarding the encoder, we vastly simplify the manifold constraint compared to current approaches which often only handle few specific manifolds. We validate our approach on three case studies -- a synthetic branching diffusion process, human migrations inferred from mitochondrial DNA, and cells undergoing a cell division cycle -- each showing that learned representations respect the prescribed geometry and capture intrinsic non-Euclidean structure. Our method requires only a decoder, is compatible with existing architectures, and yields interpretable latent spaces aligned with data geometry. Code available on https://github.com/yhsure/riemannian-generative-decoder.

翻译：黎曼表示学习通常依赖于编码器来估计选定流形上的密度。这涉及优化数值不稳定的目标函数，可能损害模型训练和质量。为彻底规避此问题，我们提出黎曼生成解码器——一种在任意黎曼流形上寻找流形值潜在空间的统一方法。潜在变量通过黎曼优化器进行学习，同时联合训练解码器网络。通过摒弃编码器，我们极大简化了流形约束，与当前通常仅能处理少数特定流形的方法形成对比。我们在三个案例研究中验证了该方法：合成分支扩散过程、线粒体DNA推断的人类迁徙，以及经历细胞分裂周期的细胞。每个案例均表明，学习到的表示能遵循预设几何结构并捕捉内在的非欧几里得特性。本方法仅需解码器，与现有架构兼容，且能产生与数据几何对齐的可解释潜在空间。代码发布于 https://github.com/yhsure/riemannian-generative-decoder。