一个模型同时学数据分布和内在维度，且有理论保证。
Distributional Autoencoders Know the Score
- 用几何关系连接数据得分与重构结构，实现可解释编码。
- 在玻尔兹曼分布下，一次训练就逼近麦塞尔-布朗势能的最低自由能路径。
- 证明超出流形维度的隐变量与数据独立，直接揭示数据内在维数。
分布主自编码器（DPA）将分布正确的重构与类似主成分的编码可解释性相结合。本文提供两方面的精确理论保证：首先，推导出最优层次集几何与数据分布得分之间的闭式关系，解释了DPA在解耦数据变化因素上的经验表现，并可直接从样本中恢复得分；当数据服从玻尔兹曼分布时，该关系在单次拟合中即可近似得到穆勒-布朗势能的最小自由能路径。其次，证明若数据位于可被编码器逼近的流形上，则超出流形维度的潜在分量与数据分布条件独立，不携带额外信息，从而揭示数据的内在维度。这些结果表明，单一模型可同时精确学习数据分布及其内在维度，统一了无监督学习中两个长期目标。
原文摘要 · Abstract (English)
The Distributional Principal Autoencoder (DPA) combines distributionally correct reconstruction with principal-component-like interpretability of the encodings. In this work, we provide exact theoretical guarantees on both fronts. First, we derive a closed-form relation linking each optimal level-set geometry to the data-distribution score. This result explains DPA's empirical ability to disentangle factors of variation of the data, as well as allows the score to be recovered directly from samples. When the data follows the Boltzmann distribution, we demonstrate that this relation yields an approximation of the minimum free-energy path for the Mueller-Brown potential in a single fit. Second, we prove that if the data lies on a manifold that can be approximated by the encoder, latent components beyond the manifold dimension are conditionally independent of the data distribution - carrying no additional information - and thus reveal the intrinsic dimension. Together, these results show that a single model can learn the data distribution and its intrinsic dimension with exact guarantees simultaneously, unifying two longstanding goals of unsupervised learning.
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