Conformal Autoencoders are a neural network architecture that imposes orthogonality conditions between the gradients of latent variables towards achieving disentangled representations of data. In this letter we show that orthogonality relations within the latent layer of the network can be leveraged to infer the intrinsic dimensionality of nonlinear manifold data sets (locally characterized by the dimension of their tangent space), while simultaneously computing encoding and decoding (embedding) maps. We outline the relevant theory relying on differential geometry, and describe the corresponding gradient-descent optimization algorithm. The method is applied to standard data sets and we highlight its applicability, advantages, and shortcomings. In addition, we demonstrate that the same computational technology can be used to build coordinate invariance to local group actions when defined only on a (reduced) submanifold of the embedding space.

本研究解决了非线性流形数据集的内在维度提取问题，采用了正交条件的自编码器架构，能够有效获取解耦表示。同时，提出了一种基于微分几何的理论框架及相应的梯度下降优化算法。研究发现该方法在标准数据集上的表现优越，并具备针对局部群体作用构建坐标不变性的潜力。

更薄的潜在空间：通过自编码器梯度约束检测维度和施加不变性