AI 中文总结
本研究通过MNIST与量子态嵌入案例，发现神经网络可学习近似线性的保结构映射，但经典核方法在精度上更优。
AI 中文摘要
我们探讨神经网络能否学习从压缩潜在空间到希尔伯特空间表示的保结构映射。使用MNIST上的8维自编码器瓶颈以及基于PCA角度编码的$n$量子位乘积态目标，我们报告了四项发现。尽管目标角度是由潜在投影的非线性sigmoid变换生成的，但所得映射在观测到的潜在分布上可被线性函数很好地近似：从$z$到真实目标角度的线性回归达到$R^{2} = 0.98$，而对MLP恢复角度的回归达到$R^{2} = 0.91$。学习映射的主方向与目标诱导方向强对齐，余弦相似度为$0.989$，同时与输入主方向几乎正交，余弦相似度为$0.002$。该映射是真正的秩4：移除任何奇异方向都会使内积保持能力降低$2.5$--$5.9$倍，尽管奇异值谱有两个主导值和两个小值。学习到的子空间与用于构建目标的PCA基不一致，不同随机种子恢复相同的主方向但在更高秩上出现分歧。最后，使用RBF核的核岭回归优于调优的MLP（内积误差$0.0144$对比$0.0197$），表明对于近似线性的保结构映射，经典核方法可能是更简单且更有效的替代方案。
英文摘要
We ask whether a neural network can learn a structure-preserving map from a compressed latent space to a Hilbert-space representation. Using an 8-dimensional autoencoder bottleneck on MNIST and $n$-qubit product-state targets from PCA-based angle encoding, we report four findings. Although the target angles are generated by a nonlinear sigmoid transformation of the latent projections, the resulting mapping is well approximated by a linear function over the observed latent distribution: linear regression from $z$ to the true target angles achieves $R^{2} = 0.98$, while regression to the MLP's recovered angles achieves $R^{2} = 0.91$. The learned map's primary direction is strongly aligned with the target-induced direction, with cosine similarity $0.989$, while remaining nearly orthogonal to the input's principal direction, with cosine similarity $0.002$. The map is genuinely rank-4: removing any singular direction degrades inner-product preservation by $2.5$--$5.9\times$ despite a singular-value spectrum with two dominant and two small values. The learned subspace does not coincide with the PCA basis used to construct the target, and different random seeds recover the same primary direction but diverge in higher ranks. Finally, kernel ridge regression with an RBF kernel outperforms a tuned MLP (IP error $0.0144$ vs.\ $0.0197$), suggesting that for approximately linear structure-preserving mappings, classical kernel methods may be a simpler and more effective alternative.
