We characterize bijective transformations on the set of all
n-dimensional subspaces of a Hilbert space that preserve orthogonality in both directions. This extends Uhlhorn's improvement of Wigner's classical theorem on symmetry transformations.
Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank-one projections). Suppose dim H ⩾ 3 . We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P ( H ) is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P ( H ) to itself is induced by a linear or conjugate-linear isometry ( H is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.
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
R2=0.98, while regression to the MLP's recovered angles achieves
R2=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× 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.