Tingley's Problem for Schatten \(p\)-Classes, $0<p\ne 2<\infty$

What the math gives to ML

The paper turns a family of Schatten-norm distance measurements between minimal partial isometries into an exact metric fingerprint of the trace overlap \({\rm Tr}(e^*v)\), while also showing that surjective isometries preserve orthogonality and consistently preserve or conjugate complex phase. This suggests a regularizer for matrix-valued neural representations that preserves rank-one overlap geometry without directly supervising inner products. The most practical transfer is to match phase-sampled Schatten-distance profiles between representations before and after a network block, optionally combined with an orthogonality penalty.