Hardy-type norms of matrices

What the math gives to ML

The paper establishes that the spherical geometric mean of the stretch induced by a matrix is itself a genuine norm, despite geometric means normally failing to preserve convexity. This provides a principled alternative to spectral-norm or Frobenius penalties: it measures typical multiplicative amplification while retaining norm-like scaling and subadditivity. A direct neural-network use is to regularize linear layers with a Monte Carlo estimate of this norm, potentially controlling average signal amplification without forcing every singular direction to be small.