Generalization of P=2, D=2 MMCR intuition to high-dimensional regimes

Determine the extent to which the closed-form intuition for Maximum Manifold Capacity Representations (MMCR) in the P=2, D=2 case—namely, that maximizing the norm of each mean vector and orthogonalizing the centers maximizes the nuclear norm—extends to general settings with arbitrary numbers of manifolds P and embedding dimensions D in the large-data, high-dimensional regime.

Background

MMCR defines a loss as the negative nuclear norm of a matrix of per-datum centers, motivating representations that increase the nuclear norm. For the special case P=2, D=2, a closed-form solution provides intuition: the nuclear norm is maximized when the mean vectors are unit-norm and orthogonal.

The paper notes that, although this low-dimensional case offers intuition, MMCR was theoretically derived and empirically implemented in regimes with large numbers of data and high embedding dimensions. The authors explicitly state uncertainty about how the P=2, D=2 intuition sheds light on general MMCR behavior, motivating their high-dimensional analysis to address this gap.