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Muon combines current and past gradients into matrix momentum. For $M=U\Sigma V^\top$, the idealized polar update $Q=UV^\top$ gives every singular direction the same weight. We refer to this as the flat profile. Several recent optimizers replace this flat profile with fine-grained spectral maps that give each direction...
CAST (Canonical Approximate Schur Tree) is introduced, which replaces this clique with a weighted random spanning tree sampled directly from it, and it is proved that its leverage-score marginals minimize the largest normalized reweighted-edge contribution among unbiased inverse-marginal one-tree estimators.
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