Muon's polar update is exact for an unweighted spectral geometry. We introduce GO-MUON, which uses a matched data-dependent geometry and reuses it across several optimization steps. Conditioned on any positive-definite left and right maps, its raw update exactly solves the corresponding weighted spectral oracle; this statement is independent of how the maps are estimated or how recently they were refreshed. For softmax cross-entropy, we quantify when the observed-label backward factor approaches the model Fisher and generalized Gauss--Newton factor. We also show that four-step refresh nearly preserves the tracking delay of slowly changing geometry while increasing stationary factor noise, making lazy geometry a compute--statistics tradeoff rather than a denoising mechanism.
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...
Meher Chaitanya, Tian-Yi Zhou, A. Gionis· 0 citations
Hyper^2 is proposed, a dual-space consistency framework that extends HyperbolicCD by reusing the identical arcosh(1+alpha d^2) functional form as a positional bias on the refinement attention (a hyperbolic distance encoding), paired with HyperbolicCD's hyperbolic Chamfer loss under a single shared curvature alpha.
We study sparse Mellin spectral recovery from geometric samples. Dilation covariance is shown to generate the Hankel structure directly, rather than only after reduction to a classical exponential-sum model. For complex exponents, incommensurate sampling scales remove the logarithmic aliasing that persists at a single...
The objective of this study is to construct and solve exactly a curvature-adapted radial Dunkl Hamiltonian associated with the coordinate-reflection group Z23 on three-dimensional hyperbolic space, and to determine how curvature and Dunkl multiplicities control its discrete spectrum and admissible radial states. The wa...
Weak perturbations of equal size can produce terminal Wasserstein errors of different orders. For Gaussian-smoothed infinitely divisible laws, we differentiate distribution functions with respect to variance-weighted L\'evy characteristics, including distributional directions generated by moving atoms. We obtain first-...
We study the squared Rayleigh-residual functional on complex projective space and its negative gradient flow. For an arbitrary complex matrix, every positive-residual critical point has a strictly descending tangent direction; consequently, the local minima are precisely the eigenlines. For normal matrices, we describe...
Eric Shen· 0 citations
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