ProPolar: Progressive Polar Decomposition for Implicit Neural Representations

Implicit neural representations (INRs) capture global low-frequency structure during the early stage of training and then refine localized high-frequency details, afterwards. However, standard optimizers are agnostic to the coarse-to-fine learning behavior of INRs. Such optimizers accumulate a gradient matrix in momentum buffer and retain components misaligned with the dominant low-frequency directions in the early stage. We propose a progressively growing rank scheduler and a rank-aware learning-rate scheduler for enhancing a subspace-based momentum optimization, where a momentum matrix is updated within a low-rank subspace. The rank schedule expands the subspace during training, and the scheduler holds the peak learning rate through rank growth so that the added subspace contributes to fine-detail recovery. Experiments on image fitting, single-image super-resolution (SISR), and neural radiance field optimization show that the proposed method improves convergence and peak reconstruction quality over competitive optimizers. The largest gains appear in the high-frequency refinement stage, mirroring the coarse-to-fine progression that the rank schedule targets. A significant study on hyperparameter optimization (HPO) on NeRF further confirms that these gains persist under matched search budgets.