Predicting When Random Low-Dimensional Reparameterizations Train Neural Networks
This work addresses the problem of selecting the latent space dimension in randomized low-dimensional reparameterization to ensure neural networks can be efficiently trained into low-loss regions. By characterizing accessibility phase transitions through conic geometry, the authors propose a directionally resolved quadratic theoretical framework that accurately predicts residual errors in random slices. Integrating structured random projections—such as Hadamard or recycled Gaussian mappings—with matrix-free curvature approximations and optimizer state compression, they develop a memory-efficient training framework. The method automatically determines the optimal dimensionality without exhaustive scanning, and empirical results on both vision and language models reveal training phase transitions that align closely with theoretical predictions, substantially outperforming existing approximation strategies that neglect directional information.