Optimized Weight Initialization on the Stiefel Manifold for Deep ReLU Neural Networks

arXiv:2509.00362v2 Announce Type: replace Abstract: Stable and efficient training of ReLU networks with large depth is highly sensitive to weight initialization. Improper initialization can cause permanent neuron inactivation dying ReLU and exacerbate gradient instability as network depth increases. Methods such as He, Xavier, and orthogonal initialization preserve variance or promote approximate isometry. However, they do not necessarily regulate the pre-activation mean or control activation sparsity, and their effectiveness often diminishes in very deep architectures. This work introduces a semi-orthogonal initialization specifically optimized for ReLU by solving an optimization problem on the Stiefel manifold, thereby preserving scale and calibrating the pre-activation statistics from the outset. A family of closed-form solutions and an efficient sampling scheme are derived. Theoretical analysis at initialization shows that the proposed initialization mitigates dying ReLU, slows the decay of activation variance, and alleviates gradient vanishing, thereby stabilizing signal and gradient flow in deep architectures. Empirically, across MNIST, Fashion-MNIST, multiple tabular datasets, few-shot settings, and ReLU-family activations, the proposed initialization maintains stable performance as network depth increases, with particularly clear advantages in few-shot learning and large-depth settings.