LDLT L-Lipschitz Network Weight Parameterization Initialization

• URL: http://arxiv.org/abs/2601.08253v1
• Date: Tue, 13 Jan 2026 06:20:21 GMT
• Title: LDLT L-Lipschitz Network Weight Parameterization Initialization
• Authors: Marius F. R. Juston, Ramavarapu S. Sreenivas, Dustin Nottage, Ahmet Soylemezoglu,
• Abstract summary: We derive the exact marginal output variance for $mathcalL$-Lipschitz layers using Isserlis-based expansion for $operatornamemathbbEleft[tr]$ and explicit truncated moments up to $k=10$.<n>The findings clarify why deep $mathcalL$-Lipschitz networks suffer rapid information loss at initialization.
• Score: 3.744861320984297
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We analyze initialization dynamics for LDLT-based $\mathcal{L}$-Lipschitz layers by deriving the exact marginal output variance when the underlying parameter matrix $W_0\in \mathbb{R}^{m\times n}$ is initialized with IID Gaussian entries $\mathcal{N}(0,σ^2)$. The Wishart distribution, $S=W_0W_0^\top\sim\mathcal{W}_m(n,σ^2 \boldsymbol{I}_m)$, used for computing the output marginal variance is derived in closed form using expectations of zonal polynomials via James' theorem and a Laplace-integral expansion of $(α\boldsymbol{I}_m+S)^{-1}$. We develop an Isserlis/Wick-based combinatorial expansion for $\operatorname{\mathbb{E}}\left[\operatorname{tr}(S^k)\right]$ and provide explicit truncated moments up to $k=10$, which yield accurate series approximations for small-to-moderate $σ^2$. Monte Carlo experiments confirm the theoretical estimates. Furthermore, empirical analysis was performed to quantify that, using current He or Kaiming initialization with scaling $1/\sqrt{n}$, the output variance is $0.41$, whereas the new parameterization with $10/ \sqrt{n}$ for $α=1$ results in an output variance of $0.9$. The findings clarify why deep $\mathcal{L}$-Lipschitz networks suffer rapid information loss at initialization and offer practical prescriptions for choosing initialization hyperparameters to mitigate this effect. However, using the Higgs boson classification dataset, a hyperparameter sweep over optimizers, initialization scale, and depth was conducted to validate the results on real-world data, showing that although the derivation ensures variance preservation, empirical results indicate He initialization still performs better.

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