Papers

The core write-ups of Schur damping. For the wider literature — antecedents, independent rediscoveries, and derived works across fields — see the bibliography.

Core

Cotton, P. (2024). Schur Complementary Allocation: A Unification of Hierarchical Risk Parity and Minimum Variance Portfolios. arXiv:2411.05807. The canonical write-up: the damped Schur complement and the recursion at arbitrary hierarchy depth, with γ interpolating HRP and minimum variance.
Cotton, P. (2026). Two Sides of Schur Damping: High-Dimensional Pseudo-Likelihoods and Portfolio Allocation. arXiv:2606.14798. The cross-field identity: the same damping, and the same closed-form reliability γ*, in portfolio allocation and in spatial pseudo-likelihoods.

Working papers & notes

Cotton, P. (2026). When the Out-of-Sample-Optimal Schur Portfolio Lies Between HRP and Minimum Variance. Working paper — PDF. Where to sit on the bridge when the covariance is estimated with error, in four layers: a structural theorem (top-level cross-block noise is invisible at HRP and its expected cost factors as γτ²H, so small noise uniformly separates the optimum from HRP whenever coupling has population value); a local theorem at an exact minimum-variance endpoint (positive marginal noise sensitivity moves the optimum inside by order τ²); and an exactly solved four-asset family whose unique optimizer is interior and slides monotonically from 1 to ½ as the noise grows. A whole-matrix section then noises every entry of the covariance: a sharp second-moment condition Ξ > 0 decides interiority, exact enumeration certifies interior optima under entrywise noise, and under the sample covariance of T Gaussian draws the optimizer sweeps the bridge monotonically with T. Exact examples bound the general claims: HRP can become locally optimal at a finite noise level (boundary derivative −1/700 + (8/189)τ²), and the surrogate frontier is a clipped monotone inverse of τ² = −V₀′/G′. All exact claims are machine-verified in rational arithmetic.
Cotton, P. (2026). Schur Damping for Perpetual Demand Lending Pools. Working paper — PDF, web note. Proposes the interior of the dial for the single-pool-vs-two-pools decision of Chitra et al.: a partially-merged DeFi lending pool that hedges against $A - \gamma B C^{-1} B^{\mathsf T}$, with γ* set by how well the cross-pool covariance can be estimated. In the undersampled on-chain regime, γ* is pulled toward 0.