Papers

Core

• Cotton, P. (2024). Schur Complementary Allocation: A Unification of Hierarchical Risk Parity and Minimum Variance Portfolios. arXiv:2411.05807. Original version: October 29, 2024; latest version: September 19, 2026 (corrected PDF, source). 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 median-optimal coupling rises with T, from 0.62 at T=15 to 0.96 at T=250. 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′. A closing section shows that every portfolio with a unit budget is the exact minimum-variance portfolio of some positive definite matrix, by a symmetric rank-two correction of the same algebraic form as Jagannathan and Ma's, so the distance from that correction to the data measures how far an allocator's implied beliefs sit from its own input: zero for minimum variance, 0.205 for HRP, 0.385 for equal weight. All exact claims are machine-verified in rational arithmetic.
• Cotton, P. (2026). Nested Clustered Optimization Is One End of a Schur Bridge, and the Interior Is Sometimes Provably Better. Working paper: SSRN 7480738, PDF, source, certificate, demos. The flat-partition counterpart of the bridge. Block inversion writes the unconstrained minimum-variance portfolio as one problem per cluster on a Schur-complemented block. Conditioning each cluster on one knot from every other cluster, a truncation in the manner of Vecchia approximations, is exact under a rank-one “gateway” model of cross-cluster dependence. Damping by γ then runs from nested clustered optimization at γ=0 to the global optimum at γ=1, with no linear solve larger than a cluster or the number of clusters. The flat and tree constructions agree at γ=1 and differ in the interior.
  Under the gateway model only the knot exposures move along the bridge, in closed form; the member positions are the same at both ends. Under estimation error the optimal γ can be strictly interior, and full coupling can remain locally optimal: a local theorem at the minimum-variance end, with exact rational examples of both. A symmetric family reduces the question to one scalar allocation, gives the optimum in closed form (γ⋆=2/3 in the simplest case), and shows both signs in one model. An exact lost-precision identity makes full coupling the unique noiseless optimum, and a final section treats singular covariances, where damping by γ itself becomes singular. All identities and both examples are machine-verified.
• Cotton, P. (2026). Hierarchical Equal Risk Contribution Is a Corner of a Schur Square, and Two Dials Lead to Minimum Variance. Working paper: PDF, source, certificate. HERC budgets clusters by inverse cluster variance, which is the rule block inversion gives the global minimum-variance portfolio. HERC lacks conditioning in two places: each cluster on the others, and each asset on its cluster mates. Damping the two Schur complements by γ and η gives a square with HERC at (0,0) and the global optimum at (1,1).
  Raffinot's split formula, read with risk contribution as a precision, telescopes down any dendrogram to a flat inverse-fitness rule; the reference implementations sum variances instead and depart from it at three or more clusters. The inner dial has a closed form interpolating the naive and Stevens quantities, a long-only frontier, and on an equicorrelated cluster is the straight segment from inverse variance to cluster minimum variance, so its noise problem is one scalar: an exact interior optimum η⋆=3/5 with gaps 1/45 and 1/20, a sign criterion at full coupling with both signs realized, and a two-dial local theorem at the minimum-variance corner with exact examples where both dials move inside and where one stays. Every claim is machine-verified in rational arithmetic.
• Cotton, P. (2026). The Thurstone Tilt Is a Schur Bridge, and Both of Its Ends Are Races. Working paper: PDF, source, certificate. Thurstone polishing dials the estimated correlation into a race with a scalar φ. Taking the reference to be the block-diagonal part of the estimate makes that blend the estimate with its cross-cluster entries scaled by φ, and for two blocks the conditional of that matrix is the damped complement at γ = φ². So the tilt dial is the Schur dial under a square root, the near end races the clusters independently, and the far end races them under the full estimate. Every other bridge here ends at a linear solve; this one ends on the simplex, splits a duplicated position rather than doubling it, and has no pole, the taper being positive semidefinite throughout. The closed-form reliability transfers as φ⋆ = √γ⋆.
• Cotton, P. (2026). The Far End of a Schur Bridge Does Not Care How Its Clusters Are Priced, and the Interior Does. Working paper: PDF, source, certificate. A two-tier allocation prices its clusters with a $k\times k$ matrix, and every construction so far builds that matrix from the same portfolios it holds. Separating the two gives a family with a trial space and a test space. At the far end the $k$ conditioned directions are the blocks of $\Sigma^{-1}u$ and sum to it, so any non-degenerate pairing recovers it and the choice of representative cannot reach the far end at all. The interior is not free: an unmatched pairing loses symmetry and can pass through singularity inside the dial, with a positive-definite $6\times6$ example where it does and a rate of about a quarter of estimated markets. Pricing against the cluster portfolios makes the matrix a Gram matrix of $\Sigma$ and removes the possibility.
• Cotton, P. (2026). Covariance Localization Is Schur Damping, and Its Optimal Strength Is a Sampling-Error Correction. Working paper: PDF, source, verification. Ensemble Kalman filters localize by an elementwise taper. A taper that is constant on blocks is Schur damping: the localized covariance is positive semidefinite, its conditional covariance of the unobserved block is A − γBD⁻¹B′, and the localized gain is √γ times the optimal gain. Under sampling error in the cross-covariance the expected analysis error is an exact quadratic in √γ, so the optimal coupling is strictly interior at every noise level, in closed form, and equals the square of a signal-to-noise shrinkage of the gain.
• 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.