Spectral Portfolio Theory: From SGD Weight Matrices to Wealth Dynamics

TL;DR

This paper introduces spectral portfolio theory linking neural network weight matrices trained on stochastic processes to portfolio dynamics, revealing how spectral properties govern wealth concentration and diversification over different time horizons.

Contribution

It establishes a spectral foundation connecting neural network weights, stochastic gradient descent dynamics, and portfolio theory, including a key invariance theorem and applications to finance and neural diagnostics.

Findings

Spectral properties transition from Marchenko-Pastur to inverse-Wishart distributions over time.

The Spectral Invariance Theorem shows isotropic perturbations preserve spectral distribution.

Applications include portfolio design, wealth inequality measurement, and tax policy analysis.

Abstract

We develop spectral portfolio theory by establishing a direct identification: neural network weight matrices trained on stochastic processes are portfolio allocation matrices, and their spectral structure encodes factor decompositions and wealth concentration patterns. The three forces governing stochastic gradient descent (SGD) - gradient signal, dimensional regularisation, and eigenvalue repulsion - translate directly into portfolio dynamics: smart money, survival constraint, and endogenous diversification. The spectral properties of SGD weight matrices transition from Marchenko-Pastur statistics (additive regime, short horizon) to inverse-Wishart via the free log-normal (multiplicative regime, long horizon), mirroring the transition from daily returns to long-run wealth compounding. We unify the cross-sectional wealth dynamics of Bouchaud and Mezard (2000), the within-portfolio dynamics of Olsen et al. (2025), and the scalar Fokker-Planck framework via a common spectral foundation. A central result is the Spectral Invariance Theorem: any isotropic perturbation to the portfolio objective preserves the singular-value distribution up to scale and shift, while anisotropic perturbations produce spectral distortion proportional to their cross-asset variance. We develop applications to portfolio design, wealth inequality measurement, tax policy, and neural network diagnostics. In the tax context, the invariance result recovers and generalises the neutrality conditions of Froseth (2026).