[Paper Review] Spectral Portfolio Theory: From SGD Weight Matrices to Wealth Dynamics

A. G. Frøseth|arXiv (Cornell University)|Mar 9, 2026

Stochastic Gradient Optimization Techniques0 citations

TL;DR

This paper establishes a direct identification between neural network weight matrices trained on stochastic processes and portfolio allocations, showing how their spectral structure encodes factor decompositions and wealth dynamics across short- and long-horizon regimes. It introduces a Spectral Invariance Theorem and unifies cross-sectional wealth models, within-portfolio dynamics, and scalar Fokker–Planck frameworks through spectral 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 Frøseth (2026).

Motivation & Objective

• Motivate and formalize the identification of neural network weight matrices as portfolio allocation matrices when trained on stochastic processes.
• Characterize the stationary spectral distribution of weight matrices and the resulting core–satellite portfolio structure.
• Explain the additive and multiplicative regimes of spectral dynamics and their relation to short- and long-horizon wealth processes.
• Unify Bouchaud–Mézard wealth dynamics, Olsen et al. within-portfolio dynamics, and scalar Fokker–Planck frameworks via spectral decomposition.
• Develop applications to portfolio design, wealth inequality measurement, tax policy, and neural network diagnostics.

Proposed method

• Model the weight matrix W as a portfolio allocation matrix with rows as states and columns as assets.
• Apply singular value decomposition W = U Σ V^T to obtain eigenportfolios and factor structures.
• Use the SGD-driven singular-value evolution dσ_k with three forces (gradient signal, survival constraint, eigenvalue repulsion) and interpret these as smart money, survival constraints, and endogenous diversification.
• Derive the stationary spectral distribution p(σ) ∝ σ^{m−n+1} exp(−(β1/4ηD) σ^2) and define core–satellite structure.
• Prove the Spectral Invariance Theorem: isotropic perturbations preserve spectral shape up to scale and shift; anisotropic perturbations distort spectra proportionally to cross-asset variance.
• Bridge matrix-level dynamics to scalar wealth via Itô projection to x = ||W||_F and connect to Pareto tails through a radial Fokker–Planck framework.

Experimental results

Research questions

• RQ1 How can neural network weight matrices trained on stochastic processes be interpreted as portfolio allocation matrices?
• RQ2 What is the stationary spectral distribution of SGD weight matrices, and how does it imply a core–satellite portfolio structure?
• RQ3 How do short-horizon additive and long-horizon multiplicative regimes manifest in the spectral properties of weight matrices and in wealth distributions?
• RQ4 What is the Spectral Invariance Theorem and how does isotropic versus anisotropic perturbations affect the spectrum?
• RQ5 How can the spectral framework unify cross-sectional wealth dynamics, within-portfolio dynamics, and scalar wealth processes, and what are the practical implications for policy and diagnostics?

Key findings

• The three SGD forces map to portfolio concepts: gradient signal as smart money, survival constraint as endogenous protection, and eigenvalue repulsion as endogenous diversification.
• The stationary spectral density has a gamma-type bulk with a power-law tail, yielding a core–satellite portfolio structure.
• There is a spectral transition from additive (Marchenko–Pastur) to multiplicative (inverse-Wishart) regimes governed by a free log-normal distribution and the matrix Kesten problem.
• An Itô projection links matrix spectra to a scalar wealth process, producing Pareto tails with exponent tied to the radial drift and effective diffusion.
• The Spectral Invariance Theorem shows isotropic perturbations preserve spectral shape up to scale/shift, while anisotropic perturbations distort the spectrum proportionally to cross-asset variance.
• Applications encompass portfolio design, wealth inequality measurement, tax policy, and neural network diagnostics, with a unifying spectral foundation across models.