Spectral Portfolio Theory: From SGD Weight Matrices to Wealth Dynamics

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).

Figures (3)

• Figure 1: The learning setup. A stochastic process \ref{['eq:data-sde']} generates trajectory data (left). A neural network with weight matrix $W$ at each layer learns the drift function $\hat{v}(x; W)$ or the score function $s_\theta(x)$ (centre). The loss function is quadratic in the prediction residual (right), whether using maximum likelihood or score matching. SGD updates $W$ iteratively, and the stationary spectral distribution of $W$ encodes the structure of the learned process.
• Figure 2: The neural network--portfolio identification. A single layer with weight matrix $W \in \mathbb{R}^{m \times n}$ maps input $x_t$ to output $y_t = W^\top x_t$ (top left). Relabelling rows as states and columns as assets gives the allocation matrix $W_{ti}$ (bottom left). The SVD decomposes $W$ into temporal patterns $u_k$, factor magnitudes $\sigma_k$, and eigenportfolio compositions $v_k$ (top right). SGD dynamics on $W$ are equivalent to adaptive portfolio rebalancing (bottom right); the stationary spectral density of $\sigma_k$ is determined by the signal-to-noise ratio $\beta_1/\eta D$.
• Figure 3: From forces to wealth distributions. The three forces in the singular-value SDE \ref{['eq:sv-sde']} --- gradient signal, survival constraint, and eigenvalue repulsion --- determine the stationary spectral density (middle). The gamma-type density has a concentrated bulk (core portfolio) and a power-law tail (satellite positions). The radial Itô projection (Section \ref{['sec:aggregation']}) maps the matrix-valued spectral density to a scalar wealth process $x = \|W\|_F$, whose stationary distribution exhibits a Pareto tail with exponent $\alpha$ determined by the signal-to-noise ratio $\beta_1/\eta D$.

Theorems & Definitions (18)

• Remark : Connection to Merton
• Remark : Adaptive dynamics
• Definition 1: Core--Satellite Portfolio
• Definition 2: Effective Spectral Rank
• Remark : Interpolating Family
• Definition 3: Ergodicity Gap
• Theorem 1: Spectral Invariance
• proof : Proof sketch
• Corollary 1: Tax Neutrality
• proof