Paper: arXiv 2603.09006

Authors: Anders G Frøseth

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).

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly novel and mathematically rich bridge between neural network spectral theory (SGD dynamics) and quantitative portfolio design. However, it is primarily theoretical, establishing formal derivations and theorems (e.g., Spectral Invariance Theorem) with very limited empirical backtesting or data validation in the provided text.

Research Flowchart

  flowchart TD
    A[Research Goal: Map SGD Weight Matrix Spectra to Portfolio & Wealth Dynamics] --> B[Input: Stochastic Processes & Neural Network Weight Matrices]
    B --> C[Methodology: Map SGD Forces to Portfolio Forces<br>Signal → Smart Money, Reg → Survival, Repulsion → Diversification]
    C --> D[Spectral Dynamics: Transition MP Statistics to Inverse-Wishart via Free Log-Normal]
    D --> E[Unification & Invariance: Integrate Bouchaud-Mezard, Olsen et al., & Prove Spectral Invariance Theorem]
    E --> F[Key Outcomes: Applications to Portfolio Design, Wealth Inequality, Tax Neutrality, & NN Diagnostics]

Cited References (53)

  1. Heterogeneous Returns and Wealth Tax Neutrality: A Fokker-Planck Framework — → 2603.16006
  2. Flow Taxes, Stock Taxes, and Portfolio Choice: A Generalised Neutrality Result — → 2603.15974
  3. Extensions to the Wealth Tax Neutrality Framework — → 2603.05277
  4. Wealth Taxation as a Drift Modification: A Fokker-Planck Approach to Tax Neutrality — → 2603.05283
  5. Asset Returns, Portfolio Choice, and Proportional Wealth Taxation — → 2603.05264
  6. From SGD to Spectra: A Theory of Neural Network Weight Dynamics (2025) — link
  7. Mean-field theory for heterogeneous random growth with redistribution. (2025) — link
  8. Free Convolution and Generalized Dyson Brownian Motion (2024) — link
and 45 more
  1. Stochastic weight matrix dynamics during learning and Dyson Brownian motion (2024) — link
  2. A First Course in Random Matrix Theory for Physicists, Engineers and Data Scientists (2023) — link
  3. In Search of the Origins of Financial Fluctuations: The Inelastic Markets Hypothesis (2021) — link
  4. A First Course in Random Matrix Theory (2020) — link
  5. Deep Learning for Portfolio Optimization (2020) — link
  6. Nonlinear random matrix theory for deep learning (2019) — link
  7. The ergodicity problem in economics (2019) — link
  8. arXiv (2019) — link
  9. Far from the madding crowd: collective wisdom in prediction markets (2019) — link
  10. Implicit Self-Regularization in Deep Neural Networks: Evidence from Random Matrix Theory and Implications for Learning (2018) — link
  11. Adaptive Markets: Financial Evolution at the Speed of Thought (2018)
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  33. Centre De Referència En Economia Analítica Barcelona Economics Working Paper Series Working Paper Nº 92 Honey, I Shrunk the Sample Covariance Matrix Honey, I Shrunk the Sample Covariance Matrix
  34. Computational cost vs. spectral complexity. Training networks on data with high intrinsic dimensionality (fat spectral tails, large r eff ) requires more computation
  35. The Spectral Invariance Theorem establishes that isotropic perturbations preserve spectral shape, while anisotropic perturbations produce distortion proportional to their cross-asset variance
  36. The cross-sectional spectral structure of the population allocation matrix reflects the multiplicative regime
  37. If we consider a single-factor portfolio ( N = 1, so the allocation matrix reduces to a vector)
  38. The stationary spectral distribution exhibits a core–satellite structure (bulk plus power-law tail) that matches empirical wealth and institutional portfolio data
  39. Neural network weight matrices are portfolio allocation matrices, and their spectral structure encodes factor decompositions faithful to the underlying data
  40. Implicit regularization in matrix sensing
  41. Nonlinear activations. We assume linear output (network is a linear projection of hidden units).
  42. Budget constraints
  43. The three forces in the singular-value evolution—gradient signal, survival constraint, and eigenvalue repulsion—translate directly into
  44. Finite-width effects
  45. Weight spectrum of trained networks. After training a neural network on wealth or return data, the weight spectrum should reflect the Pareto exponent of the target distribution. Deviation signals

5 of 53 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.