Spectral Portfolio Theory

At a given layer, a neural network weight matrix is a portfolio allocation matrix. Its spectral structure encodes factor decompositions, wealth concentration, and the conditions for tax neutrality.

Author

Anders G Frøseth

The Identification

A feedforward neural network trained on a stochastic process has a weight matrix $W \in \mathbb{R}^{m \times n}$ at each layer. At a given layer, this matrix is a portfolio allocation matrix: row $i$ specifies how capital is distributed across $n$ assets in state $i$; columns index assets. Training the network under stochastic gradient descent is equivalent to running an adaptive portfolio optimiser on that process.

This is not merely an analogy. The spectral structure of the weight matrix — its singular values and singular vectors — encodes the portfolio’s factor decomposition, the concentration–diversification tradeoff, and the transition from short-horizon to long-run wealth dynamics.

Three Forces

The singular-value evolution equation of SGD decomposes into three forces, each with a direct portfolio interpretation.

Force 1

Gradient Signal

The gradient drives singular values toward directions of high expected return. In portfolio terms, this is smart money — capital flows toward the most rewarding factor exposures.

Force 2

Dimensional Regularisation

Large singular values are pulled back by a term proportional to $\sigma^2 / n$, where $\sigma^2$ is the noise variance and $n$ the dimension. This is an endogenous survival constraint: it prevents any single factor from absorbing too much capital, without needing an explicit risk budget.

Force 3

Eigenvalue Repulsion

Nearby singular values repel one another, a universal phenomenon in random matrix theory. In portfolio terms, this is endogenous diversification: the spectral dynamics prevent factor loadings from collapsing onto a single axis.

The Spectral Invariance Theorem

The central result is a spectral invariance theorem for portfolio perturbations. Any isotropic perturbation to the portfolio objective — one that treats all assets symmetrically — preserves the singular-value distribution up to a global scale and shift. The portfolio’s factor structure is unchanged; only the overall level of risk and return adjusts.

Anisotropic perturbations, by contrast, produce spectral distortion proportional to their cross-asset variance. The theorem gives a sharp criterion for when a perturbation is neutral and when it distorts.

In the tax context, a proportional wealth tax at market value is isotropic: it applies uniformly to all holdings. The invariance theorem therefore recovers and generalises the neutrality conditions from the wealth tax framework. Non-uniform assessment or progressive brackets are anisotropic and generate spectral distortion — precisely the distortion channels identified in the earlier work.

Connection to the Wealth Tax Framework

This paper grew out of the statistical physics formulation of wealth tax neutrality. The Fokker–Planck equation governing the wealth distribution is a scalar projection of the full matrix-valued dynamics. The spectral approach lifts the analysis from scalar wealth to the full portfolio structure, revealing that tax neutrality is a special case of a broader spectral invariance.

The two research streams are complementary. The wealth tax framework asks: when does a tax distort investment decisions? Spectral portfolio theory asks the more general question: when does any perturbation to the portfolio objective preserve or destroy its factor structure? The wealth tax question is embedded in the spectral one.

Research Paper

Spectral Portfolio Theory: From SGD Dynamics to Wealth Distribution via Random Matrix Duality

Anders G Frøseth · Working Paper · First posted 9 March 2026

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 — gradient signal, dimensional regularisation, and eigenvalue repulsion — translate directly into portfolio dynamics: smart money, survival constraint, and endogenous diversification. 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.

Keywords: Spectral portfolio theory, random matrix theory, stochastic gradient descent, Marchenko–Pastur distribution, free log-normal, matrix Kesten problem, wealth distribution, tax neutrality

Download PDF arXiv

BibTeX

@article{Froeseth2026spectral,
  author        = {Fr{\o}seth, Anders G.},
  title         = {Spectral Portfolio Theory: From {SGD} Dynamics
                   to Wealth Distribution via Random Matrix Duality},
  year          = {2026},
  eprint        = {2603.09006},
  archiveprefix = {arXiv},
  primaryclass  = {q-fin.PM}
}

--- title: "Spectral Portfolio Theory" subtitle: | At a given layer, a neural network weight matrix is a portfolio allocation matrix. Its spectral structure encodes factor decompositions, wealth concentration, and the conditions for tax neutrality. author: "Anders G Frøseth" toc: true toc-depth: 2 page-layout: article --- ## The Identification {#identification} ::: {.section-rule} ::: A feedforward neural network trained on a stochastic process has a weight matrix $W \in \mathbb{R}^{m \times n}$ at each layer. At a given layer, this matrix is a portfolio allocation matrix: row $i$ specifies how capital is distributed across $n$ assets in state $i$; columns index assets. Training the network under stochastic gradient descent is equivalent to running an adaptive portfolio optimiser on that process. This is not merely an analogy. The spectral structure of the weight matrix --- its singular values and singular vectors --- encodes the portfolio's factor decomposition, the concentration--diversification tradeoff, and the transition from short-horizon to long-run wealth dynamics. ## Three Forces {#forces} ::: {.section-rule} ::: The singular-value evolution equation of SGD decomposes into three forces, each with a direct portfolio interpretation. ::: {.cards-grid} ::: {.card} [Force 1]{.card-number} ### Gradient Signal The gradient drives singular values toward directions of high expected return. In portfolio terms, this is *smart money* --- capital flows toward the most rewarding factor exposures. ::: ::: {.card} [Force 2]{.card-number} ### Dimensional Regularisation Large singular values are pulled back by a term proportional to $\sigma^2 / n$, where $\sigma^2$ is the noise variance and $n$ the dimension. This is an endogenous *survival constraint*: it prevents any single factor from absorbing too much capital, without needing an explicit risk budget. ::: ::: ::: {.cards-grid} ::: {.card} [Force 3]{.card-number} ### Eigenvalue Repulsion Nearby singular values repel one another, a universal phenomenon in random matrix theory. In portfolio terms, this is *endogenous diversification*: the spectral dynamics prevent factor loadings from collapsing onto a single axis. ::: ::: ## The Spectral Invariance Theorem {#invariance} ::: {.section-rule} ::: The central result is a spectral invariance theorem for portfolio perturbations. Any *isotropic* perturbation to the portfolio objective --- one that treats all assets symmetrically --- preserves the singular-value distribution up to a global scale and shift. The portfolio's factor structure is unchanged; only the overall level of risk and return adjusts. Anisotropic perturbations, by contrast, produce spectral distortion proportional to their cross-asset variance. The theorem gives a sharp criterion for when a perturbation is neutral and when it distorts. In the tax context, a proportional wealth tax at market value is isotropic: it applies uniformly to all holdings. The invariance theorem therefore recovers and generalises the [neutrality conditions](../wealth-tax/index.qmd) from the wealth tax framework. Non-uniform assessment or progressive brackets are anisotropic and generate spectral distortion --- precisely the distortion channels identified in the earlier work. ## Connection to the Wealth Tax Framework {#connection} ::: {.section-rule} ::: This paper grew out of the [statistical physics formulation](../wealth-tax/index.qmd#statphys) of wealth tax neutrality. The Fokker--Planck equation governing the wealth distribution is a scalar projection of the full matrix-valued dynamics. The spectral approach lifts the analysis from scalar wealth to the full portfolio structure, revealing that tax neutrality is a special case of a broader spectral invariance. The two research streams are complementary. The wealth tax framework asks: *when does a tax distort investment decisions?* Spectral portfolio theory asks the more general question: *when does any perturbation to the portfolio objective preserve or destroy its factor structure?* The wealth tax question is embedded in the spectral one. ## Research Paper {#paper} ::: {.section-rule} ::: ### Spectral Portfolio Theory: From SGD Dynamics to Wealth Distribution via Random Matrix Duality ::: {.paper-meta} Anders G Frøseth · Working Paper · First posted 9 March 2026 ::: ::: {.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 --- gradient signal, dimensional regularisation, and eigenvalue repulsion --- translate directly into portfolio dynamics: smart money, survival constraint, and endogenous diversification. 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. ::: ::: {.paper-meta} **Keywords:** Spectral portfolio theory, random matrix theory, stochastic gradient descent, Marchenko--Pastur distribution, free log-normal, matrix Kesten problem, wealth distribution, tax neutrality ::: [Download PDF](papers/froeseth_2026_spectral.pdf){.btn .btn-primary} [arXiv](https://arxiv.org/abs/2603.09006){.btn .btn-outline-primary} ::: {.callout-note collapse="true" title="BibTeX"} ```bibtex @article{Froeseth2026spectral, author = {Fr{\o}seth, Anders G.}, title = {Spectral Portfolio Theory: From {SGD} Dynamics to Wealth Distribution via Random Matrix Duality}, year = {2026}, eprint = {2603.09006}, archiveprefix = {arXiv}, primaryclass = {q-fin.PM} } ``` :::