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.
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.
The Market’s Memory, Measured
The theory above says the spectral structure of a portfolio’s covariance is where the economics lives. The empirical companion paper turns that claim into an instrument. It generalises the classic Lo–MacKinlay variance-ratio test from a single return series to every eigenmode of the cross-sectional covariance at once, in two channels — one for returns, one for volatility — with a 1000-replicate bootstrap supplying error bars for every number.
Applied to half a century of U.S. equity returns (and a European replication), the instrument reads off a five-factor multi-memory structure that reproduces seven classic stylised facts simultaneously; dates a regime transition in the market’s volatility memory to the late 1980s, with the slowest component of the volatility cascade lengthening from roughly two years to four; and shows that the cross-sections carrying return-side memory and volatility-side memory are statistically distinct. A general-audience account is in the accompanying blog post.
Research Papers
Spectral Portfolio Theory: From SGD Dynamics to Wealth Distribution via Random Matrix Duality
The theory paper: a neural network’s weight matrix is a portfolio allocation matrix — and tax neutrality turns out to be a special case of a spectral invariance.
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 Mézard (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).
@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}
}A Spectral Generalisation of the Variance Ratio: Eigenstructure of Long-Horizon Portfolio Covariance and a Multi-Memory Factor Model of U.S. Equity Returns
The measurement paper: the classic variance-ratio test generalised to every eigenmode of the market at once — five kinds of memory read off half a century of returns, with error bars on everything.
We propose a multivariate generalisation of the Lo–MacKinlay (1988) variance ratio that decomposes long-horizon equity-return dynamics into separate return-channel and volatility-channel memory components across the cross-section of asset returns. The framework identifies a parsimonious five-factor model — capturing persistent, antipersistent, and multi-scale memory in returns and volatility — that fits four U.S. portfolio sub-samples (Fama–French 49-industry universe full sample and pre/post-1998 split halves; Fama–French 100 size×book-to-market sort) and a non-U.S. replication on the Fama–French Europe 25 sort, recovering seven well-known stylised facts of long-horizon equity dynamics simultaneously across all five panels.
Three findings carry economic content. (i) The same five-factor decomposition fits all five panels, indicating a cross-sectional structure of long-horizon equity dynamics that is robust to industry vs. size-and-value sorts, to pre- vs. post-1998 sub-periods, and to U.S. vs. developed-European markets. (ii) U.S. equity volatility memory underwent a regime transition in the late 1980s — not at the static 1998 split-half boundary — with the slowest component of the volatility cascade lengthening from approximately two to four years across the transition. A 1000-replicate rolling-window bootstrap localises the transition to the late 1980s with strictly non-overlapping 90% confidence bands separating pre- and post-transition windows; the 28-year window precludes narrower dating. (iii) The cross-sectional loadings driving return-channel long memory are economically distinct from those driving volatility-channel cascade memory: a cross-channel β-inversion test finds no panel exhibits the positive cross-channel alignment that a single shared loading predicts, and rejects the shared-loading hypothesis toward anti-alignment on the two largest panels at Bonferroni p = 0.0004. Industry and size×book-to-market characteristics that predict return-momentum patterns therefore need not predict volatility-persistence patterns.
@article{Froeseth2026varianceratio,
author = {Fr{\o}seth, Anders G.},
title = {A Spectral Generalisation of the Variance Ratio:
Eigenstructure of Long-Horizon Portfolio Covariance and
a Multi-Memory Factor Model of {U.S.} Equity Returns},
year = {2026},
eprint = {2607.03858},
archiveprefix = {arXiv},
primaryclass = {q-fin.ST}
}