Wavelet–PCA mirror-flow portfolios: Multiscale risk geometry and reduced-order dynamics on the simplex

Muhammad Hilal Alkhudaydi; Muhammad Hilal Alkhudaydi

doi:10.3934/math.2026255

AIMS Mathematics

2026, Volume 11, Issue 3: 6162-6216. doi: 10.3934/math.2026255

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Wavelet–PCA mirror-flow portfolios: Multiscale risk geometry and reduced-order dynamics on the simplex

Muhammad Hilal Alkhudaydi ^,

Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia

Received: 16 January 2026 Revised: 17 February 2026 Accepted: 02 March 2026 Published: 11 March 2026
MSC : 62P05

Financial markets exhibit heterogeneous investment horizons and nonstationary dependence, so portfolio risk and comovement are intrinsically multiscale and time varying. Classical single-horizon covariance models and repeated static reoptimization can therefore be fragile in high-dimensional universes, especially considering transaction costs.
We proposed a reduced-order portfolio optimization method in which long-only allocations evolved dynamically through simplex-preserving mirror flow. The strategy was parameterized by a small number of time-varying decision variables, and exposures to a larger asset universe were obtained by combining a few investable basis portfolios. The basis was updated online from the principal components of a wavelet-based multiscale covariance estimator and mapped to admissible weights through an entropy-based softmax construction. A wavelet multiscale layer was integrated as a structural component of the model rather than as a preprocessing step, inducing both a multiscale quadratic risk geometry and a risk-normalized multiscale return signal that drove allocation dynamics.
We provided a self-contained formulation and established core properties of the scheme, including positive definiteness and stability of the multiscale covariance updates, stability of the principal component analysis (PCA)–softmax basis under perturbations, simplex invariance of the mirror update, and explicit turnover control that transferred from reduced allocations to traded allocations, including the effect of periodic basis refresh. Empirical experiments were then conducted on real equity data, including transaction costs, compared against equal-weight and classical mean–variance baselines, including sensitivity and ablation analyses. The results illustrated that the multiscale and reduced-order components produced materially different dynamic allocations and contributed to risk-adjusted performance and trading stability. We referred to the proposed method as WPROD-R, a shorthand for a wavelet-driven product (multiplicative mirror-flow) scheme with a reduced-order representation.
- multiscale portfolios,
- wavelet covariance,
- mirror flow,
- reduced-order optimization
Citation: Muhammad Hilal Alkhudaydi. Wavelet–PCA mirror-flow portfolios: Multiscale risk geometry and reduced-order dynamics on the simplex[J]. AIMS Mathematics, 2026, 11(3): 6162-6216. doi: 10.3934/math.2026255

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Abstract

Financial markets exhibit heterogeneous investment horizons and nonstationary dependence, so portfolio risk and comovement are intrinsically multiscale and time varying. Classical single-horizon covariance models and repeated static reoptimization can therefore be fragile in high-dimensional universes, especially considering transaction costs.

We proposed a reduced-order portfolio optimization method in which long-only allocations evolved dynamically through simplex-preserving mirror flow. The strategy was parameterized by a small number of time-varying decision variables, and exposures to a larger asset universe were obtained by combining a few investable basis portfolios. The basis was updated online from the principal components of a wavelet-based multiscale covariance estimator and mapped to admissible weights through an entropy-based softmax construction. A wavelet multiscale layer was integrated as a structural component of the model rather than as a preprocessing step, inducing both a multiscale quadratic risk geometry and a risk-normalized multiscale return signal that drove allocation dynamics.

We provided a self-contained formulation and established core properties of the scheme, including positive definiteness and stability of the multiscale covariance updates, stability of the principal component analysis (PCA)–softmax basis under perturbations, simplex invariance of the mirror update, and explicit turnover control that transferred from reduced allocations to traded allocations, including the effect of periodic basis refresh. Empirical experiments were then conducted on real equity data, including transaction costs, compared against equal-weight and classical mean–variance baselines, including sensitivity and ablation analyses. The results illustrated that the multiscale and reduced-order components produced materially different dynamic allocations and contributed to risk-adjusted performance and trading stability. We referred to the proposed method as WPROD-R, a shorthand for a wavelet-driven product (multiplicative mirror-flow) scheme with a reduced-order representation.

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