A multiscale differential principal-feature architecture for portfolio optimization: Mathematical formulation and empirical results

Muhammad Hilal Alkhudaydi; Muhammad Hilal Alkhudaydi

doi:10.3934/nhm.2026045

Networks and Heterogeneous Media

2026, Volume 21, Issue 3: 1099-1145. doi: 10.3934/nhm.2026045

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A multiscale differential principal-feature architecture for portfolio optimization: Mathematical formulation and empirical results

Muhammad Hilal Alkhudaydi ^,

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

Received: 27 March 2026 Revised: 01 May 2026 Accepted: 14 May 2026 Published: 25 May 2026

This paper proposed a multiscale differential principal-feature (MDPF) architecture for portfolio optimization in which the central contribution is an interpretable feature-extraction layer rather than a claim of unconditional benchmark dominance. The model combines the discrete wavelet transform (DWT), ordinary differential equation (ODE) identification, and principal component analysis (PCA) to transform rolling asset-return histories into low-dimensional dynamic signatures that are subsequently mapped into constrained portfolio overlays. In the U.S. mega-cap equity universe, the strongest MDPF overlay obtained a Sharpe ratio of 1.117 with average $ \ell_1 $ turnover of 0.132, compared with 1.138 for the equal-weight benchmark and 1.187 for raw mean–variance with substantially higher $ \ell_1 $ turnover of 0.464. In the global multi-asset ETF universe, the strongest MDPF overlay obtained a Sharpe ratio of 0.801, close to the equal-weight benchmark at 0.804 but below raw mean–variance at 0.927, again with lower $ \ell_1 $ turnover than the raw mean–variance allocation. These results indicated that the proposed architecture is most useful when it is used to extract structured multiscale dynamic information and to express that information as controlled active tilts around a stable base portfolio. Supplementary ablation, statistical, regime, sensitivity, ODE-diagnostic, PCA-interpretability, feature-to-allocation, optimization-refinement, and transaction-cost diagnostics further clarify why the representation is economically meaningful while showing that benchmark dominance is not universal and that higher-order nonlinear components require further refinement. Accordingly, the paper positions MDPF as a transparent feature-extraction framework for dynamic portfolio construction and as a basis for future work on regime-aware multiscale allocation.
Citation: Muhammad Hilal Alkhudaydi. A multiscale differential principal-feature architecture for portfolio optimization: Mathematical formulation and empirical results[J]. Networks and Heterogeneous Media, 2026, 21(3): 1099-1145. doi: 10.3934/nhm.2026045

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Abstract

This paper proposed a multiscale differential principal-feature (MDPF) architecture for portfolio optimization in which the central contribution is an interpretable feature-extraction layer rather than a claim of unconditional benchmark dominance. The model combines the discrete wavelet transform (DWT), ordinary differential equation (ODE) identification, and principal component analysis (PCA) to transform rolling asset-return histories into low-dimensional dynamic signatures that are subsequently mapped into constrained portfolio overlays. In the U.S. mega-cap equity universe, the strongest MDPF overlay obtained a Sharpe ratio of 1.117 with average $ \ell_1 $ turnover of 0.132, compared with 1.138 for the equal-weight benchmark and 1.187 for raw mean–variance with substantially higher $ \ell_1 $ turnover of 0.464. In the global multi-asset ETF universe, the strongest MDPF overlay obtained a Sharpe ratio of 0.801, close to the equal-weight benchmark at 0.804 but below raw mean–variance at 0.927, again with lower $ \ell_1 $ turnover than the raw mean–variance allocation. These results indicated that the proposed architecture is most useful when it is used to extract structured multiscale dynamic information and to express that information as controlled active tilts around a stable base portfolio. Supplementary ablation, statistical, regime, sensitivity, ODE-diagnostic, PCA-interpretability, feature-to-allocation, optimization-refinement, and transaction-cost diagnostics further clarify why the representation is economically meaningful while showing that benchmark dominance is not universal and that higher-order nonlinear components require further refinement. Accordingly, the paper positions MDPF as a transparent feature-extraction framework for dynamic portfolio construction and as a basis for future work on regime-aware multiscale allocation.

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