The Shape of Markets: Machine learning modeling and Prediction Using 2-Manifold Geometries

TL;DR

The paper introduces a Geometry Informed Model (GIM) that embeds high-dimensional market data onto constant-curvature 2-manifolds (S^2, R^2, H^2) and a torus T^2, modeling dynamics as Brownian motion with a curvature-induced drift. By combining expanding-window PCA-based eigenportfolios, intrinsic manifold mappings (log/exp) and tangent-space forecasts (VAR, RF, GP), the approach selects geometry through local curvature and persistent-homology tests, then lifts predictions back to the data space. Empirical results on a broad multi-asset dataset show that a torus-like latent geometry with curvature gating and eigenvalue-weighted sleeves yields the strongest predictive performance and trading outcomes, outperforming Euclidean baselines and standard benchmarks. The framework provides a principled way to integrate differential geometry with data-driven inference, offering macro-finance interpretability (via IS–LM-inspired cycles) and a path toward geometry-aware nowcasting and stress testing. Overall, the work demonstrates that the market’s shape—captured by curvature and topology on manifolds—contains actionable predictive content beyond conventional linear models, with practical implications for asset allocation and risk management.

Abstract

We introduce a Geometry Informed Model for financial forecasting by embedding high dimensional market data onto constant curvature 2manifolds. Guided by the uniformization theorem, we model market dynamics as Brownian motion on spherical S2, Euclidean R2, and hyperbolic H2 geometries. We further include the torus T, a compact, flat manifold admissible as a quotient space of the Euclidean plane anticipating its relevance for capturing cyclical dynamics. Manifold learning techniques infer the latent curvature from financial data, revealing the torus as the best performing geometry. We interpret this result through a macroeconomic lens, the torus circular dimensions align with endogenous cycles in output, interest rates, and inflation described by IS LM theory. Our findings demonstrate the value of integrating differential geometry with data-driven inference for financial modeling.

Figures (10)

• Figure 1: Simulated Brownian-motion scenarios: time-series panels (a–g) and the corresponding 3D PCA embedding (h).
• Figure 2: Local Gaussian curvature estimates $K$ across benchmark shapes and the real-data embedded path. The benchmarks provide sign/scale references; the finance panel shows intermittent, regime-like curvature bursts.
• Figure 3: Flow chart of the manifold-aware pipeline -- geometry ($M$) via curvature $K_t$ and persistent homology -- with an explicit native-space baseline.
• Figure 4: Manifold embedding (log), tangent-space forecasting, and lifting (exp). Labels are placed inside the geometry boxes.
• Figure 5: Sphere $S^2$: log map to tangent at $\mu$, prediction $\hat{v}$, and lift $\hat{X}=\exp_\mu(\hat{v})$.