A Homotopy-Theoretic Framework for Analyzing the Solution Space of Financial Models

Abstract

The inherent complexity and non-stationarity of financial markets pose significant challenges for traditional modeling, often resulting in opaque, black-box solutions. We introduce a novel framework that transcends single-model optimization by first using symbolic regression to generate a rich ecosystem of explicit mathematical models. Our core contribution lies in treating this solution space as a primary object of study. Leveraging principles from Homotopy Type Theory and Topological Data Analysis (TDA), we map the underlying geometric structure connecting these diverse models. This topological analysis successfully identified distinct, stable classes of solutions, while an optimized ensemble of these models demonstrated a significant leap in predictive robustness and accuracy. This research provides a powerful methodology for navigating model complexity and augmenting strategic decision-making in dynamic systems, offering a new lens to understand the entire landscape of potential solutions rather than settling for one.

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GitHub Repo : https://github.com/Apoth3osis-ai/homotopy_forex

Research Gate: https://www.researchgate.net/publication/392596553_A_Homotopy-Theoretic_Framework_for_Analyzing_the_Solution_Space_of_Financial_Models_Notice_of_Proprietary_Information

1. Introduction

Modeling financial time series is a formidable challenge. The underlying systems are adaptive, non-stationary, and characterized by complex, multi-scale interactions. In response, the field has largely trended towards increasingly complex machine learning models, such as deep neural networks. While powerful, these models often function as "black boxes," yielding predictions without providing transparent, interpretable reasoning. This opacity is a significant liability in high-stakes environments where understanding model behavior is as crucial as the predictions themselves.

This paper presents a paradigm shift away from the pursuit of a single, optimal black-box model. We propose a framework for generating and analyzing an entire solution space of transparent, mathematical models. Our central thesis is that the relationships between potential models—their similarities, differences, and classifications—contain valuable information that is lost when focusing on a single solution.

To achieve this, we introduce a novel methodology combining three key components:

Symbolic Regression (SR): To generate a diverse population of explicit, interpretable mathematical formulae that act as candidate models.

Topological Data Analysis (TDA) and Homotopy Theory: To analyze the "shape" and relational structure of the solution space. We treat each formula as a point in a high-dimensional space and use topological methods to identify clusters, equivalences, and fundamental modeling strategies.

Ensemble and Probabilistic Modeling: To leverage the insights gained from the topological analysis to construct superior predictive models that are both robust and sensitive to uncertainty.

This paper details the theoretical underpinnings and practical implementation of this framework. We demonstrate its ability to not only yield high-performing predictive models but also to provide a new, powerful lens for analyzing the fundamental nature of financial modeling itself.

2. Methodology

Our methodology unfolds in four distinct phases: generating a candidate space of models, analyzing its topological structure, constructing an optimized ensemble, and exploring a probabilistic enhancement.

2.1. Problem Formulation and Data

The experimental task is to predict the future price of a currency pair (EUR/USD). Specifically, using a 20-minute lookback window of Open, High, Low, and Close (OHLC) data, the goal is to predict the closing price 5 minutes into the future. The input is therefore an 80-dimensional feature vector (20 time steps × 4 features), and the output is a single scalar value.

2.2. Phase 1: Generating the Solution Space via Symbolic Regression

To avoid the opacity of traditional machine learning, we begin by generating a population of candidate models using Symbolic Regression (SR). SR is an evolutionary algorithm that searches the space of mathematical expressions to find the formula that best fits a given dataset. Unlike regression techniques that fit parameters to a pre-defined equation, SR discovers the equation itself.

We employed PySR, a high-performance SR library, to produce a diverse set of equations. The system was configured to allow for moderate complexity, ensuring a rich and varied population of models. This population of equations forms the foundational "solution space" that is the primary object of our analysis.

2.3. Phase 2: A Topological Lens on the Solution Space

The core innovation of our framework is the application of mathematical topology to understand the structure of the solution space. We conceptualize each generated equation as a point and investigate the geometry of their collective arrangement. This is achieved through two complementary techniques.

2.3.1. Geometric Analysis via TDA

To apply geometric tools, we first represent each symbolic equation as a numerical vector. This vectorization is a heuristic that captures the frequency of various components (operators, variables, etc.) within the formula. From this set of vectors, we compute a pairwise distance matrix, quantifying the syntactic dissimilarity between every pair of equations.

Using this distance matrix, we apply Persistent Homology, a cornerstone of TDA, via the ripser library. Persistent homology tracks the evolution of topological features (e.g., connected components, loops) as a proximity threshold is varied. The output, a persistence diagram, visualizes the "birth" and "death" of these features. Features that persist for a long duration (i.e., points far from the diagonal in the diagram) correspond to robust, statistically significant structures in the data. In our context, persistent connected components (H₀ features) reveal the existence of distinct, stable clusters of models, suggesting fundamentally different—yet equally viable—modeling strategies.

2.3.2. Symbolic Analysis via Homotopy Equivalence

As a more direct and rigorous approach, we leverage the sympy computer algebra system to test for mathematical equivalence. In the language of Homotopy Type Theory (HoTT), two objects (equations) are considered equivalent if there exists a continuous path, or transformation, between them. In our symbolic context, the most direct test is for mathematical identity.

By parsing each equation into a symbolic expression, we can computationally check if simplify(equation1 equation2) = 0. If this condition holds, the two equations are analytically identical, merely different representations of the same underlying function. They belong to the same homotopy equivalence class. This process allows us to partition the entire solution space into disjoint sets of truly unique models.

2.4. Phase 3: Ensemble Modeling via Optimization

The discovery of multiple, distinct model clusters suggests that no single equation captures the full dynamics of the market. Therefore, we hypothesize that a "committee of experts"—a weighted ensemble of diverse models—will outperform any individual model.

We construct a linear ensemble where the final prediction is the weighted sum of the outputs from the top-performing symbolic equations. To find the optimal contribution of each model, we use the SLSQP optimization algorithm to identify the set of weights that minimizes the mean absolute error on the training data, with the constraint that all weights must be non-negative and sum to one.

2.5. Phase 4: A Framework for Probabilistic Modeling

To better account for market uncertainty, we explored a final enhancement: a probabilistic feature layer. This technique transforms the original input features before they are fed into a model. Each feature is passed through the Cumulative Distribution Function (CDF) of several standard probability distributions (e.g., Normal, Gamma, Beta) with varying parameters.

This creates a new, vastly enriched feature set where each new feature represents the original data through the lens of a specific probabilistic assumption. A simple linear regression model fitted to this transformed feature space can capture complex, non-linear relationships and uncertainties with remarkable efficacy. This approach proved more tractable than our initial explorations into evolving fully probabilistic formulae with genetic programming, which faced convergence challenges in such a high-dimensional search space.

3. Results and Discussion

The application of our four-phase framework yielded several key insights:

Financial Models Inhabit a Rich Topological Space: The persistence diagram generated by TDA confirmed our primary hypothesis. The presence of multiple, highly persistent H₀ features provided strong evidence that the solution space is not a single, uniform cloud of points. Instead, it is structured into distinct clusters. This implies that there is not one "correct" way to model the financial time series, but rather several fundamentally different, stable strategies.

Ensemble Models are Superior: As hypothesized, the optimized ensemble model demonstrated significantly stronger predictive performance than the single best equation identified by the initial symbolic regression search. This underscores the practical value of exploring the full solution space and leveraging its diversity, rather than prematurely converging on a single solution.

Probabilistic Layers Enhance Predictive Power: The model built upon the probabilistic feature layer was highly effective. This demonstrates that explicitly modeling the uncertainty of input features provides a powerful mechanism for improving predictive accuracy, often allowing a simpler final model architecture.

Limitations: This research has limitations that open avenues for future work. The vectorization of equations for TDA is a heuristic and could be improved with more sophisticated representations, such as graph-based structures. Furthermore, our exploration into evolving fully probabilistic models with genetic programming was inconclusive, highlighting the immense difficulty of such an unconstrained search.

4. Implications and Applications

The implications of this framework extend beyond generating a trading signal. By providing a method to understand the structure of a model space, it offers significant value in several domains:

Financial Risk Management: By identifying distinct model clusters, analysts can better understand different market regimes and the types of strategies that are effective in each. An ensemble's shifting weights could act as an indicator of changing market dynamics.

Strategy Discovery: Analysis of the equations within different homotopy classes can reveal novel and non-obvious relationships in financial data, leading to the development of new trading or investment strategies.

General Complex Systems Modeling: The methodology is domain-agnostic. It can be applied to any field where complex systems are modeled, such as climate science, epidemiology, or systems biology. It provides a structured approach to managing model uncertainty and diversity, augmenting an expert's ability to reason about the problem space.

5. Future Work

This research lays a foundation for several promising directions of future work:

Topologically-Guided Search: The insights from TDA can be used to create a feedback loop. The symbolic regression algorithm could be actively guided to explore under-represented regions of the solution space or to search more deeply within promising clusters.

Dynamic Ensembles: The ensemble weights could be re-calibrated in real-time or made dependent on market volatility metrics, creating a model that dynamically adapts its strategy.

Advanced Topological Representations: Exploring graph-based representations of equations and leveraging techniques from geometric deep learning could provide a more nuanced understanding of the solution space's structure.

Constrained Probabilistic Search: Revisiting the direct evolution of probabilistic models using more constrained search spaces or improved genetic operators may overcome the previous convergence issues.

6. Conclusion

This paper introduces a novel framework for financial modeling that shifts the objective from finding a single "best" model to understanding the entire landscape of potential solutions. By combining Symbolic Regression with principles from Homotopy Theory and Topological Data Analysis, we have demonstrated a method to map the structure of a solution space, identify unique classes of models, and leverage this diversity to build more robust and accurate ensembles. This approach moves away from opaque, black-box systems towards a more transparent, analyzable, and ultimately more powerful methodology for augmenting human decision-making in the face of complexity. The success of this research in a client context validates its practical utility and provides a strong impetus for its continued development.

7. References

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Cranmer, M., Sanchez Gonzalez, A., Battaglia, P., Xu, R., Cranmer, K., Spergel, D., & Ho, S. (2020). Discovering symbolic models from deep learning with inductive biases. Advances in Neural Information Processing Systems, 331

Edelsbrunner, H., & Harer, J. (2010). Computational topology: an introduction. American Mathematical Society.

The Univalent Foundations Program. (2013). Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study.2