Topological Data Analysis

Class of topological portfolios: Are they better than classical portfolios? ArXiv ID: 2601.03974 “View on arXiv” Authors: Anubha Goel, Amita Sharma, Juho Kanniainen Abstract Topological Data Analysis (TDA), an emerging field in investment sciences, harnesses mathematical methods to extract data features based on shape, offering a promising alternative to classical portfolio selection methodologies. We utilize persistence landscapes, a type of summary statistics for persistent homology, to capture the topological variation of returns, blossoming a novel concept of ``Topological Risk". Our proposed topological risk then quantifies portfolio risk by tracking time-varying topological properties of assets through the $L_p$ norm of the persistence landscape. Through optimization, we derive an optimal portfolio that minimizes this topological risk. Numerical experiments conducted using nearly a decade long S&P 500 data demonstrate the superior performance of our TDA-based portfolios in comparison to the seven popular portfolio optimization models and two benchmark portfolio strategies, the naive $1/N$ portfolio and the S&P 500 market index, in terms of excess mean return, and several financial ratios. The outcome remains consistent through out the computational analysis conducted for the varying size of holding and investment time horizon. These results underscore the potential of our TDA-based topological risk metric in providing a more comprehensive understanding of portfolio dynamics than traditional statistical measures. As such, it holds significant relevance for modern portfolio management practices. ...

Integral Betti signature confirms the hyperbolic geometry of brain, climate, and financial networks ArXiv ID: 2406.15505 “View on arXiv” Authors: Unknown Abstract This paper extends the possibility to examine the underlying curvature of data through the lens of topology by using the Betti curves, tools of Persistent Homology, as key topological descriptors, building on the clique topology approach. It was previously shown that Betti curves distinguish random from Euclidean geometric matrices - i.e. distance matrices of points randomly distributed in a cube with Euclidean distance. In line with previous experiments, we consider their low-dimensional approximations named integral Betti values, or signatures that effectively distinguish not only Euclidean, but also spherical and hyperbolic geometric matrices, both from purely random matrices as well as among themselves. To prove this, we analyse the behaviour of Betti curves for various geometric matrices – i.e. distance matrices of points randomly distributed on manifolds of constant sectional curvature, considering the classical models of curvature 0, 1, -1, given by the Euclidean space, the sphere, and the hyperbolic space. We further investigate the dependence of integral Betti signatures on factors including the sample size and dimension. This is important for assessment of real-world connectivity matrices, as we show that the standard approach to network construction gives rise to (spurious) spherical geometry, with topology dependent on sample dimensions. Finally, we use the manifolds of constant curvature as comparison models to infer curvature underlying real-world datasets coming from neuroscience, finance and climate. Their associated topological features exhibit a hyperbolic character: the integral Betti signatures associated to these datasets sit in between Euclidean and hyperbolic (of small curvature). The potential confounding ``hyperbologenic effect’’ of intrinsic low-rank modular structures is also evaluated through simulations. ...

Sparse Index Tracking via Topological Learning ArXiv ID: 2310.09578 “View on arXiv” Authors: Unknown Abstract In this research, we introduce a novel methodology for the index tracking problem with sparse portfolios by leveraging topological data analysis (TDA). Utilizing persistence homology to measure the riskiness of assets, we introduce a topological method for data-driven learning of the parameters for regularization terms. Specifically, the Vietoris-Rips filtration method is utilized to capture the intricate topological features of asset movements, providing a robust framework for portfolio tracking. Our approach has the advantage of accommodating both $\ell_1$ and $\ell_2$ penalty terms without the requirement for expensive estimation procedures. We empirically validate the performance of our methodology against state-of-the-art sparse index tracking techniques, such as Elastic-Net and SLOPE, using a dataset that covers 23 years of S&P500 index and its constituent data. Our out-of-sample results show that this computationally efficient technique surpasses conventional methods across risk metrics, risk-adjusted performance, and trading expenses in varied market conditions. Furthermore, in turbulent markets, it not only maintains but also enhances tracking performance. ...