Status and attribution

This project reproduces the topological portfolio model of Goel, Sharma and Kanniainen (2026, arXiv:2601.03974). I implemented the Takens embedding, the persistence landscape and its $L^1$ norm, the asset topological risk, and the portfolio quadratic program. The scaffolding (packaging, CLI, test harness and plotting) was written with AI assistance. All 34 tests pass, and the pipeline has been characterised on real market data across FX, bond ETFs, commodities, indices and crypto

Topological risk portfolios

Using topology of data we can avoid model-based estimation errors linked to distributional assumptions or statistical inputs like mean and covariance. A 2026 January paper by Goel, Sharma, Kanniainen (arXiv: https://doi.org/10.48550/arXiv.2601.03974) reports that portfolios constructed using Topological Data Analysis methods outperformed 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 practical argument is dimensional. Authors used $462$ assets with time-series data of almost a decade. Minimum-variance needs the full covariance matrix. Estimated from a $252$-day window, the sample covariance of $462$ assets has rank at most $251$, so it is singular and cannot be inverted. Separately, the matrix carries $n(n+1)/2$ parameters, or $106, 953$ for $462$ assets, so even where it is inevitable it is estimated from far too little data to be reliable. Topological risk estimates only $n$ quantities, the per-asset risks $\Lambda_i$, because the risk matrix $Q = diag(\Lambda_i)$ is diagonal by construction. (Each $\Lambda_i$ is computed with fixed hyperparameters $d=3,\tau=1,\tilde T= 126, h=21,k=1,p=1,$ set by convention rather than tuned.)

The intuitive idea of the method is as follows. We map a one-dimensional time-series of returns into $\mathbb{R}^3$ space as data-points via Takens' delay embedding. Next we form a simplicial complex using the Vietoris-Rips method by connecting any two data-points that fall within a radius. The radius starts at $0$, in which case all points are disconnected (unless they are located at the same place, as with constant returns). We continuously increase the radius, and whenever two points fall within it we connect them via an edge. As the radius increases new connections are made, and we are interested in how well the overall shape is connected through the varying radius.

It may happen that at some radius $r_i$ we end up with $n$ distinct point-clouds not connected to each other. This is precisely what the $0$-th homology group $H_0$ measures: the number of distinct components of the space. In our case we start with distinct data-points at radius $0$, then as the radius increases new connections are formed and some points merge. This decreases the total amount of distinct point-clouds, which were just data-points. We continue increasing the radius and get a sequence of merges, each recorded as a (birth, death) pair, where death is the radius at which two components merge. A tightly clustered cloud merges quickly at small radii; a scattered cloud may have components surviving to large radii. We summarise the topology of our space via a persistence landscape (a sequence of functions $\eta_k$, $k = 1, 2, \ldots$ indexed by the filtration radius, where $\eta_k$ returns the $k$-th largest tent value at that radius).

From a topological point of view, more scattering of the point cloud means less stable returns; a more concentrated cloud, more stable. The amount of scatteredness among return observations over time can be quantified using the $L_1$ norm of the persistence landscape. Therefore this norm can effectively track changes in the state of stock return dynamics without any prior distributional assumptions. Lastly, the space of persistence landscapes forms a Banach space, so the theory of random variables can be applied to it. Given a persistence diagram, the landscape involves no parameter and is thus free from parameter tuning and over-fitting risk.[1]

[1] Bubenik, P., et al. Statistical topological data analysis using persistence landscapes. J. Mach. Learn. Res. 16(1), 77–102 (2015).

What this project covers

Implementation and verification of the method, plus characterisation of the risk measure. Portfolio performance is not evaluated here; testing the paper's performance claim requires their 462-constituent universe; a 19-name basket cannot settle it.

How Λ relates to volatility. Across 19 US large-cap tickers (18 stocks plus the SPY ETF, 2018–2023), Λ ranked assets similarly but not identically to annualised volatility (Spearman ρ ≈ 0.73). On a wider 20-instrument cross-section spanning FX, bonds, commodities, indices and crypto (2020–2026), the agreement is much stronger (ρ ≈ 0.95), suggesting the lower correlation within equities reflects the narrow volatility range of that sample rather than genuinely independent information.

Install & run

git clone https://github.com/gitiann/topological-portfolio-analysis
cd topological-portfolio-analysis
python -m venv .venv && source .venv/bin/activate
pip install -e ".[dev,qp]"
python scripts/make_sample_data.py
pytest
toporisk build --prices data/sample_prices.csv --out figures/weights.png

Example: testing Λ against volatility on arbitrary markets

Fetch a cross-asset basket (FX, bonds, commodities, indices, crypto) and compare topological risk against annualised volatility. Requires yfinance:

pip install yfinance
bash scripts/example_cross_asset.sh 2020-01-01 2026-08-07

Downloads into data/cross_asset/ and prints Λ and annualised volatility per instrument with their ranks, plus the Spearman rank correlation between the two. Edit the ticker list in the script to use your own basket.

Quote tickers containing = or ^ so the shell doesn't expand them. Do not glob data/*.csv — that would include sample_prices.csv, which is a wide multi-asset panel rather than a single series, and it will be misread as one asset.

Λ is dimensionless, since simple returns carry no units and so neither do distances in the embedding, which is what makes it comparable across instruments with entirely different price scales.

The model

For each asset independently: take its simple-return series, cut it into overlapping sub-windows, Takens-embed each sub-window into a point cloud, compute the H0 (0th homology group, connected components) persistence landscape of each cloud, and measure how much the landscape's L¹ norm varies across sub-windows. That variability is the asset's topological risk Λᵢ. Assets are then coupled only through a diagonal risk matrix Q = diag(Λᵢ) and a long-only quadratic program that minimises wᵀQw. There is no cross-asset topology in this model, by design (the paper flags cross-asset coupling as future work).

Fixed parameters, per the paper (verify against your copy): simple returns; d = 3, τ = 1; sub-window length T̃ = 126, shift h = 21; $H_0$ only; landscape layer k = 1; norm order p = 1.

Layout

src/toporisk/
   data.py        load prices, simple returns                 
   embedding.py   sub_windows, takens_embedding               
   topology.py    Gudhi H0 diagram, finite_pairs              
   landscape.py   mean_landscape, persistence_landscape, lp_norm   
   risk.py        risk_matrix, asset_topological_risk         
   portfolio.py   min_topological_risk_portfolio              
   cli.py         toporisk build ...                       
   viz.py         weight + landscape plots                    
tests/           pytest (34 passing)
scripts/         make_sample_data.py, fetch_prices.py, compare_risk_vs_vol.py, example_cross_asset.sh

Implementation notes

The pipeline is built and tested end to end. Key functions:

1. embedding.takens_embedding — length-T̃ series, d=3, τ=1 → T̃-2 points in $\mathbb{R}^3$.
2. landscape.persistence_landscape — tent functions sampled on a fixed shared grid, so landscapes from different sub-windows can be averaged pointwise.
3. landscape.lp_norm — discrete Lᵖ norm of a sampled landscape (p=1), grid-spacing aware.
4. risk.asset_topological_risk — composes 1–3 over the sub-windows. The reference term is ‖mean landscape‖. At p=1 this coincides with mean(‖landscape‖) since the L¹ norm is linear on non-negative functions, but the distinction matters if p is changed.
5. portfolio.min_topological_risk_portfolio — the convex QP; for diagonal Q it reduces to inverse-risk weighting, with zero-risk assets excluded.
6. Landscape convention: this implementation uses Bubenik's tent, peak (d−b)/2. Gudhi's representations.Landscape measures perpendicular distance to the diagonal, peak (d−b)/√2. The two agree up to that factor (verified in test_landscape_matches_gudhi), which scales Λ by 2 and leaves portfolio weights unchanged, since the QP's argmin is scale-invariant.