Fig. 4: Portfolio optimization with low rank approximation.

a Frequency histogram of eigenvalues obtained from the covariance matrix of S&P 500 stock data. There are only a few dominating eigenvalues, and most eigenvalues are orders of magnitude smaller than the dominant ones. b Equal-weighted cardinality-constrained portfolios constructed from the full rank covariance matrix S (blue), K = 20 low rank matrix \({{{{\bf{S}}}}}^{{\prime} }\) (orange), and K = 5 low rank matrix (green). The cumulative return percentage of each portfolio is calculated as \(\frac{100}{V(0)}{\sum }_{t = 1}^{T}{{{\bf{w}}}}\cdot {{{\bf{r}}}}(t)\), where V(0) is the initial value of the portfolio at t = 0. Here, λ = 0.5, η = 1, and q = 20. The portfolios were built by minimizing Eq. (21) using commercial solver Gurobi. Frequency data for each eigenvalue bin in (a) can be found in Supplementary Data 2 file. The time series data for the full rank, K = 5, K = 20 lines if (b) can be found in the Supplementary Data 3, 4, 5 files respectively.