Hi Alex, I'm not sure the scikit-learn code is doing the same thing. From what I understand, at every iteration the 'active' portion of the covariance matrix is expanded by adding an extra active feature, and the Cholesky factorization is updated to reflect the new covariance. So if C = X.T @ X, then C = L @ L.T where X is [n, d]. At each iteration I think the LARS algorithm is adding a column to X, making it [n, d + 1] which then requires expanding the covariance matrix and the Cholesky factor. The Cholesky up/downdate (proposed in the mentioned PR) instead considers a rank-1 modification to matrix X: [n, n]. Take vector z: [1, n], the updated matrix is X' = X + z @ z.T. The updated matrix has the same size as the original, and so will its Cholesky factor, but the actual values will be different - and since this is a 'small' update to the original matrix it can be done in O(n^2) operations compared to O(n^3) for a full Cholesky factorization. Do let me know if you have further comments about whether there is interest in merging this PR. Giacomo