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# Lasso on dense and sparse data

We show that linear_model.Lasso provides the same results for dense and sparse data and that in the case of sparse data the speed is improved.

    from time import time

    from scipy import linalg, sparse

    from sklearn.datasets import make_regression
    from sklearn.linear_model import Lasso

## Comparing the two Lasso implementations on Dense data

We create a linear regression problem that is suitable for the Lasso, that is to say, with more features than samples. We then store the data matrix in both dense (the usual) and sparse format, and train a Lasso on each. We compute the runtime of both and check that they learned the same model by computing the Euclidean norm of the difference between the coefficients they learned. Because the data is dense, we expect better runtime with a dense data format.

    X, y = make_regression(n_samples=200, n_features=5000, random_state=0)
    # create a copy of X in sparse format
    X_sp = sparse.coo_matrix(X)

    alpha = 1
    sparse_lasso = Lasso(alpha=alpha, fit_intercept=False, max_iter=1000)
    dense_lasso = Lasso(alpha=alpha, fit_intercept=False, max_iter=1000)

    t0 = time()
    sparse_lasso.fit(X_sp, y)
    print(f"Sparse Lasso done in {(time() - t0):.3f}s")

    t0 = time()
    dense_lasso.fit(X, y)
    print(f"Dense Lasso done in {(time() - t0):.3f}s")

    # compare the regression coefficients
    coeff_diff = linalg.norm(sparse_lasso.coef_ - dense_lasso.coef_)
    print(f"Distance between coefficients : {coeff_diff:.2e}")

    #

    Sparse Lasso done in 0.080s
    Dense Lasso done in 0.022s
    Distance between coefficients : 1.30e-13

## Comparing the two Lasso implementations on Sparse data

We make the previous problem sparse by replacing all small values with 0 and run the same comparisons as above. Because the data is now sparse, we expect the implementation that uses the sparse data format to be faster.

    # make a copy of the previous data
    Xs = X.copy()
    # make Xs sparse by replacing the values lower than 2.5 with 0s
    Xs[Xs < 2.5] = 0.0
    # create a copy of Xs in sparse format
    Xs_sp = sparse.coo_matrix(Xs)
    Xs_sp = Xs_sp.tocsc()

    # compute the proportion of non-zero coefficient in the data matrix
    print(f"Matrix density : {(Xs_sp.nnz / float(X.size) * 100):.3f}%")

    alpha = 0.1
    sparse_lasso = Lasso(alpha=alpha, fit_intercept=False, max_iter=10000)
    dense_lasso = Lasso(alpha=alpha, fit_intercept=False, max_iter=10000)

    t0 = time()
    sparse_lasso.fit(Xs_sp, y)
    print(f"Sparse Lasso done in {(time() - t0):.3f}s")

    t0 = time()
    dense_lasso.fit(Xs, y)
    print(f"Dense Lasso done in  {(time() - t0):.3f}s")

    # compare the regression coefficients
    coeff_diff = linalg.norm(sparse_lasso.coef_ - dense_lasso.coef_)
    print(f"Distance between coefficients : {coeff_diff:.2e}")

    Matrix density : 0.626%
    Sparse Lasso done in 0.091s
    Dense Lasso done in  0.400s
    Distance between coefficients : 8.56e-12

Total running time of the script: (0 minutes 0.645 seconds)

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