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# Sparse coding with a precomputed dictionary

Transform a signal as a sparse combination of Ricker wavelets. This example visually compares different sparse coding methods using the `SparseCoder` estimator. The Ricker (also known as Mexican hat or the second derivative of a Gaussian) is not a particularly good kernel to represent piecewise constant signals like this one. It can therefore be seen how much adding different widths of atoms matters and it therefore motivates learning the dictionary to best fit your type of signals.

The richer dictionary on the right is not larger in size, heavier subsampling is performed in order to stay on the same order of magnitude.

    import matplotlib.pyplot as plt
    import numpy as np

    from sklearn.decomposition import SparseCoder

    def ricker_function(resolution, center, width):
        """Discrete sub-sampled Ricker (Mexican hat) wavelet"""
        x = np.linspace(0, resolution - 1, resolution)
        x = (
            (2 / (np.sqrt(3 * width) * np.pi**0.25))
            * (1 - (x - center) ** 2 / width**2)
            * np.exp(-((x - center) ** 2) / (2 * width**2))
        )
        return x

    def ricker_matrix(width, resolution, n_components):
        """Dictionary of Ricker (Mexican hat) wavelets"""
        centers = np.linspace(0, resolution - 1, n_components)
        D = np.empty((n_components, resolution))
        for i, center in enumerate(centers):
            D[i] = ricker_function(resolution, center, width)
        D /= np.sqrt(np.sum(D**2, axis=1))[:, np.newaxis]
        return D

    resolution = 1024
    subsampling = 3  # subsampling factor
    width = 100
    n_components = resolution // subsampling

    # Compute a wavelet dictionary
    D_fixed = ricker_matrix(width=width, resolution=resolution, n_components=n_components)
    D_multi = np.r_[
        tuple(
            ricker_matrix(width=w, resolution=resolution, n_components=n_components // 5)
            for w in (10, 50, 100, 500, 1000)
        )
    ]

    # Generate a signal
    y = np.linspace(0, resolution - 1, resolution)
    first_quarter = y < resolution / 4
    y[first_quarter] = 3.0
    y[np.logical_not(first_quarter)] = -1.0

    # List the different sparse coding methods in the following format:
    # (title, transform_algorithm, transform_alpha,
    #  transform_n_nozero_coefs, color)
    estimators = [
        ("OMP", "omp", None, 15, "navy"),
        ("Lasso", "lasso_lars", 2, None, "turquoise"),
    ]
    lw = 2

    plt.figure(figsize=(13, 6))
    for subplot, (D, title) in enumerate(
        zip((D_fixed, D_multi), ("fixed width", "multiple widths"))
    ):
        plt.subplot(1, 2, subplot + 1)
        plt.title("Sparse coding against %s dictionary" % title)
        plt.plot(y, lw=lw, linestyle="--", label="Original signal")
        # Do a wavelet approximation
        for title, algo, alpha, n_nonzero, color in estimators:
            coder = SparseCoder(
                dictionary=D,
                transform_n_nonzero_coefs=n_nonzero,
                transform_alpha=alpha,
                transform_algorithm=algo,
            )
            x = coder.transform(y.reshape(1, -1))
            density = len(np.flatnonzero(x))
            x = np.ravel(np.dot(x, D))
            squared_error = np.sum((y - x) ** 2)
            plt.plot(
                x,
                color=color,
                lw=lw,
                label="%s: %s nonzero coefs,\n%.2f error" % (title, density, squared_error),
            )

        # Soft thresholding debiasing
        coder = SparseCoder(
            dictionary=D, transform_algorithm="threshold", transform_alpha=20
        )
        x = coder.transform(y.reshape(1, -1))
        _, idx = np.where(x != 0)
        x[0, idx], _, _, _ = np.linalg.lstsq(D[idx, :].T, y, rcond=None)
        x = np.ravel(np.dot(x, D))
        squared_error = np.sum((y - x) ** 2)
        plt.plot(
            x,
            color="darkorange",
            lw=lw,
            label="Thresholding w/ debiasing:\n%d nonzero coefs, %.2f error"
            % (len(idx), squared_error),
        )
        plt.axis("tight")
        plt.legend(shadow=False, loc="best")
    plt.subplots_adjust(0.04, 0.07, 0.97, 0.90, 0.09, 0.2)
    plt.show()

Total running time of the script: (0 minutes 0.216 seconds)

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