Note

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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.

Sparse coding against fixed width dictionary, Sparse coding against multiple widths dictionary

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()