MMV with non-negative norms

In this tutorial we will use the spg_mmv solver, which solves a multi-measurement vector basis pursuit denoise problem. Both the standard L12 and L12non-negative norms will be compared. The latter set of norms is used if we want our solution to have only positive values in case we have access to such a kind of prior information.

import matplotlib.pyplot as plt
import numpy as np

import spgl1

Let’s first import our data, matrix operator and weights

data = np.load("../testdata/mmvnn.npz")

A = data["A"]
B = data["B"]
weights = data["weights"]

We apply unweighted inversions with and without non-negative norms.

X, _, _, info = spgl1.spg_mmv(A, B, 0.5, iter_lim=100, verbosity=0)

XNN, _, _, infoNN = spgl1.spg_mmv(
    A,
    B,
    0.5,
    iter_lim=100,
    verbosity=0,
    project=spgl1.norm_l12nn_project,
    primal_norm=spgl1.norm_l12nn_primal,
    dual_norm=spgl1.norm_l12nn_dual,
)
print("Negative X - MMV:", np.any(X < 0))
print("Negative X - MMVNN:", np.any(XNN < 0))
print("Residual norm - MMV:", info["rnorm"])
print("Residual norm - MMVNN:", infoNN["rnorm"])

plt.figure()
plt.plot(X[:, 0], "k", label="Coefficients")
plt.plot(XNN[:, 0], "--r", label="Coefficients NN")
plt.legend()
plt.title("Unweighted Basis Pursuit with Multiple Measurement Vectors")

plt.figure()
plt.plot(X[:, 1], "k", label="Coefficients")
plt.plot(XNN[:, 1], "--r", label="Coefficients NN")
plt.legend()
plt.title("Unweighted Basis Pursuit with Multiple Measurement Vectors")

• 
• 

/home/docs/checkouts/readthedocs.org/user_builds/spgl1/checkouts/latest/spgl1/spgl1.py:363: RuntimeWarning: invalid value encountered in divide
  xc = xc / xa
Negative X - MMV: True
Negative X - MMVNN: False
Residual norm - MMV: 1.3845145951758175
Residual norm - MMVNN: 1.8882592382476573

Text(0.5, 1.0, 'Unweighted Basis Pursuit with Multiple Measurement Vectors')

We repeat the same steps with weighted norms.

X, _, _, info = spgl1.spg_mmv(
    A, B, 0.5, iter_lim=100, weights=np.array(weights), verbosity=0
)
XNN, _, _, infoNN = spgl1.spg_mmv(
    A,
    B,
    0.5,
    iter_lim=100,
    verbosity=0,
    weights=np.array(weights),
    project=spgl1.norm_l12nn_project,
    primal_norm=spgl1.norm_l12nn_primal,
    dual_norm=spgl1.norm_l12nn_dual,
)
print("Negative X - MMV:", np.any(X < 0))
print("Negative X - MMVNN:", np.any(XNN < 0))
print("Residual norm - MMV:", info["rnorm"])
print("Residual norm - MMVNN:", infoNN["rnorm"])

plt.figure()
plt.plot(X[:, 0], "k", label="Coefficients")
plt.plot(XNN[:, 0], "--r", label="Coefficients NN")
plt.legend()
plt.title("Weighted Basis Pursuit with Multiple Measurement Vectors")

plt.figure()
plt.plot(X[:, 1], "k", label="Coefficients")
plt.plot(XNN[:, 1], "--r", label="Coefficients NN")
plt.legend()
plt.title("Weighted Basis Pursuit with Multiple Measurement Vectors")

• 
• 

Negative X - MMV: True
Negative X - MMVNN: False
Residual norm - MMV: 1.5813724036787324
Residual norm - MMVNN: 1.8187503062325858

Text(0.5, 1.0, 'Weighted Basis Pursuit with Multiple Measurement Vectors')