Finalized doc of Log1 and added test

Author: mrava87

pyproximal/proximal/Log.py

@@ -100,21 +100,22 @@ def prox(self, x, tau):
         out *= np.sign(x)
         return out
 
+
 class Log1(ProxOperator):
-    r"""Logarithmic penalty.
+    r"""Logarithmic penalty 2.
 
     The logarithmic penalty (Log) is defined as
 
     .. math::
 
-        \mathrm{Log}_{\sigma,\delta}(\mathbf{x}) = \sum_i \log(|x_i| + \delta)
+        \mathrm{Log}_{\sigma,\delta}(\mathbf{x}) = \sigma \sum_i \log(|x_i| + \delta)
 
     where :math:`{\sigma>0}`, :math:`{\gamma>0}`.
 
     Parameters
     ----------
     sigma : :obj:`float`
-        Regularization parameter.
+        Multiplicative coefficient of Log norm.
     delta : :obj:`float`, optional
         Regularization parameter. Default is 1e-10.
 
@@ -129,9 +130,9 @@ class Log1(ProxOperator):
 
         \prox_{\tau \sigma log}(\mathbf{x}) =
         \begin{cases}
-        0.5 (x_i - \delta + \sqrt{(x_i+\delta)^2-2\tau \sigma}), & x_i - g_i < -x_0 \\
-        0, & -x_0 \leq x_i - g_i x_0 \\
-        0.5 (x_i + \delta - \sqrt{(x_i-\delta)^2-2\tau \sigma}),  & x_i - g_i > x_0\\
+        0.5 (x_i + \delta - \sqrt{(x_i-\delta)^2-2\tau \sigma}), & x_i < -x_0 \\
+        0, & -x_0 \leq x_i \leq  x_0 \\
+        0.5 (x_i - \delta + \sqrt{(x_i+\delta)^2-2\tau \sigma}), & x_i  > x_0\\
         \end{cases}
 
     where :math:`x_0=\sqrt{2 \tau \sigma} - \delta`.

pytests/test_concave_penalties.py

@@ -3,7 +3,7 @@
 import numpy as np
 
 from pyproximal.utils import moreau
-from pyproximal.proximal import ETP, Geman, Log, SCAD, QuadraticEnvelopeCard
+from pyproximal.proximal import ETP, Geman, Log, Log1, SCAD, QuadraticEnvelopeCard
 
 par1 = {'nx': 10, 'sigma': 1., 'a': 2.1, 'gamma': 0.5, 'mu': 0.5, 'dtype': 'float32'}  # even float32
 par2 = {'nx': 11, 'sigma': 2., 'a': 3.7, 'gamma': 5.0, 'mu': 1.5, 'dtype': 'float64'}  # odd float64
@@ -49,6 +49,19 @@ def test_Log(par):
     assert moreau(log, x, tau)
 
 
+@pytest.mark.parametrize("par", [(par1), (par2)])
+def test_Log1(par):
+    """Log1 penalty and proximal/dual proximal
+    """
+    np.random.seed(10)
+    log = Log1(sigma=par['sigma'])
+
+    # Check proximal operator
+    tau = 2.
+    x = np.random.normal(0., 10.0, par['nx']).astype(par['dtype'])
+    assert moreau(log, x, tau)
+
+
 @pytest.mark.parametrize("par", [(par1), (par2)])
 def test_ETP(par):
     """Exponential-type penalty and proximal/dual proximal