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1 # -*- coding: utf-8 -*-
2 """
3 The :mod:`parsimony.functions.nesterov.properties` module contains the
4 necessary properties for Nesterov functions.
5
6 Created on Mon Feb 3 10:51:33 2014
7
8 Copyright (c) 2013-2014, CEA/DSV/I2BM/Neurospin. All rights reserved.
9
10 @author: Tommy Löfstedt
11 @email: lofstedt.tommy@gmail.com
12 @license: BSD 3-clause.
13 """
14 #import abc
15 #
16 #import numpy as np
17 #
18 #import parsimony.utils.consts as consts
19 #
20 #__all__ = ["NesterovFunction"]
21 #
22 #
23 #class NesterovFunction(object):
24 # """Abstract superclass of Nesterov functions.
25 #
26 # Attributes:
27 # ----------
28 # l : Non-negative float. The Lagrange multiplier, or regularisation
29 # constant, of the function.
30 #
31 # mu : Non-negative float. The Nesterov function regularisation constant for
32 # the smoothing.
33 #
34 # penalty_start : Non-negative integer. The number of columns, variables
35 # etc., to except from penalisation. Equivalently, the first index
36 # to be penalised. Default is 0, all columns are included.
37 # """
38 # __metaclass__ = abc.ABCMeta
39 #
40 # def __init__(self, l, A=None, mu=consts.TOLERANCE, penalty_start=0):
41 # """
42 # Parameters
43 # ----------
44 # l : Non-negative float. The Lagrange multiplier, or regularisation
45 # constant, of the function.
46 #
47 # A : A (usually sparse) array. The linear operator for the Nesterov
48 # formulation. May not be None!
49 #
50 # mu: Non-negative float. The regularisation constant for the smoothing.
51 #
52 # penalty_start : Non-negative integer. The number of columns, variables
53 # etc., to except from penalisation. Equivalently, the first
54 # index to be penalised. Default is 0, all columns are included.
55 # """
56 # self.l = float(l)
57 # if A is None:
58 # raise ValueError("The linear operator A must not be None.")
59 # self._A = A
60 # self.mu = float(mu)
61 # self.penalty_start = int(penalty_start)
62 #
63 # def fmu(self, beta, mu=None):
64 # """Returns the smoothed function value.
65 #
66 # Parameters
67 # ----------
68 # beta : Numpy array. A weight vector.
69 #
70 # mu : Non-negative float. The regularisation constant for the smoothing.
71 # """
72 # if mu is None:
73 # mu = self.get_mu()
74 #
75 # alpha = self.alpha(beta)
76 # alpha_sqsum = 0.0
77 # for a in alpha:
78 # alpha_sqsum += np.sum(a ** 2.0)
79 #
80 # Aa = self.Aa(alpha)
81 #
82 # if self.penalty_start > 0:
83 # beta_ = beta[self.penalty_start:, :]
84 # else:
85 # beta_ = beta
86 #
87 # return self.l * (np.dot(beta_.T, Aa)[0, 0] - (mu / 2.0) * alpha_sqsum)
88 #
89 # @abc.abstractmethod
90 # def phi(self, alpha, beta):
91 # """ Function value with known alpha.
92 # """
93 # raise NotImplementedError('Abstract method "phi" must be '
94 # 'specialised!')
95 #
96 # def grad(self, beta):
97 # """ Gradient of the function at beta.
98 #
99 # Parameters
100 # ----------
101 # beta : Numpy array. The point at which to evaluate the gradient.
102 # """
103 # if self.l < consts.TOLERANCE:
104 # return 0.0
105 #
106 # # \beta need not be sliced here.
107 # alpha = self.alpha(beta)
108 #
109 # if self.penalty_start > 0:
110 # grad = self.l * np.vstack((np.zeros((self.penalty_start, 1)),
111 # self.Aa(alpha)))
112 # else:
113 # grad = self.l * self.Aa(alpha)
114 #
115 ## approx_grad = utils.approx_grad(self.f, beta, eps=1e-6)
116 ## print "NesterovFunction:", maths.norm(grad - approx_grad)
117 #
118 # return grad
119 #
120 # def get_mu(self):
121 # """Return the regularisation constant for the smoothing.
122 # """
123 # return self.mu
124 #
125 # def set_mu(self, mu):
126 # """Set the regularisation constant for the smoothing.
127 #
128 # Parameters
129 # ----------
130 # mu : Non-negative float. The regularisation constant for the smoothing
131 # to use from now on.
132 #
133 # Returns
134 # -------
135 # old_mu : Non-negative float. The old regularisation constant for the
136 # smoothing that was overwritten and no longer is used.
137 # """
138 # old_mu = self.get_mu()
139 #
140 # self.mu = mu
141 #
142 # return old_mu
143 #
144 # def alpha(self, beta):
145 # """ Dual variable of the Nesterov function.
146 #
147 # Parameters
148 # ----------
149 # beta : Numpy array (p-by-1). The variable for which to compute the dual
150 # variable alpha.
151 # """
152 # if self.penalty_start > 0:
153 # beta_ = beta[self.penalty_start:, :]
154 # else:
155 # beta_ = beta
156 #
157 # A = self.A()
158 # mu = self.get_mu()
159 # alpha = [0] * len(A)
160 # for i in xrange(len(A)):
161 # alpha[i] = A[i].dot(beta_) / mu
162 #
163 # # Apply projection
164 # alpha = self.project(alpha)
165 #
166 # return alpha
167 #
168 # def A(self):
169 # """ Linear operator of the Nesterov function.
170 # """
171 # return self._A
172 #
173 # def Aa(self, alpha):
174 # """ Compute A'*alpha.
175 #
176 # Parameters
177 # ----------
178 # alpha : Numpy array (x-by-1). The dual variable alpha.
179 # """
180 # A = self.A()
181 # Aa = A[0].T.dot(alpha[0])
182 # for i in xrange(1, len(A)):
183 # Aa += A[i].T.dot(alpha[i])
184 #
185 # return Aa
186 #
187 # @abc.abstractmethod
188 # def project(self, alpha):
189 # """ Projection onto the compact space of the Nesterov function.
190 #
191 # Parameters
192 # ----------
193 # alpha : Numpy array (x-by-1). The not-yet-projected dual variable
194 # alpha.
195 # """
196 # raise NotImplementedError('Abstract method "project" must be '
197 # 'specialised!')
198 #
199 # @abc.abstractmethod
200 # def M(self):
201 # """ The maximum value of the regularisation of the dual variable. We
202 # have
203 #
204 # M = max_{alpha in K} 0.5*|alpha|²_2.
205 # """
206 # raise NotImplementedError('Abstract method "M" must be '
207 # 'specialised!')
208 #
209 # @abc.abstractmethod
210 # def estimate_mu(self, beta):
211 # """ Compute a "good" value of mu with respect to the given beta.
212 #
213 # Parameters
214 # ----------
215 # beta : Numpy array (p-by-1). The primal variable at which to compute a
216 # feasible value of mu.
217 # """
218 # raise NotImplementedError('Abstract method "estimate_mu" must be '
219 # 'specialised!')
220
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