Source code for cvxpy.atoms.pnorm

Copyright 2013 Steven Diamond

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from cvxpy.atoms.axis_atom import AxisAtom
from cvxpy.atoms.norm1 import norm1
from cvxpy.atoms.norm_inf import norm_inf
import numpy as np
import scipy.sparse as sp
from cvxpy.utilities.power_tools import pow_high, pow_mid, pow_neg

[docs]def pnorm(x, p=2, axis=None, keepdims=False, max_denom=1024): """Factory function for a mathematical p-norm. Parameters ---------- p : numeric type or string The type of norm to construct; set this to np.inf or 'inf' to construct an infinity norm. Returns ------- Atom A norm1, norm_inf, or Pnorm object. """ if p == 1: return norm1(x, axis=axis, keepdims=keepdims) elif p in [np.inf, 'inf', 'Inf']: return norm_inf(x, axis=axis, keepdims=keepdims) else: return Pnorm(x, p=p, axis=axis, keepdims=keepdims, max_denom=max_denom)
[docs]class Pnorm(AxisAtom): r"""The vector p-norm, for p not equal to 1 or infinity. If given a matrix variable, ``pnorm`` will treat it as a vector, and compute the p-norm of the concatenated columns. Only accepts p values that are not equal to 1 or infinity; the norm1 and norm_inf classes handle those norms. For :math:`p > 1`, the p-norm is given by .. math:: \|x\|_p = \left(\sum_i |x_i|^p \right)^{1/p}, with domain :math:`x \in \mathbf{R}^n`. For :math:`p < 1,\ p \neq 0`, the p-norm is given by .. math:: \|x\|_p = \left(\sum_i x_i^p \right)^{1/p}, with domain :math:`x \in \mathbf{R}^n_+`. - Note that the "p-norm" is actually a **norm** only when :math:`p > 1`. For these cases, it is convex. - The expression is not defined when :math:`p = 0`. - Otherwise, when :math:`p < 1`, the expression is concave, but it is not a true norm. .. note:: Generally, ``p`` cannot be represented exactly, so a rational, i.e., fractional, **approximation** must be made. Internally, ``pnorm`` computes a rational approximation to the reciprocal :math:`1/p` with a denominator up to ``max_denom``. The resulting approximation can be found through the attribute ``pnorm.p``. The approximation error is given by the attribute ``pnorm.approx_error``. Increasing ``max_denom`` can give better approximations. When ``p`` is an ``int`` or ``Fraction`` object, the approximation is usually **exact**. Parameters ---------- x : cvxpy.Variable The value to take the norm of. p : int, float, or Fraction We require that :math:`p > 1`, but :math:`p \neq \infty`. See the norm1 and norm_inf classes for these norms, or use the pnorm function wrapper to instantiate them. max_denom : int The maximum denominator considered in forming a rational approximation for ``p``. axis : 0 or 1 The axis to apply the norm to. Returns ------- Expression An Expression representing the norm. """ _allow_complex = True def __init__(self, x, p=2, axis=None, keepdims=False, max_denom=1024): if p < 0: # TODO(akshayka): Why do we accept p < 0? self.p, _ = pow_neg(p, max_denom) elif 0 < p < 1: self.p, _ = pow_mid(p, max_denom) elif p > 1: self.p, _ = pow_high(p, max_denom) elif p == 1: raise ValueError('Use the norm1 class to instantiate a one norm.') elif p == 'inf' or p == 'Inf' or p == np.inf: raise ValueError('Use the norm_inf class to instantiate an ' 'infinity norm.') else: raise ValueError('Invalid p: {}'.format(p)) self.approx_error = float(abs(self.p - p)) self.original_p = p super(Pnorm, self).__init__(x, axis=axis, keepdims=keepdims) def numeric(self, values): """Returns the p-norm of x. """ if self.axis is None: values = np.array(values[0]).flatten() else: values = np.array(values[0]) if self.p < 1 and np.any(values < 0): return -np.inf if self.p < 0 and np.any(values == 0): return 0.0 return np.linalg.norm(values, float(self.p), axis=self.axis, keepdims=self.keepdims) def validate_arguments(self): super(Pnorm, self).validate_arguments() # TODO(akshayka): Why is axis not supported for other norms? if self.axis is not None and self.p != 2: raise ValueError( "The axis parameter is only supported for p=2.") if self.p < 1 and self.args[0].is_complex(): raise ValueError("pnorm(x, p) cannot have x complex for p < 1.") def sign_from_args(self): """Returns sign (is positive, is negative) of the expression. """ # Always positive. return (True, False) def is_atom_convex(self): """Is the atom convex? """ return self.p > 1 def is_atom_concave(self): """Is the atom concave? """ return self.p < 1 def is_atom_log_log_convex(self): """Is the atom log-log convex? """ return True def is_atom_log_log_concave(self): """Is the atom log-log concave? """ return False def is_incr(self, idx): """Is the composition non-decreasing in argument idx? """ return self.p < 1 or (self.p > 1 and self.args[0].is_nonneg()) def is_decr(self, idx): """Is the composition non-increasing in argument idx? """ return self.p > 1 and self.args[0].is_nonpos() def is_pwl(self): """Is the atom piecewise linear? """ return False def get_data(self): return [self.p, self.axis] def name(self): return "%s(%s, %s)" % (self.__class__.__name__, self.args[0].name(), self.p) def _domain(self): """Returns constraints describing the domain of the node. """ if self.p < 1 and self.p != 0: return [self.args[0] >= 0] else: return [] def _grad(self, values): """Gives the (sub/super)gradient of the atom w.r.t. each argument. Matrix expressions are vectorized, so the gradient is a matrix. Args: values: A list of numeric values for the arguments. Returns: A list of SciPy CSC sparse matrices or None. """ return self._axis_grad(values) def _column_grad(self, value): """Gives the (sub/super)gradient of the atom w.r.t. a column argument. Matrix expressions are vectorized, so the gradient is a matrix. Args: value: A numeric value for a column. Returns: A NumPy ndarray or None. """ rows = self.args[0].size # Outside domain. if self.p < 1 and np.any(value <= 0): return None D_null = sp.csc_matrix((rows, 1), dtype='float64') denominator = np.linalg.norm(value, float(self.p)) denominator = np.power(denominator, self.p - 1) # Subgrad is 0 when denom is 0 (or undefined). if denominator == 0: if self.p > 1: return D_null else: return None else: nominator = np.power(value, self.p - 1) frac = np.divide(nominator, denominator) return np.reshape(frac, (frac.size, 1))