def huber(x, M=1):
"""The Huber function
Huber(x, M) = 2M|x|-M^2 for |x| >= |M|
|x|^2 for |x| <= |M|
M defaults to 1.
Parameters
----------
x : Expression
A CVXPY expression.
M : int/float
"""
# TODO require that M is positive?
return square(M)*huber_pos(abs(x)/abs(M))
def huber(x, M=1):
"""The Huber function
Huber(x, M) = 2M|x|-M^2 for |x| >= |M|
|x|^2 for |x| <= |M|
M defaults to 1.
Parameters
----------
x : Expression
A CVXPY expression.
M : int/float
"""
# TODO require that M is positive?
return square(M) * huber_pos(abs(x) / abs(M))
def norm1_canon(expr, args):
x = args[0]
axis = expr.axis
# we need an absolute value constraint for the symmetric convex branches
# (p >= 1)
constraints = []
# TODO(akshayka): Express this more naturally (recursively), in terms
# of the other atoms
abs_expr = abs(x)
abs_x, abs_constraints = abs_canon(abs_expr, abs_expr.args)
constraints += abs_constraints
return sum(abs_x, axis=axis), constraints
def pnorm_canon(expr, args):
x = args[0]
p = expr.p
axis = expr.axis
shape = expr.shape
t = Variable(shape)
if p == 2:
if axis is None:
assert shape == tuple()
return t, [SOC(t, vec(x))]
else:
return t, [SOC(vec(t), x, axis)]
# we need an absolute value constraint for the symmetric convex branches
# (p > 1)
constraints = []
if p > 1:
# TODO(akshayka): Express this more naturally (recursively), in terms
# of the other atoms
abs_expr = abs(x)
abs_x, abs_constraints = abs_canon(abs_expr, abs_expr.args)
x = abs_x
constraints += abs_constraints
# now, we take care of the remaining convex and concave branches
# to create the rational powers, we need a new variable, r, and
# the constraint sum(r) == t
r = Variable(x.shape)
constraints += [sum(r) == t]
# todo: no need to run gm_constr to form the tree each time.
# we only need to form the tree once
promoted_t = Constant(np.ones(x.shape)) * t
p = Fraction(p)
if p < 0:
constraints += gm_constrs(promoted_t, [x, r],
(-p / (1 - p), 1 / (1 - p)))
if 0 < p < 1:
constraints += gm_constrs(r, [x, promoted_t], (p, 1 - p))
if p > 1:
constraints += gm_constrs(x, [r, promoted_t], (1 / p, 1 - 1 / p))
return t, constraints
def huber_canon(expr, args):
M = expr.M
x = args[0]
shape = expr.shape
n = Variable(shape)
s = Variable(shape)
# n**2 + 2*M*|s|
# TODO(akshayka): Make use of recursion inherent to canonicalization
# process and just return a power / abs expressions for readability sake
power_expr = power(n, 2)
n2, constr_sq = power_canon(power_expr, power_expr.args)
abs_expr = abs(s)
abs_s, constr_abs = abs_canon(abs_expr, abs_expr.args)
obj = n2 + 2 * M * abs_s
constraints = constr_sq + constr_abs
constraints.append(x == s + n)
return obj, constraints