def first_value(obj):
""" Return the first value
Parameters
----------
obj: dict-like object
"""
return six.next(six.itervalues(obj))
def first_key(obj):
""" Return the first key
Parameters
----------
obj: dict-like object
"""
return six.next(six.iterkeys(obj))
def first_value(obj):
""" Return the first value
Parameters
----------
obj: dict-like object
"""
return six.next(six.itervalues(obj))
def first_key(obj):
""" Return the first key
Parameters
----------
obj: dict-like object
"""
return six.next(six.iterkeys(obj))
def _feature_sign_search_single(dictionary, signal, sparsity, max_iter,
solution=None):
"""
Solve a single L1-penalized minimization problem with
feature-sign search.
Parameters
----------
dictionary : array_like, 2-dimensional
The dictionary of basis functions from which to form the
sparse linear combination.
signal : array_like, 1-dimensional
The signal being decomposed as a sparse linear combination
of the columns of the dictionary.
sparsity : float
The coefficient on the L1 penalty term of the cost function.
max_iter : int
The maximum number of iterations to run.
solution : ndarray, 1-dimensional, optional
Pre-allocated vector to use to store the solution.
Returns
-------
solution : ndarray, 1-dimensional
Vector containing the solution. If an array was passed in
as the argument `solution`, it will be updated in place
and the same object will be returned.
count : int
The number of iterations of the algorithm that were run.
Notes
-----
See the docstring of `feature_sign_search` for details on the
algorithm.
"""
# This prevents the sparsity penalty scalar from upcasting the entire
# rhs vector sent to linalg.solve().
sparsity = np.array(sparsity).astype(dictionary.dtype)
effective_zero = 1e-18
# precompute matrices for speed.
gram_matrix = np.dot(dictionary.T, dictionary)
target_correlation = np.dot(dictionary.T, signal)
# initialization goes here.
if solution is None:
solution = np.zeros(gram_matrix.shape[0], dtype=dictionary.dtype)
else:
assert solution.ndim == 1, "solution must be 1-dimensional"
assert solution.shape[0] == dictionary.shape[1], (
"solution.shape[0] does not match dictionary.shape[1]"
)
# Initialize all elements to be zero.
solution[...] = 0.
signs = np.zeros(gram_matrix.shape[0], dtype=np.int8)
active_set = set()
z_opt = np.inf
# Used to store whether max(abs(grad[nzidx] + sparsity * signs[nzidx]))
# is approximately 0.
# Set to True here to trigger a new feature activation on first iteration.
nz_optimal = True
# second term is zero on initialization.
grad = - 2 * target_correlation # + 2 * np.dot(gram_matrix, solution)
max_grad_zero = np.argmax(np.abs(grad))
# Just used to compute exact cost function.
sds = np.dot(signal.T, signal)
counter = count(0)
while z_opt > sparsity or not nz_optimal:
if six.next(counter) == max_iter:
break
if nz_optimal:
candidate = np.argmax(np.abs(grad) * (signs == 0))
log.debug("candidate feature: %d" % candidate)
if grad[candidate] > sparsity:
signs[candidate] = -1.
solution[candidate] = 0.
log.debug("added feature %d with negative sign" %
candidate)
active_set.add(candidate)
elif grad[candidate] < -sparsity:
signs[candidate] = 1.
solution[candidate] = 0.
log.debug("added feature %d with positive sign" %
candidate)
active_set.add(candidate)
if len(active_set) == 0:
break
else:
log.debug("Non-zero coefficient optimality not satisfied, "
"skipping new feature activation")
indices = np.array(sorted(active_set))
restr_gram = gram_matrix[np.ix_(indices, indices)]
restr_corr = target_correlation[indices]
restr_sign = signs[indices]
rhs = restr_corr - sparsity * restr_sign / 2
# TODO: implement footnote 3.
#
# If restr_gram becomes singular, check if rhs is in the column
# space of restr_gram.
#
# If so, use the pseudoinverse instead; if not, update to first
# zero-crossing along any direction in the null space of restr_gram
# such that it has non-zero dot product with rhs (how to choose
# this direction?).
new_solution = np.linalg.solve(np.atleast_2d(restr_gram), rhs)
new_signs = np.sign(new_solution)
restr_oldsol = solution[indices]
sign_flips = np.where(abs(new_signs - restr_sign) > 1)[0]
if len(sign_flips) > 0:
best_obj = np.inf
best_curr = None
best_curr = new_solution
best_obj = (sds + (np.dot(new_solution,
np.dot(restr_gram, new_solution))
- 2 * np.dot(new_solution, restr_corr))
+ sparsity * abs(new_solution).sum())
if log.isEnabledFor(logging.DEBUG):
# Extra computation only done if the log-level is
# sufficient.
ocost = (sds + (np.dot(restr_oldsol,
np.dot(restr_gram, restr_oldsol))
- 2 * np.dot(restr_oldsol, restr_corr))
+ sparsity * abs(restr_oldsol).sum())
cost = (sds + np.dot(new_solution,
np.dot(restr_gram, new_solution))
- 2 * np.dot(new_solution, restr_corr)
+ sparsity * abs(new_solution).sum())
log.debug("Cost before linesearch (old)\t: %e" % ocost)
log.debug("Cost before linesearch (new)\t: %e" % cost)
else:
ocost = None
for idx in sign_flips:
a = new_solution[idx]
b = restr_oldsol[idx]
prop = b / (b - a)
curr = restr_oldsol - prop * (restr_oldsol - new_solution)
cost = sds + (np.dot(curr, np.dot(restr_gram, curr))
- 2 * np.dot(curr, restr_corr)
+ sparsity * abs(curr).sum())
log.debug("Line search coefficient: %.5f cost = %e "
"zero-crossing coefficient's value = %e" %
(prop, cost, curr[idx]))
if cost < best_obj:
best_obj = cost
best_prop = prop
best_curr = curr
log.debug("Lowest cost after linesearch\t: %e" % best_obj)
if ocost is not None:
if ocost < best_obj and not np.allclose(ocost, best_obj):
log.debug("Warning: objective decreased from %e to %e" %
(ocost, best_obj))
else:
log.debug("No sign flips, not doing line search")
best_curr = new_solution
solution[indices] = best_curr
zeros = indices[np.abs(solution[indices]) < effective_zero]
solution[zeros] = 0.
signs[indices] = np.int8(np.sign(solution[indices]))
active_set.difference_update(zeros)
grad = - 2 * target_correlation + 2 * np.dot(gram_matrix, solution)
z_opt = np.max(abs(grad[signs == 0]))
nz_opt = np.max(abs(grad[signs != 0] + sparsity * signs[signs != 0]))
nz_optimal = np.allclose(nz_opt, 0)
return solution, min(six.next(counter), max_iter)
def _feature_sign_search_single(dictionary, signal, sparsity, max_iter, solution=None):
"""
Solve a single L1-penalized minimization problem with
feature-sign search.
Parameters
----------
dictionary : array_like, 2-dimensional
The dictionary of basis functions from which to form the
sparse linear combination.
signal : array_like, 1-dimensional
The signal being decomposed as a sparse linear combination
of the columns of the dictionary.
sparsity : float
The coefficient on the L1 penalty term of the cost function.
max_iter : int
The maximum number of iterations to run.
solution : ndarray, 1-dimensional, optional
Pre-allocated vector to use to store the solution.
Returns
-------
solution : ndarray, 1-dimensional
Vector containing the solution. If an array was passed in
as the argument `solution`, it will be updated in place
and the same object will be returned.
count : int
The number of iterations of the algorithm that were run.
Notes
-----
See the docstring of `feature_sign_search` for details on the
algorithm.
"""
# This prevents the sparsity penalty scalar from upcasting the entire
# rhs vector sent to linalg.solve().
sparsity = np.array(sparsity).astype(dictionary.dtype)
effective_zero = 1e-18
# precompute matrices for speed.
gram_matrix = np.dot(dictionary.T, dictionary)
target_correlation = np.dot(dictionary.T, signal)
# initialization goes here.
if solution is None:
solution = np.zeros(gram_matrix.shape[0], dtype=dictionary.dtype)
else:
assert solution.ndim == 1, "solution must be 1-dimensional"
assert solution.shape[0] == dictionary.shape[1], "solution.shape[0] does not match dictionary.shape[1]"
# Initialize all elements to be zero.
solution[...] = 0.0
signs = np.zeros(gram_matrix.shape[0], dtype=np.int8)
active_set = set()
z_opt = np.inf
# Used to store whether max(abs(grad[nzidx] + sparsity * signs[nzidx]))
# is approximately 0.
# Set to True here to trigger a new feature activation on first iteration.
nz_optimal = True
# second term is zero on initialization.
grad = -2 * target_correlation # + 2 * np.dot(gram_matrix, solution)
max_grad_zero = np.argmax(np.abs(grad))
# Just used to compute exact cost function.
sds = np.dot(signal.T, signal)
counter = count(0)
while z_opt > sparsity or not nz_optimal:
if six.next(counter) == max_iter:
break
if nz_optimal:
candidate = np.argmax(np.abs(grad) * (signs == 0))
log.debug("candidate feature: %d" % candidate)
if grad[candidate] > sparsity:
signs[candidate] = -1.0
solution[candidate] = 0.0
log.debug("added feature %d with negative sign" % candidate)
active_set.add(candidate)
elif grad[candidate] < -sparsity:
signs[candidate] = 1.0
solution[candidate] = 0.0
log.debug("added feature %d with positive sign" % candidate)
active_set.add(candidate)
if len(active_set) == 0:
break
else:
log.debug("Non-zero coefficient optimality not satisfied, " "skipping new feature activation")
indices = np.array(sorted(active_set))
restr_gram = gram_matrix[np.ix_(indices, indices)]
restr_corr = target_correlation[indices]
restr_sign = signs[indices]
rhs = restr_corr - sparsity * restr_sign / 2
# TODO: implement footnote 3.
#
# If restr_gram becomes singular, check if rhs is in the column
# space of restr_gram.
#
# If so, use the pseudoinverse instead; if not, update to first
# zero-crossing along any direction in the null space of restr_gram
# such that it has non-zero dot product with rhs (how to choose
# this direction?).
new_solution = np.linalg.solve(np.atleast_2d(restr_gram), rhs)
new_signs = np.sign(new_solution)
restr_oldsol = solution[indices]
sign_flips = np.where(abs(new_signs - restr_sign) > 1)[0]
if len(sign_flips) > 0:
best_obj = np.inf
best_curr = None
best_curr = new_solution
best_obj = (
sds
+ (np.dot(new_solution, np.dot(restr_gram, new_solution)) - 2 * np.dot(new_solution, restr_corr))
+ sparsity * abs(new_solution).sum()
)
if log.isEnabledFor(logging.DEBUG):
# Extra computation only done if the log-level is
# sufficient.
ocost = (
sds
+ (np.dot(restr_oldsol, np.dot(restr_gram, restr_oldsol)) - 2 * np.dot(restr_oldsol, restr_corr))
+ sparsity * abs(restr_oldsol).sum()
)
cost = (
sds
+ np.dot(new_solution, np.dot(restr_gram, new_solution))
- 2 * np.dot(new_solution, restr_corr)
+ sparsity * abs(new_solution).sum()
)
log.debug("Cost before linesearch (old)\t: %e" % ocost)
log.debug("Cost before linesearch (new)\t: %e" % cost)
else:
ocost = None
for idx in sign_flips:
a = new_solution[idx]
b = restr_oldsol[idx]
prop = b / (b - a)
curr = restr_oldsol - prop * (restr_oldsol - new_solution)
cost = sds + (
np.dot(curr, np.dot(restr_gram, curr)) - 2 * np.dot(curr, restr_corr) + sparsity * abs(curr).sum()
)
log.debug(
"Line search coefficient: %.5f cost = %e "
"zero-crossing coefficient's value = %e" % (prop, cost, curr[idx])
)
if cost < best_obj:
best_obj = cost
best_prop = prop
best_curr = curr
log.debug("Lowest cost after linesearch\t: %e" % best_obj)
if ocost is not None:
if ocost < best_obj and not np.allclose(ocost, best_obj):
log.debug("Warning: objective decreased from %e to %e" % (ocost, best_obj))
else:
log.debug("No sign flips, not doing line search")
best_curr = new_solution
solution[indices] = best_curr
zeros = indices[np.abs(solution[indices]) < effective_zero]
solution[zeros] = 0.0
signs[indices] = np.int8(np.sign(solution[indices]))
active_set.difference_update(zeros)
grad = -2 * target_correlation + 2 * np.dot(gram_matrix, solution)
z_opt = np.max(abs(grad[signs == 0]))
nz_opt = np.max(abs(grad[signs != 0] + sparsity * signs[signs != 0]))
nz_optimal = np.allclose(nz_opt, 0)
return solution, min(six.next(counter), max_iter)
