def calculate_Q(X: csc_matrix, update_in_place: bool = True) -> np.ndarray:
"""
:param X: a CSC matrix of shape (rows=features, cols=observations).
:param update_in_place: whether or not to return a new Numpy array or modify the input X.
:return: the word-word correlation matrix Q as a dense Numpy ndarray.
"""
n_features, n_observations = X.shape
diagonal = np.zeros(n_features)
if not update_in_place:
X = X.copy()
for col_idx in range(X.indptr.size - 1):
col_start = X.indptr[col_idx]
col_end = X.indptr[col_idx + 1]
col_entries = X.data[col_start:col_end]
col_sum = np.sum(col_entries)
row_indices = X.indices[col_start:col_end]
# TODO: figure out whether this loop and update by division can be written in more idiomatic Numpy
diagonal[row_indices] += col_entries / (col_sum * (col_sum - 1))
X.data[col_start:col_end] = col_entries / sqrt(col_sum * (col_sum - 1))
Q = X * X.T / n_observations
Q = np.array(Q.todense(), copy=False)
diagonal = diagonal / n_observations
Q = Q - np.diag(diagonal)
return Q
def adjacency2laplacian(adj: csc_matrix,
degree: csc_matrix = None,
mode: int = 0) -> csc_matrix:
"""
This function create a graph laplacian matrix from the adjacency matrix.
Parameters
----------
A (sparse matrix): Adjacency matrix.
D (Degree matrix): Optional, diagonal matrix containing the sum over the adjacency row.
mode (int): 0 Returns the standard graph Laplacian, L = D - A.
1 Returns the random walk normalized graph Laplacian, L = I - D^-1 * A.
2 Returns the symmetric normalized graph Laplacian, L = I - D^-0.5 * A D^-0.5.
Returns
-------
L (sparse matrix): graph Laplacian
"""
degree = adjacency2degree(adj) if degree is None else degree
if mode == 0: # standard graph Laplacian
return degree - adj
elif mode == 1: # random walk graph Laplacian
return eye(degree.shape[0], format='csc') - degree.power(-1) * adj
elif mode == 2: # symmetric normalized graph Laplacian
return eye(
degree.shape[0],
format='csc') - degree.power(-0.5) * adj * degree.power(-0.5)
else:
raise NotImplementedError
def csc_to_rmat(csc: sparse.csc_matrix):
csc.sort_indices()
t, conv_data, _ = get_type_conv(csc.dtype)
return methods.new(
f"{t}gCMatrix",
i=as_integer(csc.indices),
p=as_integer(csc.indptr),
x=conv_data(csc.data),
Dim=as_integer(list(csc.shape)),
)
def fit(self, X: csc_matrix, y):
print("Fit starts")
# X, X_, y, y_ = train_test_split(X, y, self.test_fraction)
e = self.predict(X) - y
q = X.dot(self.V)
n_samples, n_features = X.shape
X = X.tocsc()
for i in range(self.n_iter):
# self.evaluate(i, X_, y_)
# Global bias
w0_ = -(e - self.w0).sum() / (n_samples + self.lambda_w0)
e += w0_ - self.w0
self.w0 = w0_
# self.evaluate(i, X_, y_)
# 1-way interaction
for l in range(n_features):
Xl = X.getcol(l).toarray()
print(("\r Iteration #{} 1-way interaction "
"progress {:.2%}; train error {}").format(
i, l / n_features, err(e)),
end="")
w_ = -((e - self.w[l] * Xl) * Xl).sum() / (
np.power(Xl, 2).sum() + self.lambda_w)
e += (w_ - self.w[l]) * Xl
self.w[l] = w_
# self.evaluate(i, X_, y_)
# 2-way interaction
for f in range(self.latent_dimension):
Qf = q[:, f].reshape(-1, 1)
for l in range(n_features):
Xl = X.getcol(l)
idx = Xl.nonzero()[0]
Xl = Xl.data.reshape(-1, 1)
Vlf = self.V[l, f]
print((
"\r Iteration #{} 2-way interaction progress {:.2%};" +
"error {:.5}; validation_error NO").format(
i, (f * n_features + l) /
(self.latent_dimension * n_features), err(e)),
end="")
h = Xl * Qf[idx] - np.power(Xl, 2) * Vlf
v_ = -((e[idx] - Vlf * h) * h).sum() / (
np.power(h, 2).sum() + self.lambda_v)
e[idx] += (v_ - Vlf) * h
Qf[idx] += (v_ - Vlf) * Xl
self.V[l, f] = v_
q[:, f] = Qf.reshape(-1)
def SLIM(A: sp.csc_matrix, elanet: ElasticNet):
'''
SLIM: Sparse Linear Methods for Top-N Recommender Systems - Xia Ning ; George Karypis
https://ieeexplore.ieee.org/document/6137254
Run Sparse Linear Models over the rating matrix A.
It uses the ElasticNet object internally.
(code from https://github.com/MaurizioFD/RecSys2019_DeepLearning_Evaluation/blob/master/SLIM_ElasticNet/SLIMElasticNetRecommender.py)
:param A: Rating matrix nxm where m in the number of items. Must be a csc_matrix
:param elanet: ElasticNet object
:return: W weight matrix.
'''
warnings.simplefilter("ignore")
n_users, n_items = A.shape
# Predicting each column at time
W_rows_idxs = []
W_cols_idxs = []
W_data = []
for j in tqdm(range(n_items), leave=False):
# Target column
aj = A[:, j].toarray()
# Removing the j-th item from all users
# Need to zero the data entries related to the j-th column
st_idx = A.indptr[j]
en_idx = A.indptr[j + 1]
copy = A.data[st_idx:en_idx].copy()
A.data[st_idx:en_idx] = 0.0
# Predicting the column
elanet.fit(A, aj)
# Fetching the coefficients (sparse)
widx = elanet.sparse_coef_.indices
wdata = elanet.sparse_coef_.data
# Save information about position in the final matrix
W_rows_idxs += list(widx)
W_cols_idxs += [j] * len(widx)
W_data += list(wdata)
# reconstrucing the matrix
A.data[st_idx:en_idx] = copy
W = sp.csr_matrix((W_data, (W_rows_idxs, W_cols_idxs)),
shape=(n_items, n_items))
return W
def significance(
TTM: sp.csc_matrix,
metric: Union[Callable, KeynessMetric],
normalize: bool = False,
n_contexts=None,
n_words=None,
) -> sp.csc_matrix:
"""Computes statistical significance tf co-occurrences using `metric`.
Args:
TTM (sp.csc_matrix): [description]
normalize (bool, optional): [description]. Defaults to False.
Returns:
sp.csc_matrix: [description]
"""
metric = metric if callable(metric) else METRIC_FUNCTION.get(
metric, _undefined)
K: float = n_contexts
N: float = n_words
"""Total number of observations (counts)"""
Z: float = float(TTM.sum())
"""Number of observations per context (document, row sum)"""
Zr = np.array(TTM.sum(axis=1), dtype=np.float64).flatten()
"""Row and column indices of non-zero elements."""
ii, jj = TTM.nonzero()
Cij: np.ndarray = np.array(TTM[ii, jj], dtype=np.float64).flatten()
"""Compute weights (with optional normalize)."""
weights: np.ndarray = metric(Cij=Cij,
Z=Z,
Zr=Zr,
ii=ii,
jj=jj,
K=K,
N=N,
normalize=normalize)
np.nan_to_num(
weights,
copy=False,
posinf=0.0,
neginf=0.0,
nan=0.0,
)
nz_indices: np.ndarray = weights.nonzero()
return (weights[nz_indices], (ii[nz_indices], jj[nz_indices]))
def lsmr_annihilate(x: csc_matrix, y: ndarray, use_cache: bool = True, x_hash=None,
**lsmr_options) -> ndarray:
r"""
Removes projection of x on y from y
Parameters
----------
x : csc_matrix
Sparse array of regressors
y : ndarray
Array with shape (nobs, nvar)
use_cache : bool
Flag indicating whether results should be stored in the cache,
and retrieved if available.
x_hash : object
Hashable object representing the values in x
lsmr_options: dict
Dictionary of options to pass to scipy.sparse.linalg.lsmr
Returns
-------
resids : ndarray
Returns the residuals from regressing y on x, (nobs, nvar)
Notes
-----
Residuals are estiamted column-by-column as
.. math::
\hat{\epsilon}_{j} = y_{j} - x^\prime \hat{\beta}
where :math:`\hat{\beta}` is computed using lsmr.
"""
use_cache = use_cache and x_hash is not None
regressor_hash = x_hash if x_hash is not None else ''
default_opts = dict(atol=1e-8, btol=1e-8, show=False)
default_opts.update(lsmr_options)
resids = []
for i in range(y.shape[1]):
_y = y[:, i:i + 1]
variable_digest = ''
if use_cache:
hasher = hash_func()
hasher.update(ascontiguousarray(_y.data))
variable_digest = hasher.hexdigest()
if use_cache and variable_digest in _VARIABLE_CACHE[regressor_hash]:
resid = _VARIABLE_CACHE[regressor_hash][variable_digest]
else:
beta = lsmr(x, _y, **default_opts)[0]
resid = y[:, i:i + 1] - (x.dot(csc_matrix(beta[:, None]))).A
_VARIABLE_CACHE[regressor_hash][variable_digest] = resid
resids.append(resid)
if resids:
return column_stack(resids)
else:
return empty_like(y)
def PopularItems(A: sp.csc_matrix, limit=50):
"""
Returns the most popular items.
:param A: user-item matrix
:param limit: how many popular items should be returned. The other entries will be filled with 0s.
"""
n = A.shape[0]
# used for indexing
dummy_column = np.arange(n).reshape(n, 1)
# Counting the number of interactions
item_count = np.asarray(A.sum(axis=0)).reshape(-1)
# Partially sorted indexes
part_sort_indexes = bn.argpartition(-item_count, kth=limit)
# Focusing on the tops
unsorted_idx_tops = part_sort_indexes[:limit]
unsorted_tops = item_count[unsorted_idx_tops]
sorted_idx_tops_part = np.argsort(unsorted_tops)
# Extracting the indexes of the tops respect of the original array
sorted_idx_tops = part_sort_indexes[sorted_idx_tops_part]
recommend = sp.lil_matrix(A.shape)
# We assign real values between 0.5 and 1 to the tops so we can employ ranking metrics.
recommend[dummy_column, sorted_idx_tops] = np.linspace(start=0.5,
stop=1.0,
num=limit)
return recommend
def _calculate_neighbor_weight_matrix(self,
train_r: csc_matrix) -> np.ndarray:
l2norm = norm(train_r, ord=2, axis=1)
l2norm[l2norm == 0] = 1 # handel 0 vector
U: csc_matrix = train_r.multiply(1 / l2norm.reshape(-1, 1))
UUT = U.dot(U.transpose()).toarray() # dense
W = np.exp(self._tau * np.power(1 - UUT, self._k))
np.fill_diagonal(W, 0)
return W
def matrix_similarity(urm: sp.csc_matrix, shrink: int):
item_weights = np.sqrt(
np.sum(urm.power(2), axis=0)
).A
numerator = urm.T.dot(urm)
denominator = item_weights.T.dot(item_weights) + shrink + 1e-6
weights = numerator / denominator
np.fill_diagonal(item_similarity, 0.0)
return weights
def train(self, train_r: csc_matrix):
def cal_loss():
rhat = W_theta.dot(R) + b
rhat[rhat < 0] = 0
return np.sum(np.power(rhat - R, 2)) / (2 * n)
R = train_r.toarray()
m, n = train_r.shape
theta = np.zeros((m, m))
b = np.zeros((m, 1))
W = self._calculate_neighbor_weight_matrix(train_r)
W_theta = W * theta
alpha_W = self._alpha * W
self._training_history = [cal_loss()]
print(f"Loss before training: {self._training_history[-1]:0.8E}")
for epoch in range(self._max_epoch):
pbar = tqdm(range(math.ceil(n / self._batch_size)))
for batch_index in pbar:
start = batch_index * self._batch_size
B = train_r[:, start:start + self._batch_size].toarray()
bsize = B.shape[1]
B_hat = W_theta.dot(B) + b
B_hat_B = (B_hat - B)
B_hat_B[B_hat < 0] = 0 # multiply ReLu'(r_hat)
delta_theta = alpha_W / bsize * (B_hat_B.dot(
B.transpose())) + self._lambda * theta
delta_b = self._alpha / bsize * (np.sum(
B_hat_B, axis=1, keepdims=True))
# update theta
theta = theta - delta_theta
b = b - delta_b
# update W_theta
W_theta = W * theta
delta_eps = np.sum(
np.power(delta_theta, 2) + np.power(delta_b, 2))
pbar.set_description(
f"Epoch {epoch + 1} batch {batch_index + 1} delta={delta_eps:.8E}"
)
pbar.close()
loss = cal_loss()
self._training_history.append(loss)
print(f"Final Loss of epoch {epoch + 1}: {loss:0.8E}")
self._W_theta = W_theta
self._b = b
def get_tfidf_matrix(cnts: sp.csc_matrix):
"""Convert the word count matrix into tfidf one.
tfidf = log(tf + 1) * log((N - Nt + 0.5) / (Nt + 0.5))
* tf = term frequency in document
* N = number of documents
* Nt = number of occurences of term in all documents
"""
Ns = get_doc_freqs(cnts)
idfs = np.log((cnts.shape[1] - Ns + 0.5) / (Ns + 0.5))
idfs[idfs < 0] = 0
idfs = sp.diags(idfs, 0)
tfs = cnts.log1p()
tfidfs = idfs.dot(tfs)
return tfidfs
def __adjustTransitionMatrix(self, M: sparse.csc_matrix) \
-> sparse.csc_matrix:
"""Function to compute the adjusted Markov transition matrix, given the
unadjusted matrix. This method enforces column stochastic behavior.
Returns:
sparse.csc_matrix -- Adjusted Markov transition matrix.
"""
logging.info('Building adjusted transition matrix')
# counter
last_check = 0
logging.info('Computing sum of columns of M')
magnitues = M.sum(axis=0)
logging.info('Iterating through each column, rebalancing')
# Iterate through each column
for i in range(self.N):
# Isolating magnitude
magnitude = magnitues[0, i]
# If criteria are satisfied, redistribute probabilities
if (magnitude < 1.0) and (magnitude != 0):
count = M[:, i].nnz
# Isolate nonzero indezes
nonzero_idx = M[:, i].nonzero()[0]
# Update indexes with balanced probabilities
for idx in nonzero_idx:
M[idx, i] = 1 / count
# Log progress
last_check = logLoopProgress(i, last_check, self.N,
'Stable transition matrix')
logging.info('Built adjusted Markov transition matrix with {0} \
elements'.format(M.nnz))
return M
def SLIM_parallel(A: sp.csc_matrix, alpha: float, l1_ratio: float,
max_iter: int):
'''
SLIM: Sparse Linear Methods for Top-N Recommender Systems - Xia Ning ; George Karypis
https://ieeexplore.ieee.org/document/6137254
Run Sparse Linear Models over the rating matrix A.
(code from https://github.com/ruhan/toyslim/blob/master/slim_parallel.py)
:param A: Rating matrix nxm where m in the number of items. Must be a csc_matrix
:param alpha: a + b (a and b are the multipliers of the L1 and L2 penalties, respectively)
:param l1_ratio: a / (a + b) (a and b are the multipliers of the L1 and L2 penalties, respectively)
:param max_iter: number of iterations to run max for each item
:return: W weight matrix.
'''
warnings.simplefilter("ignore")
n_items = A.shape[1]
ranges = generate_slices(n_items)
separated_tasks = []
for from_j, to_j in ranges:
separated_tasks.append(
[from_j, to_j, A.copy(), alpha, l1_ratio, max_iter])
with multiprocessing.Pool() as pool:
results = pool.map(work, separated_tasks)
W_rows_idxs = list(itertools.chain(*[x[0] for x in results]))
W_cols_idxs = list(itertools.chain(*[x[1] for x in results]))
W_data = list(itertools.chain(*[x[2] for x in results]))
W = sp.csr_matrix((W_data, (W_rows_idxs, W_cols_idxs)),
shape=(n_items, n_items))
return W
def optimize(sparse_km: csc_matrix, gamma: float,
regcoef: float, L1: float, eps: float, max_iter: int) -> (csc_matrix, dict):
"""
Perform SVM on sparsified kernel matrix.
:param sparse_km: sparsified kernel matrix.
:param eps: epsilon value.
:param max_iter: maximal number of iterations.
:return: object weights.
"""
if sparse_km.shape[0] != sparse_km.shape[1]:
raise Exception("Kernel matrix is not a squared matrix")
log = {"grad_norm": [], "time": []}
N = sparse_km.shape[0]
def grad_f(x):
t = x.copy()
t.data -= 1 / (2 * N * regcoef)
return -csr_matrix((N, 1)) + sparse_km.dot(x) - \
gamma*sparse_clip(-x, 0, None) + gamma*sparse_clip(t, 0, None)
x0 = csr_matrix((N, 1))
x0[0, 0] = 1/2
grad_f0 = grad_f(x0)
grad_min = BasicGradientUpdater(grad_f0.T)
grad_max = BasicGradientUpdater(-grad_f0.T)
iter_counter = 0
start = timeit.default_timer()
current_point = x0
true_grad = grad_f0
while grad_min.get_norm() > eps**2 or iter_counter < max_iter:
#if true_grad == grad_min.get():
log["grad_norm"].append(grad_min.get_norm())
log["time"].append(timeit.default_timer() - start)
i_plus = grad_max.get_coordinate()
g_plus = -grad_max.get_value()
i_minus = grad_min.get_coordinate()
g_minus = grad_min.get_value()
h_val = 1/(4*L1)*(g_plus - g_minus)
h = csr_matrix((N, 1))
h[i_plus, 0] = h_val
h[i_minus, 0] = -h_val
t = current_point.copy() # had to make this turnaround 'cause "sparse vector + constant" operation hasn't been implemented yet
t.data -= 1 / (2 * N * regcoef)
delta_grad = sparse_km.dot(h)
delta_grad -= gamma*sparse_clip(-current_point - h, 0, None)
delta_grad += gamma*sparse_clip(-current_point, 0, None)
delta_grad += gamma*sparse_clip(t + h, 0, None)
delta_grad -= gamma*sparse_clip(t, 0, None)
grad_min.update(delta_grad.T)
grad_max.update(-delta_grad.T)
current_point += h
true_grad = grad_f(current_point.T)
iter_counter += 1
return h, log
def adjacency2degree(adj: csc_matrix) -> csc_matrix:
""" Compute the degree matrix for a give adjacency matrix A"""
return diags(np.asarray(adj.sum(1)).reshape(-1), format='csc')
def solve_problem(self, solver_right_mat: sps.csc_matrix) -> list:
dense_right_mat = solver_right_mat.todense()
dst_def_vts = self.inv_left_mat.solve(dense_right_mat).tolist()
return dst_def_vts
def __init__(self, x: np.ndarray, sg: np.ndarray, hess: np.ndarray,
scaling: csc_matrix, g_dscaling: csc_matrix, delta: float,
theta: float, ub: np.ndarray, lb: np.ndarray, logger: Logger):
"""
:param x:
Reference point
:param sg:
Gradient in rescaled coordinates
:param hess:
Hessian in unscaled coordinates
:param scaling:
Matrix that defines scaling transformation
:param g_dscaling:
Unscaled gradient multiplied by derivative of scaling
transformation
:param delta:
Trust region Radius in scaled coordinates
:param theta:
Stepback parameter that controls how close steps are allowed to
get to the boundary
:param ub:
Upper boundary
:param lb:
Lower boundary
"""
self.x: np.ndarray = x
self.s: Union[np.ndarray, None] = None
self.sc: Union[np.ndarray, None] = None
self.ss: Union[np.ndarray, None] = None
self.og_s: Union[np.ndarray, None] = None
self.og_sc: Union[np.ndarray, None] = None
self.og_ss: Union[np.ndarray, None] = None
self.sg: np.ndarray = sg.copy()
self.scaling: csc_matrix = scaling.copy()
self.delta: float = delta
self.theta: float = theta
self.lb: np.ndarray = lb
self.ub: np.ndarray = ub
self.br: np.ndarray = np.ones(sg.shape)
self.minbr: float = 1.0
self.alpha: float = 1.0
self.iminbr: np.ndarray = np.array([])
self.qpval: float = 0.0
# B_hat (Eq 2.5) [ColemanLi1996]
self.shess: np.ndarray = np.asarray(scaling * hess * scaling +
g_dscaling)
self.cg: Union[np.ndarray, None] = None
self.chess: Union[np.ndarray, None] = None
self.subspace: Union[np.ndarray, None] = None
self.s0: np.ndarray = np.zeros(sg.shape)
self.ss0: np.ndarray = np.zeros(sg.shape)
self.reflection_indices: set = set()
self.truncation_indices: set = set()
self.logger: Logger = logger
def trust_region(x: np.ndarray, g: np.ndarray, hess: np.ndarray,
scaling: csc_matrix, delta: float, dv: np.ndarray,
theta: float, lb: np.ndarray, ub: np.ndarray,
subspace_dim: SubSpaceDim,
stepback_strategy: StepBackStrategy, refine_stepback: bool,
logger: logging.Logger) -> Step:
"""
Compute a step according to the solution of the trust-region subproblem.
If step-back is necessary, gradient and reflected trust region step are
also evaluated in terms of their performance according to the local
quadratic approximation
:param x:
Current values of the optimization variables
:param g:
Objective function gradient at x
:param hess:
(Approximate) objective function Hessian at x
:param scaling:
Scaling transformation according to distance to boundary
:param delta:
Trust region radius, note that this applies after scaling
transformation
:param dv:
derivative of scaling transformation
:param theta:
parameter regulating stepback
:param lb:
lower optimization variable boundaries
:param ub:
upper optimization variable boundaries
:param subspace_dim:
Subspace dimension in which the subproblem will be solved. Larger
subspaces require more compute time but can yield higher quality step
proposals.
:param stepback_strategy:
Strategy that is applied when the proposed step exceeds the
optimization boundary.
:param refine_stepback:
If set to True, proposed steps that are computed via the specified
stepback_strategy will be refined via optimization.
:param logger:
logging.Logger instance to be used for logging
:return:
s: proposed step,
ss: rescaled proposed step,
qpval: expected function value according to local quadratic
approximation,
subspace: computed subspace for reuse if proposed step is not accepted,
steptype: type of step that was selected for proposal
"""
sg = scaling.dot(g)
g_dscaling = csc_matrix(np.diag(np.abs(g) * dv))
if subspace_dim == SubSpaceDim.TWO:
tr_step = TRStep2D(x, sg, hess, scaling, g_dscaling, delta, theta, ub,
lb, logger)
elif subspace_dim == SubSpaceDim.FULL:
tr_step = TRStepFull(x, sg, hess, scaling, g_dscaling, delta, theta,
ub, lb, logger)
else:
raise ValueError('Invalid choice of subspace dimension.')
tr_step.calculate()
# in case of truncation, we hit the boundary and we check both the
# gradient and the reflected step, either of which could be better than the
# TR step
steps = [tr_step]
if tr_step.alpha < 1.0 and len(g) > 1:
g_step = GradientStep(x, sg, hess, scaling, g_dscaling, delta, theta,
ub, lb, logger)
g_step.calculate()
steps.append(g_step)
if stepback_strategy == StepBackStrategy.SINGLE_REFLECT:
rtr_step = TRStepReflected(x, sg, hess, scaling, g_dscaling, delta,
theta, ub, lb, tr_step)
rtr_step.calculate()
steps.append(rtr_step)
if stepback_strategy in [
StepBackStrategy.REFLECT, StepBackStrategy.MIXED
]:
steps.extend(
stepback_reflect(tr_step, x, sg, hess, scaling, g_dscaling,
delta, theta, ub, lb))
if stepback_strategy in [
StepBackStrategy.TRUNCATE, StepBackStrategy.MIXED
]:
steps.extend(
stepback_truncate(tr_step, x, sg, hess, scaling, g_dscaling,
delta, theta, ub, lb))
if refine_stepback:
steps.extend(
stepback_refine(steps, x, sg, hess, scaling, g_dscaling, delta,
theta, ub, lb))
if len(steps) > 1:
rcountstrs = [
str(step.reflection_count) * int(step.reflection_count > 0)
for step in steps
]
logger.debug(' | '.join([
f'{step.type + rcountstr}: [qp:'
f' {step.qpval:.2E}, '
f'a: {step.alpha:.2E}]'
for rcountstr, step in zip(rcountstrs, steps)
]))
qpvals = [step.qpval for step in steps]
return steps[int(np.argmin(qpvals))]
def get_distances(node: Tuple[int, int],
nodes: sparse.csc_matrix) -> List[int]:
non_zero = [coord for coord in zip(*nodes.nonzero())]
distances = spatial.distance.cdist([node], non_zero,
metric='cityblock').flatten().tolist()
return distances
def adjacency2transition(adj: csc_matrix,
degree: csc_matrix = None) -> csc_matrix:
""" Compute the transition matrix associated with the adjacency matrix A"""
degree = adjacency2degree(adj) if degree is None else degree
return adj * degree.power(-1)
def l1_normalize_row(X: sps.csc_matrix) -> sps.csc_matrix:
result: sps.csc_matrix = X.astype(np.float64)
result.sort_indices()
l1_norms: np.ndarray = result.sum(axis=1).A1
result.data /= l1_norms[result.indices]
return result
def trust_region(x: np.ndarray, g: np.ndarray, hess: np.ndarray,
scaling: csc_matrix, delta: float, dv: np.ndarray,
theta: float, lb: np.ndarray, ub: np.ndarray,
subspace_dim: SubSpaceDim,
stepback_strategy: StepBackStrategy,
logger: logging.Logger) -> Step:
"""
Compute a step according to the solution of the trust-region subproblem.
If step-back is necessary, gradient and reflected trust region step are
also evaluated in terms of their performance according to the local
quadratic approximation
:param x:
Current values of the optimization variables
:param g:
Objective function gradient at x
:param hess:
(Approximate) objective function Hessian at x
:param scaling:
Scaling transformation according to distance to boundary
:param delta:
Trust region radius, note that this applies after scaling
transformation
:param dv:
derivative of scaling transformation
:param theta:
parameter regulating stepback
:param lb:
lower optimization variable boundaries
:param ub:
upper optimization variable boundaries
:param subspace_dim:
Subspace dimension in which the subproblem will be solved. Larger
subspaces require more compute time but can yield higher quality step
proposals.
:param stepback_strategy:
Strategy that is applied when the proposed step exceeds the
optimization boundary.
:param logger:
logging.Logger instance to be used for logging
:return:
s: proposed step,
"""
sg = scaling.dot(g)
# diag(g_k)*J^v_k Eq (2.5) [ColemanLi1994]
g_dscaling = csc_matrix(np.diag(np.abs(g) * dv))
step_options = {
SubSpaceDim.TWO: TRStep2D,
SubSpaceDim.FULL: TRStepFull,
SubSpaceDim.STEIHAUG: TRStepSteihaug,
}
tr_step = step_options[subspace_dim](x, sg, hess, scaling, g_dscaling,
delta, theta, ub, lb, logger)
tr_step.calculate()
# in case of truncation, we hit the boundary and we check both the
# gradient and the reflected step, either of which could be better than the
# TR step
steps = [tr_step]
if tr_step.alpha < 1.0 and len(g) > 1:
g_step = GradientStep(x, sg, hess, scaling, g_dscaling, delta, theta,
ub, lb, logger)
g_step.calculate()
steps.append(g_step)
if stepback_strategy == StepBackStrategy.SINGLE_REFLECT:
rtr_step = TRStepReflected(x, sg, hess, scaling, g_dscaling, delta,
theta, ub, lb, tr_step)
rtr_step.calculate()
steps.append(rtr_step)
if stepback_strategy in [
StepBackStrategy.REFLECT, StepBackStrategy.MIXED
]:
steps.extend(
stepback_reflect(tr_step, x, sg, hess, scaling, g_dscaling,
delta, theta, ub, lb))
if stepback_strategy in [
StepBackStrategy.TRUNCATE, StepBackStrategy.MIXED
]:
steps.extend(
stepback_truncate(tr_step, x, sg, hess, scaling, g_dscaling,
delta, theta, ub, lb))
if stepback_strategy == StepBackStrategy.REFINE and \
tr_step.subspace.shape[1] > 1:
ref_step = RefinedStep(x, sg, hess, scaling, g_dscaling, delta,
theta, ub, lb, tr_step)
ref_step.calculate()
steps.append(ref_step)
if len(steps) > 1:
rcountstrs = [
str(step.reflection_count) * int(step.reflection_count > 0)
for step in steps
]
logger.debug(' | '.join([
f'{step.type + rcountstr}: [qp:'
f' {step.qpval:.2E}, '
f'a: {step.alpha:.2E}]'
for rcountstr, step in zip(rcountstrs, steps)
]))
qpvals = [step.qpval for step in steps]
return steps[np.argmin(qpvals)]