def arnoldi_iteration(A: BackendGraph, b: BackendPrimitive, n: int):
"""Computes a basis of the (n + 1)-Krylov subspace of A: the space
spanned by {b, Ab, ..., A^n b}.
Source: https://en.wikipedia.org/wiki/Arnoldi_iteration
Arguments:
A: m × m array
b: initial vector (length m)
n: dimension of Krylov subspace, must be >= 1
Returns:
Q: m x (n + 1) array, the columns are an orthonormal basis of the
Krylov subspace.
h: (n + 1) x n array, A on basis Q. It is upper Hessenberg.
"""
h = [[0 for _ in range(n)] for _ in range(n + 1)]
Q = [backend.self_normalize(b)]
for k in range(1, n):
v = backend.conv(Q[k - 1], A)
for j in range(k):
h[j][k - 1] = backend.dot(Q[j], v)
v = v - h[j][k - 1] * Q[j]
h[k][k - 1] = backend.dot(v, v)**0.5
Q.append(v / h[k][k - 1] if h[k][k - 1] != 0 else v * 0)
return backend.combine_cols(Q), h
def krylov_base(M, personalization, krylov_space_degree):
warnings.warn(
"Krylov approximation is not stable yet (results may differ in future versions)"
)
# TODO: throw exception for non-symmetric matrix
personalization = backend.to_primitive(personalization)
base = [
personalization / backend.dot(personalization, personalization)**0.5
]
base_norms = []
alphas = []
for j in range(0, krylov_space_degree):
v = base[j]
w = backend.conv(v, M)
a = backend.dot(v, w)
alphas.append(a)
next_w = w - a * v
if j > 0:
next_w -= base[j - 1] * base_norms[j - 1]
next_w_norm = (backend.sum(next_w**2))**0.5
base_norms.append(next_w_norm)
if j != krylov_space_degree - 1:
base.append(next_w / next_w_norm)
H = diags([alphas, base_norms[1:], base_norms[1:]], [0, -1, 1])
V = backend.combine_cols(base) #V = np.column_stack(base)
return V, H
def evaluate(self, scores: GraphSignalData) -> BackendPrimitive:
known_scores, scores = self.to_numpy(scores)
#thresh = backend.min(scores[known_scores!=0])
#scores = 1/(1+np.exp(-scores/thresh+1))
eps = backend.epsilon()
ret = -backend.dot(known_scores,
backend.log(scores + eps)) - backend.dot(
1 - known_scores, backend.log(1 - scores + eps))
return ret
def evaluate(self, scores: GraphSignalData) -> BackendPrimitive:
sensitive, scores = self.to_numpy(scores)
p1 = backend.dot(scores, sensitive)
p2 = backend.dot(scores, 1 - sensitive)
if p1 == 0 or p2 == 0:
return 0
s = backend.sum(sensitive)
p1 = backend.safe_div(p1, s)
p2 = backend.safe_div(p2, backend.length(sensitive) - s)
if p1 <= p2: # this implementation is derivable
return p1 / p2
return p2 / p1
def evaluate(self, scores: GraphSignalData) -> BackendPrimitive:
sensitive, scores = self.to_numpy(scores)
p1 = backend.dot(scores, sensitive)
p2 = backend.dot(scores, 1 - sensitive)
s = backend.sum(sensitive)
n = backend.length(sensitive)
p1 = backend.safe_div(p1, s)
p2 = backend.safe_div(p2, n - s)
#if p1 <= p2*self.target_pRule:
# p2 *= self.target_pRule
#elif p2 <= p1*self.target_pRule:
# p1 *= self.target_pRule
#else:
# return 0
return (p1 - p2)**2 * n
def eigdegree(M):
"""
Calculates the entropy-preserving eigenvalue degree to be used in matrix normalization.
Args:
M: the adjacency matrix
"""
v = backend.repeat(1., M.shape[0])
from pygrank.algorithms import ConvergenceManager
convergence = ConvergenceManager(tol=backend.epsilon(), max_iters=1000)
convergence.start()
eig = 0
v = v / backend.dot(v, v)
while not convergence.has_converged(eig):
v = backend.conv(v, M)
v = backend.safe_div(v, backend.dot(v, v)**0.5)
eig = backend.dot(backend.conv(v, M), v)
return eig / (v * v)
def evaluate(self, scores: GraphSignalData) -> BackendPrimitive:
if len(self.get_graph(scores)) == 0:
return 0
adjacency, scores = self.to_numpy(scores)
neighbors = backend.conv(scores, adjacency)
internal_edges = backend.dot(neighbors, scores)
expected_edges = backend.sum(scores) ** 2 - backend.sum(scores ** 2) # without self-loops
return backend.safe_div(internal_edges, expected_edges)
def evaluate(self, scores: GraphSignalData) -> BackendPrimitive:
graph = self.get_graph(scores)
if len(graph) == 0:
return float('inf')
adjacency, scores = self.to_numpy(scores)
if backend.max(scores) > self.max_rank:
if self.autofix:
scores = scores * (self.max_rank / backend.max(scores))
else:
raise Exception("Normalize scores to be <= " + str(self.max_rank) + " for non-negative conductance")
neighbors = backend.conv(scores, adjacency)
internal_edges = backend.dot(neighbors, scores)
external_edges = backend.dot(neighbors, self.max_rank-scores)
if not graph.is_directed():
external_edges += backend.dot(scores, backend.conv(self.max_rank-scores, adjacency))
internal_edges *= 2
if external_edges == 0:
return float('inf')
return backend.safe_div(external_edges, internal_edges, default=float('inf'))
def krylov_error_bound(V,
H,
M,
personalization,
measure=measures.Mabs,
max_powers=1):
personalization = personalization / backend.dot(personalization,
personalization)**0.5
krylov_dims = V.shape[1]
krylov_result = backend.eye(int(krylov_dims))
errors = list()
for power in range(max_powers + 1):
errors.append(
measure(personalization)(krylov2original(V, krylov_result,
int(krylov_dims))))
if power < max_powers:
krylov_result = krylov_result @ H
personalization = backend.conv(personalization, M)
return max(errors)
def evaluate(self, scores: GraphSignalData) -> BackendPrimitive:
known_scores, scores = self.to_numpy(scores)
return backend.dot(known_scores, scores)
def evaluate(self, scores: GraphSignalData) -> BackendPrimitive:
known_scores, scores = self.to_numpy(scores)
divide = backend.dot(known_scores, known_scores) * backend.dot(
scores, scores)
return backend.safe_div(backend.dot(known_scores, scores), divide**0.5)
def _tune(self, graph=None, personalization=None, *args, **kwargs):
#graph_dropout = kwargs.get("graph_dropout", 0)
#kwargs["graph_dropout"] = 0
previous_backend = backend.backend_name()
personalization = to_signal(graph, personalization)
graph = personalization.graph
if self.tuning_backend is not None and self.tuning_backend != previous_backend:
backend.load_backend(self.tuning_backend)
backend_personalization = to_signal(
personalization, backend.to_array(personalization.np))
#training, validation = split(backend_personalization, 0.8)
#training2, validation2 = split(backend_personalization, 0.6)
#measure_weights = [1, 1, 1, 1, 1]
#propagated = [training.np, validation.np, backend_personalization.np, training2.np, validation2.np]
measure_values = [None] * (self.num_parameters + self.autoregression)
M = self.ranker_generator(measure_values).preprocessor(graph)
#for _ in range(10):
# backend_personalization.np = backend.conv(backend_personalization.np, M)
training, validation = split(backend_personalization, 0.8)
training1, training2 = split(training, 0.5)
propagated = [training1.np, training2.np]
measures = [
self.measure(backend_personalization, training1),
self.measure(backend_personalization, training2)
]
#measures = [self.measure(validation, training), self.measure(training, validation)]
if self.basis == "krylov":
for i in range(len(measure_values)):
measure_values[i] = [
measure(p) for p, measure in zip(propagated, measures)
]
propagated = [backend.conv(p, M) for p in propagated]
else:
basis = [
arnoldi_iteration(M, p, len(measure_values))[0]
for p in propagated
]
for i in range(len(measure_values)):
measure_values[i] = [
float(measure(base[:, i]))
for base, measure in zip(basis, measures)
]
measure_values = backend.to_primitive(measure_values)
mean_value = backend.mean(measure_values, axis=0)
measure_values = measure_values - mean_value
best_parameters = measure_values
measure_weights = [1] * measure_values.shape[1]
if self.autoregression != 0:
#vals2 = -measure_values-mean_value
#measure_values = np.concatenate([measure_values, vals2-np.mean(vals2, axis=0)], axis=1)
window = backend.repeat(1. / self.autoregression,
self.autoregression)
beta1 = 0.9
beta2 = 0.999
beta1t = 1
beta2t = 1
rms = window * 0
momentum = window * 0
error = float('inf')
while True:
beta1t *= beta1
beta2t *= beta2
prev_error = error
parameters = backend.copy(measure_values)
for i in range(len(measure_values) - len(window) - 2, -1, -1):
parameters[i, :] = backend.dot(
(window),
measure_values[(i + 1):(i + len(window) + 1), :])
errors = (parameters - measure_values
) * measure_weights / backend.sum(measure_weights)
for j in range(len(window)):
gradient = 0
for i in range(len(measure_values) - len(window) - 1):
gradient += backend.dot(measure_values[i + j + 1, :],
errors[i, :])
momentum[j] = beta1 * momentum[j] + (
1 - beta1) * gradient #*np.sign(window[j])
rms[j] = beta2 * rms[j] + (1 - beta2) * gradient * gradient
window[j] -= 0.01 * momentum[j] / (1 - beta1t) / (
(rms[j] / (1 - beta2t))**0.5 + 1.E-8)
#window[j] -= 0.01*gradient*np.sign(window[j])
error = backend.mean(backend.abs(errors))
if error == 0 or abs(error - prev_error) / error < 1.E-6:
best_parameters = parameters
break
best_parameters = backend.mean(best_parameters[:self.num_parameters, :]
* backend.to_primitive(measure_weights),
axis=1) + backend.mean(mean_value)
if self.tunable_offset is not None:
div = backend.max(best_parameters)
if div != 0:
best_parameters /= div
measure = self.tunable_offset(validation, training)
base = basis[0] if self.basis != "krylov" else None
best_offset = optimize(
lambda params: -measure.best_direction() * measure(
self._run(training, [(best_parameters[i] + params[
2]) * params[0]**i + params[1] for i in range(
len(best_parameters))], base, *args, **kwargs)),
#lambda params: - measure.evaluate(self._run(training, best_parameters + params[0], *args, **kwargs)),
max_vals=[1, 0, 0],
min_vals=[0, 0, 0],
deviation_tol=0.005,
parameter_tol=1,
partitions=5,
divide_range=2)
#best_parameters += best_offset[0]
best_parameters = [
(best_parameters[i] + best_offset[2]) * best_offset[0]**i +
best_offset[1] for i in range(len(best_parameters))
]
best_parameters = backend.to_primitive(best_parameters)
if backend.sum(backend.abs(best_parameters)) != 0:
best_parameters /= backend.mean(backend.abs(best_parameters))
if self.tuning_backend is not None and self.tuning_backend != previous_backend:
best_parameters = [
float(param) for param in best_parameters
] # convert parameters to backend-independent list
backend.load_backend(previous_backend)
#kwargs["graph_dropout"] = graph_dropout
if self.basis != "krylov":
return Tautology(), self._run(
personalization, best_parameters, *args,
**kwargs) # TODO: make this unecessary
return self.ranker_generator(best_parameters), personalization