def _updateB(oldB, B, W, degrees, damping, inds, backinds): # pragma: no cover
"""belief update function."""
for j, d in enumerate(degrees):
kk = inds[j]
bk = backinds[j]
if d == 0:
B[kk, bk] = -np.inf
continue
belief = W[kk, bk] + W[j]
oldBj = oldB[j]
if d == oldBj.shape[0]:
bth = quickselect(oldBj, d - 1)
bplus = -1
else:
bth, bplus = quickselect(oldBj, d - 1, d)
belief -= np.where(oldBj >= oldBj[bth], oldBj[bplus], oldBj[bth])
B[kk, bk] = damping * belief + (1 - damping) * oldB[kk, bk]
def b_matching(D, k, max_iter=1000, damping=1, conv_thresh=1e-4, verbose=False):
"""
"Belief-Propagation for Weighted b-Matchings on Arbitrary Graphs
and its Relation to Linear Programs with Integer Solutions"
Bayati et al.
Finds the minimal weight perfect b-matching using min-sum loopy-BP.
@param D pairwise distance matrix
@param k number of neighbors per vertex (scalar or array-like)
Based on the code at http://www.cs.columbia.edu/~bert/code/bmatching/bdmatch
"""
INTERVAL = 2
oscillation = 10
cbuff = np.zeros(100, dtype=float)
cbuffpos = 0
N = D.shape[0]
assert D.shape[1] == N, "Input distance matrix must be square"
mask = ~np.eye(N, dtype=bool) # Assume all nonzero except for diagonal
W = -D[mask].reshape((N, -1)).astype(float)
degrees = np.clip(np.atleast_1d(k), 0, N - 1)
if degrees.size == 1: # broadcast scalar up to length-N array
degrees = np.repeat(degrees, N)
else:
assert degrees.shape == (N,), "Input degrees must have length N"
# TODO: remove these later
inds = np.tile(np.arange(N), (N, 1))
backinds = inds.copy()
inds = inds[mask].reshape((N, -1))
backinds = backinds.T.ravel()[: (N * (N - 1))].reshape((N, -1))
# Run Belief Revision
change = 1.0
B = W.copy()
for n_iter in range(1, max_iter + 1):
oldB = B.copy()
updateB(oldB, B, W, degrees, damping, inds, backinds)
# check for convergence
if n_iter % INTERVAL == 0:
# track changes
c = np.abs(B[:, 0]).sum()
# c may be infinite here, and that's ok
with np.errstate(invalid="ignore"):
if np.any(np.abs(c - cbuff) < conv_thresh):
oscillation -= 1
cbuff[cbuffpos] = c
cbuffpos = (cbuffpos + 1) % len(cbuff)
change = update_change(B, oldB)
if np.isnan(change):
warnings.warn(
"change is NaN! " "BP will quit but solution could be invalid. " "Problem may be infeasible."
)
break
if change < conv_thresh or oscillation < 1:
break
else:
warnings.warn("Hit iteration limit (%d) before converging" % max_iter)
if verbose: # pragma: no cover
if change < conv_thresh:
print("Converged to stable beliefs in %d iterations" % n_iter)
elif oscillation < 1:
print("Stopped after reaching oscillation in %d iterations" % n_iter)
print("No feasible solution found or there are multiple maxima.")
print("Outputting best approximate solution. Try damping.")
# recover result from B
thresholds = np.zeros(N)
for i, d in enumerate(degrees):
Brow = B[i]
if d >= N - 1:
thresholds[i] = -np.inf
elif d < 1:
thresholds[i] = np.inf
else:
thresholds[i] = Brow[quickselect(Brow, d - 1)]
ii, jj = np.where(B >= thresholds[:, None])
pairs = np.column_stack((ii, inds[ii, jj]))
return Graph.from_edge_pairs(pairs, num_vertices=N)