def plot_estimators_vary_Z(n, trials=100, random=True):
'''
See how estimators compare when changing the value of Z with fixed n
Inputs:
- n: int, generate set of Z vectors in {-1,1}^n
- trials: int, solve max problem trials times and take the mean
- random: bool, true->generate random set, false->generate hypercube set
'''
Z_values = [2**i for i in range(n+1)]
print Z_values
gumbel_estimators = []
barvinok_estimators = []
barvinok_uppers = []
barvinok_lowers = []
barvinok_estimators_gaussian = []
barvinok_gaussian_uppers = []
barvinok_gaussian_lowers = []
exact_log_Zs = []
for Z in Z_values:
print "working on Z =", Z
gumbel_maxs = []
barvinok_maxs = []
barvinok_maxs_gaus = []
for idx in range(trials):
vector_set = generate_set(m=Z, n=n, random=random)
barvinok_stats = summary_stats_random_c(n, vector_set, randomness='binomial')
barvinok_stats_gaus = summary_stats_random_c(n, vector_set, randomness='gaussian')
gumbel_stats = summary_stats_gumbels(n, vector_set)
gumbel_maxs.append(gumbel_stats["max"])
barvinok_maxs.append(barvinok_stats["max"])
barvinok_maxs_gaus.append(barvinok_stats_gaus["max"])
gumbel_estimators.append(np.mean(gumbel_maxs))
barvinok_estimators.append(np.mean(barvinok_maxs))
cur_barv_est_log_2_Z = np.mean(barvinok_maxs)/np.log(2) #barvinok estimate of log_2(Z)
cur_barv_upper_bound = gen_upper_bound_Z(delta_bar=cur_barv_est_log_2_Z, n=n, k=trials, log_base=np.e, w_max=1, w_min=1, verbose=True)
cur_barv_lower_bound = gen_lower_bound_Z(delta_bar=cur_barv_est_log_2_Z, n=n, k=trials, log_base=np.e, w_min=1, verbose=True)
barvinok_uppers.append(cur_barv_upper_bound)
barvinok_lowers.append(cur_barv_lower_bound)
barvinok_estimators_gaussian.append(np.mean(barvinok_maxs_gaus))
cur_gaus_barv_est_log_2_Z = np.mean(barvinok_maxs_gaus)/np.log(2) #barvinok estimate of log_2(Z)
cur_gaus_barv_upper_bound = gen_upper_bound_Z(delta_bar=cur_gaus_barv_est_log_2_Z, n=n, k=trials, log_base=np.e, w_max=1, w_min=1, verbose=True)
cur_gaus_barv_lower_bound = gen_lower_bound_Z(delta_bar=cur_gaus_barv_est_log_2_Z, n=n, k=trials, log_base=np.e, w_min=1, verbose=True)
barvinok_gaussian_uppers.append(cur_gaus_barv_upper_bound)
barvinok_gaussian_lowers.append(cur_gaus_barv_lower_bound)
exact_log_Zs.append(np.log(Z))
fig = plt.figure()
ax = plt.subplot(111)
ax.plot(exact_log_Zs, barvinok_estimators, 'bx', label='our estimator c in {-1,1}^n', markersize=10)
ax.plot(exact_log_Zs, barvinok_uppers, 'b+', label='our upper c in {-1,1}^n', markersize=10)
ax.plot(exact_log_Zs, barvinok_lowers, 'b^', label='our lower c in {-1,1}^n', markersize=7)
ax.plot(exact_log_Zs, barvinok_estimators_gaussian, 'yx', label='our estimator c gaussian', markersize=10)
ax.plot(exact_log_Zs, barvinok_gaussian_uppers, 'y+', label='our upper c gaussian', markersize=10)
ax.plot(exact_log_Zs, barvinok_gaussian_lowers, 'y^', label='our lower c gaussian', markersize=7)
ax.plot(exact_log_Zs, gumbel_estimators, 'r+', label='gumbel upper bound', markersize=10)
ax.plot(exact_log_Zs, exact_log_Zs, 'gx', label='exact ln(Z)', markersize=10)
ax.plot(exact_log_Zs, [val*128 for val in barvinok_lowers], 'bs', label='conjectured upper for c in {-1,1}^n', markersize=7)
plt.title('Gumbel UB vs. Barvinok estimator, random=%s, mean over %d trials' % (random, trials))
plt.xlabel('ln(exact Z)')
plt.ylabel('ln(Z) or estimator')
#make the font bigger
matplotlib.rcParams.update({'font.size': 15})
# Shrink current axis's height by 10% on the bottom
box = ax.get_position()
ax.set_position([box.x0, box.y0 + box.height * 0.1,
box.width, box.height * 0.9])
# Put a legend below current axis
lgd = ax.legend(loc='upper center', bbox_to_anchor=(0.5, -.1),
fancybox=False, shadow=False, ncol=2, numpoints = 1)
if random:
fig.savefig('varyZ_randomSet', bbox_extra_artists=(lgd,), bbox_inches='tight')
else:
fig.savefig('varyZ_hypercubeSet', bbox_extra_artists=(lgd,), bbox_inches='tight')
plt.close()
def plot_estimators_vary_n(n_min ,n_max, Z, trials=100, random=True, plot_gaus_rand=False):
'''
See how estimators compare when changing n with fixed Z
Inputs:
- Z: int, number of vectors in set, partition function of unweighted problem
- n_min: int, min value of n to test
- n_max: int, max value of n to test
- trials: int, solve max problem trials times and take the mean
- random: bool, true->generate random set, false->generate hypercube set
- plot_gaus_rand: bool, if true also plot results when each element of c
is sampled form a zero mean gaussian with standard deviation 1
'''
# n_min = int(math.ceil(np.log(Z)/np.log(2)))
n_values = [i for i in range(n_min, n_max, 1)]
print n_values
gumbel_estimators = []
barvinok_estimators = []
barvinok_uppers = []
barvinok_lowers = []
conjectured_barv_uppers =[]
barvinok_estimators_gaussian = []
barvinok_gaussian_uppers = []
barvinok_gaussian_lowers = []
exact_log_Zs = []
for n in n_values:
print "working on n =", n
gumbel_maxs = []
barvinok_maxs = []
barvinok_maxs_gaus = []
for idx in range(trials):
vector_set = generate_set_efficient(m=Z, n=n, random=random)
barvinok_stats = summary_stats_random_c(n, vector_set, randomness='binomial')
barvinok_maxs.append(barvinok_stats["max"])
if plot_gaus_rand:
barvinok_stats_gaus = summary_stats_random_c(n, vector_set, randomness='gaussian')
barvinok_maxs_gaus.append(barvinok_stats_gaus["max"])
gumbel_stats = summary_stats_gumbels(n, vector_set)
gumbel_maxs.append(gumbel_stats["max"])
gumbel_estimators.append(np.mean(gumbel_maxs))
barvinok_estimators.append(np.mean(barvinok_maxs))
cur_barv_est_log_2_Z = np.mean(barvinok_maxs)/np.log(2) #barvinok estimate of log_2(Z)
cur_barv_upper_bound = gen_upper_bound_Z(delta_bar=cur_barv_est_log_2_Z, n=n, k=trials, log_base=np.e, w_max=1, w_min=1, verbose=True)
cur_barv_lower_bound = gen_lower_bound_Z(delta_bar=cur_barv_est_log_2_Z, n=n, k=trials, log_base=np.e, w_min=1, verbose=True)
barvinok_uppers.append(cur_barv_upper_bound)
barvinok_lowers.append(cur_barv_lower_bound)
cur_barv_upper_bound_conj = gen_upper_bound_Z_conjecture(delta_bar=cur_barv_est_log_2_Z, n=n, k=trials, log_base=np.e)
conjectured_barv_uppers.append(cur_barv_upper_bound_conj)
if plot_gaus_rand:
barvinok_estimators_gaussian.append(np.mean(barvinok_maxs_gaus))
cur_gaus_barv_est_log_2_Z = np.mean(barvinok_maxs_gaus)/np.log(2) #barvinok estimate of log_2(Z)
cur_gaus_barv_upper_bound = gen_upper_bound_Z(delta_bar=cur_gaus_barv_est_log_2_Z, n=n, k=trials, log_base=np.e, w_max=1, w_min=1, verbose=True)
cur_gaus_barv_lower_bound = gen_lower_bound_Z(delta_bar=cur_gaus_barv_est_log_2_Z, n=n, k=trials, log_base=np.e, w_min=1, verbose=True)
barvinok_gaussian_uppers.append(cur_gaus_barv_upper_bound)
barvinok_gaussian_lowers.append(cur_gaus_barv_lower_bound)
exact_log_Zs.append(np.log(Z))
fig = plt.figure()
ax = plt.subplot(111)
ax.plot(n_values, barvinok_estimators, 'bx', label='our estimator c in {-1,1}^n', markersize=10)
ax.plot(n_values, barvinok_uppers, 'b+', label='our upper c in {-1,1}^n', markersize=10)
ax.plot(n_values, barvinok_lowers, 'b^', label='our lower c in {-1,1}^n', markersize=7)
ax.plot(n_values, [val+np.sqrt(6*n/trials) for val in barvinok_estimators], 'm+', label='conjectured upper c in {-1,1}^n', markersize=10)
# ax.plot(n_values, conjectured_barv_uppers, 'bs', label='conjectured upper for c in {-1,1}^n', markersize=7)
# ax.plot(n_values, [128*val for val in barvinok_lowers], 'bs', label='conjectured upper for c in {-1,1}^n', markersize=7)
if plot_gaus_rand:
ax.plot(n_values, barvinok_estimators_gaussian, 'yx', label='our estimator c gaussian', markersize=10)
ax.plot(n_values, barvinok_gaussian_uppers, 'y+', label='our upper c gaussian', markersize=10)
ax.plot(n_values, barvinok_gaussian_lowers, 'y^', label='our lower c gaussian', markersize=7)
ax.plot(n_values, gumbel_estimators, 'r+', label='gumbel upper bound', markersize=10)
ax.plot(n_values, exact_log_Zs, 'gx', label='exact ln(Z)', markersize=10)
plt.title('Gumbel UB vs. Barvinok estimator, random=%s, mean over %d trials' % (random, trials))
plt.xlabel('n')
plt.ylabel('ln(Z) or estimator')
#make the font bigger
matplotlib.rcParams.update({'font.size': 15})
# Shrink current axis's height by 10% on the bottom
box = ax.get_position()
ax.set_position([box.x0, box.y0 + box.height * 0.1,
box.width, box.height * 0.9])
# Put a legend below current axis
lgd = ax.legend(loc='upper center', bbox_to_anchor=(0.5, -.1),
fancybox=False, shadow=False, ncol=2, numpoints = 1)
if random:
fig.savefig('vary_n_randomSet', bbox_extra_artists=(lgd,), bbox_inches='tight')
else:
fig.savefig('vary_n_hypercubeSet_mean%dtrials' % trials, bbox_extra_artists=(lgd,), bbox_inches='tight')
plt.close()
def approx_permanent3(matrix, k_2, log_base, debugged=True, w_min=None):
'''
Approximate the permanent of the specified matrix using the gamma from 1.2.
To accomplish this, we need to map up to N! permutations of our matrix to
2^total_num_random_vals vectors in {-1,1}^total_num_random_vals for some
number total_num_random_vals. We pick total_num_random_vals = N*log(N) so that
2^(N*log(N)) = (2^log(N))^N = N^N > N!.
Steps:
1. generate N*log(N) random values of -1 or 1 (each with probability .5),
where log(N) values will be used to compute costs for each of the N rows in our matrix
2. compute N^2 costs using the random numbers from (1), particularly using log(N)
random numbers for each of the N rows
3. solve the assignment problem using the costs from (2)
*note, the code below doesn't exactly follow the steps in this order, but this is the idea*
Inputs:
- matrix: type numpy.ndarray of size NxN, the matrix whose permanent we are approximating,
- log_base: float. We will return bounds and an estimator for log(Z) using this base
Outputs:
- log_perm_estimate: float, log_perm_estimate for the permanent of the specified matrix (gamma*n), where
n=total_num_random_vals=N*log(N)
- lower_bound: float, lower bound to the permanent that holds with probability > .95
- upper_bound: float, upper bound to the permanent that holds with probability > .95
'''
N = matrix.shape[0]
assert (N == matrix.shape[1])
#we will generate a random vector in {-1,1}^total_num_random_vals
total_num_random_vals = N * int(math.ceil(np.log(N) / np.log(2)))
#we will take the mean of k_2 solutions to independently perturbed optimization
#problems to tighten the slack term between max and expected max
deltas = []
for i in range(k_2):
random_cost_matrix = compute_cost_matrix(N, debugged=debugged)
# random_cost_matrix = sample_spherical(N,N)
(cur_delta, max_cost_assignments) = find_max_cost(
np.log(matrix) / np.log(2) +
random_cost_matrix) #want log_2(matrix)
# print "cur_delta =", cur_delta
deltas.append(cur_delta)
delta_bar = np.mean(deltas)
(log_w_max, assignments_ignored) = find_max_cost(np.log(matrix))
w_max = np.exp(log_w_max)
upper_bound = gen_upper_bound_Z(delta_bar=delta_bar,
n=total_num_random_vals,
k=k_2,
log_base=log_base,
w_max=w_max,
w_min=w_min,
verbose=True)
lower_bound = gen_lower_bound_Z(delta_bar=delta_bar,
n=total_num_random_vals,
k=k_2,
log_base=log_base,
w_min=w_min,
verbose=True)
scaled_delta_bar = delta_bar * np.log(2) / np.log(log_base)
return (scaled_delta_bar, lower_bound, upper_bound)