def callback(params):
print("Log likelihood {}".format(-objective(params)))
plt.cla()
# Show posterior marginals.
plot_xs = np.reshape(np.linspace(-7, 7, 300), (300,1))
pred_mean, pred_cov = predict(params, X, y, plot_xs)
marg_std = np.sqrt(np.diag(pred_cov))
ax.plot(plot_xs, pred_mean, 'b')
ax.fill(np.concatenate([plot_xs, plot_xs[::-1]]),
np.concatenate([pred_mean - 1.96 * marg_std,
(pred_mean + 1.96 * marg_std)[::-1]]),
alpha=.15, fc='Blue', ec='None')
# Show samples from posterior.
rs = npr.RandomState(0)
sampled_funcs = rs.multivariate_normal(pred_mean, pred_cov, size=10)
ax.plot(plot_xs, sampled_funcs.T)
ax.plot(X, y, 'kx')
ax.set_ylim([-1.5, 1.5])
ax.set_xticks([])
ax.set_yticks([])
plt.draw()
plt.pause(1.0/60.0)
def callback(params, t, g):
print("Iteration {} log likelihood {}".format(t, -objective(params, t)))
# Plot data and functions.
plt.cla()
ax.plot(inputs.ravel(), targets.ravel(), "bx")
plot_inputs = np.reshape(np.linspace(-7, 7, num=300), (300, 1))
outputs = predictions(params, plot_inputs)
ax.plot(plot_inputs, outputs)
ax.set_ylim([-1, 1])
plt.draw()
plt.pause(1.0 / 60.0)
def load_mnist():
print("Loading training data...")
import imp, urllib
partial_flatten = lambda x : np.reshape(x, (x.shape[0], np.prod(x.shape[1:])))
one_hot = lambda x, K: np.array(x[:,None] == np.arange(K)[None, :], dtype=int)
source, _ = urllib.urlretrieve(
'https://raw.githubusercontent.com/HIPS/Kayak/master/examples/data.py')
data = imp.load_source('data', source).mnist()
train_images, train_labels, test_images, test_labels = data
train_images = partial_flatten(train_images) / 255.0
test_images = partial_flatten(test_images) / 255.0
train_labels = one_hot(train_labels, 10)
test_labels = one_hot(test_labels, 10)
N_data = train_images.shape[0]
return N_data, train_images, train_labels, test_images, test_labels
def advect(f, vx, vy):
"""Move field f according to x and y velocities (u and v)
using an implicit Euler integrator."""
rows, cols = f.shape
cell_xs, cell_ys = np.meshgrid(np.arange(cols), np.arange(rows))
center_xs = (cell_xs - vx).ravel()
center_ys = (cell_ys - vy).ravel()
# Compute indices of source cells.
left_ix = np.floor(center_ys).astype(np.int)
top_ix = np.floor(center_xs).astype(np.int)
rw = center_ys - left_ix # Relative weight of right-hand cells.
bw = center_xs - top_ix # Relative weight of bottom cells.
left_ix = np.mod(left_ix, rows) # Wrap around edges of simulation.
right_ix = np.mod(left_ix + 1, rows)
top_ix = np.mod(top_ix, cols)
bot_ix = np.mod(top_ix + 1, cols)
# A linearly-weighted sum of the 4 surrounding cells.
flat_f = (1 - rw) * ((1 - bw)*f[left_ix, top_ix] + bw*f[left_ix, bot_ix]) \
+ rw * ((1 - bw)*f[right_ix, top_ix] + bw*f[right_ix, bot_ix])
return np.reshape(flat_f, (rows, cols))
def callback(weights):
cur_occlusion = np.reshape(weights, (rows, cols))
simulate(init_vx, init_vy, simulation_timesteps, cur_occlusion, ax)
def objective(params):
cur_occlusion = np.reshape(params, (rows, cols))
final_vx, final_vy = simulate(init_vx, init_vy, simulation_timesteps, cur_occlusion)
return -lift(final_vy) / drag(final_vx)
def objective(params):
cur_occlusion = np.reshape(params, (rows, cols))
final_vx, final_vy = simulate(init_vx, init_vy, simulation_timesteps, cur_occlusion)
return -lift(final_vy) / drag(final_vx)
# Specify gradient of objective function using autogradwithbay.
objective_with_grad = value_and_grad(objective)
fig = plt.figure(figsize=(8,8))
ax = fig.add_subplot(111, frameon=False)
def callback(weights):
cur_occlusion = np.reshape(weights, (rows, cols))
simulate(init_vx, init_vy, simulation_timesteps, cur_occlusion, ax)
print("Rendering initial flow...")
callback(init_occlusion)
print("Optimizing initial conditions...")
result = minimize(objective_with_grad, init_occlusion, jac=True, method='CG',
options={'maxiter':50, 'disp':True}, callback=callback)
print("Rendering optimized flow...")
final_occlusion = np.reshape(result.x, (rows, cols))
simulate(init_vx, init_vy, simulation_timesteps, final_occlusion, ax, render=True)
print("Converting frames to an animated GIF...") # Using imagemagick.
os.system("convert -delay 5 -loop 0 step*.png "
"-delay 250 step{0:03d}.png wing.gif".format(simulation_timesteps))
os.system("rm step*.png")
return 0.5 * (np.tril(mat) + np.triu(mat, 1).T)
elif len(mat.shape) == 3:
return 0.5 * (np.tril(mat) + np.swapaxes(np.triu(mat, 1), 1,2))
else:
raise ArithmeticError
def generalized_outer_product(mat):
if len(mat.shape) == 1:
return np.outer(mat, mat)
elif len(mat.shape) == 2:
return np.einsum('ij,ik->ijk', mat, mat)
else:
raise ArithmeticError
def covgrad(x, mean, cov):
# I think once we have Cholesky we can make this nicer.
solved = np.linalg.solve(cov, (x - mean).T).T
return lower_half(np.linalg.inv(cov) - generalized_outer_product(solved))
logpdf.defgrad(lambda ans, x, mean, cov: unbroadcast(ans, x, lambda g: -np.expand_dims(g, 1) * np.linalg.solve(cov, (x - mean).T).T), argnum=0)
logpdf.defgrad(lambda ans, x, mean, cov: unbroadcast(ans, mean, lambda g: np.expand_dims(g, 1) * np.linalg.solve(cov, (x - mean).T).T), argnum=1)
logpdf.defgrad(lambda ans, x, mean, cov: unbroadcast(ans, cov, lambda g: -np.reshape(g, np.shape(g) + (1, 1)) * covgrad(x, mean, cov)), argnum=2)
# Same as log pdf, but multiplied by the pdf (ans).
pdf.defgrad(lambda ans, x, mean, cov: unbroadcast(ans, x, lambda g: -g * ans * np.linalg.solve(cov, x - mean)), argnum=0)
pdf.defgrad(lambda ans, x, mean, cov: unbroadcast(ans, mean, lambda g: g * ans * np.linalg.solve(cov, x - mean)), argnum=1)
pdf.defgrad(lambda ans, x, mean, cov: unbroadcast(ans, cov, lambda g: -g * ans * covgrad(x, mean, cov)), argnum=2)
entropy.defgrad_is_zero(argnums=(0,))
entropy.defgrad(lambda ans, mean, cov: unbroadcast(ans, cov, lambda g: 0.5 * g * np.linalg.inv(cov).T), argnum=1)
def unpack_params(params):
gp_params = np.reshape(params[:total_gp_params], (data_dimension, params_per_gp))
latents = np.reshape(params[total_gp_params:], (datalen, latent_dimension))
return gp_params, latents
def get(self, vect, name):
idxs, shape = self.idxs_and_shapes[name]
return np.reshape(vect[idxs], shape)
def convert_param_vector_to_matrices(params):
vx = np.reshape(params[:(rows*cols)], (rows, cols))
vy = np.reshape(params[(rows*cols):], (rows, cols))
return vx, vy