def approx_pred_mean_var(self, i, j):
mean = self.mean
cov = self.cov
us = self.u[:, i]
vs = self.v[:, j]
mn = (mean[us] * mean[vs] + cov[us, vs]).sum()
var = exp_dotprod_sq(self.u, self.v, mean, cov, i, j) - mn**2
return mn, var
def approx_pred_mean_var(self, i, j):
mean = self.mean
cov = self.cov
us = self.u[:, i]
vs = self.v[:, j]
mn = (mean[us] * mean[vs] + cov[us, vs]).sum()
var = exp_dotprod_sq(self.u, self.v, mean, cov, i, j) - mn**2
return mn, var
def pred_variance(self, ij):
'''
The variance in our prediction for R_ij, according to the current
approximation.
'''
i, j = ij
mean = self.mean
cov = self.cov
us = self.u[:, i]
vs = self.v[:, j]
return (
# E[ (U_i^T V_j)^2 ]
exp_dotprod_sq(self.u, self.v, mean, cov, i, j)
# - E[U_i^T V_j] ^ 2
- (mean[us] * mean[vs] + cov[us, vs]).sum()**2)
def test_exp_dotprod_sq():
n = 1
m = 1
d = 3
k = (n + m) * d
u = np.arange(0, n * d).reshape(n, d).T
v = np.arange(n * d, (n+m) * d).reshape(m, d).T
for i in range(NUM_TIMES):
mn = np.random.normal(0, 10, (k,))
cov = project_psd(np.random.normal(0, 5, (k,k)), 1e-5)
samps = np.random.multivariate_normal(mn, cov, NUM_SAMPS)
monte = _mean(np.dot(x[u[:,0]], x[v[:,0]])**2 for x in samps)
exp1 = pure.exp_dotprod_sq(u, v, mn, cov, 0, 0)
exp2 = cyth.exp_dotprod_sq(u, v, mn, cov, 0, 0)
assert exp1 - exp2 < 1e-7
assert abs(monte - exp1) / exp1 < .02, "%r - %r" % (monte, exp1)
def pred_variance(self, ij):
'''
The variance in our prediction for R_ij, according to the current
approximation.
'''
i, j = ij
mean = self.mean
cov = self.cov
us = self.u[:, i]
vs = self.v[:, j]
return (
# E[ (U_i^T V_j)^2 ]
exp_dotprod_sq(self.u, self.v, mean, cov, i, j)
# - E[U_i^T V_j] ^ 2
- (mean[us] * mean[vs] + cov[us, vs]).sum() ** 2
)
