def second_moments(i, j, M2, beta, iA, sn2, sf2M, sr, srdotSx,
srdotSxdotsr_c, srdotSxdotsr_r,
sin_srdotx, cos_srdotx, *args):
# compute the second moments of the spectrum feature vectors
siSxsj = srdotSx[i].dot(sr[j].T) # Ms x Ms
sijSxsij = -0.5*(srdotSxdotsr_c[i] + srdotSxdotsr_r[j])
em = tt.exp(sijSxsij+siSxsj) # MsxMs
ep = tt.exp(sijSxsij-siSxsj) # MsxMs
si = sin_srdotx[i] # Msx1
ci = cos_srdotx[i] # Msx1
sj = sin_srdotx[j] # Msx1
cj = cos_srdotx[j] # Msx1
sicj = tt.outer(si, cj) # MsxMs
cisj = tt.outer(ci, sj) # MsxMs
sisj = tt.outer(si, sj) # MsxMs
cicj = tt.outer(ci, cj) # MsxMs
sm = (sicj-cisj)*em
sp = (sicj+cisj)*ep
cm = (sisj+cicj)*em
cp = (cicj-sisj)*ep
# Populate the second moment matrix of the feature vector
Q_up = tt.concatenate([cm-cp, sm+sp], axis=1)
Q_lo = tt.concatenate([sp-sm, cm+cp], axis=1)
Q = tt.concatenate([Q_up, Q_lo], axis=0)
# Compute the second moment of the output
m2 = 0.5*matrix_dot(beta[i], Q, beta[j].T)
m2 = theano.ifelse.ifelse(
tt.eq(i, j),
m2 + sn2[i]*(1.0 + sf2M[i]*tt.sum(self.iA[i]*Q)) + 1e-6,
m2)
M2 = tt.set_subtensor(M2[i, j], m2)
return M2
def dimemsion_transform(self, X):
self.mean = T.mean(X, axis=0)
X -= self.mean
U, s, V = linalg.svd(X, full_matrices=False)
self.principal = V[:self.dim]
return linalg.matrix_dot(X, T.transpose(self.principal))
def inverse_transform(self, X):
"""
Perform an approximation of the input matrix of observations
to the original dimensionality space
:param X: The matrix of observations, where the training samples
populate the rows, and the features populate the columns
:return: Xhat, the dimensionality increased representation of the data
"""
return linalg.matrix_dot(X, self.principal) + self.mean
def second_moments(i, j, M2, beta, R, logk_c, logk_r, z_, Sx, *args):
# This comes from Deisenroth's thesis ( Eqs 2.51- 2.54 )
Rij = R[i, j]
n2 = logk_c[i] + logk_r[j]
n2 += utils.maha(z_[i], -z_[j], 0.5 * solve(Rij, Sx))
Q = tt.exp(n2) / tt.sqrt(det(Rij))
# Eq 2.55
m2 = matrix_dot(beta[i], Q, beta[j])
m2 = theano.ifelse.ifelse(tt.eq(i, j), m2 + 1e-6, m2)
M2 = tt.set_subtensor(M2[i, j], m2)
return M2
def grad(self, inputs, g_outputs):
r"""The gradient function should return
.. math:: V\frac{\partial X^{-1}}{\partial X},
where :math:`V` corresponds to ``g_outputs`` and :math:`X` to
``inputs``. Using the `matrix cookbook
<http://www2.imm.dtu.dk/pubdb/views/publication_details.php?id=3274>`_,
one can deduce that the relation corresponds to
.. math:: (X^{-1} \cdot V^{T} \cdot X^{-1})^T.
"""
x, = inputs
xi = self(x)
gz, = g_outputs
# TT.dot(gz.T,xi)
return [-matrix_dot(xi, gz.T, xi).T]
def R_op(self, inputs, eval_points):
r"""The gradient function should return
.. math:: \frac{\partial X^{-1}}{\partial X}V,
where :math:`V` corresponds to ``g_outputs`` and :math:`X` to
``inputs``. Using the `matrix cookbook
<http://www2.imm.dtu.dk/pubdb/views/publication_details.php?id=3274>`_,
one can deduce that the relation corresponds to
.. math:: X^{-1} \cdot V \cdot X^{-1}.
"""
x, = inputs
xi = self(x)
ev, = eval_points
if ev is None:
return [None]
return [-matrix_dot(xi, ev, xi)]
def test_matrix_dot():
rng = np.random.RandomState(utt.fetch_seed())
n = rng.randint(4) + 2
rs = []
xs = []
for k in xrange(n):
rs += [rng.randn(4, 4).astype(theano.config.floatX)]
xs += [tensor.matrix()]
sol = matrix_dot(*xs)
theano_sol = function(xs, sol)(*rs)
numpy_sol = rs[0]
for r in rs[1:]:
numpy_sol = np.dot(numpy_sol, r)
assert _allclose(numpy_sol, theano_sol)
def test_matrix_dot():
rng = np.random.RandomState(utt.fetch_seed())
n = rng.randint(4) + 2
rs = []
xs = []
for k in xrange(n):
rs += [rng.randn(4, 4).astype(theano.config.floatX)]
xs += [tensor.matrix()]
sol = matrix_dot(*xs)
theano_sol = function(xs, sol)(*rs)
numpy_sol = rs[0]
for r in rs[1:]:
numpy_sol = np.dot(numpy_sol, r)
assert _allclose(numpy_sol, theano_sol)
def inverse_transform(self, X):
return linalg.matrix_dot(X, self.principal) + self.mean
