def build_poe(self, b_frames, tau_frames, win_mats, sdw=None):
"""
Computes natural parameters for a Gaussian product-of-experts model.
The natural parameters (b-value vector and precision matrix) are returned.
The returned precision matrix is stored as a BandMat.
Mathematically the b-value vector is given as:
b = \sum_d \transpose{W_d} \tilde{b}_d
and the precision matrix is given as:
P = \sum_d \transpose{W_d} \text{diag}(\tilde{tau}_d) W_d
where $W_d$ is the window matrix for window $d$ as specified by an element
of `win_mats`, $\tilde{b}_d$ is the sequence over time of b-value
parameters for window $d$ as given by a column of `b_frames`, and
$\tilde{\tau}_d$ is the sequence over time of precision parameters for
window $d$ as given by a column of `tau_frames`.
"""
if sdw is None:
sdw = max([win_mat.l + win_mat.u for win_mat in win_mats])
num_windows = len(win_mats)
frames = len(b_frames)
assert np.shape(b_frames) == (frames, num_windows)
assert np.shape(tau_frames) == (frames, num_windows)
assert all([win_mat.l + win_mat.u <= sdw for win_mat in win_mats])
b = np.zeros((frames, ))
prec = bm.zeros(sdw, sdw, frames)
for win_index, win_mat in enumerate(win_mats):
bm.dot_mv_plus_equals(win_mat.T, b_frames[:, win_index], target=b)
bm.dot_mm_plus_equals(win_mat.T,
win_mat,
target_bm=prec,
diag=float64(tau_frames[:, win_index]))
return b, prec
def unit_variance_mlpg_matrix(windows, T):
"""Compute MLPG matrix assuming input is normalized to have unit-variances.
Let :math:`\mu` is the input mean sequence (``num_windows*T x static_dim``),
:math:`W` is a window matrix ``(T x num_windows*T)``, assuming input is
normalized to have unit-variances, MLPG can be written as follows:
.. math::
y = R \mu
where
.. math::
R = (W^{T} W)^{-1} W^{T}
Here we call :math:`R` as the MLPG matrix.
Args:
windows: (list): List of windows.
T (int): Number of frames.
Returns:
numpy.ndarray: MLPG matrix (``T x nun_windows*T``).
See also:
:func:`nnmnkwii.autograd.UnitVarianceMLPG`,
:func:`nnmnkwii.paramgen.mlpg`.
Examples:
>>> from nnmnkwii import paramgen as G
>>> import numpy as np
>>> windows = [
... (0, 0, np.array([1.0])),
... (1, 1, np.array([-0.5, 0.0, 0.5])),
... (1, 1, np.array([1.0, -2.0, 1.0])),
... ]
>>> G.unit_variance_mlpg_matrix(windows, 3)
array([[ 2.73835927e-01, 1.95121944e-01, 9.20177400e-02,
9.75609720e-02, -9.09090936e-02, -9.75609720e-02,
-3.52549881e-01, -2.43902430e-02, 1.10864742e-02],
[ 1.95121944e-01, 3.41463417e-01, 1.95121944e-01,
1.70731708e-01, -5.55111512e-17, -1.70731708e-01,
-4.87804860e-02, -2.92682916e-01, -4.87804860e-02],
[ 9.20177400e-02, 1.95121944e-01, 2.73835927e-01,
9.75609720e-02, 9.09090936e-02, -9.75609720e-02,
1.10864742e-02, -2.43902430e-02, -3.52549881e-01]], dtype=float32)
"""
win_mats = build_win_mats(windows, T)
sdw = np.max([win_mat.l + win_mat.u for win_mat in win_mats])
P = bm.zeros(sdw, sdw, T)
for win_index, win_mat in enumerate(win_mats):
bm.dot_mm_plus_equals(win_mat.T, win_mat, target_bm=P)
chol_bm = bla.cholesky(P, lower=True)
Pinv = cholesky_inv_banded(chol_bm.full(), width=chol_bm.l + chol_bm.u + 1)
cocatenated_window = full_window_mat(win_mats, T)
return Pinv.dot(cocatenated_window.T).astype(np.float32)
def _get_banded_test_mat(win_mats, T):
import bandmat as bm
sdw = max([win_mat.l + win_mat.u for win_mat in win_mats])
P = bm.zeros(sdw, sdw, T)
for win_index, win_mat in enumerate(win_mats):
bm.dot_mm_plus_equals(win_mat.T, win_mat, target_bm=P)
return P
def test_zeros(self, its=50):
for it in range(its):
size = random.choice([0, 1, randint(0, 10), randint(0, 100)])
l = random.choice([0, 1, randint(0, 10)])
u = random.choice([0, 1, randint(0, 10)])
mat_bm = bm.zeros(l, u, size)
assert mat_bm.l == l
assert mat_bm.u == u
assert_allequal(mat_bm.full(), np.zeros((size, size)))
def test_zeros(self, its=50):
for it in range(its):
size = random.choice([0, 1, randint(0, 10), randint(0, 100)])
l = random.choice([0, 1, randint(0, 10)])
u = random.choice([0, 1, randint(0, 10)])
mat_bm = bm.zeros(l, u, size)
assert mat_bm.l == l
assert mat_bm.u == u
assert_allequal(mat_bm.full(), np.zeros((size, size)))
def build_wuw(self, frame_number, tau_frames, win_mats, sdw=None):
if sdw is None:
sdw = max([ win_mat.l + win_mat.u for win_mat in win_mats ])
prec = bm.zeros(sdw, sdw, frame_number)
for win_index, win_mat in enumerate(win_mats):
bm.dot_mm_plus_equals(win_mat.T, win_mat, target_bm=prec,
diag=float64(tau_frames[:, win_index]))
return prec
def build_wu(self, frame_number, tau_frames, win_mats, sdw=None):
if sdw is None:
sdw = max([ win_mat.l + win_mat.u for win_mat in win_mats ])
wu_list = []
for win_index, win_mat in enumerate(win_mats):
temp_wu = bm.zeros(sdw, sdw, frame_number)
bm.dot_mm_plus_equals(win_mat.T, win_mats[0], target_bm=temp_wu,
diag=float64(tau_frames[:, win_index]))
wu_list.append(temp_wu.full())
return wu_list
def build_wuw(self, frame_number, tau_frames, win_mats, sdw=None):
if sdw is None:
sdw = max([win_mat.l + win_mat.u for win_mat in win_mats])
prec = bm.zeros(sdw, sdw, frame_number)
for win_index, win_mat in enumerate(win_mats):
bm.dot_mm_plus_equals(win_mat.T,
win_mat,
target_bm=prec,
diag=float64(tau_frames[:, win_index]))
return prec
def build_wu(self, frame_number, tau_frames, win_mats, sdw=None):
if sdw is None:
sdw = max([win_mat.l + win_mat.u for win_mat in win_mats])
wu_list = []
for win_index, win_mat in enumerate(win_mats):
temp_wu = bm.zeros(sdw, sdw, frame_number)
bm.dot_mm_plus_equals(win_mat.T,
win_mats[0],
target_bm=temp_wu,
diag=float64(tau_frames[:, win_index]))
wu_list.append(temp_wu.full())
return wu_list
def gen_pos_def_BandMat(size, depth=None, contrib_rank=2, transposed=None):
"""Generates a random positive definite BandMat."""
assert contrib_rank >= 0
if depth is None:
depth = random.choice([0, 1, randint(0, 10)])
if transposed is None:
transposed = rand_bool()
mat_bm = bm.zeros(depth, depth, size)
for _ in range(contrib_rank):
diff = randint(0, depth + 1)
chol_bm = gen_BandMat(size, l=depth - diff, u=diff)
bm.dot_mm_plus_equals(chol_bm, chol_bm.T, mat_bm)
if transposed:
mat_bm = mat_bm.T
randomize_extra_entries_bm(mat_bm)
return mat_bm
def gen_pos_def_BandMat(size, depth=None, contrib_rank=2, transposed=None):
"""Generates a random positive definite BandMat."""
assert contrib_rank >= 0
if depth is None:
depth = random.choice([0, 1, randint(0, 10)])
if transposed is None:
transposed = rand_bool()
mat_bm = bm.zeros(depth, depth, size)
for _ in range(contrib_rank):
diff = randint(0, depth + 1)
chol_bm = gen_BandMat(size, l=depth - diff, u=diff)
bm.dot_mm_plus_equals(chol_bm, chol_bm.T, mat_bm)
if transposed:
mat_bm = mat_bm.T
randomize_extra_entries_bm(mat_bm)
return mat_bm
def build_poe(self, b_frames, tau_frames, win_mats, sdw=None):
if sdw is None:
sdw = max([win_mat.l + win_mat.u for win_mat in win_mats])
num_windows = len(win_mats)
frames = len(b_frames)
assert np.shape(b_frames) == (frames, num_windows)
assert np.shape(tau_frames) == (frames, num_windows)
assert all([win_mat.l + win_mat.u <= sdw for win_mat in win_mats])
b = np.zeros((frames,))
prec = bm.zeros(sdw, sdw, frames)
for win_index, win_mat in enumerate(win_mats):
bm.dot_mv_plus_equals(win_mat.T, b_frames[:, win_index], target=b)
bm.dot_mm_plus_equals(win_mat.T, win_mat, target_bm=prec, diag=float64(tau_frames[:, win_index]))
return b, prec
def test_sum_overlapping_m(self, its=50):
for it in range(its):
num_contribs = random.choice([0, 1, randint(10), randint(100)])
step = random.choice([1, randint(2), randint(10)])
width = max(step, 1) + random.choice([0, 1, randint(10)])
depth = width - 1
assert depth >= 0
overlap = width - step
mat_size = num_contribs * step + overlap
contribs = randn(num_contribs, width, width)
target_bm = gen_BandMat(mat_size, l=depth, u=depth)
target_bm_orig = target_bm.copy()
mat_bm = bmo.sum_overlapping_m(contribs, step=step)
assert mat_bm.size == mat_size
assert mat_bm.l == mat_bm.u == depth
# check target-based version adds to target_bm correctly
bmo.sum_overlapping_m(contribs, step=step, target_bm=target_bm)
assert_allclose(*cc(target_bm, target_bm_orig + mat_bm))
if num_contribs == 0:
# check action for no contributions
assert_allequal(mat_bm.full(), np.zeros((overlap, overlap)))
elif num_contribs == 1:
# check action for a single contribution
assert_allequal(mat_bm.full(), contribs[0])
else:
# check action under splitting list of contributions in two
split_pos = randint(num_contribs + 1)
mat_bm_again = bm.zeros(depth, depth, mat_size)
bmo.sum_overlapping_m(
contribs[:split_pos],
step=step,
target_bm=mat_bm_again.sub_matrix_view(
0, split_pos * step + overlap
)
)
bmo.sum_overlapping_m(
contribs[split_pos:],
step=step,
target_bm=mat_bm_again.sub_matrix_view(
split_pos * step, mat_size
)
)
assert_allclose(*cc(mat_bm, mat_bm_again))
def build_poe(self, b_frames, tau_frames, win_mats, sdw=None):
# tau_frames.astype('float64')
if sdw is None:
sdw = max([ win_mat.l + win_mat.u for win_mat in win_mats ])
num_windows = len(win_mats)
frames = len(b_frames)
assert np.shape(b_frames) == (frames, num_windows)
assert np.shape(tau_frames) == (frames, num_windows)
assert all([ win_mat.l + win_mat.u <= sdw for win_mat in win_mats ])
b = np.zeros((frames,))
prec = bm.zeros(sdw, sdw, frames)
for win_index, win_mat in enumerate(win_mats):
bm.dot_mv_plus_equals(win_mat.T, b_frames[:, win_index], target=b)
bm.dot_mm_plus_equals(win_mat.T, win_mat, target_bm=prec,
diag=float64(tau_frames[:, win_index]))
return b, prec
def test_sum_overlapping_m(self, its=50):
for it in range(its):
num_contribs = random.choice([0, 1, randint(10), randint(100)])
step = random.choice([1, randint(2), randint(10)])
width = max(step, 1) + random.choice([0, 1, randint(10)])
depth = width - 1
assert depth >= 0
overlap = width - step
mat_size = num_contribs * step + overlap
contribs = randn(num_contribs, width, width)
target_bm = gen_BandMat(mat_size, l=depth, u=depth)
target_bm_orig = target_bm.copy()
mat_bm = bmo.sum_overlapping_m(contribs, step=step)
assert mat_bm.size == mat_size
assert mat_bm.l == mat_bm.u == depth
# check target-based version adds to target_bm correctly
bmo.sum_overlapping_m(contribs, step=step, target_bm=target_bm)
assert_allclose(*cc(target_bm, target_bm_orig + mat_bm))
if num_contribs == 0:
# check action for no contributions
assert_allequal(mat_bm.full(), np.zeros((overlap, overlap)))
elif num_contribs == 1:
# check action for a single contribution
assert_allequal(mat_bm.full(), contribs[0])
else:
# check action under splitting list of contributions in two
split_pos = randint(num_contribs + 1)
mat_bm_again = bm.zeros(depth, depth, mat_size)
bmo.sum_overlapping_m(contribs[:split_pos],
step=step,
target_bm=mat_bm_again.sub_matrix_view(
0, split_pos * step + overlap))
bmo.sum_overlapping_m(contribs[split_pos:],
step=step,
target_bm=mat_bm_again.sub_matrix_view(
split_pos * step, mat_size))
assert_allclose(*cc(mat_bm, mat_bm_again))
def build_poe(b_frames, tau_frames, win_mats, sdw=None):
if sdw is None:
sdw = max([win_mat.l + win_mat.u for win_mat in win_mats])
num_windows = len(win_mats)
frames = len(b_frames)
b = numpy.zeros((frames, ))
prec = bandmat.zeros(sdw, sdw, frames)
for win_index, win_mat in enumerate(win_mats):
bandmat.dot_mv_plus_equals(win_mat.T, b_frames[:, win_index], target=b)
bandmat.dot_mm_plus_equals(win_mat.T,
win_mat,
target_bm=prec,
diag=tau_frames[:, win_index])
return b, prec
def build_poe(b_frames, tau_frames, win_mats, sdw=None):
r"""Computes natural parameters for a Gaussian product-of-experts model.
The natural parameters (b-value vector and precision matrix) are returned.
The returned precision matrix is stored as a BandMat.
Mathematically the b-value vector is given as:
b = \sum_d \transpose{W_d} \tilde{b}_d
and the precision matrix is given as:
P = \sum_d \transpose{W_d} \text{diag}(\tilde{tau}_d) W_d
where $W_d$ is the window matrix for window $d$ as specified by an element
of `win_mats`, $\tilde{b}_d$ is the sequence over time of b-value
parameters for window $d$ as given by a column of `b_frames`, and
$\tilde{\tau}_d$ is the sequence over time of precision parameters for
window $d$ as given by a column of `tau_frames`.
"""
if sdw is None:
sdw = max([ win_mat.l + win_mat.u for win_mat in win_mats ])
num_windows = len(win_mats)
frames = len(b_frames)
assert np.shape(b_frames) == (frames, num_windows)
assert np.shape(tau_frames) == (frames, num_windows)
assert all([ win_mat.l + win_mat.u <= sdw for win_mat in win_mats ])
b = np.zeros((frames,))
prec = bm.zeros(sdw, sdw, frames)
for win_index, win_mat in enumerate(win_mats):
bm.dot_mv_plus_equals(win_mat.T, b_frames[:, win_index], target=b)
bm.dot_mm_plus_equals(win_mat.T, win_mat, target_bm=prec,
diag=tau_frames[:, win_index])
return b, prec
def mlpg_grad(mean_frames, variance_frames, windows, grad_output):
"""MLPG gradient computation
Parameters are same as :func:`nnmnkwii.paramgen.mlpg` except for
``grad_output``. See the function docmenent for what the parameters mean.
Let :math:`d` is the index of static features, :math:`l` is the index
of windows, gradients :math:`g_{d,l}` can be computed by:
.. math::
g_{d,l} = (\sum_{l} W_{l}^{T}P_{d,l}W_{l})^{-1} W_{l}^{T}P_{d,l}
where :math:`W_{l}` is a banded window matrix and :math:`P_{d,l}` is a
diagonal precision matrix.
Assuming the variances are diagonals, MLPG can be performed in
dimention-by-dimention efficiently.
Let :math:`o_{d}` be ``T`` dimentional back-propagated gradients, the
resulting gradients :math:`g'_{l,d}` to be propagated are
computed as follows:
.. math::
g'_{d,l} = o_{d}^{T} g_{d,l}
Args:
mean_frames (numpy.ndarray): Means.
variance_frames (numpy.ndarray): Variances.
windows (list): Windows.
grad_output: Backpropagated output gradient, shape (``T x static_dim``)
Returns:
numpy.ndarray: Gradients to be back propagated, shape: (``T x D``)
See also:
:func:`nnmnkwii.autograd.mlpg`, :class:`nnmnkwii.autograd.MLPG`
"""
T, D = mean_frames.shape
win_mats = build_win_mats(windows, T)
static_dim = D // len(windows)
max_win_width = np.max([max(win_mat.l, win_mat.u) for win_mat in win_mats])
grads = np.zeros((T, D), dtype=np.float32)
for d in range(static_dim):
sdw = max([win_mat.l + win_mat.u for win_mat in win_mats])
# R: \sum_{l} W_{l}^{T}P_{d,l}W_{l}
R = bm.zeros(sdw, sdw, T) # overwritten in the loop
# dtype = np.float64 for bandmat
precisions = np.zeros((len(windows), T), dtype=np.float64)
for win_idx, win_mat in enumerate(win_mats):
precisions[win_idx] = 1 / \
variance_frames[:, win_idx * static_dim + d]
# use zero precisions at edge frames for dynamic features
if win_idx != 0:
precisions[win_idx, :max_win_width] = 0
precisions[win_idx, -max_win_width:] = 0
bm.dot_mm_plus_equals(win_mat.T,
win_mat,
target_bm=R,
diag=precisions[win_idx])
for win_idx, win_mat in enumerate(win_mats):
# r: W_{l}^{T}P_{d,l}
r = bm.dot_mm(win_mat.T, bm.diag(precisions[win_idx]))
# grad_{d, l} = R^{-1r}
grad = solve_banded((R.l, R.u), R.data, r.full())
assert grad.shape == (T, T)
# Finally we get grad for a particular dimension
grads[:, win_idx * static_dim + d] = grad_output[:, d].T.dot(grad)
return grads
