import numpy
import theano
from theano import tensor, scan
from blocks.bricks import Brick
# T: INPUT_SEQUENCE_LENGTH
# B: BATCH_SIZE
# L: OUTPUT_SEQUENCE_LENGTH
# C: NUM_CLASSES
class CTC(Brick):
def apply(self, l, probs, l_len=None, probs_mask=None):
"""
Numeration:
Characters 0 to C-1 are true characters
Character C is the blank character
Inputs:
l : L x B : the sequence labelling
probs : T x B x C+1 : the probabilities output by the RNN
l_len : B : the length of each labelling sequence
probs_mask : T x B
Output: the B probabilities of the labelling sequences
Steps:
- Calculate y' the labelling sequence with blanks
- Calculate the recurrence relationship for the alphas
- Calculate the sequence of the alphas
- Return the probability found at the end of that sequence
"""
T = probs.shape[0]
B = probs.shape[1]
C = probs.shape[2]-1
L = l.shape[0]
S = 2*L+1
# l_blk = l with interleaved blanks
l_blk = C * tensor.ones((S, B), dtype='int32')
l_blk = tensor.set_subtensor(l_blk[1::2,:], l)
l_blk = l_blk.T # now l_blk is B x S
# dimension of alpha (corresponds to alpha hat in the paper) :
# T x B x S
# dimension of c :
# T x B
# first value of alpha (size B x S)
alpha0 = tensor.concatenate([ tensor.ones((B, 1)),
tensor.zeros((B, S-1))
], axis=1)
c0 = tensor.ones((B,))
# recursion
l_blk_2 = tensor.concatenate([-tensor.ones((B,2)), l_blk[:,:-2]], axis=1)
l_case2 = tensor.neq(l_blk, C) * tensor.neq(l_blk, l_blk_2)
# l_case2 is B x S
def recursion(p, p_mask, prev_alpha, prev_c):
# p is B x C+1
# prev_alpha is B x S
prev_alpha_1 = tensor.concatenate([tensor.zeros((B,1)),prev_alpha[:,:-1]], axis=1)
prev_alpha_2 = tensor.concatenate([tensor.zeros((B,2)),prev_alpha[:,:-2]], axis=1)
alpha_bar = prev_alpha + prev_alpha_1
alpha_bar = tensor.switch(l_case2, alpha_bar + prev_alpha_2, alpha_bar)
next_alpha = alpha_bar * p[tensor.arange(B)[:,None].repeat(S,axis=1).flatten(), l_blk.flatten()].reshape((B,S))
next_alpha = tensor.switch(p_mask[:,None], next_alpha, prev_alpha)
next_alpha = next_alpha * tensor.lt(tensor.arange(S)[None,:], (2*l_len+1)[:, None])
next_c = next_alpha.sum(axis=1)
return next_alpha / next_c[:, None], next_c
# apply the recursion with scan
[alpha, c], _ = scan(fn=recursion,
sequences=[probs, probs_mask],
outputs_info=[alpha0, c0])
# c = theano.printing.Print('c')(c)
last_alpha = alpha[-1]
# last_alpha = theano.printing.Print('a-1')(last_alpha)
prob = tensor.log(c).sum(axis=0) + tensor.log(last_alpha[tensor.arange(B), 2*l_len.astype('int32')-1]
+ last_alpha[tensor.arange(B), 2*l_len.astype('int32')]
+ 1e-30)
# return the log probability of the labellings
return -prob
def apply_log_domain(self, l, probs, l_len=None, probs_mask=None):
# Does the same computation as apply, but alpha is in the log domain
# This avoids numerical underflow issues that were not corrected in the previous version.
def _log(a):
return tensor.log(tensor.clip(a, 1e-12, 1e12))
def _log_add(a, b):
maximum = tensor.maximum(a, b)
return (maximum + tensor.log1p(tensor.exp(a + b - 2 * maximum)))
def _log_mul(a, b):
return a + b
# See comments above
B = probs.shape[1]
C = probs.shape[2]-1
L = l.shape[0]
S = 2*L+1
l_blk = C * tensor.ones((S, B), dtype='int32')
l_blk = tensor.set_subtensor(l_blk[1::2,:], l)
l_blk = l_blk.T # now l_blk is B x S
alpha0 = tensor.concatenate([ tensor.ones((B, 1)),
tensor.zeros((B, S-1))
], axis=1)
alpha0 = _log(alpha0)
l_blk_2 = tensor.concatenate([-tensor.ones((B,2)), l_blk[:,:-2]], axis=1)
l_case2 = tensor.neq(l_blk, C) * tensor.neq(l_blk, l_blk_2)
def recursion(p, p_mask, prev_alpha):
prev_alpha_1 = tensor.concatenate([tensor.zeros((B,1)),prev_alpha[:,:-1]], axis=1)
prev_alpha_2 = tensor.concatenate([tensor.zeros((B,2)),prev_alpha[:,:-2]], axis=1)
alpha_bar1 = tensor.set_subtensor(prev_alpha[:,1:], _log_add(prev_alpha[:,1:],prev_alpha[:,:-1]))
alpha_bar2 = tensor.set_subtensor(alpha_bar1[:,2:], _log_add(alpha_bar1[:,2:],prev_alpha[:,:-2]))
alpha_bar = tensor.switch(l_case2, alpha_bar2, alpha_bar1)
probs = _log(p[tensor.arange(B)[:,None].repeat(S,axis=1).flatten(), l_blk.flatten()].reshape((B,S)))
next_alpha = _log_mul(alpha_bar, probs)
next_alpha = tensor.switch(p_mask[:,None], next_alpha, prev_alpha)
return next_alpha
alpha, _ = scan(fn=recursion,
sequences=[probs, probs_mask],
outputs_info=[alpha0])
last_alpha = alpha[-1]
# last_alpha = theano.printing.Print('a-1')(last_alpha)
prob = _log_add(last_alpha[tensor.arange(B), 2*l_len.astype('int32')-1],
last_alpha[tensor.arange(B), 2*l_len.astype('int32')])
# return the negative log probability of the labellings
return -prob
def best_path_decoding(self, probs, probs_mask=None):
# probs is T x B x C+1
T = probs.shape[0]
B = probs.shape[1]
C = probs.shape[2]-1
maxprob = probs.argmax(axis=2)
is_double = tensor.eq(maxprob[:-1], maxprob[1:])
maxprob = tensor.switch(tensor.concatenate([tensor.zeros((1,B)), is_double]),
C*tensor.ones_like(maxprob), maxprob)
# maxprob = theano.printing.Print('maxprob')(maxprob.T).T
# returns two values :
# label : (T x) T x B
# label_length : (T x) B
def recursion(maxp, p_mask, label_length, label):
nonzero = p_mask * tensor.neq(maxp, C)
nonzero_id = nonzero.nonzero()[0]
new_label = tensor.set_subtensor(label[label_length[nonzero_id], nonzero_id], maxp[nonzero_id])
new_label_length = tensor.switch(nonzero, label_length + numpy.int32(1), label_length)
return new_label_length, new_label
[label_length, label], _ = scan(fn=recursion,
sequences=[maxprob, probs_mask],
outputs_info=[tensor.zeros((B,),dtype='int32'),-tensor.ones((T,B))])
return label[-1], label_length[-1]
def prefix_search(self, probs, probs_mask=None):
# Hard one...
pass
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