1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
|
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
# vim: set sts=4 ts=4 sw=4 sw=4 tw=0 et:
|
