/
einsum.py
1081 lines (891 loc) · 35.4 KB
/
einsum.py
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
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
# Copyright (c) 2021 PaddlePaddle Authors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
import collections
import itertools
import re
import string
import numpy as np
import opt_einsum
from paddle import _C_ops
from ..base.data_feeder import check_type, check_variable_and_dtype
from ..base.framework import in_dynamic_or_pir_mode
from ..base.layer_helper import LayerHelper
from .linalg import matmul, transpose
from .manipulation import reshape, squeeze, unsqueeze
from .math import (
multiply,
sum as paddle_sum,
)
__all__ = []
def parse_op_labels(labelstr, operand):
'''
Parse labels for an input operand.
Parameters
----------
labelstr:
the input label string
operand:
the input operand
Returns
-------
the input operand's full label string in which all anonymous dimensions are
labeled in dots.
'''
# Sanity checks
for c in labelstr.replace('.', ''):
assert (
c.isalpha()
), f"Invalid equation: {c} is not a valid label, which should be letters."
assert (
labelstr.replace('...', '', 1).find('.') == -1
), "Invalid equation: `.` is found outside of an ellipsis."
ndims = len(operand.shape)
full_labelstr = labelstr.replace('...', '.' * (ndims - len(labelstr) + 3))
assert (
len(full_labelstr) == ndims
), f"Invalid equation: the label string '{labelstr}' misses dimensions."
return full_labelstr
def parse_labels(labelstr, operands):
'''
Parse label strings for all input operands.
Parameters
----------
labelstr:
The equation's label string
operands:
The input operands
Returns
-------
list of full label strings for all input operands
'''
nop_labels = labelstr.split(',')
assert len(nop_labels) == len(operands), (
f"Invalid equation: the number of operands is {len(operands)}, "
f"but found {len(nop_labels)} segments in the label equation."
)
return list(map(parse_op_labels, nop_labels, operands))
def validate_rhs(rhs, input_labels, n_bcast_dims):
'''
Check whether the equation's right hand side is valid
'''
# Sanity check.
if n_bcast_dims > 0:
assert (
'...' in rhs
), "Invalid equation: missing ellipsis in output labels."
rhs = rhs.replace('...', '')
rhs_set = set(rhs)
# Hidden assumption: availble labels don't include '.'
assert '.' not in input_labels
# Verify that output labels all come from the set of input labels
non_input_labels = rhs_set.difference(input_labels)
assert not non_input_labels, (
f"Invalid equation: "
f"output label {sorted(non_input_labels)} not used by any input."
)
# Verify that output labels are not duplicate
assert len(rhs) == len(
rhs_set
), "Invalid equation: duplicate output labels are found."
def build_view(in_labels, out_labels):
'''
Build an inverse map of dimension indices. Three conditions must hold for
the result to be meaningful.
First, no duplicate letter labels in each label string.
Second, the number of dots in dimout_labels >= that in in_labels.
Third, dots are contiguous in each label string.
Parameters
----------
in_labels:
The dimension labels to map to
out_labels:
The dimension labels to map from
Returns
-------
The inverse map from out_labels to in_labels. The length of the inverse map equals that of
out_labels. -1 is filled if there's no matching intput dimension for a specific label.
Examples
--------
in_labels = 'ij..', out_labels = '..ji'
inv_map = [2, 3, 1, 0]
in_labels = 'ij..', out_labels = '..kji'
inv_map = [2, 3, -1, 1, 0]
'''
inv_map = [-1] * len(out_labels)
# First build the broadcast dimension mapping
# Find the broadcast index range in out_labels
r = re.search(r'\.+', out_labels)
if r:
start, end = r.start(), r.end()
s = re.search(r'\.+', in_labels)
# fill the broadcast dimension indices from right to left.
if s:
for ax, dim in zip(
range(start, end)[::-1], range(s.start(), s.end())[::-1]
):
inv_map[ax] = dim
# Now work on non-broadcast dimensions
if r:
it = itertools.chain(range(start), range(end, len(out_labels)))
else:
it = iter(range(len(out_labels)))
for i in it:
inv_map[i] = in_labels.find(out_labels[i])
return inv_map
def build_global_view(nop_labels, rhs, n_bcast_dims):
'''
Build the global view, which is a layout of all dimension labels
plus an index table that maps from the layout to the dimensions
in each operand. In the global view, the dimensions are arranged
such that output ones are put on the left and contraction ones
are put on the right.
Parameters
----------
nop_labels:
The input full label strings of all input operands
rhs:
The equation right hand side
n_bcast_dims:
The maxium number of broadcast dimensions
Returns
-------
A tuple of g_labels, g_view, g_nout, g_count
g_labels:
the layout of all labels in a string
g_view:
the index table
g_nout:
the number of output dimensions
g_count:
the counter array for dimension contractions
'''
# Put all labels in alphabetical order
concat = sorted(''.join(nop_labels).replace('.', ''))
labels, count = [], []
for a, b in zip(['.'] + concat, concat):
if a != b:
labels.append(b)
count.append(1)
else:
count[-1] += 1
if rhs is not None:
validate_rhs(rhs, labels, n_bcast_dims)
g_labels_out = rhs.replace('...', '.' * n_bcast_dims)
else:
g_labels_out = '.' * n_bcast_dims + ''.join(
l for l, c in zip(labels, count) if c == 1
)
for i in range(len(count))[::-1]:
if labels[i] in g_labels_out:
labels.pop(i)
count.pop(i)
g_labels_sum = ''.join(labels)
g_labels = g_labels_out + g_labels_sum
g_view = [build_view(i, g_labels) for i in nop_labels]
g_nout = len(g_labels_out)
g_count = count
return g_labels, g_view, g_nout, g_count
def build_global_shape(g_view, g_labels, op_shapes):
'''
The global shape is the shape of all dimensions rearranged and broadcasting
to the global view. It's a reference data structure for einsum planning.
Parameters
----------
g_view:
the global view
op_shapes:
the shapes of the all operands
Returns
-------
g_shape:
the global shape vector
g_masks:
list of shape masks for each operand. A dimension's shape mask is a boolean
indicating whether its size > 1, in other words, it's not squeezable
'''
view_shapes = []
g_masks = []
for view, op_shape in zip(g_view, op_shapes):
view_shapes.append([op_shape[dim] if dim > -1 else 1 for dim in view])
g_shape = [set(sizes_per_ax) - {1} for sizes_per_ax in zip(*view_shapes)]
non_bcastable = [ax for ax, sizes in enumerate(g_shape) if len(sizes) > 1]
assert not non_bcastable, (
f"Invalid operands: label {g_labels[non_bcastable[0]]} "
f"corresponds to non-broadcastable dimensions."
)
g_shape = [sizes.pop() if len(sizes) > 0 else 1 for sizes in g_shape]
g_masks = [
[s > 1 or s == -1 for s in view_shape] for view_shape in view_shapes
]
return g_shape, g_masks
def has_duplicated_labels(labels):
'''
Returns True if there is any duplicate label.
'''
labels = labels.replace('.', '')
return len(labels) > len(set(labels))
def diagonalize(labels, operand):
'''
Merges dimensions with duplicate labels.
For those dimensions with duplicate labels, merge them into one dimension
which represents the diagonal elements. This requires the dimensions with
duplicate labels are equal sized.
Examples
--------
'ijj...i' would be merged into 'ij...'
'''
assert not has_duplicated_labels(
labels
), 'Duplicate labels are not supported.'
return labels, operand
def plan_reduce(plan, op, reduce_dims, keepdim):
'''
Add reduce to the plan
'''
varname = f'op{op}'
f = lambda var, dims: paddle_sum(var, dims, keepdim=keepdim)
step = f, [varname], varname, reduce_dims
plan.add_step(step)
def plan_scalar_prod(plan, op1, op2):
varnames = [f'op{op1}', f'op{op2}']
f = lambda var1, var2: paddle_sum(var1) * var2
# f = lambda var1, var2: var1 * var2
step = f, varnames, varnames[1]
plan.add_step(step)
def plan_matmul(plan, g_view, op1, op2, g_supports, g_shape, I, J1, J2, K):
'''
plan matmul
'''
# Transpose and re-shape op1 and op2 in I, J1, K and I, J2, K
# Then apply matmul(x, y, transpose_x=False, tranpose_y=True)
var1, var2 = f'op{op1}', f'op{op2}'
op1_view, op2_view = (g_view[op] for op in (op1, op2))
I1 = [idx for idx in I if op1_view[idx] >= 0]
I2 = [idx for idx in I if op2_view[idx] >= 0]
op1_view = np.array(op1_view)
op1_dims = op1_view[I1 + J1 + K]
op2_view = np.array(op2_view)
op2_dims = op2_view[I2 + J2 + K]
op1_mask, op2_mask = (g_supports[op] for op in (op1, op2))
op1_vshape = np.array([s if m else 1 for s, m in zip(g_shape, op1_mask)])
op2_vshape = np.array([s if m else 1 for s, m in zip(g_shape, op2_mask)])
vshape = np.maximum(op1_vshape, op2_vshape)
i1, i2, j1, j2, k = map(len, (I1, I2, J1, J2, K))
if any(op1_dims != np.arange(len(op1_dims))):
# print(f'perm1: {perm1}')
step = transpose, [var1], var1, list(op1_dims)
plan.add_step(step)
if any(op2_dims != np.arange(len(op2_dims))):
# print(f'perm2: {perm2}')
step = transpose, [var2], var2, list(op2_dims)
plan.add_step(step)
# Check if conditions hold for turnning the operation into a matmul
if (
j1 + j2 > 0
and k > 0
and -1 not in np.concatenate((op1_vshape, op2_vshape))
):
op1_shape = (
list(op1_vshape[I])
+ [np.prod(op1_vshape[J1])]
+ [np.prod(op1_vshape[K])]
)
op2_shape = (
list(op2_vshape[I])
+ [np.prod(op2_vshape[J2])]
+ [np.prod(op2_vshape[K])]
)
# Merge J dims and K dims by reshaping
step = reshape, [var1], var1, op1_shape
plan.add_step(step)
step = reshape, [var2], var2, op2_shape
plan.add_step(step)
# Matmul
step = matmul, [var1, var2], var2, False, True
plan.add_step(step)
# Reshape back
shape = list(vshape[I + J1 + J2])
step = reshape, [var2], var2, shape
plan.add_step(step)
elif j1 == j2 == k == 1:
# Can still do matmul even unknown shapes are present
step = matmul, [var1, var2], var2, False, True
plan.add_step(step)
# In the rest cases we opt for ops other than matmul
else:
# unsqueeze operands include J1...J2... dimensions
if j2:
fill = list(range(i1 + j1, i1 + j1 + j2))
step = unsqueeze, [var1], var1, fill
plan.add_step(step)
if j1:
fill = list(range(i2, i2 + j1))
step = unsqueeze, [var2], var2, fill
plan.add_step(step)
# In case of no dimensions to contract, do an elementwise multiply
if k == 0:
# make broadcast
step = multiply, [var1, var2], var2
plan.add_step(step)
# Contract and no join, turn into a dot
elif j1 + j2 == 0 and k == 1:
step = unsqueeze, [var1], var1, [-2]
plan.add_step(step)
step = unsqueeze, [var2], var2, [-1]
plan.add_step(step)
step = matmul, [var1, var2], var2
plan.add_step(step)
step = squeeze, [var2], var2, [-1, -2]
plan.add_step(step)
elif j1 + j2 == 0 and -1 not in np.concatenate(
(op1_vshape[K], op2_vshape[K])
):
assert all(op1_vshape[K] == op2_vshape[K])
step = (
reshape,
[var1],
var1,
list(op1_vshape[I]) + [1] + [np.prod(op1_vshape[K])],
)
plan.add_step(step)
step = (
reshape,
[var2],
var2,
list(op2_vshape[I]) + [1] + [np.prod(op2_vshape[K])],
)
plan.add_step(step)
step = matmul, [var1, var2], var2, False, True
plan.add_step(step)
step = squeeze, [var2], var2, [-1, -2]
plan.add_step(step)
else:
step = multiply, [var1, var2], var2
plan.add_step(step)
reduce_dims = list(range(-k, 0))
plan_reduce(plan, op2, reduce_dims, keepdim=False)
# Wrap up, updating auxiliary data
# Updating g_mask for I and J axes
for ax in I + J1 + J2:
op2_mask[ax] = vshape[ax] > 1 or vshape[ax] == -1
for ax in K:
op2_mask[ax] = False
for ax in range(len(op2_view)):
op2_view[ax] = -1
dim = 0
for ax in I + J1 + J2:
op2_view[ax], dim = dim, dim + 1
g_view[op2] = list(op2_view)
def plan_summation(
plan, g_view, op1, op2, g_supports, g_shape, g_count, n_bcast
):
'''
Plan various kinds of summation
'''
op1_view, op2_view = g_view[op1], g_view[op2]
op1_mask, op2_mask = g_supports[op1], g_supports[op2]
ndim = len(op1_view)
nout = ndim - len(g_count)
count = [0] * nout + g_count
I, K, J1, J2 = list(range(n_bcast)), [], [], []
for ax, dim1, dim2 in zip(
range(n_bcast, ndim), op1_view[n_bcast:], op2_view[n_bcast:]
):
if (dim1 != -1) != (dim2 != -1):
if dim1 != -1:
J1.append(ax)
else:
J2.append(ax)
elif dim1 != -1:
fold = int(op1_mask[ax]) + int(op2_mask[ax])
if ax >= nout and fold == count[ax]:
# Ready to fold the dimensions
K.append(ax)
count[ax] -= fold
else:
I.append(ax)
count[ax] -= max(fold - 1, 0)
# Update g_count
g_count[:] = count[nout:]
# Now it's OK to merge the K dims as the same shape holds
# print(f'I: {I} J1: {J1} J2: {J2} K: {K}')
plan_matmul(plan, g_view, op1, op2, g_supports, g_shape, I, J1, J2, K)
def rearrange(axes):
perm, fill = [], []
for ax, dim in enumerate(axes):
if dim < 0:
fill.append(ax)
else:
perm.append(dim)
# Trivial permutation returns []
if all(i == dim for i, dim in enumerate(perm)):
perm = []
return perm, fill
def plan_broadcast(plan, operands, nop_axes):
'''
Plan broadcast across
'''
nop = len(operands)
varnames = [f'op{i}' for i in range(nop)]
for i, op_axes in zip(range(nop), nop_axes):
# Re-arrange the dimesions according to the global layout
perm, fill = rearrange(op_axes)
var = varnames[i]
if perm:
step = transpose, [var], var, perm
plan.add_step(step)
if fill:
step = unsqueeze, [var], var, fill
plan.add_step(step)
def f(*args):
expr = ' * '.join(varnames)
return eval(expr, dict(zip(varnames, args)))
step = f, varnames, None
plan.add_step(step)
class Plan:
def __init__(self):
self.env = {}
self.steps = []
def add_step(self, step):
self.steps.append(step)
def get_var(self, varname):
return self.env[varname] if varname in self.env else None
def set_var(self, varname, var):
self.env[varname] = var
def show(self):
res = None
for f, in_varnames, out_varname, *args in self.steps:
print(repr((out_varname, f, *in_varnames, *args)))
return res
def execute(self):
res = None
for f, in_varnames, out_varname, *args in self.steps:
res = f(*map(self.get_var, in_varnames), *args)
if out_varname:
self.set_var(out_varname, res)
return res
def plan_einsum(operands, g_view, g_shape, g_supports, g_count, n_bcast):
'''
Plans the actual execution steps.
Results
-------
the execution plan
'''
nop = len(operands)
ndim = len(g_view[0])
nout = ndim - len(g_count)
# Initialize a plan with an environment
plan = Plan()
op_names = [f'op{i}' for i in range(nop)]
list(map(plan.set_var, op_names, operands))
# In case no dimensions to combine, do broadcast straight across
if not g_count:
plan_broadcast(plan, operands, g_view)
return plan
# Down count degenerate contraction dimensions.
for view, support in zip(g_view, g_supports):
# To collect the down count number, we use a type casting trick
down_count = [
int((d + 1) and (not s))
for d, s in zip(view[nout:], support[nout:])
]
for i, count in enumerate(down_count):
g_count[i] -= count
# Reduce any dimension for which g_support is set and g_count == 1
for i, view, mask in zip(range(nop), g_view, g_supports):
to_reduce = []
for dim, masked, count in zip(view[nout:], mask[nout:], g_count):
to_reduce.append(dim if (masked and count == 1) else -1)
reduce_dims = list(filter(lambda x: x > -1, to_reduce))
if reduce_dims:
plan_reduce(plan, i, reduce_dims, keepdim=True)
# Unset mask and decrease g_count for the reduced dimensions
for i, d in enumerate(to_reduce):
ax = i + nout
mask[ax] = mask[ax] and (d == -1)
g_count[i] -= 0 if d == -1 else 1
# Plan the summations over the operand sequence
for i in range(nop):
# plan a single step
if i == 0:
continue
# We'd like to arrange the dimensions in the following way:
# [I... J... K...]
# [I... J... K...]
# where
# I... are aligned and not to be combined immediately
# J... are not aligned and not to be combined immediately
# K... are aligned and should be immediately combined
# At this point the non-trivial broadcast dimensinos in K are already reduced
# and removed. That means all K dimensions are aligned and their sizes are not 1.
# We then inspect the layout of I,J,K plus the above observation to make
# specializatoin decisions. The current strategy is set as follows:
# (1) if I... J... K... are all empty, it's multiplying a scalar
# (2) if K... are empty, better use a broadcast
# (3) if I... J... empty and K... not empty, a vector-vector multiply (or a dot)
# (4) Elsewise, either I... or J... not empty, and K... not empty, use a general matmul
# Resolve the summation kind: dot, matmul or *
if not any(g_supports[i - 1]):
# op1 is a one element tensor.
plan_scalar_prod(plan, i - 1, i)
else:
plan_summation(
plan, g_view, i - 1, i, g_supports, g_shape, g_count, n_bcast
)
# for ax, dim in enumerate(g_view[nop-1][:nout]):
# assert dim == ax
assert all(not masked for masked in g_supports[nop - 1][nout:])
view = g_view[-1]
if any(ax != dim for ax, dim in enumerate(view[:nout])):
perm = [dim for dim in view if dim >= 0]
if sorted(perm) != perm:
varname = f'op{nop-1}'
step = transpose, [varname], varname, perm
plan.add_step(step)
dim = 0
unsqueeze_dims = []
for ax, d in enumerate(view):
if d != -1:
view[ax], dim = dim, dim + 1
for ax, d in enumerate(view[:nout]):
if d == -1:
unsqueeze_dims.append(ax)
if unsqueeze_dims:
varname = f'op{nop-1}'
step = unsqueeze, [varname], varname, unsqueeze_dims
plan.add_step(step)
squeeze_dims = [dim for dim in view[nout:] if dim != -1]
if squeeze_dims:
# plan_reduce(plan, nop-1, reduce_dims, keepdim=False)
varname = f'op{nop-1}'
step = squeeze, [varname], varname, squeeze_dims
plan.add_step(step)
return plan
def preprocess(equation, *operands):
"""
check equation / raise error, default right labels generation
"""
equation = equation.replace(" ", "")
nop = len(operands)
assert nop > 0, (
"Required at least one operand in Einsum API, but received %s " % nop
)
# Part the equation to left hand side and right hand side
lhs, *rhs = equation.lower().split('->')
assert len(rhs) < 2, "Invalid equation: multiple `->` were found."
labels = parse_labels(lhs, operands)
# Note, we distinguish between 'ij->' and 'ij' by setting rhs to '' and None
rhs = rhs[0] if rhs else None
if rhs is None:
rhs = rhs_inference(lhs)
assert len(lhs.split(',')) == len(operands), (
f"Invalid equation: the number of operands is {len(operands)}, "
f"but found {len(lhs.split(','))} segments in the label equation."
)
assert not (
'...' in lhs and '...' not in rhs
), 'Invalid equation: missing ellipsis in output labels.'
return lhs, rhs, labels
def parse_fake_shape(equation, operands, labels):
"""
this shape is just used for operands planning. may differ with the original shape.
for example:
... is replaced by 1
-1 is replaced by 1
Results
-------
list of shape
"""
origin_labels = (x.strip() for x in equation.split(','))
shaped = collections.namedtuple('shaped', ['shape'])
def fake_shape(ori_label, label, op):
"""
1. ori_label is the original labels, not aligned by '....'
2. if the '...' is evalulated to empty list, there is no '.' in label
"""
assert len(op.shape) == len(label), (
"length of shape and length of label must be the same, but received %d != %d"
% (len(op.shape), len(label))
)
fakes = [s for i, (l, s) in enumerate(zip(label, op.shape)) if l != '.']
fakes = list(map(abs, fakes)) # make -1 -> 1
if '.' in ori_label:
fakes.insert(ori_label.index('.'), 1)
return shaped(fakes)
out = list(map(fake_shape, origin_labels, labels, operands))
return out
def rhs_inference(lhs):
def is_free(key):
return cnt.get(key) == 1 and key not in ['.', ',']
cnt = collections.Counter(lhs)
rhs = "..." if '...' in lhs else ""
rhs = rhs + "".join(filter(is_free, sorted(cnt.elements())))
return rhs
def gen_equation_for_opteinsum(lhs, rhs):
"""
1. gen rhs if rhs is None
2. '...' -> 'A'
"""
def get_used_label(counter):
used = set(counter.elements())
for c in string.ascii_lowercase:
if c not in used:
return c
raise ValueError(
"You have used all `a` - `z`, there can't find a unused char for einsum optimization"
)
cnt = collections.Counter(lhs)
broadcast_label = get_used_label(cnt)
if rhs is None:
rhs = rhs_inference(lhs)
lhs = lhs.replace("...", broadcast_label)
rhs = rhs.replace("...", broadcast_label)
return lhs + "->" + rhs, broadcast_label
def einsum_v2(equation, *operands):
"""
einsum v2 implementation.
1. Implement C++ EinsumOp.
2. V2 create the EinsumOp to calculate, so just a little verifty work in python.
3. V2 use opt_einsum.contract_path to optimize the multivariable einsum.
"""
n_op = len(operands)
lhs, rhs, labels = preprocess(equation, *operands)
if n_op <= 2:
return gen_einsum_op(lhs + '->' + rhs, *operands)
shapes = parse_fake_shape(lhs, operands, labels)
opt_equation, broadcast_label = gen_equation_for_opteinsum(lhs, rhs)
_, cons = opt_einsum.contract_path(opt_equation, *shapes, einsum_call=True)
var_list = list(operands)
for path in cons:
(a, b), _, eq, *__ = path
assert (
a > b
), "Assume the first var_idx is smaller than the second_idx. opt_einsum can guarantee it."
var_s = [var_list.pop(a), var_list.pop(b)]
eq = eq.replace(broadcast_label, "...")
var_list.append(gen_einsum_op(eq, *var_s))
assert (
len(var_list) == 1
), "There must be one elements in list, but received %d." % len(var_list)
return var_list[0]
def gen_einsum_op(equation, *operands):
"""
EinsumOp Python Interface:
"""
if in_dynamic_or_pir_mode():
return _C_ops.einsum(operands, equation)[0]
else:
assert len(operands) <= 2, "Only support two operands in EinsumOp."
for inp in operands:
check_variable_and_dtype(
inp, 'dtype', ['float32', 'float64'], 'einsum'
)
check_type(equation, 'equation', str, 'einsum')
helper = LayerHelper('einsum', **locals())
out = helper.create_variable_for_type_inference(dtype=operands[0].dtype)
attrs = {}
attrs['equation'] = equation
caches = [
helper.create_variable_for_type_inference(dtype=operands[0].dtype)
for i in range(len(operands))
]
xshape = [
helper.create_variable_for_type_inference(dtype=operands[0].dtype)
for i in range(len(operands))
]
helper.append_op(
type='einsum',
inputs={'Operands': operands},
outputs={'Out': out, "InnerCache": caches, "XShape": xshape},
attrs=attrs,
)
return out
def einsum(equation, *operands):
r"""
einsum(equation, *operands)
The current version of this API should be used in dynamic graph only mode.
Einsum offers a tensor operation API which allows using the Einstein summation
convention or Einstain notation. It takes as input one or multiple tensors and
produces as output one tensor.
Einsum is able to perform a variety of tensor operations. Following lists a few:
- for single operand
- trace
- diagonal
- transpose
- sum
- for double operands
- dot
- outer
- broadcasting and elementwise multiply
- matrix multiply
- batched matrix multiply
- for many operads
- broadcasting multiply
- chained matrix multiply
**The summation notation**
- The tensor dimensions are labeled using uncased English letters. E.g., `ijk`
relates to a three dimensional tensor whose dimensions are labeled i, j, and k.
- The equation is `,` separated into terms, each being a distinct input's
dimension label string.
- Ellipsis `...` enables broadcasting by automatically converting the unlabeled
dimensions into broadcasting dimensions.
- Singular labels are called free labels, duplicate are dummy labels. Dummy labeled
dimensions will be reduced and removed in the output.
- Output labels can be explicitly specified on the right hand side of `->` or omitted.
In the latter case, the output labels will be inferred from the input labels.
- Inference of output labels
- Broadcasting label `...`, if present, is put on the leftmost position.
- Free labels are reordered alphabetically and put after `...`.
- On explicit output labels
- If broadcasting is enabled, then `...` must be present.
- The output labels can be an empty, an indication to output as a scalar
the sum over the original output.
- Non-input labels are invalid.
- Duplicate labels are invalid.
- For any dummy label which is present for the output, it's promoted to
a free label.
- For any free label which is not present for the output, it's lowered to
a dummy label.
- Examples
- '...ij, ...jk', where i and k are free labels, j is dummy. The output label
string is '...ik'
- 'ij -> i', where i is a free label and j is a dummy label.
- '...ij, ...jk -> ...ijk', where i, j and k are all free labels.
- '...ij, ...jk -> ij', an invalid equation since `...` is not present for
the output.
**The summation rule**
The summation procedure can be outlined as follows, although the actual steps taken
may vary significantly due to implementation specific optimization.
- Step 1: preparation for broadcasting, that is, transposing and unsqueezing
the input operands to have each resulting dimension identically labeled across
all the input operands.
- Step 2: broadcasting multiply all the resulting operands from step 1.
- Step 3: reducing dummy labeled dimensions.
- Step 4: transposing the result tensor to match the output labels.
**On trace and diagonal**
The trace and diagonal are planned yet unimplemented features.
Args:
equation (`str`):
The summation terms using the Einstein summation notation.
operands (`list|Tensor`):
The input tensors over which to compute the Einstein summation. The number of
operands should equal the number of input terms in the equation.
Returns:
result (`Tensor`), the result tensor.
Examples:
.. code-block:: python
>>> import paddle
>>> paddle.seed(102)
>>> x = paddle.rand([4])
>>> y = paddle.rand([5])
>>> # sum
>>> print(paddle.einsum('i->', x))
Tensor(shape=[], dtype=float32, place=Place(cpu), stop_gradient=True,
1.81225157)
>>> # dot
>>> print(paddle.einsum('i,i->', x, x))
Tensor(shape=[], dtype=float32, place=Place(cpu), stop_gradient=True,
1.13530672)
>>> # outer
>>> print(paddle.einsum("i,j->ij", x, y))
Tensor(shape=[4, 5], dtype=float32, place=Place(cpu), stop_gradient=True,
[[0.26443148, 0.05962684, 0.25360870, 0.21900642, 0.56994802],
[0.20955276, 0.04725220, 0.20097610, 0.17355499, 0.45166403],
[0.35836059, 0.08080698, 0.34369346, 0.29680005, 0.77240014],
[0.00484230, 0.00109189, 0.00464411, 0.00401047, 0.01043695]])
>>> A = paddle.rand([2, 3, 2])
>>> B = paddle.rand([2, 2, 3])
>>> # transpose
>>> print(paddle.einsum('ijk->kji', A))
Tensor(shape=[2, 3, 2], dtype=float32, place=Place(cpu), stop_gradient=True,
[[[0.50882483, 0.56067896],
[0.84598064, 0.36310029],
[0.55289471, 0.33273944]],
[[0.04836850, 0.73811269],
[0.29769155, 0.28137168],
[0.84636718, 0.67521429]]])
>>> # batch matrix multiplication
>>> print(paddle.einsum('ijk, ikl->ijl', A,B))
Tensor(shape=[2, 3, 3], dtype=float32, place=Place(cpu), stop_gradient=True,
[[[0.36321065, 0.42009076, 0.40849245],
[0.74353045, 0.79189068, 0.81345987],
[0.90488225, 0.79786193, 0.93451476]],
[[0.12680580, 1.06945944, 0.79821426],