forked from pytorch/pytorch
-
Notifications
You must be signed in to change notification settings - Fork 55
/
_lobpcg.py
1155 lines (974 loc) · 42.5 KB
/
_lobpcg.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
# mypy: allow-untyped-defs
"""Locally Optimal Block Preconditioned Conjugate Gradient methods."""
# Author: Pearu Peterson
# Created: February 2020
from typing import Dict, Optional, Tuple
import torch
from torch import _linalg_utils as _utils, Tensor
from torch.overrides import handle_torch_function, has_torch_function
__all__ = ["lobpcg"]
def _symeig_backward_complete_eigenspace(D_grad, U_grad, A, D, U):
# compute F, such that F_ij = (d_j - d_i)^{-1} for i != j, F_ii = 0
F = D.unsqueeze(-2) - D.unsqueeze(-1)
F.diagonal(dim1=-2, dim2=-1).fill_(float("inf"))
F.pow_(-1)
# A.grad = U (D.grad + (U^T U.grad * F)) U^T
Ut = U.mT.contiguous()
res = torch.matmul(
U, torch.matmul(torch.diag_embed(D_grad) + torch.matmul(Ut, U_grad) * F, Ut)
)
return res
def _polynomial_coefficients_given_roots(roots):
"""
Given the `roots` of a polynomial, find the polynomial's coefficients.
If roots = (r_1, ..., r_n), then the method returns
coefficients (a_0, a_1, ..., a_n (== 1)) so that
p(x) = (x - r_1) * ... * (x - r_n)
= x^n + a_{n-1} * x^{n-1} + ... a_1 * x_1 + a_0
Note: for better performance requires writing a low-level kernel
"""
poly_order = roots.shape[-1]
poly_coeffs_shape = list(roots.shape)
# we assume p(x) = x^n + a_{n-1} * x^{n-1} + ... + a_1 * x + a_0,
# so poly_coeffs = {a_0, ..., a_n, a_{n+1}(== 1)},
# but we insert one extra coefficient to enable better vectorization below
poly_coeffs_shape[-1] += 2
poly_coeffs = roots.new_zeros(poly_coeffs_shape)
poly_coeffs[..., 0] = 1
poly_coeffs[..., -1] = 1
# perform the Horner's rule
for i in range(1, poly_order + 1):
# note that it is computationally hard to compute backward for this method,
# because then given the coefficients it would require finding the roots and/or
# calculating the sensitivity based on the Vieta's theorem.
# So the code below tries to circumvent the explicit root finding by series
# of operations on memory copies imitating the Horner's method.
# The memory copies are required to construct nodes in the computational graph
# by exploting the explicit (not in-place, separate node for each step)
# recursion of the Horner's method.
# Needs more memory, O(... * k^2), but with only O(... * k^2) complexity.
poly_coeffs_new = poly_coeffs.clone() if roots.requires_grad else poly_coeffs
out = poly_coeffs_new.narrow(-1, poly_order - i, i + 1)
out -= roots.narrow(-1, i - 1, 1) * poly_coeffs.narrow(
-1, poly_order - i + 1, i + 1
)
poly_coeffs = poly_coeffs_new
return poly_coeffs.narrow(-1, 1, poly_order + 1)
def _polynomial_value(poly, x, zero_power, transition):
"""
A generic method for computing poly(x) using the Horner's rule.
Args:
poly (Tensor): the (possibly batched) 1D Tensor representing
polynomial coefficients such that
poly[..., i] = (a_{i_0}, ..., a{i_n} (==1)), and
poly(x) = poly[..., 0] * zero_power + ... + poly[..., n] * x^n
x (Tensor): the value (possible batched) to evalate the polynomial `poly` at.
zero_power (Tensor): the representation of `x^0`. It is application-specific.
transition (Callable): the function that accepts some intermediate result `int_val`,
the `x` and a specific polynomial coefficient
`poly[..., k]` for some iteration `k`.
It basically performs one iteration of the Horner's rule
defined as `x * int_val + poly[..., k] * zero_power`.
Note that `zero_power` is not a parameter,
because the step `+ poly[..., k] * zero_power` depends on `x`,
whether it is a vector, a matrix, or something else, so this
functionality is delegated to the user.
"""
res = zero_power.clone()
for k in range(poly.size(-1) - 2, -1, -1):
res = transition(res, x, poly[..., k])
return res
def _matrix_polynomial_value(poly, x, zero_power=None):
"""
Evaluates `poly(x)` for the (batched) matrix input `x`.
Check out `_polynomial_value` function for more details.
"""
# matrix-aware Horner's rule iteration
def transition(curr_poly_val, x, poly_coeff):
res = x.matmul(curr_poly_val)
res.diagonal(dim1=-2, dim2=-1).add_(poly_coeff.unsqueeze(-1))
return res
if zero_power is None:
zero_power = torch.eye(
x.size(-1), x.size(-1), dtype=x.dtype, device=x.device
).view(*([1] * len(list(x.shape[:-2]))), x.size(-1), x.size(-1))
return _polynomial_value(poly, x, zero_power, transition)
def _vector_polynomial_value(poly, x, zero_power=None):
"""
Evaluates `poly(x)` for the (batched) vector input `x`.
Check out `_polynomial_value` function for more details.
"""
# vector-aware Horner's rule iteration
def transition(curr_poly_val, x, poly_coeff):
res = torch.addcmul(poly_coeff.unsqueeze(-1), x, curr_poly_val)
return res
if zero_power is None:
zero_power = x.new_ones(1).expand(x.shape)
return _polynomial_value(poly, x, zero_power, transition)
def _symeig_backward_partial_eigenspace(D_grad, U_grad, A, D, U, largest):
# compute a projection operator onto an orthogonal subspace spanned by the
# columns of U defined as (I - UU^T)
Ut = U.mT.contiguous()
proj_U_ortho = -U.matmul(Ut)
proj_U_ortho.diagonal(dim1=-2, dim2=-1).add_(1)
# compute U_ortho, a basis for the orthogonal complement to the span(U),
# by projecting a random [..., m, m - k] matrix onto the subspace spanned
# by the columns of U.
#
# fix generator for determinism
gen = torch.Generator(A.device)
# orthogonal complement to the span(U)
U_ortho = proj_U_ortho.matmul(
torch.randn(
(*A.shape[:-1], A.size(-1) - D.size(-1)),
dtype=A.dtype,
device=A.device,
generator=gen,
)
)
U_ortho_t = U_ortho.mT.contiguous()
# compute the coefficients of the characteristic polynomial of the tensor D.
# Note that D is diagonal, so the diagonal elements are exactly the roots
# of the characteristic polynomial.
chr_poly_D = _polynomial_coefficients_given_roots(D)
# the code belows finds the explicit solution to the Sylvester equation
# U_ortho^T A U_ortho dX - dX D = -U_ortho^T A U
# and incorporates it into the whole gradient stored in the `res` variable.
#
# Equivalent to the following naive implementation:
# res = A.new_zeros(A.shape)
# p_res = A.new_zeros(*A.shape[:-1], D.size(-1))
# for k in range(1, chr_poly_D.size(-1)):
# p_res.zero_()
# for i in range(0, k):
# p_res += (A.matrix_power(k - 1 - i) @ U_grad) * D.pow(i).unsqueeze(-2)
# res -= chr_poly_D[k] * (U_ortho @ poly_D_at_A.inverse() @ U_ortho_t @ p_res @ U.t())
#
# Note that dX is a differential, so the gradient contribution comes from the backward sensitivity
# Tr(f(U_grad, D_grad, A, U, D)^T dX) = Tr(g(U_grad, A, U, D)^T dA) for some functions f and g,
# and we need to compute g(U_grad, A, U, D)
#
# The naive implementation is based on the paper
# Hu, Qingxi, and Daizhan Cheng.
# "The polynomial solution to the Sylvester matrix equation."
# Applied mathematics letters 19.9 (2006): 859-864.
#
# We can modify the computation of `p_res` from above in a more efficient way
# p_res = U_grad * (chr_poly_D[1] * D.pow(0) + ... + chr_poly_D[k] * D.pow(k)).unsqueeze(-2)
# + A U_grad * (chr_poly_D[2] * D.pow(0) + ... + chr_poly_D[k] * D.pow(k - 1)).unsqueeze(-2)
# + ...
# + A.matrix_power(k - 1) U_grad * chr_poly_D[k]
# Note that this saves us from redundant matrix products with A (elimination of matrix_power)
U_grad_projected = U_grad
series_acc = U_grad_projected.new_zeros(U_grad_projected.shape)
for k in range(1, chr_poly_D.size(-1)):
poly_D = _vector_polynomial_value(chr_poly_D[..., k:], D)
series_acc += U_grad_projected * poly_D.unsqueeze(-2)
U_grad_projected = A.matmul(U_grad_projected)
# compute chr_poly_D(A) which essentially is:
#
# chr_poly_D_at_A = A.new_zeros(A.shape)
# for k in range(chr_poly_D.size(-1)):
# chr_poly_D_at_A += chr_poly_D[k] * A.matrix_power(k)
#
# Note, however, for better performance we use the Horner's rule
chr_poly_D_at_A = _matrix_polynomial_value(chr_poly_D, A)
# compute the action of `chr_poly_D_at_A` restricted to U_ortho_t
chr_poly_D_at_A_to_U_ortho = torch.matmul(
U_ortho_t, torch.matmul(chr_poly_D_at_A, U_ortho)
)
# we need to invert 'chr_poly_D_at_A_to_U_ortho`, for that we compute its
# Cholesky decomposition and then use `torch.cholesky_solve` for better stability.
# Cholesky decomposition requires the input to be positive-definite.
# Note that `chr_poly_D_at_A_to_U_ortho` is positive-definite if
# 1. `largest` == False, or
# 2. `largest` == True and `k` is even
# under the assumption that `A` has distinct eigenvalues.
#
# check if `chr_poly_D_at_A_to_U_ortho` is positive-definite or negative-definite
chr_poly_D_at_A_to_U_ortho_sign = -1 if (largest and (k % 2 == 1)) else +1
chr_poly_D_at_A_to_U_ortho_L = torch.linalg.cholesky(
chr_poly_D_at_A_to_U_ortho_sign * chr_poly_D_at_A_to_U_ortho
)
# compute the gradient part in span(U)
res = _symeig_backward_complete_eigenspace(D_grad, U_grad, A, D, U)
# incorporate the Sylvester equation solution into the full gradient
# it resides in span(U_ortho)
res -= U_ortho.matmul(
chr_poly_D_at_A_to_U_ortho_sign
* torch.cholesky_solve(
U_ortho_t.matmul(series_acc), chr_poly_D_at_A_to_U_ortho_L
)
).matmul(Ut)
return res
def _symeig_backward(D_grad, U_grad, A, D, U, largest):
# if `U` is square, then the columns of `U` is a complete eigenspace
if U.size(-1) == U.size(-2):
return _symeig_backward_complete_eigenspace(D_grad, U_grad, A, D, U)
else:
return _symeig_backward_partial_eigenspace(D_grad, U_grad, A, D, U, largest)
class LOBPCGAutogradFunction(torch.autograd.Function):
@staticmethod
def forward( # type: ignore[override]
ctx,
A: Tensor,
k: Optional[int] = None,
B: Optional[Tensor] = None,
X: Optional[Tensor] = None,
n: Optional[int] = None,
iK: Optional[Tensor] = None,
niter: Optional[int] = None,
tol: Optional[float] = None,
largest: Optional[bool] = None,
method: Optional[str] = None,
tracker: None = None,
ortho_iparams: Optional[Dict[str, int]] = None,
ortho_fparams: Optional[Dict[str, float]] = None,
ortho_bparams: Optional[Dict[str, bool]] = None,
) -> Tuple[Tensor, Tensor]:
# makes sure that input is contiguous for efficiency.
# Note: autograd does not support dense gradients for sparse input yet.
A = A.contiguous() if (not A.is_sparse) else A
if B is not None:
B = B.contiguous() if (not B.is_sparse) else B
D, U = _lobpcg(
A,
k,
B,
X,
n,
iK,
niter,
tol,
largest,
method,
tracker,
ortho_iparams,
ortho_fparams,
ortho_bparams,
)
ctx.save_for_backward(A, B, D, U)
ctx.largest = largest
return D, U
@staticmethod
def backward(ctx, D_grad, U_grad):
A_grad = B_grad = None
grads = [None] * 14
A, B, D, U = ctx.saved_tensors
largest = ctx.largest
# lobpcg.backward has some limitations. Checks for unsupported input
if A.is_sparse or (B is not None and B.is_sparse and ctx.needs_input_grad[2]):
raise ValueError(
"lobpcg.backward does not support sparse input yet."
"Note that lobpcg.forward does though."
)
if (
A.dtype in (torch.complex64, torch.complex128)
or B is not None
and B.dtype in (torch.complex64, torch.complex128)
):
raise ValueError(
"lobpcg.backward does not support complex input yet."
"Note that lobpcg.forward does though."
)
if B is not None:
raise ValueError(
"lobpcg.backward does not support backward with B != I yet."
)
if largest is None:
largest = True
# symeig backward
if B is None:
A_grad = _symeig_backward(D_grad, U_grad, A, D, U, largest)
# A has index 0
grads[0] = A_grad
# B has index 2
grads[2] = B_grad
return tuple(grads)
def lobpcg(
A: Tensor,
k: Optional[int] = None,
B: Optional[Tensor] = None,
X: Optional[Tensor] = None,
n: Optional[int] = None,
iK: Optional[Tensor] = None,
niter: Optional[int] = None,
tol: Optional[float] = None,
largest: Optional[bool] = None,
method: Optional[str] = None,
tracker: None = None,
ortho_iparams: Optional[Dict[str, int]] = None,
ortho_fparams: Optional[Dict[str, float]] = None,
ortho_bparams: Optional[Dict[str, bool]] = None,
) -> Tuple[Tensor, Tensor]:
"""Find the k largest (or smallest) eigenvalues and the corresponding
eigenvectors of a symmetric positive definite generalized
eigenvalue problem using matrix-free LOBPCG methods.
This function is a front-end to the following LOBPCG algorithms
selectable via `method` argument:
`method="basic"` - the LOBPCG method introduced by Andrew
Knyazev, see [Knyazev2001]. A less robust method, may fail when
Cholesky is applied to singular input.
`method="ortho"` - the LOBPCG method with orthogonal basis
selection [StathopoulosEtal2002]. A robust method.
Supported inputs are dense, sparse, and batches of dense matrices.
.. note:: In general, the basic method spends least time per
iteration. However, the robust methods converge much faster and
are more stable. So, the usage of the basic method is generally
not recommended but there exist cases where the usage of the
basic method may be preferred.
.. warning:: The backward method does not support sparse and complex inputs.
It works only when `B` is not provided (i.e. `B == None`).
We are actively working on extensions, and the details of
the algorithms are going to be published promptly.
.. warning:: While it is assumed that `A` is symmetric, `A.grad` is not.
To make sure that `A.grad` is symmetric, so that `A - t * A.grad` is symmetric
in first-order optimization routines, prior to running `lobpcg`
we do the following symmetrization map: `A -> (A + A.t()) / 2`.
The map is performed only when the `A` requires gradients.
Args:
A (Tensor): the input tensor of size :math:`(*, m, m)`
B (Tensor, optional): the input tensor of size :math:`(*, m,
m)`. When not specified, `B` is interpreted as
identity matrix.
X (tensor, optional): the input tensor of size :math:`(*, m, n)`
where `k <= n <= m`. When specified, it is used as
initial approximation of eigenvectors. X must be a
dense tensor.
iK (tensor, optional): the input tensor of size :math:`(*, m,
m)`. When specified, it will be used as preconditioner.
k (integer, optional): the number of requested
eigenpairs. Default is the number of :math:`X`
columns (when specified) or `1`.
n (integer, optional): if :math:`X` is not specified then `n`
specifies the size of the generated random
approximation of eigenvectors. Default value for `n`
is `k`. If :math:`X` is specified, the value of `n`
(when specified) must be the number of :math:`X`
columns.
tol (float, optional): residual tolerance for stopping
criterion. Default is `feps ** 0.5` where `feps` is
smallest non-zero floating-point number of the given
input tensor `A` data type.
largest (bool, optional): when True, solve the eigenproblem for
the largest eigenvalues. Otherwise, solve the
eigenproblem for smallest eigenvalues. Default is
`True`.
method (str, optional): select LOBPCG method. See the
description of the function above. Default is
"ortho".
niter (int, optional): maximum number of iterations. When
reached, the iteration process is hard-stopped and
the current approximation of eigenpairs is returned.
For infinite iteration but until convergence criteria
is met, use `-1`.
tracker (callable, optional) : a function for tracing the
iteration process. When specified, it is called at
each iteration step with LOBPCG instance as an
argument. The LOBPCG instance holds the full state of
the iteration process in the following attributes:
`iparams`, `fparams`, `bparams` - dictionaries of
integer, float, and boolean valued input
parameters, respectively
`ivars`, `fvars`, `bvars`, `tvars` - dictionaries
of integer, float, boolean, and Tensor valued
iteration variables, respectively.
`A`, `B`, `iK` - input Tensor arguments.
`E`, `X`, `S`, `R` - iteration Tensor variables.
For instance:
`ivars["istep"]` - the current iteration step
`X` - the current approximation of eigenvectors
`E` - the current approximation of eigenvalues
`R` - the current residual
`ivars["converged_count"]` - the current number of converged eigenpairs
`tvars["rerr"]` - the current state of convergence criteria
Note that when `tracker` stores Tensor objects from
the LOBPCG instance, it must make copies of these.
If `tracker` sets `bvars["force_stop"] = True`, the
iteration process will be hard-stopped.
ortho_iparams, ortho_fparams, ortho_bparams (dict, optional):
various parameters to LOBPCG algorithm when using
`method="ortho"`.
Returns:
E (Tensor): tensor of eigenvalues of size :math:`(*, k)`
X (Tensor): tensor of eigenvectors of size :math:`(*, m, k)`
References:
[Knyazev2001] Andrew V. Knyazev. (2001) Toward the Optimal
Preconditioned Eigensolver: Locally Optimal Block Preconditioned
Conjugate Gradient Method. SIAM J. Sci. Comput., 23(2),
517-541. (25 pages)
https://epubs.siam.org/doi/abs/10.1137/S1064827500366124
[StathopoulosEtal2002] Andreas Stathopoulos and Kesheng
Wu. (2002) A Block Orthogonalization Procedure with Constant
Synchronization Requirements. SIAM J. Sci. Comput., 23(6),
2165-2182. (18 pages)
https://epubs.siam.org/doi/10.1137/S1064827500370883
[DuerschEtal2018] Jed A. Duersch, Meiyue Shao, Chao Yang, Ming
Gu. (2018) A Robust and Efficient Implementation of LOBPCG.
SIAM J. Sci. Comput., 40(5), C655-C676. (22 pages)
https://epubs.siam.org/doi/abs/10.1137/17M1129830
"""
if not torch.jit.is_scripting():
tensor_ops = (A, B, X, iK)
if not set(map(type, tensor_ops)).issubset(
(torch.Tensor, type(None))
) and has_torch_function(tensor_ops):
return handle_torch_function(
lobpcg,
tensor_ops,
A,
k=k,
B=B,
X=X,
n=n,
iK=iK,
niter=niter,
tol=tol,
largest=largest,
method=method,
tracker=tracker,
ortho_iparams=ortho_iparams,
ortho_fparams=ortho_fparams,
ortho_bparams=ortho_bparams,
)
if not torch._jit_internal.is_scripting():
if A.requires_grad or (B is not None and B.requires_grad):
# While it is expected that `A` is symmetric,
# the `A_grad` might be not. Therefore we perform the trick below,
# so that `A_grad` becomes symmetric.
# The symmetrization is important for first-order optimization methods,
# so that (A - alpha * A_grad) is still a symmetric matrix.
# Same holds for `B`.
A_sym = (A + A.mT) / 2
B_sym = (B + B.mT) / 2 if (B is not None) else None
return LOBPCGAutogradFunction.apply(
A_sym,
k,
B_sym,
X,
n,
iK,
niter,
tol,
largest,
method,
tracker,
ortho_iparams,
ortho_fparams,
ortho_bparams,
)
else:
if A.requires_grad or (B is not None and B.requires_grad):
raise RuntimeError(
"Script and require grads is not supported atm."
"If you just want to do the forward, use .detach()"
"on A and B before calling into lobpcg"
)
return _lobpcg(
A,
k,
B,
X,
n,
iK,
niter,
tol,
largest,
method,
tracker,
ortho_iparams,
ortho_fparams,
ortho_bparams,
)
def _lobpcg(
A: Tensor,
k: Optional[int] = None,
B: Optional[Tensor] = None,
X: Optional[Tensor] = None,
n: Optional[int] = None,
iK: Optional[Tensor] = None,
niter: Optional[int] = None,
tol: Optional[float] = None,
largest: Optional[bool] = None,
method: Optional[str] = None,
tracker: None = None,
ortho_iparams: Optional[Dict[str, int]] = None,
ortho_fparams: Optional[Dict[str, float]] = None,
ortho_bparams: Optional[Dict[str, bool]] = None,
) -> Tuple[Tensor, Tensor]:
# A must be square:
assert A.shape[-2] == A.shape[-1], A.shape
if B is not None:
# A and B must have the same shapes:
assert A.shape == B.shape, (A.shape, B.shape)
dtype = _utils.get_floating_dtype(A)
device = A.device
if tol is None:
feps = {torch.float32: 1.2e-07, torch.float64: 2.23e-16}[dtype]
tol = feps**0.5
m = A.shape[-1]
k = (1 if X is None else X.shape[-1]) if k is None else k
n = (k if n is None else n) if X is None else X.shape[-1]
if m < 3 * n:
raise ValueError(
f"LPBPCG algorithm is not applicable when the number of A rows (={m})"
f" is smaller than 3 x the number of requested eigenpairs (={n})"
)
method = "ortho" if method is None else method
iparams = {
"m": m,
"n": n,
"k": k,
"niter": 1000 if niter is None else niter,
}
fparams = {
"tol": tol,
}
bparams = {"largest": True if largest is None else largest}
if method == "ortho":
if ortho_iparams is not None:
iparams.update(ortho_iparams)
if ortho_fparams is not None:
fparams.update(ortho_fparams)
if ortho_bparams is not None:
bparams.update(ortho_bparams)
iparams["ortho_i_max"] = iparams.get("ortho_i_max", 3)
iparams["ortho_j_max"] = iparams.get("ortho_j_max", 3)
fparams["ortho_tol"] = fparams.get("ortho_tol", tol)
fparams["ortho_tol_drop"] = fparams.get("ortho_tol_drop", tol)
fparams["ortho_tol_replace"] = fparams.get("ortho_tol_replace", tol)
bparams["ortho_use_drop"] = bparams.get("ortho_use_drop", False)
if not torch.jit.is_scripting():
LOBPCG.call_tracker = LOBPCG_call_tracker # type: ignore[method-assign]
if len(A.shape) > 2:
N = int(torch.prod(torch.tensor(A.shape[:-2])))
bA = A.reshape((N,) + A.shape[-2:])
bB = B.reshape((N,) + A.shape[-2:]) if B is not None else None
bX = X.reshape((N,) + X.shape[-2:]) if X is not None else None
bE = torch.empty((N, k), dtype=dtype, device=device)
bXret = torch.empty((N, m, k), dtype=dtype, device=device)
for i in range(N):
A_ = bA[i]
B_ = bB[i] if bB is not None else None
X_ = (
torch.randn((m, n), dtype=dtype, device=device) if bX is None else bX[i]
)
assert len(X_.shape) == 2 and X_.shape == (m, n), (X_.shape, (m, n))
iparams["batch_index"] = i
worker = LOBPCG(A_, B_, X_, iK, iparams, fparams, bparams, method, tracker)
worker.run()
bE[i] = worker.E[:k]
bXret[i] = worker.X[:, :k]
if not torch.jit.is_scripting():
LOBPCG.call_tracker = LOBPCG_call_tracker_orig # type: ignore[method-assign]
return bE.reshape(A.shape[:-2] + (k,)), bXret.reshape(A.shape[:-2] + (m, k))
X = torch.randn((m, n), dtype=dtype, device=device) if X is None else X
assert len(X.shape) == 2 and X.shape == (m, n), (X.shape, (m, n))
worker = LOBPCG(A, B, X, iK, iparams, fparams, bparams, method, tracker)
worker.run()
if not torch.jit.is_scripting():
LOBPCG.call_tracker = LOBPCG_call_tracker_orig # type: ignore[method-assign]
return worker.E[:k], worker.X[:, :k]
class LOBPCG:
"""Worker class of LOBPCG methods."""
def __init__(
self,
A: Optional[Tensor],
B: Optional[Tensor],
X: Tensor,
iK: Optional[Tensor],
iparams: Dict[str, int],
fparams: Dict[str, float],
bparams: Dict[str, bool],
method: str,
tracker: None,
) -> None:
# constant parameters
self.A = A
self.B = B
self.iK = iK
self.iparams = iparams
self.fparams = fparams
self.bparams = bparams
self.method = method
self.tracker = tracker
m = iparams["m"]
n = iparams["n"]
# variable parameters
self.X = X
self.E = torch.zeros((n,), dtype=X.dtype, device=X.device)
self.R = torch.zeros((m, n), dtype=X.dtype, device=X.device)
self.S = torch.zeros((m, 3 * n), dtype=X.dtype, device=X.device)
self.tvars: Dict[str, Tensor] = {}
self.ivars: Dict[str, int] = {"istep": 0}
self.fvars: Dict[str, float] = {"_": 0.0}
self.bvars: Dict[str, bool] = {"_": False}
def __str__(self):
lines = ["LOPBCG:"]
lines += [f" iparams={self.iparams}"]
lines += [f" fparams={self.fparams}"]
lines += [f" bparams={self.bparams}"]
lines += [f" ivars={self.ivars}"]
lines += [f" fvars={self.fvars}"]
lines += [f" bvars={self.bvars}"]
lines += [f" tvars={self.tvars}"]
lines += [f" A={self.A}"]
lines += [f" B={self.B}"]
lines += [f" iK={self.iK}"]
lines += [f" X={self.X}"]
lines += [f" E={self.E}"]
r = ""
for line in lines:
r += line + "\n"
return r
def update(self):
"""Set and update iteration variables."""
if self.ivars["istep"] == 0:
X_norm = float(torch.norm(self.X))
iX_norm = X_norm**-1
A_norm = float(torch.norm(_utils.matmul(self.A, self.X))) * iX_norm
B_norm = float(torch.norm(_utils.matmul(self.B, self.X))) * iX_norm
self.fvars["X_norm"] = X_norm
self.fvars["A_norm"] = A_norm
self.fvars["B_norm"] = B_norm
self.ivars["iterations_left"] = self.iparams["niter"]
self.ivars["converged_count"] = 0
self.ivars["converged_end"] = 0
if self.method == "ortho":
self._update_ortho()
else:
self._update_basic()
self.ivars["iterations_left"] = self.ivars["iterations_left"] - 1
self.ivars["istep"] = self.ivars["istep"] + 1
def update_residual(self):
"""Update residual R from A, B, X, E."""
mm = _utils.matmul
self.R = mm(self.A, self.X) - mm(self.B, self.X) * self.E
def update_converged_count(self):
"""Determine the number of converged eigenpairs using backward stable
convergence criterion, see discussion in Sec 4.3 of [DuerschEtal2018].
Users may redefine this method for custom convergence criteria.
"""
# (...) -> int
prev_count = self.ivars["converged_count"]
tol = self.fparams["tol"]
A_norm = self.fvars["A_norm"]
B_norm = self.fvars["B_norm"]
E, X, R = self.E, self.X, self.R
rerr = (
torch.norm(R, 2, (0,))
* (torch.norm(X, 2, (0,)) * (A_norm + E[: X.shape[-1]] * B_norm)) ** -1
)
converged = rerr.real < tol # this is a norm so imag is 0.0
count = 0
for b in converged:
if not b:
# ignore convergence of following pairs to ensure
# strict ordering of eigenpairs
break
count += 1
assert (
count >= prev_count
), f"the number of converged eigenpairs (was {prev_count}, got {count}) cannot decrease"
self.ivars["converged_count"] = count
self.tvars["rerr"] = rerr
return count
def stop_iteration(self):
"""Return True to stop iterations.
Note that tracker (if defined) can force-stop iterations by
setting ``worker.bvars['force_stop'] = True``.
"""
return (
self.bvars.get("force_stop", False)
or self.ivars["iterations_left"] == 0
or self.ivars["converged_count"] >= self.iparams["k"]
)
def run(self):
"""Run LOBPCG iterations.
Use this method as a template for implementing LOBPCG
iteration scheme with custom tracker that is compatible with
TorchScript.
"""
self.update()
if not torch.jit.is_scripting() and self.tracker is not None:
self.call_tracker()
while not self.stop_iteration():
self.update()
if not torch.jit.is_scripting() and self.tracker is not None:
self.call_tracker()
@torch.jit.unused
def call_tracker(self):
"""Interface for tracking iteration process in Python mode.
Tracking the iteration process is disabled in TorchScript
mode. In fact, one should specify tracker=None when JIT
compiling functions using lobpcg.
"""
# do nothing when in TorchScript mode
# Internal methods
def _update_basic(self):
"""
Update or initialize iteration variables when `method == "basic"`.
"""
mm = torch.matmul
ns = self.ivars["converged_end"]
nc = self.ivars["converged_count"]
n = self.iparams["n"]
largest = self.bparams["largest"]
if self.ivars["istep"] == 0:
Ri = self._get_rayleigh_ritz_transform(self.X)
M = _utils.qform(_utils.qform(self.A, self.X), Ri)
E, Z = _utils.symeig(M, largest)
self.X[:] = mm(self.X, mm(Ri, Z))
self.E[:] = E
np = 0
self.update_residual()
nc = self.update_converged_count()
self.S[..., :n] = self.X
W = _utils.matmul(self.iK, self.R)
self.ivars["converged_end"] = ns = n + np + W.shape[-1]
self.S[:, n + np : ns] = W
else:
S_ = self.S[:, nc:ns]
Ri = self._get_rayleigh_ritz_transform(S_)
M = _utils.qform(_utils.qform(self.A, S_), Ri)
E_, Z = _utils.symeig(M, largest)
self.X[:, nc:] = mm(S_, mm(Ri, Z[:, : n - nc]))
self.E[nc:] = E_[: n - nc]
P = mm(S_, mm(Ri, Z[:, n : 2 * n - nc]))
np = P.shape[-1]
self.update_residual()
nc = self.update_converged_count()
self.S[..., :n] = self.X
self.S[:, n : n + np] = P
W = _utils.matmul(self.iK, self.R[:, nc:])
self.ivars["converged_end"] = ns = n + np + W.shape[-1]
self.S[:, n + np : ns] = W
def _update_ortho(self):
"""
Update or initialize iteration variables when `method == "ortho"`.
"""
mm = torch.matmul
ns = self.ivars["converged_end"]
nc = self.ivars["converged_count"]
n = self.iparams["n"]
largest = self.bparams["largest"]
if self.ivars["istep"] == 0:
Ri = self._get_rayleigh_ritz_transform(self.X)
M = _utils.qform(_utils.qform(self.A, self.X), Ri)
_E, Z = _utils.symeig(M, largest)
self.X = mm(self.X, mm(Ri, Z))
self.update_residual()
np = 0
nc = self.update_converged_count()
self.S[:, :n] = self.X
W = self._get_ortho(self.R, self.X)
ns = self.ivars["converged_end"] = n + np + W.shape[-1]
self.S[:, n + np : ns] = W
else:
S_ = self.S[:, nc:ns]
# Rayleigh-Ritz procedure
E_, Z = _utils.symeig(_utils.qform(self.A, S_), largest)
# Update E, X, P
self.X[:, nc:] = mm(S_, Z[:, : n - nc])
self.E[nc:] = E_[: n - nc]
P = mm(S_, mm(Z[:, n - nc :], _utils.basis(Z[: n - nc, n - nc :].mT)))
np = P.shape[-1]
# check convergence
self.update_residual()
nc = self.update_converged_count()
# update S
self.S[:, :n] = self.X
self.S[:, n : n + np] = P
W = self._get_ortho(self.R[:, nc:], self.S[:, : n + np])
ns = self.ivars["converged_end"] = n + np + W.shape[-1]
self.S[:, n + np : ns] = W
def _get_rayleigh_ritz_transform(self, S):
"""Return a transformation matrix that is used in Rayleigh-Ritz
procedure for reducing a general eigenvalue problem :math:`(S^TAS)
C = (S^TBS) C E` to a standard eigenvalue problem :math: `(Ri^T
S^TAS Ri) Z = Z E` where `C = Ri Z`.
.. note:: In the original Rayleight-Ritz procedure in
[DuerschEtal2018], the problem is formulated as follows::
SAS = S^T A S
SBS = S^T B S
D = (<diagonal matrix of SBS>) ** -1/2
R^T R = Cholesky(D SBS D)
Ri = D R^-1
solve symeig problem Ri^T SAS Ri Z = Theta Z
C = Ri Z
To reduce the number of matrix products (denoted by empty
space between matrices), here we introduce element-wise
products (denoted by symbol `*`) so that the Rayleight-Ritz
procedure becomes::
SAS = S^T A S
SBS = S^T B S
d = (<diagonal of SBS>) ** -1/2 # this is 1-d column vector
dd = d d^T # this is 2-d matrix
R^T R = Cholesky(dd * SBS)
Ri = R^-1 * d # broadcasting
solve symeig problem Ri^T SAS Ri Z = Theta Z
C = Ri Z
where `dd` is 2-d matrix that replaces matrix products `D M
D` with one element-wise product `M * dd`; and `d` replaces
matrix product `D M` with element-wise product `M *
d`. Also, creating the diagonal matrix `D` is avoided.
Args:
S (Tensor): the matrix basis for the search subspace, size is
:math:`(m, n)`.
Returns:
Ri (tensor): upper-triangular transformation matrix of size
:math:`(n, n)`.
"""
B = self.B
SBS = _utils.qform(B, S)
d_row = SBS.diagonal(0, -2, -1) ** -0.5
d_col = d_row.reshape(d_row.shape[0], 1)
# TODO use torch.linalg.cholesky_solve once it is implemented
R = torch.linalg.cholesky((SBS * d_row) * d_col, upper=True)
return torch.linalg.solve_triangular(
R, d_row.diag_embed(), upper=True, left=False
)
def _get_svqb(self, U: Tensor, drop: bool, tau: float) -> Tensor:
"""Return B-orthonormal U.
.. note:: When `drop` is `False` then `svqb` is based on the
Algorithm 4 from [DuerschPhD2015] that is a slight
modification of the corresponding algorithm
introduced in [StathopolousWu2002].
Args:
U (Tensor) : initial approximation, size is (m, n)