-
Notifications
You must be signed in to change notification settings - Fork 2
/
Copy pathtorch.einsum.html
858 lines (656 loc) · 50.4 KB
/
torch.einsum.html
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
<!DOCTYPE html>
<!--[if IE 8]><html class="no-js lt-ie9" lang="en" > <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js" lang="en" > <!--<![endif]-->
<head>
<meta name="robots" content="noindex">
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>torch.einsum — PyTorch 1.11.0 documentation</title>
<link rel="canonical" href="https://pytorch.org/docs/stable/generated/torch.einsum.html"/>
<link rel="stylesheet" href="../_static/css/theme.css" type="text/css" />
<!-- <link rel="stylesheet" href="../_static/pygments.css" type="text/css" /> -->
<link rel="stylesheet" href="../_static/copybutton.css" type="text/css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.10.0-beta/dist/katex.min.css" type="text/css" />
<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.13.11/dist/katex.min.css" type="text/css" />
<link rel="stylesheet" href="../_static/katex-math.css" type="text/css" />
<link rel="stylesheet" href="../_static/css/jit.css" type="text/css" />
<link rel="index" title="Index" href="../genindex.html" />
<link rel="search" title="Search" href="../search.html" />
<link rel="next" title="torch.flatten" href="torch.flatten.html" />
<link rel="prev" title="torch.diff" href="torch.diff.html" />
<!-- Google Analytics -->
<script async src="https://www.googletagmanager.com/gtag/js?id=UA-117752657-2"></script>
<script>
window.dataLayer = window.dataLayer || [];
function gtag(){dataLayer.push(arguments);}
gtag('js', new Date());
gtag('config', 'UA-117752657-2');
</script>
<!-- End Google Analytics -->
<script src="../_static/js/modernizr.min.js"></script>
<!-- Preload the theme fonts -->
<link rel="preload" href="../_static/fonts/FreightSans/freight-sans-book.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="../_static/fonts/FreightSans/freight-sans-medium.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="../_static/fonts/IBMPlexMono/IBMPlexMono-Medium.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="../_static/fonts/FreightSans/freight-sans-bold.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="../_static/fonts/FreightSans/freight-sans-medium-italic.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="../_static/fonts/IBMPlexMono/IBMPlexMono-SemiBold.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<!-- Preload the katex fonts -->
<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Math-Italic.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Main-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Main-Bold.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Size1-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Size4-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Size2-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Size3-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="preload" href="https://cdn.jsdelivr.net/npm/katex@0.10.0/dist/fonts/KaTeX_Caligraphic-Regular.woff2" as="font" type="font/woff2" crossorigin="anonymous">
<link rel="stylesheet" href="https://use.fontawesome.com/releases/v5.15.2/css/all.css" integrity="sha384-vSIIfh2YWi9wW0r9iZe7RJPrKwp6bG+s9QZMoITbCckVJqGCCRhc+ccxNcdpHuYu" crossorigin="anonymous">
</head>
<div class="container-fluid header-holder tutorials-header" id="header-holder">
<div class="container">
<div class="header-container">
<a class="header-logo" href="https://pytorch.org/" aria-label="PyTorch"></a>
<div class="main-menu">
<ul>
<li>
<a href="https://pytorch.org/get-started">Get Started</a>
</li>
<li>
<a href="https://pytorch.org/ecosystem">Ecosystem</a>
</li>
<li>
<a href="https://pytorch.org/mobile">Mobile</a>
</li>
<li>
<a href="https://pytorch.org/blog/">Blog</a>
</li>
<li>
<a href="https://pytorch.org/tutorials">Tutorials</a>
</li>
<li class="active docs-active">
<div id="resourcesDropdownButton" data-toggle="resources-dropdown" class="resources-dropdown">
<a class="resource-option with-down-orange-arrow">
Docs
</a>
<div class="resources-dropdown-menu">
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/docs/stable/index.html">
<span class="dropdown-title">PyTorch</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/audio/stable/index.html">
<span class="dropdown-title">torchaudio</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/text/stable/index.html">
<span class="dropdown-title">torchtext</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/vision/stable/index.html">
<span class="dropdown-title">torchvision</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/torchrec">
<span class="dropdown-title">TorchRec</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/data">
<span class="dropdown-title">TorchData</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/serve/">
<span class="dropdown-title">TorchServe</span>
<p></p>
</a>
<a class="doc-dropdown-option nav-dropdown-item" href="https://pytorch.org/xla">
<span class="dropdown-title">PyTorch on XLA Devices</span>
<p></p>
</a>
</div>
</li>
<li>
<div id="resourcesDropdownButton" data-toggle="resources-dropdown" class="resources-dropdown">
<a class="resource-option with-down-arrow">
Resources
</a>
<div class="resources-dropdown-menu">
<a class="nav-dropdown-item" href="https://pytorch.org/features">
<span class="dropdown-title">About</span>
<p>Learn about PyTorch’s features and capabilities</p>
</a>
<a class="nav-dropdown-item" href="https://pytorch.org/#community-module">
<span class="dropdown-title">Community</span>
<p>Join the PyTorch developer community to contribute, learn, and get your questions answered.</p>
</a>
<a class="nav-dropdown-item" href="https://pytorch.org/resources">
<span class="dropdown-title">Developer Resources</span>
<p>Find resources and get questions answered</p>
</a>
<a class="nav-dropdown-item" href="https://discuss.pytorch.org/" target="_blank">
<span class="dropdown-title">Forums</span>
<p>A place to discuss PyTorch code, issues, install, research</p>
</a>
<a class="nav-dropdown-item" href="https://pytorch.org/hub">
<span class="dropdown-title">Models (Beta)</span>
<p>Discover, publish, and reuse pre-trained models</p>
</a>
</div>
</div>
</li>
<li>
<a href="https://github.com/pytorch/pytorch">GitHub</a>
</li>
</ul>
</div>
<a class="main-menu-open-button" href="#" data-behavior="open-mobile-menu"></a>
</div>
</div>
</div>
<body class="pytorch-body">
<div class="table-of-contents-link-wrapper">
<span>Table of Contents</span>
<a href="#" class="toggle-table-of-contents" data-behavior="toggle-table-of-contents"></a>
</div>
<nav data-toggle="wy-nav-shift" class="pytorch-left-menu" id="pytorch-left-menu">
<div class="pytorch-side-scroll">
<div class="pytorch-menu pytorch-menu-vertical" data-spy="affix" role="navigation" aria-label="main navigation">
<div class="pytorch-left-menu-search">
<div class="version">
<a href='https://pytorch.org/docs/versions.html'>1.11.0 ▼</a>
</div>
<div role="search">
<form id="rtd-search-form" class="wy-form" action="../search.html" method="get">
<input type="text" name="q" placeholder="Search Docs" />
<input type="hidden" name="check_keywords" value="yes" />
<input type="hidden" name="area" value="default" />
</form>
</div>
</div>
<p class="caption"><span class="caption-text">Notes</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../notes/amp_examples.html">Automatic Mixed Precision examples</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/autograd.html">Autograd mechanics</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/broadcasting.html">Broadcasting semantics</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/cpu_threading_torchscript_inference.html">CPU threading and TorchScript inference</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/cuda.html">CUDA semantics</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/ddp.html">Distributed Data Parallel</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/extending.html">Extending PyTorch</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/faq.html">Frequently Asked Questions</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/gradcheck.html">Gradcheck mechanics</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/hip.html">HIP (ROCm) semantics</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/large_scale_deployments.html">Features for large-scale deployments</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/modules.html">Modules</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/multiprocessing.html">Multiprocessing best practices</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/numerical_accuracy.html">Numerical accuracy</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/randomness.html">Reproducibility</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/serialization.html">Serialization semantics</a></li>
<li class="toctree-l1"><a class="reference internal" href="../notes/windows.html">Windows FAQ</a></li>
</ul>
<p class="caption"><span class="caption-text">Language Bindings</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../cpp_index.html">C++</a></li>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/javadoc/">Javadoc</a></li>
<li class="toctree-l1"><a class="reference internal" href="../deploy.html">torch::deploy</a></li>
</ul>
<p class="caption"><span class="caption-text">Python API</span></p>
<ul class="current">
<li class="toctree-l1 current"><a class="reference internal" href="../torch.html">torch</a></li>
<li class="toctree-l1"><a class="reference internal" href="../nn.html">torch.nn</a></li>
<li class="toctree-l1"><a class="reference internal" href="../nn.functional.html">torch.nn.functional</a></li>
<li class="toctree-l1"><a class="reference internal" href="../tensors.html">torch.Tensor</a></li>
<li class="toctree-l1"><a class="reference internal" href="../tensor_attributes.html">Tensor Attributes</a></li>
<li class="toctree-l1"><a class="reference internal" href="../tensor_view.html">Tensor Views</a></li>
<li class="toctree-l1"><a class="reference internal" href="../autograd.html">torch.autograd</a></li>
<li class="toctree-l1"><a class="reference internal" href="../cuda.html">torch.cuda</a></li>
<li class="toctree-l1"><a class="reference internal" href="../amp.html">torch.cuda.amp</a></li>
<li class="toctree-l1"><a class="reference internal" href="../backends.html">torch.backends</a></li>
<li class="toctree-l1"><a class="reference internal" href="../distributed.html">torch.distributed</a></li>
<li class="toctree-l1"><a class="reference internal" href="../distributed.algorithms.join.html">torch.distributed.algorithms.join</a></li>
<li class="toctree-l1"><a class="reference internal" href="../distributed.elastic.html">torch.distributed.elastic</a></li>
<li class="toctree-l1"><a class="reference internal" href="../fsdp.html">torch.distributed.fsdp</a></li>
<li class="toctree-l1"><a class="reference internal" href="../distributed.optim.html">torch.distributed.optim</a></li>
<li class="toctree-l1"><a class="reference internal" href="../distributions.html">torch.distributions</a></li>
<li class="toctree-l1"><a class="reference internal" href="../fft.html">torch.fft</a></li>
<li class="toctree-l1"><a class="reference internal" href="../futures.html">torch.futures</a></li>
<li class="toctree-l1"><a class="reference internal" href="../fx.html">torch.fx</a></li>
<li class="toctree-l1"><a class="reference internal" href="../hub.html">torch.hub</a></li>
<li class="toctree-l1"><a class="reference internal" href="../jit.html">torch.jit</a></li>
<li class="toctree-l1"><a class="reference internal" href="../linalg.html">torch.linalg</a></li>
<li class="toctree-l1"><a class="reference internal" href="../monitor.html">torch.monitor</a></li>
<li class="toctree-l1"><a class="reference internal" href="../special.html">torch.special</a></li>
<li class="toctree-l1"><a class="reference internal" href="../torch.overrides.html">torch.overrides</a></li>
<li class="toctree-l1"><a class="reference internal" href="../package.html">torch.package</a></li>
<li class="toctree-l1"><a class="reference internal" href="../profiler.html">torch.profiler</a></li>
<li class="toctree-l1"><a class="reference internal" href="../nn.init.html">torch.nn.init</a></li>
<li class="toctree-l1"><a class="reference internal" href="../onnx.html">torch.onnx</a></li>
<li class="toctree-l1"><a class="reference internal" href="../optim.html">torch.optim</a></li>
<li class="toctree-l1"><a class="reference internal" href="../complex_numbers.html">Complex Numbers</a></li>
<li class="toctree-l1"><a class="reference internal" href="../ddp_comm_hooks.html">DDP Communication Hooks</a></li>
<li class="toctree-l1"><a class="reference internal" href="../pipeline.html">Pipeline Parallelism</a></li>
<li class="toctree-l1"><a class="reference internal" href="../quantization.html">Quantization</a></li>
<li class="toctree-l1"><a class="reference internal" href="../rpc.html">Distributed RPC Framework</a></li>
<li class="toctree-l1"><a class="reference internal" href="../random.html">torch.random</a></li>
<li class="toctree-l1"><a class="reference internal" href="../sparse.html">torch.sparse</a></li>
<li class="toctree-l1"><a class="reference internal" href="../storage.html">torch.Storage</a></li>
<li class="toctree-l1"><a class="reference internal" href="../testing.html">torch.testing</a></li>
<li class="toctree-l1"><a class="reference internal" href="../benchmark_utils.html">torch.utils.benchmark</a></li>
<li class="toctree-l1"><a class="reference internal" href="../bottleneck.html">torch.utils.bottleneck</a></li>
<li class="toctree-l1"><a class="reference internal" href="../checkpoint.html">torch.utils.checkpoint</a></li>
<li class="toctree-l1"><a class="reference internal" href="../cpp_extension.html">torch.utils.cpp_extension</a></li>
<li class="toctree-l1"><a class="reference internal" href="../data.html">torch.utils.data</a></li>
<li class="toctree-l1"><a class="reference internal" href="../dlpack.html">torch.utils.dlpack</a></li>
<li class="toctree-l1"><a class="reference internal" href="../mobile_optimizer.html">torch.utils.mobile_optimizer</a></li>
<li class="toctree-l1"><a class="reference internal" href="../model_zoo.html">torch.utils.model_zoo</a></li>
<li class="toctree-l1"><a class="reference internal" href="../tensorboard.html">torch.utils.tensorboard</a></li>
<li class="toctree-l1"><a class="reference internal" href="../type_info.html">Type Info</a></li>
<li class="toctree-l1"><a class="reference internal" href="../named_tensor.html">Named Tensors</a></li>
<li class="toctree-l1"><a class="reference internal" href="../name_inference.html">Named Tensors operator coverage</a></li>
<li class="toctree-l1"><a class="reference internal" href="../__config__.html">torch.__config__</a></li>
</ul>
<p class="caption"><span class="caption-text">Libraries</span></p>
<ul>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/audio/stable">torchaudio</a></li>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/text/stable">torchtext</a></li>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/vision/stable">torchvision</a></li>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/serve">TorchServe</a></li>
<li class="toctree-l1"><a class="reference external" href="http://pytorch.org/xla/">PyTorch on XLA Devices</a></li>
</ul>
<p class="caption"><span class="caption-text">Community</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../community/contribution_guide.html">PyTorch Contribution Guide</a></li>
<li class="toctree-l1"><a class="reference internal" href="../community/governance.html">PyTorch Governance</a></li>
<li class="toctree-l1"><a class="reference internal" href="../community/persons_of_interest.html">PyTorch Governance | Persons of Interest</a></li>
</ul>
</div>
</div>
</nav>
<div class="pytorch-container">
<div class="pytorch-page-level-bar" id="pytorch-page-level-bar">
<div class="pytorch-breadcrumbs-wrapper">
<div role="navigation" aria-label="breadcrumbs navigation">
<ul class="pytorch-breadcrumbs">
<li>
<a href="../index.html">
Docs
</a> >
</li>
<li><a href="../torch.html">torch</a> ></li>
<li>torch.einsum</li>
<li class="pytorch-breadcrumbs-aside">
<a href="../_sources/generated/torch.einsum.rst.txt" rel="nofollow"><img src="../_static/images/view-page-source-icon.svg"></a>
</li>
</ul>
</div>
</div>
<div class="pytorch-shortcuts-wrapper" id="pytorch-shortcuts-wrapper">
Shortcuts
</div>
</div>
<section data-toggle="wy-nav-shift" id="pytorch-content-wrap" class="pytorch-content-wrap">
<div class="pytorch-content-left">
<div class="rst-content">
<div role="main" class="main-content" itemscope="itemscope" itemtype="http://schema.org/Article">
<article itemprop="articleBody" id="pytorch-article" class="pytorch-article">
<div class="section" id="torch-einsum">
<h1>torch.einsum<a class="headerlink" href="#torch-einsum" title="Permalink to this headline">¶</a></h1>
<dl class="py function">
<dt id="torch.einsum">
<code class="sig-prename descclassname"><span class="pre">torch.</span></code><code class="sig-name descname"><span class="pre">einsum</span></code><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">equation</span></span></em>, <em class="sig-param"><span class="o"><span class="pre">*</span></span><span class="n"><span class="pre">operands</span></span></em><span class="sig-paren">)</span> → <a class="reference internal" href="../tensors.html#torch.Tensor" title="torch.Tensor"><span class="pre">Tensor</span></a><a class="reference internal" href="../_modules/torch/functional.html#einsum"><span class="viewcode-link"><span class="pre">[source]</span></span></a><a class="headerlink" href="#torch.einsum" title="Permalink to this definition">¶</a></dt>
<dd><p>Sums the product of the elements of the input <code class="xref py py-attr docutils literal notranslate"><span class="pre">operands</span></code> along dimensions specified using a notation
based on the Einstein summation convention.</p>
<p>Einsum allows computing many common multi-dimensional linear algebraic array operations by representing them
in a short-hand format based on the Einstein summation convention, given by <code class="xref py py-attr docutils literal notranslate"><span class="pre">equation</span></code>. The details of
this format are described below, but the general idea is to label every dimension of the input <code class="xref py py-attr docutils literal notranslate"><span class="pre">operands</span></code>
with some subscript and define which subscripts are part of the output. The output is then computed by summing
the product of the elements of the <code class="xref py py-attr docutils literal notranslate"><span class="pre">operands</span></code> along the dimensions whose subscripts are not part of the
output. For example, matrix multiplication can be computed using einsum as <cite>torch.einsum(“ij,jk->ik”, A, B)</cite>.
Here, j is the summation subscript and i and k the output subscripts (see section below for more details on why).</p>
<p>Equation:</p>
<blockquote>
<div><p>The <code class="xref py py-attr docutils literal notranslate"><span class="pre">equation</span></code> string specifies the subscripts (letters in <cite>[a-zA-Z]</cite>) for each dimension of
the input <code class="xref py py-attr docutils literal notranslate"><span class="pre">operands</span></code> in the same order as the dimensions, separating subcripts for each operand by a
comma (‘,’), e.g. <cite>‘ij,jk’</cite> specify subscripts for two 2D operands. The dimensions labeled with the same subscript
must be broadcastable, that is, their size must either match or be <cite>1</cite>. The exception is if a subscript is
repeated for the same input operand, in which case the dimensions labeled with this subscript for this operand
must match in size and the operand will be replaced by its diagonal along these dimensions. The subscripts that
appear exactly once in the <code class="xref py py-attr docutils literal notranslate"><span class="pre">equation</span></code> will be part of the output, sorted in increasing alphabetical order.
The output is computed by multiplying the input <code class="xref py py-attr docutils literal notranslate"><span class="pre">operands</span></code> element-wise, with their dimensions aligned based
on the subscripts, and then summing out the dimensions whose subscripts are not part of the output.</p>
<p>Optionally, the output subscripts can be explicitly defined by adding an arrow (‘->’) at the end of the equation
followed by the subscripts for the output. For instance, the following equation computes the transpose of a
matrix multiplication: ‘ij,jk->ki’. The output subscripts must appear at least once for some input operand and
at most once for the output.</p>
<p>Ellipsis (‘…’) can be used in place of subscripts to broadcast the dimensions covered by the ellipsis.
Each input operand may contain at most one ellipsis which will cover the dimensions not covered by subscripts,
e.g. for an input operand with 5 dimensions, the ellipsis in the equation <cite>‘ab…c’</cite> cover the third and fourth
dimensions. The ellipsis does not need to cover the same number of dimensions across the <code class="xref py py-attr docutils literal notranslate"><span class="pre">operands</span></code> but the
‘shape’ of the ellipsis (the size of the dimensions covered by them) must broadcast together. If the output is not
explicitly defined with the arrow (‘->’) notation, the ellipsis will come first in the output (left-most dimensions),
before the subscript labels that appear exactly once for the input operands. e.g. the following equation implements
batch matrix multiplication <cite>‘…ij,…jk’</cite>.</p>
<p>A few final notes: the equation may contain whitespaces between the different elements (subscripts, ellipsis,
arrow and comma) but something like <cite>‘…’</cite> is not valid. An empty string <cite>‘’</cite> is valid for scalar operands.</p>
</div></blockquote>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p><code class="docutils literal notranslate"><span class="pre">torch.einsum</span></code> handles ellipsis (‘…’) differently from NumPy in that it allows dimensions
covered by the ellipsis to be summed over, that is, ellipsis are not required to be part of the output.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This function does not optimize the given expression, so a different formula for the same computation may
run faster or consume less memory. Projects like opt_einsum (<a class="reference external" href="https://optimized-einsum.readthedocs.io/en/stable/">https://optimized-einsum.readthedocs.io/en/stable/</a>)
can optimize the formula for you.</p>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>As of PyTorch 1.10 <a class="reference internal" href="#torch.einsum" title="torch.einsum"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.einsum()</span></code></a> also supports the sublist format (see examples below). In this format,
subscripts for each operand are specified by sublists, list of integers in the range [0, 52). These sublists
follow their operands, and an extra sublist can appear at the end of the input to specify the output’s
subscripts., e.g. <cite>torch.einsum(op1, sublist1, op2, sublist2, …, [subslist_out])</cite>. Python’s <cite>Ellipsis</cite> object
may be provided in a sublist to enable broadcasting as described in the Equation section above.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>equation</strong> (<em>string</em>) – The subscripts for the Einstein summation.</p></li>
<li><p><strong>operands</strong> (<em>List</em><em>[</em><a class="reference internal" href="../tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a><em>]</em>) – The tensors to compute the Einstein summation of.</p></li>
</ul>
</dd>
</dl>
<p>Examples:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># trace</span>
<span class="o">>>></span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'ii'</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<span class="n">tensor</span><span class="p">(</span><span class="o">-</span><span class="mf">1.2104</span><span class="p">)</span>
<span class="c1"># diagonal</span>
<span class="o">>>></span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'ii->i'</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<span class="n">tensor</span><span class="p">([</span><span class="o">-</span><span class="mf">0.1034</span><span class="p">,</span> <span class="mf">0.7952</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2433</span><span class="p">,</span> <span class="mf">0.4545</span><span class="p">])</span>
<span class="c1"># outer product</span>
<span class="o">>>></span> <span class="n">x</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">5</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">y</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'i,j->ij'</span><span class="p">,</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
<span class="n">tensor</span><span class="p">([[</span> <span class="mf">0.1156</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2897</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.3918</span><span class="p">,</span> <span class="mf">0.4963</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">0.3744</span><span class="p">,</span> <span class="mf">0.9381</span><span class="p">,</span> <span class="mf">1.2685</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.6070</span><span class="p">],</span>
<span class="p">[</span> <span class="mf">0.7208</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.8058</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.4419</span><span class="p">,</span> <span class="mf">3.0936</span><span class="p">],</span>
<span class="p">[</span> <span class="mf">0.1713</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.4291</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5802</span><span class="p">,</span> <span class="mf">0.7350</span><span class="p">],</span>
<span class="p">[</span> <span class="mf">0.5704</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.4290</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.9323</span><span class="p">,</span> <span class="mf">2.4480</span><span class="p">]])</span>
<span class="c1"># batch matrix multiplication</span>
<span class="o">>>></span> <span class="n">As</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">Bs</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'bij,bjk->bik'</span><span class="p">,</span> <span class="n">As</span><span class="p">,</span> <span class="n">Bs</span><span class="p">)</span>
<span class="n">tensor</span><span class="p">([[[</span><span class="o">-</span><span class="mf">1.0564</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.5904</span><span class="p">,</span> <span class="mf">3.2023</span><span class="p">,</span> <span class="mf">3.1271</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.6706</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.8097</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.8025</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.1183</span><span class="p">]],</span>
<span class="p">[[</span> <span class="mf">4.2239</span><span class="p">,</span> <span class="mf">0.3107</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5756</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2354</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.4558</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.3460</span><span class="p">,</span> <span class="mf">1.5087</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.8530</span><span class="p">]],</span>
<span class="p">[[</span> <span class="mf">2.8153</span><span class="p">,</span> <span class="mf">1.8787</span><span class="p">,</span> <span class="o">-</span><span class="mf">4.3839</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.2112</span><span class="p">],</span>
<span class="p">[</span> <span class="mf">0.3728</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.1131</span><span class="p">,</span> <span class="mf">0.0921</span><span class="p">,</span> <span class="mf">0.8305</span><span class="p">]]])</span>
<span class="c1"># with sublist format and ellipsis</span>
<span class="o">>>></span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="n">As</span><span class="p">,</span> <span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">Bs</span><span class="p">,</span> <span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="o">...</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
<span class="n">tensor</span><span class="p">([[[</span><span class="o">-</span><span class="mf">1.0564</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.5904</span><span class="p">,</span> <span class="mf">3.2023</span><span class="p">,</span> <span class="mf">3.1271</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.6706</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.8097</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.8025</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.1183</span><span class="p">]],</span>
<span class="p">[[</span> <span class="mf">4.2239</span><span class="p">,</span> <span class="mf">0.3107</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.5756</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.2354</span><span class="p">],</span>
<span class="p">[</span><span class="o">-</span><span class="mf">1.4558</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.3460</span><span class="p">,</span> <span class="mf">1.5087</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.8530</span><span class="p">]],</span>
<span class="p">[[</span> <span class="mf">2.8153</span><span class="p">,</span> <span class="mf">1.8787</span><span class="p">,</span> <span class="o">-</span><span class="mf">4.3839</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.2112</span><span class="p">],</span>
<span class="p">[</span> <span class="mf">0.3728</span><span class="p">,</span> <span class="o">-</span><span class="mf">2.1131</span><span class="p">,</span> <span class="mf">0.0921</span><span class="p">,</span> <span class="mf">0.8305</span><span class="p">]]])</span>
<span class="c1"># batch permute</span>
<span class="o">>>></span> <span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">5</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'...ij->...ji'</span><span class="p">,</span> <span class="n">A</span><span class="p">)</span><span class="o">.</span><span class="n">shape</span>
<span class="n">torch</span><span class="o">.</span><span class="n">Size</span><span class="p">([</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">])</span>
<span class="c1"># equivalent to torch.nn.functional.bilinear</span>
<span class="o">>>></span> <span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span><span class="mi">5</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">l</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">5</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">r</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">randn</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span><span class="mi">4</span><span class="p">)</span>
<span class="o">>>></span> <span class="n">torch</span><span class="o">.</span><span class="n">einsum</span><span class="p">(</span><span class="s1">'bn,anm,bm->ba'</span><span class="p">,</span> <span class="n">l</span><span class="p">,</span> <span class="n">A</span><span class="p">,</span> <span class="n">r</span><span class="p">)</span>
<span class="n">tensor</span><span class="p">([[</span><span class="o">-</span><span class="mf">0.3430</span><span class="p">,</span> <span class="o">-</span><span class="mf">5.2405</span><span class="p">,</span> <span class="mf">0.4494</span><span class="p">],</span>
<span class="p">[</span> <span class="mf">0.3311</span><span class="p">,</span> <span class="mf">5.5201</span><span class="p">,</span> <span class="o">-</span><span class="mf">3.0356</span><span class="p">]])</span>
</pre></div>
</div>
</dd></dl>
</div>
</article>
</div>
<footer>
<div class="rst-footer-buttons" role="navigation" aria-label="footer navigation">
<a href="torch.flatten.html" class="btn btn-neutral float-right" title="torch.flatten" accesskey="n" rel="next">Next <img src="../_static/images/chevron-right-orange.svg" class="next-page"></a>
<a href="torch.diff.html" class="btn btn-neutral" title="torch.diff" accesskey="p" rel="prev"><img src="../_static/images/chevron-right-orange.svg" class="previous-page"> Previous</a>
</div>
<hr>
<div role="contentinfo">
<p>
© Copyright 2019, Torch Contributors.
</p>
</div>
<div>
Built with <a href="http://sphinx-doc.org/">Sphinx</a> using a <a href="https://github.com/rtfd/sphinx_rtd_theme">theme</a> provided by <a href="https://readthedocs.org">Read the Docs</a>.
</div>
</footer>
</div>
</div>
<div class="pytorch-content-right" id="pytorch-content-right">
<div class="pytorch-right-menu" id="pytorch-right-menu">
<div class="pytorch-side-scroll" id="pytorch-side-scroll-right">
<ul>
<li><a class="reference internal" href="#">torch.einsum</a></li>
</ul>
</div>
</div>
</div>
</section>
</div>
<script type="text/javascript" id="documentation_options" data-url_root="../" src="../_static/documentation_options.js"></script>
<script src="../_static/jquery.js"></script>
<script src="../_static/underscore.js"></script>
<script src="../_static/doctools.js"></script>
<script src="../_static/clipboard.min.js"></script>
<script src="../_static/copybutton.js"></script>
<script type="text/javascript" src="../_static/js/vendor/popper.min.js"></script>
<script type="text/javascript" src="../_static/js/vendor/bootstrap.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/list.js/1.5.0/list.min.js"></script>
<script type="text/javascript" src="../_static/js/theme.js"></script>
<script type="text/javascript">
jQuery(function () {
SphinxRtdTheme.Navigation.enable(true);
});
</script>
<script script type="text/javascript">
var collapsedSections = ['Notes', 'Language Bindings', 'Libraries', 'Community'];
</script>
<img height="1" width="1" style="border-style:none;" alt="" src="https://www.googleadservices.com/pagead/conversion/795629140/?label=txkmCPmdtosBENSssfsC&guid=ON&script=0"/>
<!-- Begin Footer -->
<div class="container-fluid docs-tutorials-resources" id="docs-tutorials-resources">
<div class="container">
<div class="row">
<div class="col-md-4 text-center">
<h2>Docs</h2>
<p>Access comprehensive developer documentation for PyTorch</p>
<a class="with-right-arrow" href="https://pytorch.org/docs/stable/index.html">View Docs</a>
</div>
<div class="col-md-4 text-center">
<h2>Tutorials</h2>
<p>Get in-depth tutorials for beginners and advanced developers</p>
<a class="with-right-arrow" href="https://pytorch.org/tutorials">View Tutorials</a>
</div>
<div class="col-md-4 text-center">
<h2>Resources</h2>
<p>Find development resources and get your questions answered</p>
<a class="with-right-arrow" href="https://pytorch.org/resources">View Resources</a>
</div>
</div>
</div>
</div>
<footer class="site-footer">
<div class="container footer-container">
<div class="footer-logo-wrapper">
<a href="https://pytorch.org/" class="footer-logo"></a>
</div>
<div class="footer-links-wrapper">
<div class="footer-links-col">
<ul>
<li class="list-title"><a href="https://pytorch.org/">PyTorch</a></li>
<li><a href="https://pytorch.org/get-started">Get Started</a></li>
<li><a href="https://pytorch.org/features">Features</a></li>
<li><a href="https://pytorch.org/ecosystem">Ecosystem</a></li>
<li><a href="https://pytorch.org/blog/">Blog</a></li>
<li><a href="https://github.com/pytorch/pytorch/blob/master/CONTRIBUTING.md">Contributing</a></li>
</ul>
</div>
<div class="footer-links-col">
<ul>
<li class="list-title"><a href="https://pytorch.org/resources">Resources</a></li>
<li><a href="https://pytorch.org/tutorials">Tutorials</a></li>
<li><a href="https://pytorch.org/docs/stable/index.html">Docs</a></li>
<li><a href="https://discuss.pytorch.org" target="_blank">Discuss</a></li>
<li><a href="https://github.com/pytorch/pytorch/issues" target="_blank">Github Issues</a></li>
<li><a href="https://pytorch.org/assets/brand-guidelines/PyTorch-Brand-Guidelines.pdf" target="_blank">Brand Guidelines</a></li>
</ul>
</div>
<div class="footer-links-col follow-us-col">
<ul>
<li class="list-title">Stay Connected</li>
<li>
<div id="mc_embed_signup">
<form
action="https://twitter.us14.list-manage.com/subscribe/post?u=75419c71fe0a935e53dfa4a3f&id=91d0dccd39"
method="post"
id="mc-embedded-subscribe-form"
name="mc-embedded-subscribe-form"
class="email-subscribe-form validate"
target="_blank"
novalidate>
<div id="mc_embed_signup_scroll" class="email-subscribe-form-fields-wrapper">
<div class="mc-field-group">
<label for="mce-EMAIL" style="display:none;">Email Address</label>
<input type="email" value="" name="EMAIL" class="required email" id="mce-EMAIL" placeholder="Email Address">
</div>
<div id="mce-responses" class="clear">
<div class="response" id="mce-error-response" style="display:none"></div>
<div class="response" id="mce-success-response" style="display:none"></div>
</div> <!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups-->
<div style="position: absolute; left: -5000px;" aria-hidden="true"><input type="text" name="b_75419c71fe0a935e53dfa4a3f_91d0dccd39" tabindex="-1" value=""></div>
<div class="clear">
<input type="submit" value="" name="subscribe" id="mc-embedded-subscribe" class="button email-subscribe-button">
</div>
</div>
</form>
</div>
</li>
</ul>
<div class="footer-social-icons">
<a href="https://www.facebook.com/pytorch" target="_blank" class="facebook"></a>
<a href="https://twitter.com/pytorch" target="_blank" class="twitter"></a>
<a href="https://www.youtube.com/pytorch" target="_blank" class="youtube"></a>
</div>
</div>
</div>
</div>
</footer>
<div class="cookie-banner-wrapper">
<div class="container">
<p class="gdpr-notice">To analyze traffic and optimize your experience, we serve cookies on this site. By clicking or navigating, you agree to allow our usage of cookies. As the current maintainers of this site, Facebook’s Cookies Policy applies. Learn more, including about available controls: <a href="https://www.facebook.com/policies/cookies/">Cookies Policy</a>.</p>
<img class="close-button" src="../_static/images/pytorch-x.svg">
</div>
</div>
<!-- End Footer -->
<!-- Begin Mobile Menu -->
<div class="mobile-main-menu">
<div class="container-fluid">
<div class="container">
<div class="mobile-main-menu-header-container">
<a class="header-logo" href="https://pytorch.org/" aria-label="PyTorch"></a>
<a class="main-menu-close-button" href="#" data-behavior="close-mobile-menu"></a>
</div>
</div>
</div>
<div class="mobile-main-menu-links-container">
<div class="main-menu">
<ul>
<li>
<a href="https://pytorch.org/get-started">Get Started</a>
</li>
<li>
<a href="https://pytorch.org/ecosystem">Ecosystem</a>
</li>
<li>
<a href="https://pytorch.org/mobile">Mobile</a>
</li>
<li>
<a href="https://pytorch.org/hub">PyTorch Hub</a>
</li>
<li>
<a href="https://pytorch.org/blog/">Blog</a>
</li>
<li>
<a href="https://pytorch.org/tutorials">Tutorials</a>
</li>
<li class="resources-mobile-menu-title" class="active">
Docs
</li>
<ul class="resources-mobile-menu-items">
<li>
<a href="https://pytorch.org/docs/stable/index.html">PyTorch</a>
</li>
<li>
<a href="https://pytorch.org/audio/stable/index.html">torchaudio</a>
</li>
<li>
<a href="https://pytorch.org/text/stable/index.html">torchtext</a>
</li>
<li>
<a href="https://pytorch.org/vision/stable/index.html">torchvision</a>
</li>
<li>
<a href="https://pytorch.org/elastic/">TorchElastic</a>
</li>
<li>
<a href="https://pytorch.org/serve/">TorchServe</a>
</li>
<li>
<a href="https://pytorch.org/xla">PyTorch on XLA Devices</a>
</li>
</ul>
<li class="resources-mobile-menu-title">
Resources
</li>
<ul class="resources-mobile-menu-items">
<li>
<a href="https://pytorch.org/resources">Developer Resources</a>
</li>
<li>
<a href="https://pytorch.org/features">About</a>
</li>
<li>
<a href="https://pytorch.org/hub">Models (Beta)</a>
</li>
<li>
<a href="https://pytorch.org/#community-module">Community</a>
</li>
<li>
<a href="https://discuss.pytorch.org/">Forums</a>
</li>
</ul>
<li>
<a href="https://github.com/pytorch/pytorch">Github</a>
</li>
</ul>
</div>
</div>
</div>
<!-- End Mobile Menu -->
<script type="text/javascript" src="../_static/js/vendor/anchor.min.js"></script>
<script type="text/javascript">
$(document).ready(function() {
mobileMenu.bind();
mobileTOC.bind();
pytorchAnchors.bind();
sideMenus.bind();
scrollToAnchor.bind();
highlightNavigation.bind();
mainMenuDropdown.bind();
filterTags.bind();
// Add class to links that have code blocks, since we cannot create links in code blocks
$("article.pytorch-article a span.pre").each(function(e) {
$(this).closest("a").addClass("has-code");
});
})
</script>
</body>
</html>