-
Notifications
You must be signed in to change notification settings - Fork 81
/
Copy pathdecomposition.html
1675 lines (1445 loc) · 127 KB
/
decomposition.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
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
<!DOCTYPE html>
<html lang="en" data-content_root="../" >
<head>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="viewport" content="width=device-width, initial-scale=1" />
<meta property="og:title" content="2.5. Decomposing signals in components (matrix factorization problems)" />
<meta property="og:type" content="website" />
<meta property="og:url" content="https://scikit-learn/stable/modules/decomposition.html" />
<meta property="og:site_name" content="scikit-learn" />
<meta property="og:description" content="Principal component analysis (PCA): Exact PCA and probabilistic interpretation: PCA is used to decompose a multivariate dataset in a set of successive orthogonal components that explain a maximum a..." />
<meta property="og:image" content="https://scikit-learn/stable/_images/sphx_glr_plot_pca_vs_lda_001.png" />
<meta property="og:image:alt" content="scikit-learn" />
<meta name="description" content="Principal component analysis (PCA): Exact PCA and probabilistic interpretation: PCA is used to decompose a multivariate dataset in a set of successive orthogonal components that explain a maximum a..." />
<title>2.5. Decomposing signals in components (matrix factorization problems) — scikit-learn 1.7.dev0 documentation</title>
<script data-cfasync="false">
document.documentElement.dataset.mode = localStorage.getItem("mode") || "";
document.documentElement.dataset.theme = localStorage.getItem("theme") || "";
</script>
<!--
this give us a css class that will be invisible only if js is disabled
-->
<noscript>
<style>
.pst-js-only { display: none !important; }
</style>
</noscript>
<!-- Loaded before other Sphinx assets -->
<link href="../_static/styles/theme.css?digest=8878045cc6db502f8baf" rel="stylesheet" />
<link href="../_static/styles/pydata-sphinx-theme.css?digest=8878045cc6db502f8baf" rel="stylesheet" />
<link rel="stylesheet" type="text/css" href="../_static/pygments.css?v=8f2a1f02" />
<link rel="stylesheet" type="text/css" href="../_static/copybutton.css?v=76b2166b" />
<link rel="stylesheet" type="text/css" href="../_static/plot_directive.css" />
<link rel="stylesheet" type="text/css" href="https://fonts.googleapis.com/css?family=Vibur" />
<link rel="stylesheet" type="text/css" href="../_static/jupyterlite_sphinx.css?v=2c9f8f05" />
<link rel="stylesheet" type="text/css" href="../_static/sg_gallery.css?v=d2d258e8" />
<link rel="stylesheet" type="text/css" href="../_static/sg_gallery-binder.css?v=f4aeca0c" />
<link rel="stylesheet" type="text/css" href="../_static/sg_gallery-dataframe.css?v=2082cf3c" />
<link rel="stylesheet" type="text/css" href="../_static/sg_gallery-rendered-html.css?v=1277b6f3" />
<link rel="stylesheet" type="text/css" href="../_static/sphinx-design.min.css?v=95c83b7e" />
<link rel="stylesheet" type="text/css" href="../_static/styles/colors.css?v=cc94ab7d" />
<link rel="stylesheet" type="text/css" href="../_static/styles/custom.css?v=8f525996" />
<!-- So that users can add custom icons -->
<script src="../_static/scripts/fontawesome.js?digest=8878045cc6db502f8baf"></script>
<!-- Pre-loaded scripts that we'll load fully later -->
<link rel="preload" as="script" href="../_static/scripts/bootstrap.js?digest=8878045cc6db502f8baf" />
<link rel="preload" as="script" href="../_static/scripts/pydata-sphinx-theme.js?digest=8878045cc6db502f8baf" />
<script src="../_static/documentation_options.js?v=473747f4"></script>
<script src="../_static/doctools.js?v=9bcbadda"></script>
<script src="../_static/sphinx_highlight.js?v=dc90522c"></script>
<script src="../_static/clipboard.min.js?v=a7894cd8"></script>
<script src="../_static/copybutton.js?v=97f0b27d"></script>
<script src="../_static/jupyterlite_sphinx.js?v=96e329c5"></script>
<script src="../_static/design-tabs.js?v=f930bc37"></script>
<script data-domain="scikit-learn.org" defer="defer" src="https://views.scientific-python.org/js/script.js"></script>
<script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script>
<script>DOCUMENTATION_OPTIONS.pagename = 'modules/decomposition';</script>
<script>
DOCUMENTATION_OPTIONS.theme_version = '0.16.1';
DOCUMENTATION_OPTIONS.theme_switcher_json_url = 'https://scikit-learn.org/dev/_static/versions.json';
DOCUMENTATION_OPTIONS.theme_switcher_version_match = '1.7.dev0';
DOCUMENTATION_OPTIONS.show_version_warning_banner =
true;
</script>
<script src="../_static/scripts/dropdown.js?v=d6825577"></script>
<script src="../_static/scripts/version-switcher.js?v=a6dd8357"></script>
<script src="../_static/scripts/sg_plotly_resize.js?v=2167d4db"></script>
<link rel="canonical" href="https://scikit-learn.org/stable/modules/decomposition.html" />
<link rel="icon" href="../_static/favicon.ico"/>
<link rel="author" title="About these documents" href="../about.html" />
<link rel="search" title="Search" href="../search.html" />
<link rel="next" title="2.6. Covariance estimation" href="covariance.html" />
<link rel="prev" title="2.4. Biclustering" href="biclustering.html" />
<meta name="viewport" content="width=device-width, initial-scale=1"/>
<meta name="docsearch:language" content="en"/>
<meta name="docsearch:version" content="1.7" />
</head>
<body data-bs-spy="scroll" data-bs-target=".bd-toc-nav" data-offset="180" data-bs-root-margin="0px 0px -60%" data-default-mode="">
<div id="pst-skip-link" class="skip-link d-print-none"><a href="#main-content">Skip to main content</a></div>
<div id="pst-scroll-pixel-helper"></div>
<button type="button" class="btn rounded-pill" id="pst-back-to-top">
<i class="fa-solid fa-arrow-up"></i>Back to top</button>
<dialog id="pst-search-dialog">
<form class="bd-search d-flex align-items-center"
action="../search.html"
method="get">
<i class="fa-solid fa-magnifying-glass"></i>
<input type="search"
class="form-control"
name="q"
placeholder="Search the docs ..."
aria-label="Search the docs ..."
autocomplete="off"
autocorrect="off"
autocapitalize="off"
spellcheck="false"/>
<span class="search-button__kbd-shortcut"><kbd class="kbd-shortcut__modifier">Ctrl</kbd>+<kbd>K</kbd></span>
</form>
</dialog>
<div class="pst-async-banner-revealer d-none">
<aside id="bd-header-version-warning" class="d-none d-print-none" aria-label="Version warning"></aside>
</div>
<header class="bd-header navbar navbar-expand-lg bd-navbar d-print-none">
<div class="bd-header__inner bd-page-width">
<button class="pst-navbar-icon sidebar-toggle primary-toggle" aria-label="Site navigation">
<span class="fa-solid fa-bars"></span>
</button>
<div class=" navbar-header-items__start">
<div class="navbar-item">
<a class="navbar-brand logo" href="../index.html">
<img src="../_static/scikit-learn-logo-small.png" class="logo__image only-light" alt="scikit-learn homepage"/>
<img src="../_static/scikit-learn-logo-small.png" class="logo__image only-dark pst-js-only" alt="scikit-learn homepage"/>
</a></div>
</div>
<div class=" navbar-header-items">
<div class="me-auto navbar-header-items__center">
<div class="navbar-item">
<nav>
<ul class="bd-navbar-elements navbar-nav">
<li class="nav-item ">
<a class="nav-link nav-internal" href="../install.html">
Install
</a>
</li>
<li class="nav-item current active">
<a class="nav-link nav-internal" href="../user_guide.html">
User Guide
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../api/index.html">
API
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../auto_examples/index.html">
Examples
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-external" href="https://blog.scikit-learn.org/">
Community
</a>
</li>
<li class="nav-item dropdown">
<button class="btn dropdown-toggle nav-item" type="button"
data-bs-toggle="dropdown" aria-expanded="false"
aria-controls="pst-nav-more-links">
More
</button>
<ul id="pst-nav-more-links" class="dropdown-menu">
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../getting_started.html">
Getting Started
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../whats_new.html">
Release History
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../glossary.html">
Glossary
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../developers/index.html">
Development
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../faq.html">
FAQ
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../support.html">
Support
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../related_projects.html">
Related Projects
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../roadmap.html">
Roadmap
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../governance.html">
Governance
</a>
</li>
<li class=" ">
<a class="nav-link dropdown-item nav-internal" href="../about.html">
About us
</a>
</li>
</ul>
</li>
</ul>
</nav></div>
</div>
<div class="navbar-header-items__end">
<div class="navbar-item navbar-persistent--container">
<button class="btn btn-sm pst-navbar-icon search-button search-button__button pst-js-only" title="Search" aria-label="Search" data-bs-placement="bottom" data-bs-toggle="tooltip">
<i class="fa-solid fa-magnifying-glass fa-lg"></i>
</button>
</div>
<div class="navbar-item">
<button class="btn btn-sm nav-link pst-navbar-icon theme-switch-button pst-js-only" aria-label="Color mode" data-bs-title="Color mode" data-bs-placement="bottom" data-bs-toggle="tooltip">
<i class="theme-switch fa-solid fa-sun fa-lg" data-mode="light" title="Light"></i>
<i class="theme-switch fa-solid fa-moon fa-lg" data-mode="dark" title="Dark"></i>
<i class="theme-switch fa-solid fa-circle-half-stroke fa-lg" data-mode="auto" title="System Settings"></i>
</button></div>
<div class="navbar-item"><ul class="navbar-icon-links"
aria-label="Icon Links">
<li class="nav-item">
<a href="https://github.com/scikit-learn/scikit-learn" title="GitHub" class="nav-link pst-navbar-icon" rel="noopener" target="_blank" data-bs-toggle="tooltip" data-bs-placement="bottom"><i class="fa-brands fa-square-github fa-lg" aria-hidden="true"></i>
<span class="sr-only">GitHub</span></a>
</li>
</ul></div>
<div class="navbar-item">
<div class="version-switcher__container dropdown pst-js-only">
<button id="pst-version-switcher-button-2"
type="button"
class="version-switcher__button btn btn-sm dropdown-toggle"
data-bs-toggle="dropdown"
aria-haspopup="listbox"
aria-controls="pst-version-switcher-list-2"
aria-label="Version switcher list"
>
Choose version <!-- this text may get changed later by javascript -->
<span class="caret"></span>
</button>
<div id="pst-version-switcher-list-2"
class="version-switcher__menu dropdown-menu list-group-flush py-0"
role="listbox" aria-labelledby="pst-version-switcher-button-2">
<!-- dropdown will be populated by javascript on page load -->
</div>
</div></div>
</div>
</div>
<div class="navbar-persistent--mobile">
<button class="btn btn-sm pst-navbar-icon search-button search-button__button pst-js-only" title="Search" aria-label="Search" data-bs-placement="bottom" data-bs-toggle="tooltip">
<i class="fa-solid fa-magnifying-glass fa-lg"></i>
</button>
</div>
<button class="pst-navbar-icon sidebar-toggle secondary-toggle" aria-label="On this page">
<span class="fa-solid fa-outdent"></span>
</button>
</div>
</header>
<div class="bd-container">
<div class="bd-container__inner bd-page-width">
<dialog id="pst-primary-sidebar-modal"></dialog>
<div id="pst-primary-sidebar" class="bd-sidebar-primary bd-sidebar">
<div class="sidebar-header-items sidebar-primary__section">
<div class="sidebar-header-items__center">
<div class="navbar-item">
<nav>
<ul class="bd-navbar-elements navbar-nav">
<li class="nav-item ">
<a class="nav-link nav-internal" href="../install.html">
Install
</a>
</li>
<li class="nav-item current active">
<a class="nav-link nav-internal" href="../user_guide.html">
User Guide
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../api/index.html">
API
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../auto_examples/index.html">
Examples
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-external" href="https://blog.scikit-learn.org/">
Community
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../getting_started.html">
Getting Started
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../whats_new.html">
Release History
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../glossary.html">
Glossary
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../developers/index.html">
Development
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../faq.html">
FAQ
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../support.html">
Support
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../related_projects.html">
Related Projects
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../roadmap.html">
Roadmap
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../governance.html">
Governance
</a>
</li>
<li class="nav-item ">
<a class="nav-link nav-internal" href="../about.html">
About us
</a>
</li>
</ul>
</nav></div>
</div>
<div class="sidebar-header-items__end">
<div class="navbar-item">
<button class="btn btn-sm nav-link pst-navbar-icon theme-switch-button pst-js-only" aria-label="Color mode" data-bs-title="Color mode" data-bs-placement="bottom" data-bs-toggle="tooltip">
<i class="theme-switch fa-solid fa-sun fa-lg" data-mode="light" title="Light"></i>
<i class="theme-switch fa-solid fa-moon fa-lg" data-mode="dark" title="Dark"></i>
<i class="theme-switch fa-solid fa-circle-half-stroke fa-lg" data-mode="auto" title="System Settings"></i>
</button></div>
<div class="navbar-item"><ul class="navbar-icon-links"
aria-label="Icon Links">
<li class="nav-item">
<a href="https://github.com/scikit-learn/scikit-learn" title="GitHub" class="nav-link pst-navbar-icon" rel="noopener" target="_blank" data-bs-toggle="tooltip" data-bs-placement="bottom"><i class="fa-brands fa-square-github fa-lg" aria-hidden="true"></i>
<span class="sr-only">GitHub</span></a>
</li>
</ul></div>
<div class="navbar-item">
<div class="version-switcher__container dropdown pst-js-only">
<button id="pst-version-switcher-button-3"
type="button"
class="version-switcher__button btn btn-sm dropdown-toggle"
data-bs-toggle="dropdown"
aria-haspopup="listbox"
aria-controls="pst-version-switcher-list-3"
aria-label="Version switcher list"
>
Choose version <!-- this text may get changed later by javascript -->
<span class="caret"></span>
</button>
<div id="pst-version-switcher-list-3"
class="version-switcher__menu dropdown-menu list-group-flush py-0"
role="listbox" aria-labelledby="pst-version-switcher-button-3">
<!-- dropdown will be populated by javascript on page load -->
</div>
</div></div>
</div>
</div>
<div class="sidebar-primary-items__start sidebar-primary__section">
<div class="sidebar-primary-item">
<nav class="bd-docs-nav bd-links"
aria-label="Section Navigation">
<p class="bd-links__title" role="heading" aria-level="1">Section Navigation</p>
<div class="bd-toc-item navbar-nav"><ul class="current nav bd-sidenav">
<li class="toctree-l1 has-children"><a class="reference internal" href="../supervised_learning.html">1. Supervised learning</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="linear_model.html">1.1. Linear Models</a></li>
<li class="toctree-l2"><a class="reference internal" href="lda_qda.html">1.2. Linear and Quadratic Discriminant Analysis</a></li>
<li class="toctree-l2"><a class="reference internal" href="kernel_ridge.html">1.3. Kernel ridge regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="svm.html">1.4. Support Vector Machines</a></li>
<li class="toctree-l2"><a class="reference internal" href="sgd.html">1.5. Stochastic Gradient Descent</a></li>
<li class="toctree-l2"><a class="reference internal" href="neighbors.html">1.6. Nearest Neighbors</a></li>
<li class="toctree-l2"><a class="reference internal" href="gaussian_process.html">1.7. Gaussian Processes</a></li>
<li class="toctree-l2"><a class="reference internal" href="cross_decomposition.html">1.8. Cross decomposition</a></li>
<li class="toctree-l2"><a class="reference internal" href="naive_bayes.html">1.9. Naive Bayes</a></li>
<li class="toctree-l2"><a class="reference internal" href="tree.html">1.10. Decision Trees</a></li>
<li class="toctree-l2"><a class="reference internal" href="ensemble.html">1.11. Ensembles: Gradient boosting, random forests, bagging, voting, stacking</a></li>
<li class="toctree-l2"><a class="reference internal" href="multiclass.html">1.12. Multiclass and multioutput algorithms</a></li>
<li class="toctree-l2"><a class="reference internal" href="feature_selection.html">1.13. Feature selection</a></li>
<li class="toctree-l2"><a class="reference internal" href="semi_supervised.html">1.14. Semi-supervised learning</a></li>
<li class="toctree-l2"><a class="reference internal" href="isotonic.html">1.15. Isotonic regression</a></li>
<li class="toctree-l2"><a class="reference internal" href="calibration.html">1.16. Probability calibration</a></li>
<li class="toctree-l2"><a class="reference internal" href="neural_networks_supervised.html">1.17. Neural network models (supervised)</a></li>
</ul>
</details></li>
<li class="toctree-l1 current active has-children"><a class="reference internal" href="../unsupervised_learning.html">2. Unsupervised learning</a><details open="open"><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul class="current">
<li class="toctree-l2"><a class="reference internal" href="mixture.html">2.1. Gaussian mixture models</a></li>
<li class="toctree-l2"><a class="reference internal" href="manifold.html">2.2. Manifold learning</a></li>
<li class="toctree-l2"><a class="reference internal" href="clustering.html">2.3. Clustering</a></li>
<li class="toctree-l2"><a class="reference internal" href="biclustering.html">2.4. Biclustering</a></li>
<li class="toctree-l2 current active"><a class="current reference internal" href="#">2.5. Decomposing signals in components (matrix factorization problems)</a></li>
<li class="toctree-l2"><a class="reference internal" href="covariance.html">2.6. Covariance estimation</a></li>
<li class="toctree-l2"><a class="reference internal" href="outlier_detection.html">2.7. Novelty and Outlier Detection</a></li>
<li class="toctree-l2"><a class="reference internal" href="density.html">2.8. Density Estimation</a></li>
<li class="toctree-l2"><a class="reference internal" href="neural_networks_unsupervised.html">2.9. Neural network models (unsupervised)</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../model_selection.html">3. Model selection and evaluation</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="cross_validation.html">3.1. Cross-validation: evaluating estimator performance</a></li>
<li class="toctree-l2"><a class="reference internal" href="grid_search.html">3.2. Tuning the hyper-parameters of an estimator</a></li>
<li class="toctree-l2"><a class="reference internal" href="classification_threshold.html">3.3. Tuning the decision threshold for class prediction</a></li>
<li class="toctree-l2"><a class="reference internal" href="model_evaluation.html">3.4. Metrics and scoring: quantifying the quality of predictions</a></li>
<li class="toctree-l2"><a class="reference internal" href="learning_curve.html">3.5. Validation curves: plotting scores to evaluate models</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../metadata_routing.html">4. Metadata Routing</a></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../inspection.html">5. Inspection</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="partial_dependence.html">5.1. Partial Dependence and Individual Conditional Expectation plots</a></li>
<li class="toctree-l2"><a class="reference internal" href="permutation_importance.html">5.2. Permutation feature importance</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../visualizations.html">6. Visualizations</a></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../data_transforms.html">7. Dataset transformations</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="compose.html">7.1. Pipelines and composite estimators</a></li>
<li class="toctree-l2"><a class="reference internal" href="feature_extraction.html">7.2. Feature extraction</a></li>
<li class="toctree-l2"><a class="reference internal" href="preprocessing.html">7.3. Preprocessing data</a></li>
<li class="toctree-l2"><a class="reference internal" href="impute.html">7.4. Imputation of missing values</a></li>
<li class="toctree-l2"><a class="reference internal" href="unsupervised_reduction.html">7.5. Unsupervised dimensionality reduction</a></li>
<li class="toctree-l2"><a class="reference internal" href="random_projection.html">7.6. Random Projection</a></li>
<li class="toctree-l2"><a class="reference internal" href="kernel_approximation.html">7.7. Kernel Approximation</a></li>
<li class="toctree-l2"><a class="reference internal" href="metrics.html">7.8. Pairwise metrics, Affinities and Kernels</a></li>
<li class="toctree-l2"><a class="reference internal" href="preprocessing_targets.html">7.9. Transforming the prediction target (<code class="docutils literal notranslate"><span class="pre">y</span></code>)</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../datasets.html">8. Dataset loading utilities</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../datasets/toy_dataset.html">8.1. Toy datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/real_world.html">8.2. Real world datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/sample_generators.html">8.3. Generated datasets</a></li>
<li class="toctree-l2"><a class="reference internal" href="../datasets/loading_other_datasets.html">8.4. Loading other datasets</a></li>
</ul>
</details></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../computing.html">9. Computing with scikit-learn</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="../computing/scaling_strategies.html">9.1. Strategies to scale computationally: bigger data</a></li>
<li class="toctree-l2"><a class="reference internal" href="../computing/computational_performance.html">9.2. Computational Performance</a></li>
<li class="toctree-l2"><a class="reference internal" href="../computing/parallelism.html">9.3. Parallelism, resource management, and configuration</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../model_persistence.html">10. Model persistence</a></li>
<li class="toctree-l1"><a class="reference internal" href="../common_pitfalls.html">11. Common pitfalls and recommended practices</a></li>
<li class="toctree-l1 has-children"><a class="reference internal" href="../dispatching.html">12. Dispatching</a><details><summary><span class="toctree-toggle" role="presentation"><i class="fa-solid fa-chevron-down"></i></span></summary><ul>
<li class="toctree-l2"><a class="reference internal" href="array_api.html">12.1. Array API support (experimental)</a></li>
</ul>
</details></li>
<li class="toctree-l1"><a class="reference internal" href="../machine_learning_map.html">13. Choosing the right estimator</a></li>
<li class="toctree-l1"><a class="reference internal" href="../presentations.html">14. External Resources, Videos and Talks</a></li>
</ul>
</div>
</nav></div>
</div>
<div class="sidebar-primary-items__end sidebar-primary__section">
</div>
</div>
<main id="main-content" class="bd-main" role="main">
<div class="bd-content">
<div class="bd-article-container">
<div class="bd-header-article d-print-none">
<div class="header-article-items header-article__inner">
<div class="header-article-items__start">
<div class="header-article-item">
<nav aria-label="Breadcrumb" class="d-print-none">
<ul class="bd-breadcrumbs">
<li class="breadcrumb-item breadcrumb-home">
<a href="../index.html" class="nav-link" aria-label="Home">
<i class="fa-solid fa-home"></i>
</a>
</li>
<li class="breadcrumb-item"><a href="../user_guide.html" class="nav-link">User Guide</a></li>
<li class="breadcrumb-item"><a href="../unsupervised_learning.html" class="nav-link"><span class="section-number">2. </span>Unsupervised learning</a></li>
<li class="breadcrumb-item active" aria-current="page"><span class="ellipsis"><span class="section-number">2.5. </span>Decomposing signals in components (matrix factorization problems)</span></li>
</ul>
</nav>
</div>
</div>
</div>
</div>
<div id="searchbox"></div>
<article class="bd-article">
<section id="decomposing-signals-in-components-matrix-factorization-problems">
<span id="decompositions"></span><h1><span class="section-number">2.5. </span>Decomposing signals in components (matrix factorization problems)<a class="headerlink" href="#decomposing-signals-in-components-matrix-factorization-problems" title="Link to this heading">#</a></h1>
<section id="principal-component-analysis-pca">
<span id="pca"></span><h2><span class="section-number">2.5.1. </span>Principal component analysis (PCA)<a class="headerlink" href="#principal-component-analysis-pca" title="Link to this heading">#</a></h2>
<section id="exact-pca-and-probabilistic-interpretation">
<h3><span class="section-number">2.5.1.1. </span>Exact PCA and probabilistic interpretation<a class="headerlink" href="#exact-pca-and-probabilistic-interpretation" title="Link to this heading">#</a></h3>
<p>PCA is used to decompose a multivariate dataset in a set of successive
orthogonal components that explain a maximum amount of the variance. In
scikit-learn, <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> is implemented as a <em>transformer</em> object
that learns <span class="math notranslate nohighlight">\(n\)</span> components in its <code class="docutils literal notranslate"><span class="pre">fit</span></code> method, and can be used on new
data to project it on these components.</p>
<p>PCA centers but does not scale the input data for each feature before
applying the SVD. The optional parameter <code class="docutils literal notranslate"><span class="pre">whiten=True</span></code> makes it
possible to project the data onto the singular space while scaling each
component to unit variance. This is often useful if the models down-stream make
strong assumptions on the isotropy of the signal: this is for example the case
for Support Vector Machines with the RBF kernel and the K-Means clustering
algorithm.</p>
<p>Below is an example of the iris dataset, which is comprised of 4
features, projected on the 2 dimensions that explain most variance:</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_pca_vs_lda.html"><img alt="../_images/sphx_glr_plot_pca_vs_lda_001.png" src="../_images/sphx_glr_plot_pca_vs_lda_001.png" style="width: 480.0px; height: 360.0px;" />
</a>
</figure>
<p>The <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> object also provides a
probabilistic interpretation of the PCA that can give a likelihood of
data based on the amount of variance it explains. As such it implements a
<a class="reference internal" href="../glossary.html#term-score"><span class="xref std std-term">score</span></a> method that can be used in cross-validation:</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_pca_vs_fa_model_selection.html"><img alt="../_images/sphx_glr_plot_pca_vs_fa_model_selection_001.png" src="../_images/sphx_glr_plot_pca_vs_fa_model_selection_001.png" style="width: 480.0px; height: 360.0px;" />
</a>
</figure>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_pca_iris.html#sphx-glr-auto-examples-decomposition-plot-pca-iris-py"><span class="std std-ref">Principal Component Analysis (PCA) on Iris Dataset</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_pca_vs_lda.html#sphx-glr-auto-examples-decomposition-plot-pca-vs-lda-py"><span class="std std-ref">Comparison of LDA and PCA 2D projection of Iris dataset</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_pca_vs_fa_model_selection.html#sphx-glr-auto-examples-decomposition-plot-pca-vs-fa-model-selection-py"><span class="std std-ref">Model selection with Probabilistic PCA and Factor Analysis (FA)</span></a></p></li>
</ul>
</section>
<section id="incremental-pca">
<span id="incrementalpca"></span><h3><span class="section-number">2.5.1.2. </span>Incremental PCA<a class="headerlink" href="#incremental-pca" title="Link to this heading">#</a></h3>
<p>The <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> object is very useful, but has certain limitations for
large datasets. The biggest limitation is that <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> only supports
batch processing, which means all of the data to be processed must fit in main
memory. The <a class="reference internal" href="generated/sklearn.decomposition.IncrementalPCA.html#sklearn.decomposition.IncrementalPCA" title="sklearn.decomposition.IncrementalPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">IncrementalPCA</span></code></a> object uses a different form of
processing and allows for partial computations which almost
exactly match the results of <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> while processing the data in a
minibatch fashion. <a class="reference internal" href="generated/sklearn.decomposition.IncrementalPCA.html#sklearn.decomposition.IncrementalPCA" title="sklearn.decomposition.IncrementalPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">IncrementalPCA</span></code></a> makes it possible to implement
out-of-core Principal Component Analysis either by:</p>
<ul class="simple">
<li><p>Using its <code class="docutils literal notranslate"><span class="pre">partial_fit</span></code> method on chunks of data fetched sequentially
from the local hard drive or a network database.</p></li>
<li><p>Calling its fit method on a memory mapped file using
<code class="docutils literal notranslate"><span class="pre">numpy.memmap</span></code>.</p></li>
</ul>
<p><a class="reference internal" href="generated/sklearn.decomposition.IncrementalPCA.html#sklearn.decomposition.IncrementalPCA" title="sklearn.decomposition.IncrementalPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">IncrementalPCA</span></code></a> only stores estimates of component and noise variances,
in order to update <code class="docutils literal notranslate"><span class="pre">explained_variance_ratio_</span></code> incrementally. This is why
memory usage depends on the number of samples per batch, rather than the
number of samples to be processed in the dataset.</p>
<p>As in <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a>, <a class="reference internal" href="generated/sklearn.decomposition.IncrementalPCA.html#sklearn.decomposition.IncrementalPCA" title="sklearn.decomposition.IncrementalPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">IncrementalPCA</span></code></a> centers but does not scale the
input data for each feature before applying the SVD.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_incremental_pca.html"><img alt="../_images/sphx_glr_plot_incremental_pca_001.png" src="../_images/sphx_glr_plot_incremental_pca_001.png" style="width: 600.0px; height: 600.0px;" />
</a>
</figure>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_incremental_pca.html"><img alt="../_images/sphx_glr_plot_incremental_pca_002.png" src="../_images/sphx_glr_plot_incremental_pca_002.png" style="width: 600.0px; height: 600.0px;" />
</a>
</figure>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_incremental_pca.html#sphx-glr-auto-examples-decomposition-plot-incremental-pca-py"><span class="std std-ref">Incremental PCA</span></a></p></li>
</ul>
</section>
<section id="pca-using-randomized-svd">
<span id="randomizedpca"></span><h3><span class="section-number">2.5.1.3. </span>PCA using randomized SVD<a class="headerlink" href="#pca-using-randomized-svd" title="Link to this heading">#</a></h3>
<p>It is often interesting to project data to a lower-dimensional
space that preserves most of the variance, by dropping the singular vector
of components associated with lower singular values.</p>
<p>For instance, if we work with 64x64 pixel gray-level pictures
for face recognition,
the dimensionality of the data is 4096 and it is slow to train an
RBF support vector machine on such wide data. Furthermore we know that
the intrinsic dimensionality of the data is much lower than 4096 since all
pictures of human faces look somewhat alike.
The samples lie on a manifold of much lower
dimension (say around 200 for instance). The PCA algorithm can be used
to linearly transform the data while both reducing the dimensionality
and preserving most of the explained variance at the same time.</p>
<p>The class <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> used with the optional parameter
<code class="docutils literal notranslate"><span class="pre">svd_solver='randomized'</span></code> is very useful in that case: since we are going
to drop most of the singular vectors it is much more efficient to limit the
computation to an approximated estimate of the singular vectors we will keep
to actually perform the transform.</p>
<p>For instance, the following shows 16 sample portraits (centered around
0.0) from the Olivetti dataset. On the right hand side are the first 16
singular vectors reshaped as portraits. Since we only require the top
16 singular vectors of a dataset with size <span class="math notranslate nohighlight">\(n_{samples} = 400\)</span>
and <span class="math notranslate nohighlight">\(n_{features} = 64 \times 64 = 4096\)</span>, the computation time is
less than 1s:</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="orig_img" src="../_images/sphx_glr_plot_faces_decomposition_001.png" style="width: 360.0px; height: 275.4px;" /></a> <a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="pca_img" src="../_images/sphx_glr_plot_faces_decomposition_002.png" style="width: 360.0px; height: 275.4px;" /></a></strong></p><p>If we note <span class="math notranslate nohighlight">\(n_{\max} = \max(n_{\mathrm{samples}}, n_{\mathrm{features}})\)</span> and
<span class="math notranslate nohighlight">\(n_{\min} = \min(n_{\mathrm{samples}}, n_{\mathrm{features}})\)</span>, the time complexity
of the randomized <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> is <span class="math notranslate nohighlight">\(O(n_{\max}^2 \cdot n_{\mathrm{components}})\)</span>
instead of <span class="math notranslate nohighlight">\(O(n_{\max}^2 \cdot n_{\min})\)</span> for the exact method
implemented in <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a>.</p>
<p>The memory footprint of randomized <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> is also proportional to
<span class="math notranslate nohighlight">\(2 \cdot n_{\max} \cdot n_{\mathrm{components}}\)</span> instead of <span class="math notranslate nohighlight">\(n_{\max}
\cdot n_{\min}\)</span> for the exact method.</p>
<p>Note: the implementation of <code class="docutils literal notranslate"><span class="pre">inverse_transform</span></code> in <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> with
<code class="docutils literal notranslate"><span class="pre">svd_solver='randomized'</span></code> is not the exact inverse transform of
<code class="docutils literal notranslate"><span class="pre">transform</span></code> even when <code class="docutils literal notranslate"><span class="pre">whiten=False</span></code> (default).</p>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/applications/plot_face_recognition.html#sphx-glr-auto-examples-applications-plot-face-recognition-py"><span class="std std-ref">Faces recognition example using eigenfaces and SVMs</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_faces_decomposition.html#sphx-glr-auto-examples-decomposition-plot-faces-decomposition-py"><span class="std std-ref">Faces dataset decompositions</span></a></p></li>
</ul>
<p class="rubric">References</p>
<ul class="simple">
<li><p>Algorithm 4.3 in
<a class="reference external" href="https://arxiv.org/abs/0909.4061">“Finding structure with randomness: Stochastic algorithms for
constructing approximate matrix decompositions”</a>
Halko, et al., 2009</p></li>
<li><p><a class="reference external" href="https://arxiv.org/abs/1412.3510">“An implementation of a randomized algorithm for principal component
analysis”</a> A. Szlam et al. 2014</p></li>
</ul>
</section>
<section id="sparse-principal-components-analysis-sparsepca-and-minibatchsparsepca">
<span id="sparsepca"></span><h3><span class="section-number">2.5.1.4. </span>Sparse principal components analysis (SparsePCA and MiniBatchSparsePCA)<a class="headerlink" href="#sparse-principal-components-analysis-sparsepca-and-minibatchsparsepca" title="Link to this heading">#</a></h3>
<p><a class="reference internal" href="generated/sklearn.decomposition.SparsePCA.html#sklearn.decomposition.SparsePCA" title="sklearn.decomposition.SparsePCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">SparsePCA</span></code></a> is a variant of PCA, with the goal of extracting the
set of sparse components that best reconstruct the data.</p>
<p>Mini-batch sparse PCA (<a class="reference internal" href="generated/sklearn.decomposition.MiniBatchSparsePCA.html#sklearn.decomposition.MiniBatchSparsePCA" title="sklearn.decomposition.MiniBatchSparsePCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">MiniBatchSparsePCA</span></code></a>) is a variant of
<a class="reference internal" href="generated/sklearn.decomposition.SparsePCA.html#sklearn.decomposition.SparsePCA" title="sklearn.decomposition.SparsePCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">SparsePCA</span></code></a> that is faster but less accurate. The increased speed is
reached by iterating over small chunks of the set of features, for a given
number of iterations.</p>
<p>Principal component analysis (<a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a>) has the disadvantage that the
components extracted by this method have exclusively dense expressions, i.e.
they have non-zero coefficients when expressed as linear combinations of the
original variables. This can make interpretation difficult. In many cases,
the real underlying components can be more naturally imagined as sparse
vectors; for example in face recognition, components might naturally map to
parts of faces.</p>
<p>Sparse principal components yield a more parsimonious, interpretable
representation, clearly emphasizing which of the original features contribute
to the differences between samples.</p>
<p>The following example illustrates 16 components extracted using sparse PCA from
the Olivetti faces dataset. It can be seen how the regularization term induces
many zeros. Furthermore, the natural structure of the data causes the non-zero
coefficients to be vertically adjacent. The model does not enforce this
mathematically: each component is a vector <span class="math notranslate nohighlight">\(h \in \mathbf{R}^{4096}\)</span>, and
there is no notion of vertical adjacency except during the human-friendly
visualization as 64x64 pixel images. The fact that the components shown below
appear local is the effect of the inherent structure of the data, which makes
such local patterns minimize reconstruction error. There exist sparsity-inducing
norms that take into account adjacency and different kinds of structure; see
<a class="reference internal" href="#jen09" id="id1"><span>[Jen09]</span></a> for a review of such methods.
For more details on how to use Sparse PCA, see the Examples section, below.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="pca_img" src="../_images/sphx_glr_plot_faces_decomposition_002.png" style="width: 360.0px; height: 275.4px;" /></a> <a class="reference external" href="../auto_examples/decomposition/plot_faces_decomposition.html"><img alt="spca_img" src="../_images/sphx_glr_plot_faces_decomposition_005.png" style="width: 360.0px; height: 275.4px;" /></a></strong></p><p>Note that there are many different formulations for the Sparse PCA
problem. The one implemented here is based on <a class="reference internal" href="#mrl09" id="id2"><span>[Mrl09]</span></a> . The optimization
problem solved is a PCA problem (dictionary learning) with an
<span class="math notranslate nohighlight">\(\ell_1\)</span> penalty on the components:</p>
<div class="math notranslate nohighlight">
\[\begin{split}(U^*, V^*) = \underset{U, V}{\operatorname{arg\,min\,}} & \frac{1}{2}
||X-UV||_{\text{Fro}}^2+\alpha||V||_{1,1} \\
\text{subject to } & ||U_k||_2 \leq 1 \text{ for all }
0 \leq k < n_{components}\end{split}\]</div>
<p><span class="math notranslate nohighlight">\(||.||_{\text{Fro}}\)</span> stands for the Frobenius norm and <span class="math notranslate nohighlight">\(||.||_{1,1}\)</span>
stands for the entry-wise matrix norm which is the sum of the absolute values
of all the entries in the matrix.
The sparsity-inducing <span class="math notranslate nohighlight">\(||.||_{1,1}\)</span> matrix norm also prevents learning
components from noise when few training samples are available. The degree
of penalization (and thus sparsity) can be adjusted through the
hyperparameter <code class="docutils literal notranslate"><span class="pre">alpha</span></code>. Small values lead to a gently regularized
factorization, while larger values shrink many coefficients to zero.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>While in the spirit of an online algorithm, the class
<a class="reference internal" href="generated/sklearn.decomposition.MiniBatchSparsePCA.html#sklearn.decomposition.MiniBatchSparsePCA" title="sklearn.decomposition.MiniBatchSparsePCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">MiniBatchSparsePCA</span></code></a> does not implement <code class="docutils literal notranslate"><span class="pre">partial_fit</span></code> because
the algorithm is online along the features direction, not the samples
direction.</p>
</div>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_faces_decomposition.html#sphx-glr-auto-examples-decomposition-plot-faces-decomposition-py"><span class="std std-ref">Faces dataset decompositions</span></a></p></li>
</ul>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="mrl09" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id2">Mrl09</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://www.di.ens.fr/~fbach/mairal_icml09.pdf">“Online Dictionary Learning for Sparse Coding”</a>
J. Mairal, F. Bach, J. Ponce, G. Sapiro, 2009</p>
</div>
<div class="citation" id="jen09" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">Jen09</a><span class="fn-bracket">]</span></span>
<p><a class="reference external" href="https://www.di.ens.fr/~fbach/sspca_AISTATS2010.pdf">“Structured Sparse Principal Component Analysis”</a>
R. Jenatton, G. Obozinski, F. Bach, 2009</p>
</div>
</div>
</section>
</section>
<section id="kernel-principal-component-analysis-kpca">
<span id="kernel-pca"></span><h2><span class="section-number">2.5.2. </span>Kernel Principal Component Analysis (kPCA)<a class="headerlink" href="#kernel-principal-component-analysis-kpca" title="Link to this heading">#</a></h2>
<section id="exact-kernel-pca">
<h3><span class="section-number">2.5.2.1. </span>Exact Kernel PCA<a class="headerlink" href="#exact-kernel-pca" title="Link to this heading">#</a></h3>
<p><a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelPCA</span></code></a> is an extension of PCA which achieves non-linear
dimensionality reduction through the use of kernels (see <a class="reference internal" href="metrics.html#metrics"><span class="std std-ref">Pairwise metrics, Affinities and Kernels</span></a>) <a class="reference internal" href="#scholkopf1997" id="id3"><span>[Scholkopf1997]</span></a>. It
has many applications including denoising, compression and structured
prediction (kernel dependency estimation). <a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelPCA</span></code></a> supports both
<code class="docutils literal notranslate"><span class="pre">transform</span></code> and <code class="docutils literal notranslate"><span class="pre">inverse_transform</span></code>.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/decomposition/plot_kernel_pca.html"><img alt="../_images/sphx_glr_plot_kernel_pca_002.png" src="../_images/sphx_glr_plot_kernel_pca_002.png" style="width: 1050.0px; height: 300.0px;" />
</a>
</figure>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p><a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA.inverse_transform" title="sklearn.decomposition.KernelPCA.inverse_transform"><code class="xref py py-meth docutils literal notranslate"><span class="pre">KernelPCA.inverse_transform</span></code></a> relies on a kernel ridge to learn the
function mapping samples from the PCA basis into the original feature
space <a class="reference internal" href="#bakir2003" id="id4"><span>[Bakir2003]</span></a>. Thus, the reconstruction obtained with
<a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA.inverse_transform" title="sklearn.decomposition.KernelPCA.inverse_transform"><code class="xref py py-meth docutils literal notranslate"><span class="pre">KernelPCA.inverse_transform</span></code></a> is an approximation. See the example
linked below for more details.</p>
</div>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/decomposition/plot_kernel_pca.html#sphx-glr-auto-examples-decomposition-plot-kernel-pca-py"><span class="std std-ref">Kernel PCA</span></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/applications/plot_digits_denoising.html#sphx-glr-auto-examples-applications-plot-digits-denoising-py"><span class="std std-ref">Image denoising using kernel PCA</span></a></p></li>
</ul>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="scholkopf1997" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id3">Scholkopf1997</a><span class="fn-bracket">]</span></span>
<p>Schölkopf, Bernhard, Alexander Smola, and Klaus-Robert Müller.
<a class="reference external" href="https://people.eecs.berkeley.edu/~wainwrig/stat241b/scholkopf_kernel.pdf">“Kernel principal component analysis.”</a>
International conference on artificial neural networks.
Springer, Berlin, Heidelberg, 1997.</p>
</div>
<div class="citation" id="bakir2003" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id4">Bakir2003</a><span class="fn-bracket">]</span></span>
<p>Bakır, Gökhan H., Jason Weston, and Bernhard Schölkopf.
<a class="reference external" href="https://papers.nips.cc/paper/2003/file/ac1ad983e08ad3304a97e147f522747e-Paper.pdf">“Learning to find pre-images.”</a>
Advances in neural information processing systems 16 (2003): 449-456.</p>
</div>
</div>
</section>
<section id="choice-of-solver-for-kernel-pca">
<span id="kpca-solvers"></span><h3><span class="section-number">2.5.2.2. </span>Choice of solver for Kernel PCA<a class="headerlink" href="#choice-of-solver-for-kernel-pca" title="Link to this heading">#</a></h3>
<p>While in <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a> the number of components is bounded by the number of
features, in <a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelPCA</span></code></a> the number of components is bounded by the
number of samples. Many real-world datasets have large number of samples! In
these cases finding <em>all</em> the components with a full kPCA is a waste of
computation time, as data is mostly described by the first few components
(e.g. <code class="docutils literal notranslate"><span class="pre">n_components<=100</span></code>). In other words, the centered Gram matrix that
is eigendecomposed in the Kernel PCA fitting process has an effective rank that
is much smaller than its size. This is a situation where approximate
eigensolvers can provide speedup with very low precision loss.</p>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="eigensolvers">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Eigensolvers<a class="headerlink" href="#eigensolvers" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">The optional parameter <code class="docutils literal notranslate"><span class="pre">eigen_solver='randomized'</span></code> can be used to
<em>significantly</em> reduce the computation time when the number of requested
<code class="docutils literal notranslate"><span class="pre">n_components</span></code> is small compared with the number of samples. It relies on
randomized decomposition methods to find an approximate solution in a shorter
time.</p>
<p class="sd-card-text">The time complexity of the randomized <a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelPCA</span></code></a> is
<span class="math notranslate nohighlight">\(O(n_{\mathrm{samples}}^2 \cdot n_{\mathrm{components}})\)</span>
instead of <span class="math notranslate nohighlight">\(O(n_{\mathrm{samples}}^3)\)</span> for the exact method
implemented with <code class="docutils literal notranslate"><span class="pre">eigen_solver='dense'</span></code>.</p>
<p class="sd-card-text">The memory footprint of randomized <a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelPCA</span></code></a> is also proportional to
<span class="math notranslate nohighlight">\(2 \cdot n_{\mathrm{samples}} \cdot n_{\mathrm{components}}\)</span> instead of
<span class="math notranslate nohighlight">\(n_{\mathrm{samples}}^2\)</span> for the exact method.</p>
<p class="sd-card-text">Note: this technique is the same as in <a class="reference internal" href="#randomizedpca"><span class="std std-ref">PCA using randomized SVD</span></a>.</p>
<p class="sd-card-text">In addition to the above two solvers, <code class="docutils literal notranslate"><span class="pre">eigen_solver='arpack'</span></code> can be used as
an alternate way to get an approximate decomposition. In practice, this method
only provides reasonable execution times when the number of components to find
is extremely small. It is enabled by default when the desired number of
components is less than 10 (strict) and the number of samples is more than 200
(strict). See <a class="reference internal" href="generated/sklearn.decomposition.KernelPCA.html#sklearn.decomposition.KernelPCA" title="sklearn.decomposition.KernelPCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">KernelPCA</span></code></a> for details.</p>
<p class="rubric">References</p>
<ul class="simple">
<li><p class="sd-card-text"><em>dense</em> solver:
<a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.linalg.eigh.html">scipy.linalg.eigh documentation</a></p></li>
<li><p class="sd-card-text"><em>randomized</em> solver:</p>
<ul>
<li><p class="sd-card-text">Algorithm 4.3 in
<a class="reference external" href="https://arxiv.org/abs/0909.4061">“Finding structure with randomness: Stochastic
algorithms for constructing approximate matrix decompositions”</a>
Halko, et al. (2009)</p></li>
<li><p class="sd-card-text"><a class="reference external" href="https://arxiv.org/abs/1412.3510">“An implementation of a randomized algorithm
for principal component analysis”</a>
A. Szlam et al. (2014)</p></li>
</ul>
</li>
<li><p class="sd-card-text"><em>arpack</em> solver:
<a class="reference external" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.linalg.eigsh.html">scipy.sparse.linalg.eigsh documentation</a>
R. B. Lehoucq, D. C. Sorensen, and C. Yang, (1998)</p></li>
</ul>
</div>
</details></section>
</section>
<section id="truncated-singular-value-decomposition-and-latent-semantic-analysis">
<span id="lsa"></span><h2><span class="section-number">2.5.3. </span>Truncated singular value decomposition and latent semantic analysis<a class="headerlink" href="#truncated-singular-value-decomposition-and-latent-semantic-analysis" title="Link to this heading">#</a></h2>
<p><a class="reference internal" href="generated/sklearn.decomposition.TruncatedSVD.html#sklearn.decomposition.TruncatedSVD" title="sklearn.decomposition.TruncatedSVD"><code class="xref py py-class docutils literal notranslate"><span class="pre">TruncatedSVD</span></code></a> implements a variant of singular value decomposition
(SVD) that only computes the <span class="math notranslate nohighlight">\(k\)</span> largest singular values,
where <span class="math notranslate nohighlight">\(k\)</span> is a user-specified parameter.</p>
<p><a class="reference internal" href="generated/sklearn.decomposition.TruncatedSVD.html#sklearn.decomposition.TruncatedSVD" title="sklearn.decomposition.TruncatedSVD"><code class="xref py py-class docutils literal notranslate"><span class="pre">TruncatedSVD</span></code></a> is very similar to <a class="reference internal" href="generated/sklearn.decomposition.PCA.html#sklearn.decomposition.PCA" title="sklearn.decomposition.PCA"><code class="xref py py-class docutils literal notranslate"><span class="pre">PCA</span></code></a>, but differs
in that the matrix <span class="math notranslate nohighlight">\(X\)</span> does not need to be centered.
When the columnwise (per-feature) means of <span class="math notranslate nohighlight">\(X\)</span>
are subtracted from the feature values,