34:(PCA) that is used to analyze M-way arrays, also informally referred to as "data tensors". M-way arrays may be modeled by linear tensor models, such as CANDECOMP/Parafac, or by multilinear tensor models, such as multilinear principal component analysis (MPCA) or multilinear independent component analysis (MICA). The origin of MPCA can be traced back to the
89:
MPCA computes a set of orthonormal matrices associated with each mode of the data tensor which are analogous to the orthonormal row and column space of a matrix computed by the matrix SVD. This transformation aims to capture as high a variance as possible, accounting for as much of the variability
85:
Multilinear PCA may be applied to compute the causal factors of data formation, or as signal processing tool on data tensors whose individual observation have either been vectorized, or whose observations are treated as a collection of column/row observations, "data matrix" and concatenated into a
81:
introduced the
Multilinear PCA terminology as a way to better differentiate between linear and multilinear tensor decomposition, as well as, to better differentiate between the work that computed 2nd order statistics associated with each data tensor mode(axis), and subsequent work on Multilinear
65:
Circa 2001, Vasilescu and
Terzopoulos reframed the data analysis, recognition and synthesis problems as multilinear tensor problems. Tensor factor analysis is the compositional consequence of several causal factors of data formation, and are well suited for multi-modal data tensor analysis. The
317:"Multilinear Analysis of Image Ensembles: TensorFaces," Proc. 7th European Conference on Computer Vision (ECCV'02), Copenhagen, Denmark, May, 2002, in Computer Vision – ECCV 2002, Lecture Notes in Computer Science, Vol. 2350, A. Heyden et al. (Eds.), Springer-Verlag, Berlin, 2002, 447–460.
66:
power of the tensor framework was showcased by analyzing human motion joint angles, facial images or textures in terms of their causal factors of data formation in the following works: Human Motion
Signatures (CVPR 2001, ICPR 2002), face recognition –
354:"TensorTextures: Multilinear Image-Based Rendering", M. A. O. Vasilescu and D. Terzopoulos, Proc. ACM SIGGRAPH 2004 Conference Los Angeles, CA, August, 2004, in Computer Graphics Proceedings, Annual Conference Series, 2004, 336–342.
98:
The MPCA solution follows the alternating least square (ALS) approach. It is iterative in nature. As in PCA, MPCA works on centered data. Centering is a little more complicated for tensors, and it is problem dependent.
47:
46:; and to Peter Kroonenberg's "3-mode PCA" work. In 2000, De Lathauwer et al. restated Tucker and Kroonenberg's work in clear and concise numerical computational terms in their SIAM paper entitled "
107:
MPCA features: Supervised MPCA is employed in causal factor analysis that facilitates object recognition while a semi-supervised MPCA feature selection is employed in visualization tasks.
296:"Human Motion Signatures: Analysis, Synthesis, Recognition," Proceedings of International Conference on Pattern Recognition (ICPR 2002), Vol. 3, Quebec City, Canada, Aug, 2002, 456–460.
406:
K. Inoue, K. Hara, K. Urahama, "Robust multilinear principal component analysis", Proc. IEEE Conference on
Computer Vision, 2009, pp. 591–597.
71:
77:
Historically, MPCA has been referred to as "M-mode PCA", a terminology which was coined by Peter
Kroonenberg in 1980. In 2005, Vasilescu and
67:
336:
M. A. O. Vasilescu, D. Terzopoulos, Proc. Computer Vision and
Pattern Recognition Conf. (CVPR '03), Vol.2, Madison, WI, June, 2003, 93–99.
371:, "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’05), San Diego, CA, June 2005, vol.1, 547–553."
397:," in Proceedings of the 19th ACM Conference on Information and Knowledge Management (CIKM 2010), Toronto, ON, Canada, October, 2010.
145:
384:, "Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR’03), Madison, WI, June, 2003"
86:
data tensor. The main disadvantage of this approach is that rather than computing all possible combinations
82:
Independent
Component Analysis that computed higher order statistics associated with each tensor mode/axis.
502:
295:
31:
394:
353:
416:
Khan, Suleiman A.; Leppäaho, Eemeli; Kaski, Samuel (2016-06-10). "Bayesian multi-tensor factorization".
35:
316:
216:
Principal component analysis of three-mode data by means of alternating least squares algorithms
39:
478:
468:
43:
8:
381:
312:
78:
27:
333:
122:
Multi-Tensor
Factorization, that also finds the number of components automatically (MTF)
425:
261:"On the best rank-1 and rank-(R1, R2, ..., RN ) approximation of higher-order tensors"
443:
194:
435:
272:
241:
186:
172:
154:
368:
140:
487:
260:
229:
439:
276:
245:
496:
447:
177:
175:(September 1966). "Some mathematical notes on three-mode factor analysis".
198:
158:
143:(1927). "The expression of a tensor or a polyadic as a sum of products".
215:
190:
395:
Visualization and
Clustering of Crowd Video Content in MPCA Subspace
430:
50:", (HOSVD) and in their paper "On the Best Rank-1 and Rank-(R
90:
in the data associated with each data tensor mode(axis).
258:
227:
70:, (ECCV 2002, CVPR 2003, etc.) and computer graphics –
393:
H. Lu, H.-L. Eng, M. Thida, and K.N. Plataniotis, "
259:
Lathauwer, L. D.; Moor, B. D.; Vandewalle, J. (2000).
16:
Multilinear extension of principal component analysis
228:
Lathauwer, L.D.; Moor, B.D.; Vandewalle, J. (2000).
334:"Multilinear Subspace Analysis for Image Ensembles,
382:"Multilinear Subspace Analysis of Image Ensembles"
415:
210:
208:
494:
363:
361:
307:
305:
303:
290:
288:
286:
265:SIAM Journal on Matrix Analysis and Applications
234:SIAM Journal on Matrix Analysis and Applications
348:
346:
344:
328:
326:
324:
252:
221:
139:
205:
358:
300:
283:
369:"Multilinear Independent Component Analysis"
341:
321:
230:"A multilinear singular value decomposition"
380:M. A. O. Vasilescu, D. Terzopoulos (2003)
367:M. A. O. Vasilescu, D. Terzopoulos (2005)
62:) Approximation of Higher-order Tensors".
429:
352:M.A.O. Vasilescu, D. Terzopoulos (2004)
332:M.A.O. Vasilescu, D. Terzopoulos (2003)
48:Multilinear Singular Value Decomposition
20:Multilinear principal component analysis
495:
218:, Psychometrika, 45 (1980), pp. 69–97.
171:
374:
102:
214:P. M. Kroonenberg and J. de Leeuw,
13:
146:Journal of Mathematics and Physics
14:
514:
458:
93:
409:
400:
387:
165:
133:
1:
126:
110:
32:principal component analysis
7:
115:Various extension of MPCA:
10:
519:
440:10.1007/s10994-016-5563-y
277:10.1137/s0895479898346995
246:10.1137/s0895479896305696
36:tensor rank decomposition
294:M.A.O. Vasilescu (2002)
479:UMPCA (including data)
40:Frank Lauren Hitchcock
159:10.1002/sapm192761164
44:Tucker decomposition
503:Dimension reduction
119:Robust MPCA (RMPCA)
311:M.A.O. Vasilescu,
191:10.1007/BF02289464
173:Tucker, Ledyard R
103:Feature selection
74:(Siggraph 2004).
510:
452:
451:
433:
418:Machine Learning
413:
407:
404:
398:
391:
385:
378:
372:
365:
356:
350:
339:
330:
319:
309:
298:
292:
281:
280:
271:(4): 1324–1342.
256:
250:
249:
240:(4): 1253–1278.
225:
219:
212:
203:
202:
169:
163:
162:
153:(1–4): 164–189.
137:
42:in 1927; to the
518:
517:
513:
512:
511:
509:
508:
507:
493:
492:
461:
456:
455:
414:
410:
405:
401:
392:
388:
379:
375:
366:
359:
351:
342:
331:
322:
310:
301:
293:
284:
257:
253:
226:
222:
213:
206:
170:
166:
141:F. L. Hitchcock
138:
134:
129:
113:
105:
96:
61:
57:
53:
17:
12:
11:
5:
516:
506:
505:
491:
490:
482:
472:
460:
459:External links
457:
454:
453:
424:(2): 233–253.
408:
399:
386:
373:
357:
340:
320:
313:D. Terzopoulos
299:
282:
251:
220:
204:
185:(3): 279–311.
164:
131:
130:
128:
125:
124:
123:
120:
112:
109:
104:
101:
95:
92:
72:TensorTextures
59:
55:
51:
38:introduced by
15:
9:
6:
4:
3:
2:
515:
504:
501:
500:
498:
489:
486:
483:
480:
476:
473:
470:
466:
463:
462:
449:
445:
441:
437:
432:
427:
423:
419:
412:
403:
396:
390:
383:
377:
370:
364:
362:
355:
349:
347:
345:
338:
337:
329:
327:
325:
318:
314:
308:
306:
304:
297:
291:
289:
287:
278:
274:
270:
266:
262:
255:
247:
243:
239:
235:
231:
224:
217:
211:
209:
200:
196:
192:
188:
184:
180:
179:
178:Psychometrika
174:
168:
160:
156:
152:
148:
147:
142:
136:
132:
121:
118:
117:
116:
108:
100:
94:The algorithm
91:
87:
83:
80:
75:
73:
69:
63:
49:
45:
41:
37:
33:
30:extension of
29:
25:
21:
484:
474:
464:
421:
417:
411:
402:
389:
376:
335:
268:
264:
254:
237:
233:
223:
182:
176:
167:
150:
144:
135:
114:
106:
97:
88:
84:
76:
64:
23:
19:
18:
475:Matlab code
465:Matlab code
79:Terzopoulos
68:TensorFaces
28:multilinear
127:References
111:Extensions
448:0885-6125
431:1412.4679
497:Category
58:, ..., R
26:) is a
485:R code:
315:(2002)
199:5221127
446:
197:
426:arXiv
469:MPCA
444:ISSN
195:PMID
24:MPCA
488:MTF
436:doi
422:105
273:doi
242:doi
187:doi
155:doi
54:, R
499::
477::
467::
442:.
434:.
420:.
360:^
343:^
323:^
302:^
285:^
269:21
267:.
263:.
238:21
236:.
232:.
207:^
193:.
183:31
181:.
149:.
481:.
471:.
450:.
438::
428::
279:.
275::
248:.
244::
201:.
189::
161:.
157::
151:6
60:N
56:2
52:1
22:(
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.