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2. : 3. 1 Time Sensors 4. 2 Star Wars Titanic BladeRunnerUser 15 2 4User 21 4 2User 35 ? ? 5. 6. X1 X2 X3 X 4 () 7. 8. x1 x2 x3 x4 1 2 9. (2006-2009) (2010-) (?) 10. (2006-2009) (2010-) 11. 2006-2009Right or left? Tucker2010- U(1) U(3) XCU(2) 12. Brain-computer interface Aims to decode thoughts or commands fromhuman brain signal [Wolpaw+ 2002] EncodingThoughts CommandsDecodingSignal Acquisition(EEG, MEG, ) 13. P300 speller system Evoked ResponseFarwell & Donchin 1988 14. P300 speller systemA B C D E FG H I J K LM N O P Q RS T U V W XY Z 1 2 3 4 ER detected!5 6 7 8 9 _A B C D E FG H I J K LM N O P Q RS T U V W XY Z 1 2 3 4 ER detected!5 6 7 8 9 _The character must be P 15. (X1, y1), , (Xn,yn) X1X2 X3 X4 Xn Xi ( x ) yi = +1 or -1 () 16. Schatten 1- (nuclear norm / trace norm) () 17. vc: uc: 18. Modeling P300 speller (decoding) Suppose that we have a detector f(X) that detectsthe P300 response in signal X. f1 f2 f3 f4 f5 f6 f7 f8 f9 f10This is nothing but learning 2 x 6-class classifier f11 f12 19. How we do this 12 2 8 1 3 4 11 9 56 10 7 Multinomial likelihood f.Multinomial likelihood f. 20. Experiment Two subjects (A&B) from BCI competition III 64 channels x 37 time-points (600ms @ 60Hz) 12 epochs x 15 repetitions x 85 letters = 15300 epochs in training set 100 letters for test Linear detector function (bias is irrelevant) 21. (36) 15 5 Tomioka & Mller 2009 22. (Subject A) 300ms 300ms 23. (Subject B) 2 3 24. Farwell & Donchin (2x6) Schatten 1- 25. 26. (K) 1 R X R NP NP CANDECOMP / PARAFAC (CP) 27. 1 X=ABT CP (Kruskal 77)kA A k-rankkk3 28. 2 Kolda & Bader 2009X3Y2 29. Tucker Tucker [Tucker 66] n3n2r1 r2r3r3n1n1n2n3Xr1 CU(1)U(2)U(3) r2 CP (r1=r2=r3) Tucker 30. -k () -1 X ( 1) I2 I2I2I1I1 I2 I3 I3 -2 X ( 2) I3 I3I3I1I2 I2 I3 I1 31. -k-k-1 rank(X(1))=r1-2 rank(X(2))=r2-3 TuckerX(k) rank(X(3))=r3 32. CP / Tucker CP Tucker NP SVD) NP r1 x r2 x r3Tucker 33. overlapped Schatten 1--k Schatten 1- k (Cf. Liu+09, Signoretto+10, Tomioka+10, Gandy+11) 34. 35. Donoho-Tanner Phase Transition 1A: n x N (n N () C 39. 1 - 2 40. : Hlder -k 41. 2 (X) M M- Negahban & Wainwright M- 42. (M=N) (OK) 43. 2 (M=N) 44. : ( =0.01) 2 2 45. : ( =0.1)2 2 46. (Xi ) : OK ( =1/64) (M: , N: ) 47. 31. ()2. 48. () 0 49. 50. TuckerSVD Tucker Normalized rank 51. Wolpaw, Birbaumer, McFarland, Pfurtscheller, Vaughan (2002) Brain-computer interfaces for communication andcontrol. Clin. Neurophysiol. 113, 767791. Farwell & Donchin (1988) Talking off the top of your head: toward a mental prosthesis utilizing event-related brainpotentials. Electroencephalogr. Clin. Neurophysiol. 70 (6), 510523. Tomiok a & Mller (2009) A regularized discriminative framework for EEG analysis with application to brain-computerinterface. Neuroimage, 49 (1), 415-432. Kolda & Bader (2009) Tensor decompositions and applications. SIAM Review, 51(3):455500. Tucker (1966) Some mathematical notes on three-mode factor analysis. Psychometrika, 31(3):279311. Gandy, Recht, & Yamada (2011) Tensor completion and low-n-rank tensor recovery via convex optimization. InverseProblems, 27:025010. Liu, Musialski, Wonka, & Ye. (2009) Tensor completion for estimating missing values in visual data. In Prof. ICCV. Signoretto, de Lathauwer, & Suykens (2010) Nuclear norms for tensors and their use for convex multilinear estimation.Tech Report 10-186, K.U.Leuven. Donoho & Tanner (2010) Precise undersampling theorems. Proceedings of the IEEE, 98(6):913924. Recht, Fazel, & Parrilo (2010) Guaranteed minimum-rank solutions of linear matrix equations via nuclear normminimization. SIAM Review, 52(3):471501. Tomioka, Hayashi, & Kashima (2011) Estimation of low-rank tensors via convex optimization. Technical report,arXiv:1010.0789, 2011. Tomioka, Suzuki, Hayashi, & Kashima (2011) Statistical performance of convex tensor decomposition. Advances in NIPS24. 2011, Granada, Spain.