B. .. Modèle,

.. .. Conclusions,

M. Koyuturk, A. Grama, and N. Ramakrishnan, Compression, clustering, and pattern discovery in very high-dimensional discrete-attribute data sets, IEEE Transactions on Knowledge and Data Engineering, vol.17, issue.4, pp.447-461, 2005.

B. Shen, S. Ji, and J. Ye, Mining discrete patterns via binary matrix factorization, Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pp.757-766, 2009.

E. Nenova, I. Dmitry, A. V. Ignatov, and . Konstantinov, An FCA-based boolean matrix factorisation for collaborative filtering, 2013.

S. Talwar and A. Paulraj, Blind separation of synchronous cochannel digital signals using an antenna array. ii. performance analysis, IEEE Transactions on Signal Processing, vol.45, issue.3, pp.706-718, 1997.

. A-jan-van-der-veen, Analytical method for blind binary signal separation, 13th International Conference on Digital Signal Processing Proceedings, DSP 97, vol.1, pp.399-402, 1997.

S. Wold, K. Esbensen, and P. Geladi, Principal component analysis. Chemometrics and intelligent laboratory systems, vol.2, pp.37-52, 1987.

D. Daniel, H. Lee, and . Sebastian-seung, Learning the parts of objects by non-negative matrix factorization, Nature, vol.401, issue.6755, pp.788-791, 1999.

D. Daniel, H. Lee, and . Sebastian-seung, Algorithms for non-negative matrix factorization, Advances in neural information processing systems, pp.556-562, 2001.

. Richard-a-harshman, Foundations of the parafac procedure : Models and conditions for an" explanatory" multimodal factor analysis, 1970.

Y. Age-k-smilde, B. Wang, and . Kowalski, Theory of medium-rank second-order calibration with restricted-tucker models, Journal of Chemometrics, vol.8, issue.1, pp.21-36, 1994.

R. Henrion, Body diagonalization of core matrices in three-way principal components analysis : Theoretical bounds and simulation, Journal of chemometrics, vol.7, issue.6, pp.477-494, 1993.

K. Henk-al, A three-step algorithm for candecomp/parafac analysis of large data sets with multicollinearity, J. Chemometrics, vol.12, pp.155-171, 1998.

R. Bro, Parafac. tutorial and applications. Chemometrics and intelligent laboratory systems, vol.38, pp.149-171, 1997.
URL : https://hal.archives-ouvertes.fr/hal-02141162

R. Bro-jorgensen, Multiway analysis in the food industry. models, algorithms and applications, 1998.

A. Smilde, R. Bro, and P. Geladi, Multi-way analysis : applications in the chemical sciences, 2005.

J. Lieven-de-lathauwer, J. Castaing, and . Cardoso, Fourth-order cumulant-based blind identification of underdetermined mixtures, IEEE Transactions on Signal Processing, vol.55, issue.6, pp.2965-2973, 2007.

A. Lieven-de-lathauwer and . De-baynast, Blind deconvolution of DS-CDMA signals by means of decomposition in rank-(1, l, l) terms, IEEE Transactions on Signal Processing, vol.56, issue.4, pp.1562-1571, 2008.

B. Lieven-de-lathauwer and . De-moor, From matrix to tensor : Multilinear algebra and signal processing, Institute of Mathematics and its Applications Conference Series, vol.67, pp.1-16, 1998.

J. Lieven-de-lathauwer and . Vandewalle, Dimensionality reduction in higherorder signal processing and rank-(r1, r2, ..., rn) reduction in multilinear algebra, Linear Algebra and its Applications, vol.391, pp.31-55, 2004.

D. Fitzgerald, M. Cranitch, and E. Coyle, Non-negative tensor factorisation for sound source separation, 2005.

B. D. Lieven-de-lathauwer, J. Moor, and . Vandewalle, A multilinear singular value decomposition, SIAM journal on Matrix Analysis and Applications, vol.21, issue.4, pp.1253-1278, 2000.

B. D. Lieven-de-lathauwer, J. Moor, and . Vandewalle, On the best rank-1 and rank-(r 1, r 2,..., rn) approximation of higher-order tensors, SIAM journal on Matrix Analysis and Applications, vol.21, issue.4, pp.1324-1342, 2000.

S. Victor, P. A. Grigorascu, and . Regalia, Tensor displacement structures and polyspectral matching, Fast reliable algorithms for matrices with structure, pp.245-276, 1999.

G. Tamara and . Kolda, Orthogonal tensor decompositions. SIAM Journal on Matrix Analysis and Applications, vol.23, issue.1, pp.243-255, 2001.

G. Beylkin, J. Martin, and . Mohlenkamp, Numerical operator calculus in higher dimensions, Proceedings of the National Academy of Sciences, vol.99, issue.16, pp.10246-10251, 2002.
DOI : 10.1073/pnas.112329799

URL : http://www.pnas.org/content/99/16/10246.full.pdf

G. Beylkin, J. Martin, and . Mohlenkamp, Algorithms for numerical analysis in high dimensions, SIAM Journal on Scientific Computing, vol.26, issue.6, pp.2133-2159, 2005.
URL : https://hal.archives-ouvertes.fr/hal-02076682

W. Hackbusch, N. Boris, and . Khoromskij, Tensor-product approximation to operators and functions in high dimensions, Journal of Complexity, vol.23, issue.4-6, pp.697-714, 2007.

E. Acar, A. Seyit, S. Mukkai, B. Krishnamoorthy, and . Yener, Modeling and multiway analysis of chatroom tensors, International Conference on Intelligence and Security Informatics, pp.256-268, 2005.
DOI : 10.1007/11427995_21

E. Acar, A. Seyit, and B. Yener, Collective sampling and analysis of high order tensors for chatroom communications, International Conference on Intelligence and Security Informatics, pp.213-224, 2006.

W. Brett, . Bader, W. Michael, M. Berry, and . Browne, Discussion tracking in enron email using PARAFAC, Survey of Text Mining II, pp.147-163, 2008.

T. Kolda and B. Bader, The tophits model for higher-order web link analysis, Workshop on link analysis, counterterrorism and security, vol.7, pp.26-29, 2006.

G. Tamara, . Kolda, W. Brett, J. Bader, and . Kenny, Higher-order web link analysis using multilinear algebra, Fifth IEEE International Conference on Data Mining

W. Brett, R. A. Bader, T. G. Harshman, and . Kolda, Temporal analysis of semantic graphs using ASALSAN, IEEE International Conference on Data Mining, pp.33-42, 2007.

F. Christian, . Beckmann, M. Stephen, and . Smith, Tensorial extensions of independent component analysis for multisubject fmri analysis, Neuroimage, vol.25, issue.1, pp.294-311, 2005.

. Maarten-de-vos, L. Vergult, W. De-lathauwer, S. V. De-clercq, P. Huffel et al., Canonical decomposition of ictal scalp EEG reliably detects the seizure onset zone, NeuroImage, vol.37, issue.3, pp.844-854, 2007.

. Andrew-i-schein, K. Lawrence, L. H. Saul, and . Ungar, A generalized linear model for principal component analysis of binary data, AISTATS, vol.3, p.10, 2003.

J. De-leeuw, Principal component analysis of binary data by iterated singular value decomposition, Computational statistics & data analysis, vol.50, issue.1, pp.21-39, 2006.

L. Kozma, A. Ilin, and T. Raiko, Binary principal component analysis in the netflix collaborative filtering task, IEEE International Workshop on Machine Learning for Signal Processing, pp.1-6, 2009.

S. Lee, Z. Jianhua, J. Huang, and . Hu, Sparse logistic principal components analysis for binary data, The annals of applied statistics, vol.4, issue.3, p.1579, 2010.

Z. Kang, J. Costas, and . Spanos, Sequential logistic principal component analysis (slpca) : Dimensional reduction in streaming multivariate binary-state system, 13th International Conference on Machine Learning and Applications (ICMLA), pp.171-177, 2014.

P. Pajunen, Blind separation of binary sources with less sensors than sources, International Conference on Neural Networks, vol.3, pp.1994-1997, 1997.

T. Zhong-yuan-zhang, C. Li, . Ding, X. Xian-wen-ren, and . Zhang, Binary matrix factorization for analyzing gene expression data, Data Mining and Knowledge Discovery, vol.20, issue.1, p.28, 2010.

P. Miettinen and J. Vreeken, Model order selection for boolean matrix factorization, Proceedings of the 17th ACM SIGKDD international conference on Knowledge discovery and data mining, pp.51-59, 2011.

R. Belohlavek and V. Vychodil, Optimal factorization of three-way binary data, IEEE International Conference on Granular Computing (GrC), pp.61-66, 2010.

P. Miettinen, Boolean tensor factorizations, IEEE 11th International Conference on Data Mining (ICDM), pp.447-456, 2011.

M. Diop, A. Larue, S. Miron, and D. Brie, Factorisation en matrices binaires par modèle de mélange post non-linéaire, XXVIe Colloque GRETSI Traitement du Signal & des Images, 2017.

M. Diop, A. Larue, S. Miron, and D. Brie, A post-nonlinear mixture model approach to binary matrix factorization, 25th European Signal Processing Conference, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01558843

M. Diop, S. Miron, A. Larue, and D. Brie, Boolean decomposition of binary matrices using a post-nonlinear mixture approach, IEEE Transactions on Signal Processing

M. Diop, S. Miron, A. Souloumiac, and D. Brie, Boolean CP decomposition of binary tensors : uniqueness and algorithm, ICASSP 2019, pp.12-17, 2019.

G. Tamara, . Kolda, W. Brett, and . Bader, Tensor decompositions and applications, SIAM review, vol.51, pp.455-500, 2009.

G. Tamara, J. Kolda, and . Sun, Scalable tensor decompositions for multi-aspect data mining, Eighth IEEE International Conference on Data Mining, 2008. ICDM'08, pp.363-372, 2008.

G. Tamara and . Kolda, Multilinear operators for higher-order decompositions, 2006.

P. Comon, Tensors : a brief introduction, IEEE Signal Processing Magazine, vol.31, issue.3, pp.44-53, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00923279

L. Frank and . Hitchcock, The expression of a tensor or a polyadic as a sum of products, Studies in Applied Mathematics, vol.6, issue.1-4, pp.164-189, 1927.

B. Raymond and . Cattell, Parallel proportional profiles and other principles for determining the choice of factors by rotation, Psychometrika, vol.9, issue.4, pp.267-283, 1944.

D. Carroll and J. Chang, Analysis of individual differences in multidimensional scaling via an N-way generalization of " eckart-young" decomposition, Psychometrika, 1970.

M. Alex, O. Vasilescu, and D. Terzopoulos, Multilinear analysis of image ensembles : Tensorfaces, European Conference on Computer Vision, pp.447-460

. Springer, , 2002.

M. Alex, O. Vasilescu, and D. Terzopoulos, Multilinear subspace analysis of image ensembles, IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol.2, p.93, 2003.

R. Ledyard and . Tucker, Some mathematical notes on three-mode factor analysis, Psychometrika, vol.31, issue.3, pp.279-311, 1966.

L. Tucker, Implications of factor analysis of three-way matrices for measurement of changes, problems of measuring change, 1963.

R. Ledyard and . Tucker, The extension of factor analysis to three-dimensional matrices. Contributions to mathematical psychology, p.110119, 1964.

B. D. Lieven-de-lathauwer, J. Moor, and . Vandewalle, On the best rank-1 and rank-(r1, r2, ..., rn) approximation of higher-order tensors

L. Nicholas-d-sidiropoulos, X. De-lathauwer, K. Fu, . Huang, E. Evangelos et al., Tensor decomposition for signal processing and machine learning, IEEE Transactions on Signal Processing, vol.65, issue.13, pp.3551-3582, 2017.

H. Becker, L. Albera, P. Comon, M. Haardt, G. Birot et al., Eeg extended source localization : tensor-based vs. conventional methods, NeuroImage, vol.96, pp.143-157, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01011856

G. André-lf-de-almeida, J. Favier, M. Cesar, R. Mota, and . Lacerda, Estimation of frequency-selective block-fading MIMO channels using PARAFAC modeling and alternating least squares, Fortieth Asilomar Conference on Signals, Systems and Computers, 2006. ACSSC'06

P. Comon and C. Jutten, Handbook of Blind Source Separation : Independent component analysis and applications, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00460653

J. Hastad, Tensor rank is NP-Complete, Journal of Algorithms, vol.11, issue.4, pp.644-654, 1990.

T. Petros, R. G. Boufounos, and . Baraniuk, 1-bit compressive sensing, p.42

, Annual Conference on Information Sciences and Systems, pp.16-21, 2008.

L. Jacques, J. N. Laska, T. Petros, R. G. Boufounos, and . Baraniuk, Robust 1-bit compressive sensing via binary stable embeddings of sparse vectors, IEEE Transactions on Information Theory, vol.59, issue.4, pp.2082-2102, 2013.

S. Talwar, M. Viberg, and A. Paulraj, Blind separation of synchronous co-channel digital signals using an antenna array. i. algorithms, IEEE Transactions on Signal Processing, vol.44, issue.5, pp.1184-1197, 1996.

Y. Cheng, M. George, and . Church, Biclustering of expression data, ISMB, vol.8, pp.93-103, 2000.

T. Li, A general model for clustering binary data, Proceedings of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining, pp.188-197, 2005.

M. Koyutürk and A. Grama, Proximus : a framework for analyzing very high dimensional discrete-attributed datasets, Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, pp.147-156, 2003.

E. Meeds, Z. Ghahramani, M. Radford, S. T. Neal, and . Roweis, Modeling dyadic data with binary latent factors, Advances in neural information processing systems, pp.977-984, 2006.

A. Cichocki and R. Zdunek, Anh Huy Phan, and Shun-ichi Amari. Nonnegative matrix and tensor factorizations : applications to exploratory multi-way data analysis and blind source separation, 2009.

T. Watson, Nonnegative rank vs. binary rank, Chicago Journal OF Theoretical Computer Science, vol.2, pp.1-13, 2016.

L. R-duncan, A note on boolean matrix theory, Proceedings of the American Mathematical Society, vol.3, issue.3, pp.382-388, 1952.

R. Belohlavek and V. Vychodil, Discovery of optimal factors in binary data via a novel method of matrix decomposition, Journal of Computer and System Sciences, vol.76, issue.1, pp.3-20, 2010.

D. Cheng, Y. Zhao, and X. Xu, Matrix approach to boolean calculus, 50th IEEE Conference on Decision and Control and European Control Conference, pp.6950-6955, 2011.

A. Yeredor, Ica in boolean xor mixtures, Independent Component Analysis and Signal Separation, pp.827-835, 2007.

L. Valerie and . Watts, Boolean rank of Kronecker products, Linear Algebra and its Applications, vol.336, issue.1-3, pp.261-264, 2001.

H. Kim-ki, Boolean matrix theory and applications. Pure and Applied Mathematics, vol.70, 1982.

P. Miettinen and T. Mielikainen, Aristides Gionis, Gautam Das, and Heikki Mannila. The discrete basis problem, IEEE Transactions on Knowledge and Data Engineering, vol.20, issue.10, pp.1348-1362, 2008.

Y. Shen, M. Mardani, and G. Giannakis, Online categorical subspace learning for sketching big data with misses, IEEE Transactions on Signal Processing, vol.65, issue.15, pp.4004-4018, 2017.

T. Cai and W. Zhou, A max-norm constrained minimization approach to 1-bit matrix completion, The Journal of Machine Learning Research, vol.14, issue.1, pp.3619-3647, 2013.

A. Mark, Y. Davenport, and . Plan, Ewout Van Den Berg, and Mary Wootters. 1-bit matrix completion. Information and Inference : A, Journal of the IMA, vol.3, issue.3, pp.189-223, 2014.

A. Sonia, A. Bhaskar, and . Javanmard, 1-bit matrix completion under exact lowrank constraint, 49th Annual Conference on Information Sciences and Systems (CISS), pp.1-6, 2015.

V. Cottet and P. Alquier, 1-bit matrix completion : Pac-bayesian analysis of a variational approximation, Machine Learning, vol.107, issue.3, pp.579-603, 2018.

M. Koyutürk, A. Grama, and N. Ramakrishnan, Algebraic techniques for analysis of large discrete-valued datasets, European Conference on Principles of Data Mining and Knowledge Discovery, pp.311-324, 2002.

H. Lu, J. Vaidya, and V. Atluri, Optimal boolean matrix decomposition : Application to role engineering, IEEE 24th International Conference on Data Engineering, pp.297-306, 2008.

H. Nguyen and R. Zheng, Binary independent component analysis with or mixtures, IEEE Transactions on Signal Processing, vol.59, issue.7, pp.3168-3181, 2011.

Z. Zhang, C. Ding, T. Li, and X. Zhang, Binary matrix factorization with applications, Seventh IEEE International Conference on Data Mining, pp.391-400, 2007.

L. Mukherjee, N. Sathya, . Ravi, K. Vamsi, T. Ithapu et al., An NMF perspective on binary hashing, Proceedings of the IEEE International Conference on Computer Vision, pp.4184-4192, 2015.

Y. Koren, R. Bell, and C. Volinsky, Matrix factorization techniques for recommender systems, Computer, vol.42, issue.8, 2009.

S. Rendle, L. B. Marinho, A. Nanopoulos, and L. Schmidtthieme, Learning optimal ranking with tensor factorization for tag recommendation, Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining, pp.727-736, 2009.

S. Liu, Matrix results on the Khatri-rao and tracy-singh products, Linear Algebra and its Applications, vol.289, issue.1-3, pp.267-277, 1999.

. Wing-kin, T. Ma, C. Hsieh, and . Chi, Doa estimation of quasistationary signals with less sensors than sources and unknown spatial noise covariance : a Khatri-rao subspace approach, IEEE Transactions on Signal Processing, vol.58, issue.4, pp.2168-2180, 2010.

B. Ganter, G. Stumme, and R. Wille, Formal concept analysis : foundations and applications, vol.3626, 2005.

S. Andrews, In-close, a fast algorithm for computing formal concepts, 2009.

A. Taleb and C. Jutten, Source separation in post-nonlinear mixtures, IEEE transactions on Signal Processing, vol.47, issue.10, pp.2807-2820, 1999.

M. Robert, Y. Bell, C. Koren, and . Volinsky, The bellkor solution to the Netflix prize, Ref Type : Internet Communication, 2007.

J. Bennett and S. Lanning, The Netflix prize, Proceedings of KDD cup and workshop, p.35, 2007.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers. Foundations and Trends in Machine learning, vol.3, pp.1-122, 2011.

K. Huang, D. Nicholas, A. Sidiropoulos, and . Liavas, A flexible and efficient algorithmic framework for constrained matrix and tensor factorization, IEEE Transactions on Signal Processing, vol.64, issue.19, pp.5052-5065, 2016.

M. Welling and M. Weber, Positive tensor factorization, Pattern Recognition Letters, vol.22, issue.12, pp.1255-1261, 2001.

J. Stehlé, N. Voirin, A. Barrat, C. Cattuto, L. Isella et al., High-resolution measurements of face-to-face contact patterns in a primary school, PloS one, vol.6, issue.8, p.23176, 2011.

L. Gauvin, A. Panisson, and C. Cattuto, Detecting the community structure and activity patterns of temporal networks : a non-negative tensor factorization approach, PloS one, vol.9, issue.1, p.86028, 2014.

, 16 1.2 Illustration du dépliement du tenseur X suivant les 3 modes, p.17

, Décomposition de TUCKER d'un tenseur d'ordre 3, X ? R I 1 ×I 2 ×···×I N, p.19

, Décomposition de deux sources corrélées avec les différents produits matriciels booléens

, Représentation des différents tenseurs de rang 1 estimés, p.38

. .. , , p.44

, Cas de deux sources non corrélées (N = 80, M = 100, p.47

. .. Asso)), Cas de deux sources corrélées (N = 100, M = 80, p.48

. .. , Cas de trois sources corrélées (N = 80, M = 100), p.48

, Décomposition de 3 sources non-corrélées

, Unicité de la décomposition de 3 sources corrélées

, Non-unicité de la décomposition de 3 sources corrélées, p.52

. .. , Comparaison des algorithmes PNL-PF et C-PNL-PF pour une décomposition non-unique de 3 sources (N = 80, M = 100, ? = 5 × 10 5 ), p.53

, Erreur d'estimation sur W, H et de reconstruction sur X en fonction du taux de bruit b, pour p = 25% (N = 80, M = 100, ? = 10)

. .. , Erreur d'estimation sur W, H et de reconstruction sur X en fonction du taux de bruit b, pour p = 60% (N = 80, M = 100, ? = 6 × 10 5 ), p.57

, Données Netflix binarisées et approximation de rang 4

. .. , Erreur d'approximation de X en fonction du rang k, p.59

, Réorganisation de la matrice d'approximation de X (figure 2.11b) selon les 4 groupes (utilisateurs, films) estimés, p.61

, Nb iter = 100, Nb interne = 10) pour p = 0.6, Taux d'erreur de reconstruction de X en fonction du paramètre ? (N = 12, M = 9, vol.3, p.82

, Taux de reconstruction de X en fonction du taux de bruit rajouté b (N = 20, M = 30, P = 10, K = 3, ? = 10 9, p.0

, Taux d'estimation de W H et V en fonction du taux de bruit rajouté b (N = 20, M = 30, p.0

, Taux de reconstruction de X en fonction du taux de bruit rajouté b (N = 20, M = 30, P = 10, K = 3, ? = 10 9, p.0

, Taux d'estimation de W H et V en fonction du taux de bruit rajouté b (N = 12, M = 9, vol.3, p.0

. .. , Erreur d'approximation de X 1 en fonction du rang booléen K, vol.87

, 36 2.1 La moyenne, le minimum et le maximum du taux d'erreur d'estimation, Tableau des compétences ou qualités deséì eves dans divers cours

. De-w,

, ? = 5 × 10 6 )

. De-w,

, ? = 5 × 10 6 )

. Le-snr-binaire, SNR B ) en fonction du taux de bruit rajouté b pour p = 25%, p.55

, Le cardinal des groupes d'utilisateurs et de leurs intersections, p.60

, Le cardinal des groupes de films et de leurs intersections, p.60

D. Le-nombre, ´eì eves de chaque classe et de enseignants qui ont participéparticipé`participéà l'´ etude

, Tableau récapitulatif des contacts entre les classes pendant les 2 journées, p.90