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Boolean matrix factorization meets consecutive ones property Nikolaj T atti & Pauli Miettinen Boolean matrix factorisation Given a matrix A , find matrices B and C s.t. A B C T SVD, NMF, PCA, ICA, In

  1. Boolean matrix factorization 
 meets 
 consecutive ones property Nikolaj T atti & Pauli Miettinen

  2. Boolean matrix factorisation • Given a matrix A , find matrices B and C s.t. 
 A ≈ B ∘ C T • SVD, NMF, PCA, ICA, … • In Boolean matrix factorisation (BMF) 1. the input matrix A is binary 2. the factor matrices B and C are binary 3. the algebra is Boolean (1 + 1 = 1) • Sparse factors, good interpretability, sometimes you just need binary factors UEF// School of Computing � 2 Pauli Miettinen

  3. <latexit sha1_base64="uFm7AXRhUEMkistYXsyqPDczZ3o=">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</latexit> Example of BMF b c 1 2 3 4 �     1 1 0 1 1 1 1 0 0 1 2 3 4 2  1 1 1 1  2  1 1 0        d 1 0 1 1     3 0 1 1 1 3 0 1 0       e 0 1 1 1 � =       4 0 1 1 1 4 0 1 0     ƒ     0 1 1 0     5 0 1 1 0 5 0 0 1     6 1 0 1 1 6 1 0 0 UEF// School of Computing � 3 Pauli Miettinen

  4. <latexit sha1_base64="uFm7AXRhUEMkistYXsyqPDczZ3o=">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</latexit> Example of BMF 4th column = a + b b c 1 2 3 4 �     1 1 0 1 1 1 1 0 0 1 2 3 4 2  1 1 1 1  2  1 1 0        d 1 0 1 1     3 0 1 1 1 3 0 1 0       e 0 1 1 1 � =       4 0 1 1 1 4 0 1 0     ƒ     0 1 1 0     5 0 1 1 0 5 0 0 1     6 1 0 1 1 6 1 0 0 UEF// School of Computing � 3 Pauli Miettinen

  5. <latexit sha1_base64="uFm7AXRhUEMkistYXsyqPDczZ3o=">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</latexit> Example of BMF 4th column = a + b b c 1 2 3 4 �     1 1 0 1 1 1 1 0 0 1 2 3 4 2  1 1 1 1  2  1 1 0        d 1 0 1 1     3 0 1 1 1 3 0 1 0       e 0 1 1 1 � =       4 0 1 1 1 4 0 1 0     ƒ     0 1 1 0     5 0 1 1 0 5 0 0 1     6 1 0 1 1 6 1 0 0 (2, 4) = 1 = 1 + 1 = (2, a ) + (2, b ) UEF// School of Computing � 3 Pauli Miettinen

  6. Consecutive ones property • A binary vector x has the consecutive ones property (C1P) if all of its 1s are consecutive • e.g. x = (0, 0, 1, 1, 1, 1, 1, 0) • A binary matrix X has the C1P if its rows can be permuted so that all of its columns have C1P simultaneously • A BMF A ≈ B ∘ C T has the C1P if both B and C have the C1P UEF// School of Computing � 4 Pauli Miettinen

  7. Cyclic vectors and matrices • A binary vector x is cyclic if it’s complement has the consecutive ones property • e.g. x = (1, 1, 0, 0, 0, 0, 0, 1) • A binary matrix is cyclic if its columns have either the C1P or the cyclic ones property • A BMF A ≈ B ∘ C T is cyclic if both B and C are cyclic UEF// School of Computing � 5 Pauli Miettinen

  8. <latexit sha1_base64="uFm7AXRhUEMkistYXsyqPDczZ3o=">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</latexit> Example b c 1 2 3 4 �     1 1 0 1 1 1 1 0 0 1 2 3 4 2  1 1 1 1  2  1 1 0        d 1 0 1 1     3 0 1 1 1 3 0 1 0       e 0 1 1 1 � =       4 0 1 1 1 4 0 1 0     ƒ     0 1 1 0     5 0 1 1 0 5 0 0 1     6 1 0 1 1 6 1 0 0 UEF// School of Computing � 6 Pauli Miettinen

  9. <latexit sha1_base64="uFm7AXRhUEMkistYXsyqPDczZ3o=">AD03icbZLNbtQwEMe9CR9L+GrhLhYrIo4oJWzu1AOICpxKbcisW2l3WjlOJPWqhNHtOyRL0grwMb8Pb4GyiqlnH0iT34wz/5lMXAiuDSH/Bp5/5+69+8MHwcNHj583dl9dqxlqRjMmRSncZUg+A5zA03Ak4LBTSLBZzEF1/q+MklKM1l/t2sC4gyepbzlDNqLFrtDv4uY6kSUBk1iv+oAoxf49DaxNrU2gzjJVNB2GLSPsMGT25eO3jaybzBs378bguTBr/vL3kdfMJBj2hqLbGHL3EkUoclcQRSDraGroti7SKlowr1ieqO8k6OXG6qik4k6lp6gxmU261MyJjsjnYdcLWGaH2HNm/PFwmkpUZ5IYJqvUiJIWJKqoMZwKs/lJDQdkFPYOFdXOagY6qzXZd4z1LEpxKZS03eENv36hopvU6i2mbf1cb8dq2BdblCb9EFU8L0oDOWsKpaXARuJ6VXHCFTAj1tahTHGrFbNzqigzdqGDvdtl6o8X9KfsthJnb6mCy6gqaCn4Koerenbh9qRc53gyDqfjybfZ6OBjO8UheoleoTcoRPvoAB2iIzRHzHvhfYOva/+3K/8X/7vJtUbtHeo87x/wHYX36jA=</latexit> Example cyclic b c 1 2 3 4 �     1 1 0 1 1 1 1 0 0 1 2 3 4 2  1 1 1 1  2  1 1 0        d 1 0 1 1     3 0 1 1 1 3 0 1 0       e 0 1 1 1 � =       4 0 1 1 1 4 0 1 0     ƒ     0 1 1 0     5 0 1 1 0 5 0 0 1     6 1 0 1 1 6 1 0 0 UEF// School of Computing � 6 Pauli Miettinen

  10. <latexit sha1_base64="uFm7AXRhUEMkistYXsyqPDczZ3o=">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</latexit> Example cyclic C1P b c 1 2 3 4 �     1 1 0 1 1 1 1 0 0 1 2 3 4 2  1 1 1 1  2  1 1 0        d 1 0 1 1     3 0 1 1 1 3 0 1 0       e 0 1 1 1 � =       4 0 1 1 1 4 0 1 0     ƒ     0 1 1 0     5 0 1 1 0 5 0 0 1     6 1 0 1 1 6 1 0 0 UEF// School of Computing � 6 Pauli Miettinen

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