BOOLEAN MATRIX AND TENSOR DECOMPOSITIONS. Pauli Miettinen TML September 2013

Transcription

1 BOOLEAN MATRIX AND TENSOR DECOMPOSITIONS Pauli Miettinen TML September 2013

2 BOOLEAN MATRIX AND TENSOR DECOMPOSITIONS Boolean decompositions are like normal decompositions, except that Input is a binary matrix or tensor Factors are binary Arithmetic is Boolean (so reconstructions are binary) Error measure is (usually) Hamming distance (L1)

3 BOOLEAN ARITHMETIC Idenpotent, anti-negative semi-ring ({0,1},, ) Like normal arithmetic, but addition is defined as 1+1 = 1 A Boolean matrix is a binary (0/1) matrix endowned with Boolean arithmetic The Boolean matrix product is defined as (A B) ij = R_ r=1 b il c lj

4 WHY BOOLEAN ARITHMETIC? Boolean decompositions find different type of structure than decompositions under normal arithmetic Not better, not worse, just different Normal decomposition: value is a sum of values from rank-1 components Boolean decomposition: value is 1 if there is any rank-1 component with 1 in this location

5 WHY BOOLEAN CONT D Boolean artihmetic can be interpret as set operations ( ) A B C A C 2 B Pauli Miettinen 24 September 2012

6 EXAMPLE Real analysis Discr. Contacts math. Programming Internet A B C ( ) ( ) ( ) =

7 RESULTS ON BOOLEAN MATRIX FACTORIZATION Computing the Boolean rank is NP-hard As hard to approximate as minimum chromatic number Minimum-error decomposition is NP-hard And hard to approximate in both additive and multiplicative sense Given A and B, finding C such that B C is close to A is hard even to approximate Alternating updates are hard!

8 SOME MORE RESULTS Boolean rank can be a logarithm of the real rank Sparse matrices have sparse (exact) factorizations The rank of the decomposition can be defined automatically using the MDL principle Planted rank-1 matrix can be recovered under XOR noise (under certain assumptions)

9 SOME ALGORITHMS Alternating least-squares Proposed in psychometrical litterature in early 1980 s Asso [M. et al & 2008] Builds candidate factors based on correlation matrix, and greedily selects them Panda [Lucchese et al. 2010] Expands monochromatic core patterns (tiles) based on MDL-esque rule Various tiling algorithms Do not allow expressing 0 in data as 1 in factorization (false positives) Binary factorizations Normal algebra but binary factors

10 SOME APPLICATIONS Explorative data analysis Psychometrics Role mining Pattern mining Bipartite community detection Binary matrix completion But requires {0, 1,?} data Co-clustering-y applications

11 RANK-1 (BOOLEAN) TENSORS c b X X = b a X = a 1 bb 2 c

12 THE BOOLEAN CP TENSOR DECOMPOSITION c 1 c 2 c R X b 1 _ b 2 _ _ b R a 1 a 2 a R R_ x ijk a ir b jr c kr r=1

13 THE BOOLEAN CP TENSOR DECOMPOSITION C X A B R_ x ijk a ir b jr c kr r=1

14 FREQUENT TRI-ITEMSET MINING Rank-1 N-way binary tensors define an N-way itemset Particularly, rank-1 binary matrices define an itemset In itemset mining the induced sub-tensor must be full of 1s Here, the items can have holes Boolean CP decomposition = lossy N-way tiling

15 BOOLEAN TENSOR RANK The Boolean rank of a binary tensor is the minimum number of binary rank-1 tensors needed to represent the tensor exactly using Boolean arithmetic. c 1 c 2 c R X = b 1 _ b 2 _ _ b R a 1 a 2 a R

16 SOME RESULTS ON RANKS Normal tensor rank is NPhard to compute Normal tensor rank of n-by-m-by-k tensor can be more than min{n, m, k} But no more than min{nm, nk, mk} Boolean tensor rank is NP-hard to compute Boolean tensor rank of n-by-m-by-k tensor can be more than min{n, m, k} But no more than min{nm, nk, mk}

17 SPARSITY Binary N-way tensor Xof Boolean tensor rank R has Boolean rank-r CP-decomposition with factor matrices A 1, A 2,, A N such that i A i N X Binary matrix X of Boolean rank R and X 1s has Boolean rank-r decomposition A o B such that A + B 2 X Both results are existential only and extend to approximate decompositions

18 SIMPLE ALGORITHM We can use typical alternating algorithm with Boolean algebra Finding the optimal projection is NP-hard even to approximate Good initial values are needed due to multiple local minima X (1) = A (C B) T X (2) = B (C A) T X (3) = C (B A) T Obtained using Boolean matrix factorization to matricizations

19 THE BOOLEAN TUCKER TENSOR DECOMPOSITION C B X A G P_ Q_ R_ x ijk g pqr a ip b jq c kr p=1 q=1 r=1

20 THE SIMPLE ALGORITHM WITH TUCKER The core tensor has global effects C Updates are hard X A G B Factors are not orthogonal Assume core tensor is small We can afford more time per element x ijk P_ p=1 Q_ q=1 R_ r=1 g pqr a ip b jq c kr In Boolean case many changes make no difference

21 WALK N MERGE: MORE SCALABLE ALGORITHM Idea: For exact decomposition, we could find all N-way tiles Then we only need to find the ones we need among them Problem: For approximate decompositions, there might not be any big tiles We need to find tiles with holes, i.e. dense rank-1 subtensors

22 TENSORS AS GRAPHS Create a graph from the tensor Each 1 in the tensor: one vertex in the graph Edge between two vertices if they differ in at most one coordinate Idea: If two vertices are in the same all-1s rank-1 N-way subtensor, they are at most N steps from each other Small-diameter subgraphs dense rank-1 subtensors

23 @ A EXAMPLE 1,1,1 1,1,2 1,2,1 1,2,2 1,4,1 1,4,2 2,1,1 2,1,2 2,2,1 2,2,2 2,3,2 3,1,2 3,3,1

24 RANDOM WALKS We can identify the small-diameter subgraphs by random walks If many (short) random walks re-visit the same nodes often, they re on a small-diameter subgraph Problem: The random walks might return many overlapping dense areas and miss the smallest rank-1 decompositions

25 MERGE We can exhaustively look for all small (e.g. 2-by-2-by-2) all-1s sub-tensors outside the already-found dense subtensors We can now merge all partially overlapping rank-1 subtensors if the resulting subtensor is dense enough Result: A Boolean CP-decomposition of some rank False positive rate controlled by the density, false negative by the exhaustive search

26 MDL STRIKES AGAIN We have a decomposition with some rank, but what would be a good rank? Normally: pre-defined by the user (but how does she know) MDL principle: The best model to describe your data is the one that does it with the least number of bits We can use MDL to choose the rank

27 HOW YOU COUNT THE BITS? MDL asks for an exact representation of the data In case of Boolean CP, we represent the tensor X with Factor matrices Error tensor E The bit-strings representing these are encoded to compute the description length

28 WHY MDL AND TUCKER DECOMPOSITION Balance between accuracy and complexity High rank: more bits in factor matrices, less in error tensor Small rank: less bits in factor matrices, more in error tensor If one mode uses the same factor multiple times, CP contains it multiple times The Tucker decomposition needs to have that factor only once

29 FROM CP TO TUCKER WITH MDL CP is Tucker with hyper-diagonal core tensor If we can remove a repeated column from a factor matrix and adjust the core accordingly, our encoding is more efficient Algorithm: Try mergin similar factors and see if that reduces the encoding length

30 APPLICATION: FACT DISCOVERY Input: noun phrase verbal phrase noun phrase triples Non-disambiguated E.g. from OpenIE Goal: Find the facts (entity relation entity triples) underlying the observed data and mappings from surface forms to entities and relations

31 CONNECTION TO BOOLEAN TENSORS We should see an np 1 vp np 2 triple if there exists at least one fact e 1 r e 2 such that np 1 is the surface form of e 1 vp is the surface form of r np 2 is the surface form of e 2

32 CONNECTION TO BOOLEAN TENSORS What we want is Boolean Tucker3 decomposition Core tensor contains the facts Factors contain the mappings from entities and relations to surface forms x ijk P_ p=1 Q_ q=1 R_ r=1 g pqr a ip b jq c kr

33 PROS & CONS Pros: Naturally sparse core tensor Core will be huge must be sparse Natural interpretation Cons: No levels of certainity Either is or not Can only handle binary data

34 EXAMPLE RESULT Subject: claude de lorimier, de lorimier, louis, jean-baptiste Relation: was born, [[det]] born in Object: borough of lachine, villa st. pierre, lachine quebec 39,500-by-8,000-by-21,000 tensor with non-zeros

35 CONCLUSIONS Boolean factorizations are more combinatorial in their flavour Interpretations as sets or graphs Boolean matrix factorization is computationally harder than most normal factorizations With tensors the difference is not so big

36 FUTURE DIRECTIONS L Thank You! L When should one apply Boolean factorizations? More education is needed Better algorithms & implementations I ve been asking for this for 7 years now