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Using Using BlindBlind S Searchearchaandnd

FFormormal Conceptsal Conceptsforfor

Binary Factor AnalysisBinary Factor Analysis

Aleš Keprtales.keprt@vsb.cz

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Synopsis Binary Factor Analysis (BFA)

- introduction to BFA- exact solution of BFA- quality checking

Possible optimizations Blind Search method Method based on Formal Concepts Tests and the comparison of

methods Possible future work

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Binary Factor Analysis (BFA) Factor analysis of binary data Using boolean arithmetic Trying to express matrix X as a

product of two matrices is binary matrix multiplication Initial conditions:

we know X, dimensions of all matrices,number of one’s per row of A

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Exact BFA Solution(i.e. reference factorizer) For checking other algorithms Searches for the best (optimal)

solution Exact = opposite to approximate

solution

The key:Perform all possible optimizations to avoid checking of all bit-combinations

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Boolean arithmetic Classical arithmetic:

Boolean arithmetic:

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Quality check Discrepancy (česky „odchylka”)

Our goal is to minimize discrepancy We distinguish between positive

and negative discrepancy

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Possible optimizations Empty rows and columns – skip Duplicate rows and columns –

merge Order factor loadings (rows of A) Constant number of one’s per row

of A Use the knowledge of A to get F Parallel (distributed) computaion

using multiple computers

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Quality check When merging duplicate

rows/columns:

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Blind Search Blind Search = „slepé hledání”

The strategy:1. Build up particular candidate for A2. Find the best F for this A3. Compute discrepancy4. Remember the best A,F pair so far5. Back to step 1

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Blind Search (2.) The key tasks:

1. Building up the candidates for A2. How to get F when knowing A

Building up the candidates for A:- row by row (one row one factor)- bit-coded matrices can be much faster- factors cannot repeat on several rows

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Blind Search (3.) How to get F when knowing X and A:

- bit-coded matrices may help (yet again)- going row by row (better than cassical „rows x columns” multiplication algo)

How to get the particular row of F:(a) blind search (b) do some preprocessing, then blind search

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Formal context

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Formal concept

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Using Formal Concepts Concepts are our

candidates for the rows of matrix A

Example: set “p3”- data matrix size 100x100- 10 concepts (see pict.)

Searching for 5 factors:- only 8 candidates- 56 possible combinations

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Details Concepts generate no negative

discrepancy higher computation error higher semantic value (for us)

We can get negative discrepancy when computing F (as discussed before)

Performed tests gave promising results

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Final Comparison The presented algo’s are very similar Giving the same results in our tests Computation times are very different

- times of set “p2” (mentioned before): blind search: approx. 3x109 years concepts: 7 seconds

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Possible Future Work Current implementaion uses

independent application to compute concepts lattices

Integration to a single application may speed up the computation

We don’t need to compute whole concept lattice, even don’t need to know all concepts

Need to find better algorithm for binary matrix pseudo-division F = X/A