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Matrix Factorisation (and Dimensionality Reduction)

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Slides from the "Matrix Factorisation" presentation by Thierry Silbermann at the Sao Paulo Machine Learning Meetup.

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Matrix Factorisation (and Dimensionality Reduction)

  1. 1. Matrix Factorisation Thierry Silbermann, Data Scientist at Nubank (and dimensionality reduction)
  2. 2. What is Matrix Factorisation? • For some domain, we have matrix that are: very ‘big’ sparse and don’t have any order • Factoring it yield a set of: More manageable compact and ordered matrices
  3. 3. Matrix Factorisation / Decomposition • Singular Value Decomposition (SVD) • Principal Component Analysis (PCA) • Non-negative Matrix Factorisation (NMF) • LU/QR/Cholskey decomposition • etc…
  4. 4. SVD (Singular Value Decomposition) • The singular value decomposition of an m × n matrix M is a factorisation of the form : M = UΣV∗, where U is an m × m unitary matrix, Σ is an m × n rectangular diagonal matrix with non-negative real numbers on the diagonal (known as the singular values of M), and V∗ (the transpose of V) is an n × n real or complex unitary matrix. The m columns of U and the n columns of V are called the left- singular vectors and right-singular vectors of M, respectively.
  5. 5. Limitation SVD • One of the most general algorithm for decomposition. SVD is an amelioration from eigenvalue decomposition (was only used with n x n matrix) • Didn’t really reduce the space use to store our data. • From a m x n (M) matrix, we end up with m x m (U), m x n (Σ) and n x n (V) matrices…
  6. 6. Matrix Factorisation • Using Non-Negative Matrix Factorisation V ≈ WH • V, W and H are all non-negative • V is a n x m matrix • W is a n x r matrix, H is a r x m matrix • and r ≪ min(m, n)
  7. 7. How to decompose? • Minimize with respect to W and H, subject to the constraints W, H ≥ 0. • Multiplication Update Algorithm • Alternating Least Square Algorithm
  8. 8. Matrix Factorisation
  9. 9. Toy example
  10. 10. Back to the math • We have a matrix of ratings we want to approximate: • We need to construct P and Q by minimising: • We have an optimisation problem: • Update are then give by:
  11. 11. Easy to implement (for toy example)
  12. 12. Easy to implement
  13. 13. Example: Recommend given names to user of the name search engine ”Nameling*”
  14. 14. Example: Recommend given names to user of the name search engine ”Nameling*”
  15. 15. Example: Recommend given names to user of the name search engine ”Nameling*” *
  16. 16. My own advertisement • libFM ( • 2015/02/11/simple-libfm-example-part1/
  17. 17. Thank you