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Amari distance

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Similarity measure between two invertible matrices

The Amari distance, also known as Amari index and Amari metric is a similarity measure between two invertible matrices, useful for checking for convergence in independent component analysis algorithms and for comparing solutions. It is named after Japanese information theorist Shun'ichi Amari and was originally introduced as a performance index for blind source separation.

For two invertible matrices A , B ∈ R n × n {\displaystyle A,B\in \mathbb {R} ^{n\times n}} , it is defined as:

d ( A , B ) = ∑ i = 1 n ( ∑ j = 1 n | p i j | max k | p i k | − 1 ) + ∑ j = 1 n ( ∑ i = 1 n | p i j | max k | p k j | − 1 ) , P = A − 1 B {\displaystyle d(A,B)=\sum _{i=1}^{n}\left(\sum _{j=1}^{n}{\frac {|p_{ij}|}{\max _{k}|p_{ik}|}}-1\right)+\sum _{j=1}^{n}\left(\sum _{i=1}^{n}{\frac {|p_{ij}|}{\max _{k}|p_{kj}|}}-1\right),P=A^{-1}B}

It is non-negative and cancels if and only if A − 1 B {\displaystyle A^{-1}B} is a scale and permutation matrix, i.e. the product of a diagonal matrix and a permutation matrix. The Amari distance is invariant to permutation and scaling of the columns of A {\displaystyle A} and B {\displaystyle B} .

References

  1. Póczos, Barnabás; Takács, Bálint; Lőrincz, András (2005). Gama, João; Camacho, Rui; Brazdil, Pavel B.; Jorge, Alípio Mário; Torgo, Luís (eds.). "Independent Subspace Analysis on Innovations". Machine Learning: ECML 2005. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer: 698–706. doi:10.1007/11564096_71. ISBN 978-3-540-31692-3.
  2. "R Graphical Manual – Compute the 'Amari' distance between two matrices". Archived from the original on 2015-01-09. Retrieved 2019-05-16.
  3. Sobhani, Elaheh; Comon, Pierre; Jutten, Christian; Babaie-Zadeh, Massoud (2022-06-01). "CorrIndex: A permutation invariant performance index" (PDF). Signal Processing. 195: 108457. doi:10.1016/j.sigpro.2022.108457. ISSN 0165-1684.
  4. Hastie, Trevor; Friedman, Jerome; Tibshirani, Robert (2009). The Elements of Statistical Learning: Data Mining, Inference, and Prediction (PDF). Springer Series in Statistics (2nd ed.). Springer New York. doi:10.1007/978-0-387-84858-7.
  5. Amari, Shun-ichi; Cichocki, Andrzej; Yang, Howard (1995). "A New Learning Algorithm for Blind Signal Separation" (PDF). Advances in Neural Information Processing Systems. 8. MIT Press.
  6. Bach, Francis R.; Jordan, Michael I. (2002). "Kernel Independent Component Analysis". Journal of Machine Learning Research. 3 (Jul): 1–48. ISSN 1533-7928.


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