Amari distance explained

The Amari distance,^{[1] [2]} also known as Amari index^[3] and Amari metric^[4] 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.^[5]

For two invertible matrices

A, B \in R ^{n x} , it is defined as:d(A, B) = \sum ^{n} _{i =1} \left(\sum ^{n (|p _{ij} |)/(max _{k} |p _{ik} |)} _{j =1} - 1 \right) + \sum ^{n} _{j =1} \left(\sum ^{n (|p _{ij} |)/(max _{k} |p _{kj} |)} _{i =1} - 1 \right), P = A ^{-1} B

It is non-negative and cancels if and only if

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 and B .^[6]

Notes and References

1. Póczos . Barnabás . Takács . Bálint . Lőrincz . András . 2005 . Gama . João . Camacho . Rui . Brazdil . Pavel B. . Jorge . Alípio Mário . Torgo . Luís . Independent Subspace Analysis on Innovations . Machine Learning: ECML 2005 . Lecture Notes in Computer Science . en . Berlin, Heidelberg . Springer . 698–706 . 10.1007/11564096_71 . 978-3-540-31692-3. free .
2. Web site: R Graphical Manual – Compute the 'Amari' distance between two matrices. https://web.archive.org/web/20150109133357/http://rgm3.lab.nig.ac.jp/RGM/R_rdfile?f=ProDenICA%2Fman%2Famari.Rd&d=R_CC. dead. 2015-01-09. 2019-05-16.
3. Sobhani . Elaheh . Comon . Pierre . Jutten . Christian . Babaie-Zadeh . Massoud . 2022-06-01 . CorrIndex: A permutation invariant performance index . Signal Processing . en . 195 . 108457 . 10.1016/j.sigpro.2022.108457 . 0165-1684. free .
4. Book: Hastie . Trevor . Friedman . Jerome . Tibshirani . Robert . 10.1007/978-0-387-84858-7 . 2nd . Springer Series in Statistics . Springer New York . The Elements of Statistical Learning: Data Mining, Inference, and Prediction . 2009.
5. Amari . Shun-ichi . Cichocki . Andrzej . Yang . Howard . 1995 . A New Learning Algorithm for Blind Signal Separation . Advances in Neural Information Processing Systems . MIT Press . 8.
6. Bach . Francis R. . Jordan . Michael I. . 2002 . Kernel Independent Component Analysis . Journal of Machine Learning Research . 3 . Jul . 1–48 . 1533-7928.