PMID- 17206874
OWN - NLM
STAT- MEDLINE
DCOM- 20070313
LR  - 20070108
IS  - 0899-7667 (Print)
IS  - 0899-7667 (Linking)
VI  - 19
IP  - 2
DP  - 2007 Feb
TI  - One-bit-matching theorem for ICA, convex-concave programming on polyhedral set, 
      and distribution approximation for combinatorics.
PG  - 546-69
AB  - According to the proof by Liu, Chiu, and Xu (2004) on the so-called 
      one-bit-matching conjecture (Xu, Cheung, and Amari, 1998a), all the sources can 
      be separated as long as there is an one-to-one same-sign correspondence between 
      the kurtosis signs of all source probability density functions (pdf's) and the 
      kurtosis signs of all model pdf's, which is widely believed and implicitly 
      supported by many empirical studies. However, this proof is made only in a weak 
      sense that the conjecture is true when the global optimal solution of an 
      independent component analysis criterion is reached. Thus, it cannot support the 
      successes of many existing iterative algorithms that usually converge at one of 
      the local optimal solutions. This article presents a new mathematical proof that 
      is obtained in a strong sense that the conjecture is also true when any one of 
      local optimal solutions is reached in helping to investigating convex-concave 
      programming on a polyhedral set. Theorems are also provided not only on partial 
      separation of sources when there is a partial matching between the kurtosis 
      signs, but also on an interesting duality of maximization and minimization on 
      source separation. Moreover, corollaries are obtained on an interesting duality, 
      with supergaussian sources separated by maximization and subgaussian sources 
      separated by minimization. Also, a corollary is obtained to confirm the symmetric 
      orthogonalization implementation of the kurtosis extreme approach for separating 
      multiple sources in parallel, which works empirically but lacks mathematical 
      proof. Furthermore, a linkage has been set up to combinatorial optimization from 
      a distribution approximation perspective and a Stiefel manifold perspective, with 
      algorithms that guarantee convergence as well as satisfaction of constraints.
FAU - Xu, Lei
AU  - Xu L
AD  - Department of Computer Science and Engineering, Chinese University of Hong Kong, 
      Shatin, NT, Hong Kong, PRC. lxu@cse.cuhk.edu.hk
LA  - eng
PT  - Letter
PT  - Research Support, Non-U.S. Gov't
PL  - United States
TA  - Neural Comput
JT  - Neural computation
JID - 9426182
SB  - IM
MH  - *Algorithms
MH  - *Models, Statistical
MH  - *Principal Component Analysis
MH  - Signal Processing, Computer-Assisted
EDAT- 2007/01/09 09:00
MHDA- 2007/03/14 09:00
CRDT- 2007/01/09 09:00
PHST- 2007/01/09 09:00 [pubmed]
PHST- 2007/03/14 09:00 [medline]
PHST- 2007/01/09 09:00 [entrez]
AID - 10.1162/neco.2007.19.2.546 [doi]
PST - ppublish
SO  - Neural Comput. 2007 Feb;19(2):546-69. doi: 10.1162/neco.2007.19.2.546.