TY - JOUR
T1 - A self-consistent-field iteration for MAXBET with an application to multi-view feature extraction
AU - Ma, Xijun
AU - Shen, Chungen
AU - Wang, Li
AU - Zhang, Lei Hong
AU - Li, Ren Cang
N1 - The work of the third author was supported partially by NSF DMS-2009689 and NIH R01AG075582; the work of the fourth author was supported partially by the National Natural Science Foundation of China NSFC-12071332; and the work of the fifth author was supported partially by NSF DMS-1719620 and DMS-2009689.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2022/4
Y1 - 2022/4
N2 - As an extension of the traditional principal component analysis, the multi-view canonical correlation analysis (MCCA) aims at reducing m high dimensional random variables si∈ℝni(i=1,2,…,m) by proper projection matrices Xi∈ℝni×ℓ so that the m reduced ones yi=XiTsi∈ℝℓ have the “maximal correlation.” Various measures of the correlation for yi (i = 1,2,…,m) in MCCA have been proposed. One of the earliest criteria is the sum of all traces of pair-wise correlation matrices between yi and yj subject to the orthogonality constraints on Xi, i = 1,2,…,m. The resulting problem is to maximize a homogeneous quadratic function over the product of Stiefel manifolds and is referred to as the MAXBET problem. In this paper, the problem is first reformulated as a coupled nonlinear eigenvalue problem with eigenvector dependency (NEPv) and then solved by a novel self-consistent-field (SCF) iteration. Global and local convergences of the SCF iteration are studied and proven computational techniques in the standard eigenvalue problem are incorporated to yield more practical implementations. Besides the preliminary numerical evaluations on various types of synthetic problems, the efficiency of the SCF iteration is also demonstrated in an application to multi-view feature extraction for unsupervised learning.
AB - As an extension of the traditional principal component analysis, the multi-view canonical correlation analysis (MCCA) aims at reducing m high dimensional random variables si∈ℝni(i=1,2,…,m) by proper projection matrices Xi∈ℝni×ℓ so that the m reduced ones yi=XiTsi∈ℝℓ have the “maximal correlation.” Various measures of the correlation for yi (i = 1,2,…,m) in MCCA have been proposed. One of the earliest criteria is the sum of all traces of pair-wise correlation matrices between yi and yj subject to the orthogonality constraints on Xi, i = 1,2,…,m. The resulting problem is to maximize a homogeneous quadratic function over the product of Stiefel manifolds and is referred to as the MAXBET problem. In this paper, the problem is first reformulated as a coupled nonlinear eigenvalue problem with eigenvector dependency (NEPv) and then solved by a novel self-consistent-field (SCF) iteration. Global and local convergences of the SCF iteration are studied and proven computational techniques in the standard eigenvalue problem are incorporated to yield more practical implementations. Besides the preliminary numerical evaluations on various types of synthetic problems, the efficiency of the SCF iteration is also demonstrated in an application to multi-view feature extraction for unsupervised learning.
KW - MAXBET
KW - Multi-view canonical correlation analysis
KW - Multi-view feature extraction
KW - Nonlinear eigenvalue problem
KW - Self-consistent-field iteration
KW - Stiefel manifold
UR - http://www.scopus.com/inward/record.url?scp=85126274815&partnerID=8YFLogxK
U2 - 10.1007/s10444-022-09929-3
DO - 10.1007/s10444-022-09929-3
M3 - Journal article
AN - SCOPUS:85126274815
SN - 1019-7168
VL - 48
JO - Advances in Computational Mathematics
JF - Advances in Computational Mathematics
IS - 2
M1 - 13
ER -