  The ``curse of dimensionality'' has remained a challenge for high-dimensional
data analysis in statistics. The sliced inverse regression (SIR) and canonical
correlation (CANCOR) methods aim to reduce the dimensionality of data by
replacing the explanatory variables with a small number of composite directions
without losing much information. However, the estimated composite directions
generally involve all of the variables, making their interpretation difficult.
To simplify the direction estimates, Ni, Cook and Tsai [Biometrika 92 (2005)
242--247] proposed the shrinkage sliced inverse regression (SSIR) based on SIR.
In this paper, we propose the constrained canonical correlation ($C^3$) method
based on CANCOR, followed by a simple variable filtering method. As a result,
each composite direction consists of a subset of the variables for
interpretability as well as predictive power. The proposed method aims to
identify simple structures without sacrificing the desirable properties of the
unconstrained CANCOR estimates. The simulation studies demonstrate the
performance advantage of the proposed $C^3$ method over the SSIR method. We
also use the proposed method in two examples for illustration.
