Sparse CCA

Canonical Correlation Analysis (CCA) is a common statistical learning method for analyzing pairwise data. It learns a projection for both representations such that they are maximally correlated in the dimensionality-reduced space. Suppose X∈Rn×p with n samples and p features and Y∈Rn×q with n samples and q features represent two omics data from a single cancer and their columns of X and Y are centered and scaled with zero mean and unit variance. Then, the CCA model can be written as follows:

Suppose X^T X = I and Y^T Y = I, where I is the identity matrix. Then the above model reduces to:

which was called as the diagonal CCA whose performance is usually better than the traditional CCA in high-dimensional data [31, 32]. However, the classical CCA leads to non-sparse canonical vectors. It is difficult to select features and interpret in biology. To this end, a large number of sparse CCA models have been proposed to obtain sparse canonical vectors by using different penalty functions [16, 33–37]. Specifically, a sparse CCA (SCCA) with ℓ₀-norm constraint [35] can be formulated into the following optimization problem:

where k_u and k_v are two parameters to control the sparsity of canonical vectors (u and v), and ‖u‖₀ is the number of non-zero elements in the u.