Robust Two-Dimensional T$\ell _{1}$-Norm Linear Discriminant Analysis for Image Recognition

Two-dimensional linear discriminant analysis (2DLDA) is an effective supervised feature extraction technique without converting image data to vectors. It can not only alleviate the small sample size (SSS) problem but also make it full of structural information. However, 2DLDA behaves badly when processing the data with noise and outliers since it employs squared F-norm as the distance metric. Considering T
$\ell _{1}$
-norm is bounded and Lipschitz-continuous, we combine T
$\ell _{1}$
-norm with 2DLDA to overcome this problem and propose a novel method, named 2DLDA-T
$\ell _{1}$
. As far as we know, this is the first time to introduce T
$\ell _{1}$
-norm into 2DLDA. Since 2DLDA-T
$\ell _{1}$
is a non-convex and non-smooth optimization model, it is difficult to solve. So, we design a spherical gradient ascent method, which can guarantee the unit constraint and the convergence of the objective function. Due to the boundedness and Lipschitz-continuity of T
$\ell _{1}$
-norm, it can be expected that 2DLDA-T
$\ell _{1}$
should have better performance than the existing related methods, including 2DLDA,
$\ell _{1}$
-2DLDA,
$\ell _{p}$
-2DLDA, F2DLDA, and 2DBLDA. Moreover, experimental results on two real image datasets verify the effectiveness of 2DLDA-T
$\ell _{1}$
.