  In kernel methods, the kernels are often required to be positive definite,
which restricts the use of many indefinite kernels. To consider those
non-positive definite kernels, in this paper, we aim to build an indefinite
kernel learning framework for kernel logistic regression. The proposed
indefinite kernel logistic regression (IKLR) model is analysed in the
Reproducing Kernel Kre\u{\i}n Spaces (RKKS) and then becomes non-convex. Using
the positive decomposition of a non-positive definite kernel, the derived IKLR
model can be decomposed into the difference of two convex functions.
Accordingly, a concave-convex procedure is introduced to solve the non-convex
optimization problem. Since the concave-convex procedure has to solve a
sub-problem in each iteration, we propose a concave-inexact-convex procedure
(CCICP) algorithm with an inexact solving scheme to accelerate the solving
process. Besides, we propose a stochastic variant of CCICP to efficiently
obtain a proximal solution, which achieves the similar purpose with the inexact
solving scheme in CCICP. The convergence analyses of the above two variants of
concave-convex procedure are conducted. By doing so, our method works
effectively not only under a deterministic setting but also under a stochastic
setting. Experimental results on several benchmarks suggest that the proposed
IKLR model performs favorably against the standard (positive-definite) kernel
logistic regression and other competitive indefinite learning based algorithms.
