  Many applications in signal processing benefit from the sparsity of signals
in a certain transform domain or dictionary. Synthesis sparsifying dictionaries
that are directly adapted to data have been popular in applications such as
image denoising, inpainting, and medical image reconstruction. In this work, we
focus instead on the sparsifying transform model, and study the learning of
well-conditioned square sparsifying transforms. The proposed algorithms
alternate between a $\ell_0$ "norm"-based sparse coding step, and a non-convex
transform update step. We derive the exact analytical solution for each of
these steps. The proposed solution for the transform update step achieves the
global minimum in that step, and also provides speedups over iterative
solutions involving conjugate gradients. We establish that our alternating
algorithms are globally convergent to the set of local minimizers of the
non-convex transform learning problems. In practice, the algorithms are
insensitive to initialization. We present results illustrating the promising
performance and significant speed-ups of transform learning over synthesis
K-SVD in image denoising.
