  Sparse dictionary learning is a popular method for representing signals as
linear combinations of a few elements from a dictionary that is learned from
the data. In the classical setting, signals are represented as vectors and the
dictionary learning problem is posed as a matrix factorization problem where
the data matrix is approximately factorized into a dictionary matrix and a
sparse matrix of coefficients. However, in many applications in computer vision
and medical imaging, signals are better represented as matrices or tensors
(e.g. images or videos), where it may be beneficial to exploit the
multi-dimensional structure of the data to learn a more compact representation.
One such approach is separable dictionary learning, where one learns separate
dictionaries for different dimensions of the data. However, typical
formulations involve solving a non-convex optimization problem; thus
guaranteeing global optimality remains a challenge. In this work, we propose a
framework that builds upon recent developments in matrix factorization to
provide theoretical and numerical guarantees of global optimality for separable
dictionary learning. We propose an algorithm to find such a globally optimal
solution, which alternates between following local descent steps and checking a
certificate for global optimality. We illustrate our approach on diffusion
magnetic resonance imaging (dMRI) data, a medical imaging modality that
measures water diffusion along multiple angular directions in every voxel of an
MRI volume. State-of-the-art methods in dMRI either learn dictionaries only for
the angular domain of the signals or in some cases learn spatial and angular
dictionaries independently. In this work, we apply the proposed separable
dictionary learning framework to learn spatial and angular dMRI dictionaries
jointly and provide preliminary validation on denoising phantom and real dMRI
brain data.
