Global Optimality in Separable Dictionary Learning with Applications to the Analysis of Diffusion MRI
This work addresses the problem of ensuring global optimality in separable dictionary learning for researchers in medical imaging and computer vision, though it appears incremental as it builds on existing matrix factorization methods.
The authors tackled the non-convex optimization challenge in separable dictionary learning by proposing a framework with theoretical and numerical guarantees for global optimality, applying it to jointly learn spatial and angular dictionaries in diffusion MRI data for denoising tasks.
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.