Online Orthogonal Dictionary Learning Based on Frank–Wolfe Method

Dictionary learning is a widely used unsupervised learning method in signal processing and machine learning. Most existing works on dictionary learning adopt an off-line approach, and there are two main off-line ways of conducting it. One is to alternately optimize both the dictionary and the sparse code, while the other is to optimize the dictionary by restricting it over the orthogonal group. The latter, called orthogonal dictionary learning (ODL), has a lower implementation complexity and, hence, is more favorable for low-cost devices. However, existing schemes for ODL only work with batch data and cannot be implemented online, making them inapplicable for real-time applications. This article, thus, proposes a novel online orthogonal dictionary scheme to dynamically learn the dictionary from streaming data, without storing the historical data. The proposed scheme includes a novel problem formulation and an efficient online algorithm design with convergence analysis. In the problem formulation, we relax the orthogonal constraint to enable an efficient online algorithm. We then propose the design of a new Frank–Wolfe-based online algorithm with a convergence rate of
$\mathcal {O}(\ln t/t^{1/4})$
. The convergence rate in terms of key system parameters is also derived. Experiments with synthetic data and real-world Internet of things (IoT) sensor readings demonstrate the effectiveness and efficiency of the proposed online ODL scheme.