Dictionary Learning with Almost Sure Error Constraints
This work addresses dictionary learning for applications in machine learning and signal processing, but it is incremental as it builds on existing methods by adding constraints.
The authors tackled the dictionary learning problem by imposing almost sure error constraints to ensure every sample is reconstructed within a predefined threshold, resulting in a reformulation as a convex-concave min-max problem solved with gradient descent-ascent algorithms.
A dictionary is a database of standard vectors, so that other vectors / signals are expressed as linear combinations of dictionary vectors, and the task of learning a dictionary for a given data is to find a good dictionary so that the representation of data points has desirable features. Dictionary learning and the related matrix factorization methods have gained significant prominence recently due to their applications in Wide variety of fields like machine learning, signal processing, statistics etc. In this article we study the dictionary learning problem for achieving desirable features in the representation of a given data with almost sure recovery constraints. We impose the constraint that every sample is reconstructed properly to within a predefined threshold. This problem formulation is more challenging than the conventional dictionary learning, which is done by minimizing a regularised cost function. We make use of the duality results for linear inverse problems to obtain an equivalent reformulation in the form of a convex-concave min-max problem. The resulting min-max problem is then solved using gradient descent-ascent like algorithms.