  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.
