Dictionary Learning Algorithms and Applications pp 115-144 | Cite as

# Other Views on the DL Problem

## Abstract

The dictionary learning problem can be posed in different ways, as we have already seen. In this chapter we first take a look at the DL problem where the sparsity level is not bounded for each training signal; instead, we bound the average sparsity level. This allows better overall representation power, due to the ability to place the nonzeros where they are most needed. The simplest way to pose the problem is to combine the error objective with an *ℓ*_{1} penalty that encourages sparsity in the whole representation matrix * X*. Several algorithms can solve this problem; we present those based on coordinate descent in AK-SVD style, on majorization and on proximal gradient. The latter approach can also be used with a 0-norm penalization. Other modifications of the objective include the addition of a regularization term (elastic net) or of a coherence penalty. Another view is given by task-driven DL, where the optimization objective is taken directly from the application and the sparse representation is only an intermediary tool. Returning to the standard DL problem, we present two new types of algorithms. One is based on selection: the atoms are chosen from a pool of candidates and so are no longer free variables. The other is online DL, where the training signals are assumed to be available in small bunches and the dictionary is updated for each bunch; online DL can thus adapt the dictionary to a time-varying set of signals, following the behavior of the generating source. Two online algorithms are presented, one based on coordinate descent, the other inspired by the classic recursive least squares (RLS). Finally, we tackle the DL problem with incomplete data, where some of the signals elements are missing, and present a version of AK-SVD suited to this situation.

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