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### When should I write a matrix-vector function to handle the sparse matrix vector multiplication?

Multiplying matrix and a vector using sparse matrix format, such as CSR is not hard, and is a basic and common operation if you write a linear solvers library for sparse linear systems. Take SPARSEKIT ...
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Very interesting question. One option is to relax the cardinality regularizer to the reciprocal of an infinity norm, i.e., replace $\mathrm{card}(x)$ with $$1 / \|x\|_{\infty}.$$ This yields a non-...