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We consider the factorization of a rectangular matrix $X $ into a positivelinear combination of rank-one factors of the form $u v^\top$, where $u$ and$v$ belongs to certain sets $\mathcal{U}$ and $\mathcal{V}$, that may encodespecific structures regarding the factors, such as positivity or sparsity. Inthis paper, we show that computing the optimal decomposition is equivalent tocomputing a certain gauge function of $X$ and we provide a detailed analysis ofthese gauge functions and their polars. Since these gauge functions aretypically hard to compute, we present semi-definite relaxations and severalalgorithms that may recover approximate decompositions with approximationguarantees. We illustrate our results with simulations on findingdecompositions with elements in $\{0,1\}$. As side contributions, we present adetailed analysis of variational quadratic representations of norms as well asa new iterative basis pursuit algorithm that can deal with inexact first-orderoracles.

Convex relaxations of structured matrix factorizations