Log-linear models are popular tools to analyze contingency tables, particularly to model row and column effects as well as row-column interactions in two-way tables. In this paper, we introduce a regularized log-linear model designed for denoising and visualizing count data, which can incorporate side information such as row and column features. The estimation is performed through a convex optimization problem where we minimize a negative Poisson log-likelihood penalized by the nuclear norm of the interaction matrix. We derive an upper bound on the Frobenius estimation error, which improves previous rates for Poisson matrix recovery, and an algorithm based on the alternating direction method of multipliers to compute our estimator. To propose a complete methodology to users, we also address automatic selection of the regularization parameter. A Monte Carlo simulation reveals that our estimator is particularly well suited to estimate the rank of the interaction in low signal to noise ratio regimes. We illustrate with two data analyses that the results can be easily interpreted through biplot vizualization. The method is available as an R code.