  We study the problem of bivariate discrete or continuous probability density
estimation under low-rank constraints.For discrete distributions, we assume
that the two-dimensional array to estimate is a low-rank probability matrix. In
the continuous case, we assume that the density with respect to the Lebesgue
measure satisfies a generalized multi-view model, meaning that it is
$\beta$-H{\"o}lder and can be decomposed as a sum of $K$ components, each of
which is a product of one-dimensional functions. In both settings, we propose
estimators that achieve, up to logarithmic factors, the minimax optimal
convergence rates under such low-rank constraints. In the discrete case, the
proposed estimator is adaptive to the rank $K$. In the continuous case, our
estimator converges with the $L_1$ rate $\min((K/n)^{\beta/(2\beta+1)},
n^{-\beta/(2\beta+2)})$ up to logarithmic factors, and it is adaptive to the
unknown support as well as to the smoothness $\beta$ and to the unknown number
of separable components $K$. We present efficient algorithms for computing our
estimators.
