Image recovery in optical interferometry is an ill-posed nonlinear inverse problem arising from incomplete power spectrum and bispectrum measurements. We reformulate this nonlinear problem as a linear problem for the supersymmetric rank-1 order-3 tensor formed by the tensor product of the vector representing the image under scrutiny with itself. We propose a convex approach for tensor recovery with built-in supersymmetry, and regularising the inverse problem through a nuclear norm relaxation of a low-rank constraint. For comparison, and in the line of the current state of the art, we also study a nonlinear nonconvex approach. Keeping our tensor perspective, the problem is formulated for the tensor product of 3 vectors, where supersymmetry is relaxed while the rank-1 constraint is built-in. Linear convex minimisation problems are solved alternately and iteratively for these vectors. Simulation results show that the convex scheme provides significantly superior and more stable imaging quality than the nonconvex approach, both for randomly generated signals and realistic images. Code and test data are available a
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