Complex simulations and numerical experiments typically rely on a number of parameters and have an associated score function, e.g. with the goal of maximizing accuracy or minimizing computation time. However, the influence of each individual parameter is often poorly understood a priori and the joint parameter space can be difficult to explore, visualize and optimize. We model this space as an N-dimensional black-box tensor and apply a cross approximation strategy to sample it. Upon learning and compactly expressing this space as a surrogate visualization model, informative subspaces are interactively reconstructed and navigated in the form of charts, images, surface plots, etc. By exploiting efficient operations in the tensor train format, we are able to produce diagrams such as parallel coordinates, bivariate projections and dimensional stacking out of highly-compressed parameter spaces. We demonstrate the proposed framework with several scientific simulations that contain up to 6 parameters and billions of tensor grid points.