We consider tensor factorizations based on sparse measurements of the tensor components. The measurements are designed in a way that the underlying graph of interactions is a random graph. The setup will be useful in cases where a substantial amount of data is missing, as in recommendation systems heavily used in social network services. In order to obtain theoretical insights on the setup, we consider statistical inference of the tensor factorization in a high dimensional limit, which we call as dense limit, where the graphs are large and dense but not fully connected. We build message-passing algorithms and test them in a Bayes optimal teacher-student setting. We also develop a replica theory, which becomes exact in the dense limit,to examine the performance of statistical inference.

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