The heterogeneity of cancer necessitates developing a multitude of targeted therapies. We propose the view that cancer drug discovery is a low rank tensor completion problem. We implement this vision by using heterogeneous public data to construct a tensor of drug-target-disease associations. We show the validity of this approach computationally by simulations, and experimentally by testing drug candidates. Specifically, we show that a novel drug candidate, SU11652, controls melanoma tumor growth, including BRAF WT melanoma. Independently, we show that another molecule, TC-E 5008, controls tumor proliferation on ex vivo ER+ human breast cancer. Most importantly, we identify these chemicals with only a few computationally selected experiments as opposed to brute-force screens. The efficiency of our approach enables use of ex vivo human tumor assays as a primary screening tool. We provide a web server, the Cancer Vulnerability Explorer (accessible at https://cavu.biohpc.swmed.edu), to facilitate the use of our methodology.