Tensor factorization is a methodology that is applied in a variety of fields, ranging from climate modeling to medical informatics. A tensor is an n-way array that captures the relationship between n objects. These multiway arrays can be factored to study the underlying bases present in the data. Two challenges arising in tensor factorization are 1) the resulting factors can be noisy and highly overlapping with one another and 2) they may not map to insights within a domain. However, incorporating supervision to increase the number of insightful factors can be costly in terms of the time and domain expertise necessary for gathering labels or domain-specific constraints. To meet these challenges, we introduce CANDECOMP/PARAFAC (CP) tensor factorization with Cannot-Link Intermode Constraints (CP-CLIC), a framework that achieves succinct, diverse, interpretable factors. This is accomplished by gradually learning constraints that are verified with auxiliary information during the decomposition process. We demonstrate CP-CLIC’s potential to extract sparse, diverse, and interpretable factors through experiments on simulated data and a real-world application in medical informatics.