Hardness of Maximum Likelihood Learning of DPPs

Determinantal Point Processes (DPPs) are a widely used probabilistic model for negatively correlated sets. DPPs have been successfully employed in Machine Learning applications to select a diverse, yet representative subset of data. In these applications, a set of parameters that maximize the likelihood of the data is typically desirable. The algorithms used for this task to date either optimize over a limited family of DPPs, or use local improvement heuristics that do not provide theoretical guarantees of optimality. In his seminal work on DPPs in Machine Learning, Kulesza (2011) conjectured that the problem is NP-complete. The lack of a formal proof prompted Brunel et al. (COLT 2017) to suggest that, in opposition to Kulesza's conjecture, there might exist a polynomial-time algorithm for computing a maximum-likelihood DPP. They also presented some preliminary evidence supporting a conjecture that they suggested might lead to such an algorithm. In this work we prove Kulesza's conjecture. In fact, we prove the following stronger hardness of approximation result: even computing a $\left(1-O(\frac{1}{\log^9{N}})\right)$-approximation to the maximum log-likelihood of a DPP on a ground set of $N$ elements is NP-complete. From a technical perspective, we reduce the problem of approximating the maximum log-likelihood of a DPP to solving a gap instance of a \textsc{$3$-Coloring} problem on a hypergraph. This hypergraph is based on the bounded-degree construction of Bogdanov et al. (FOCS 2002), which we enhance using the strong expanders of Alon and Capalbo (FOCS 2007). We demonstrate that if a rank-$3$ DPP achieves near-optimal log-likelihood, its marginal kernel must encode an almost perfect ``vector-coloring" of the hypergraph. Finally, we show that these continuous vectors can be decoded into a proper $3$-coloring after removing a small fraction of ``noisy" edges.