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, the parameters of the DPP need to be fitted to match the data; typically, we seek a set of parameters that maximize the likelihood of the data. 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.
It is natural to ask if there exist efficient algorithms for finding a maximum likelihood DPP model for a given data set. In seminal work on DPPs in Machine Learning, Kulesza conjectured in his PhD Thesis (2011) that the problem is NP-complete.
The lack of a formal proof prompted Brunel, Moitra, Rigollet and Urschel (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. At the same time, we also obtain the first polynomial-time algorithm that achieves a nontrivial worst-case approximation
to the optimal log-likelihood: the approximation factor is $\frac{1}{(1+o(1))\log{m}}$ unconditionally (for data sets that consist of $m$ subsets), and can be improved to $1-\frac{1+o(1)}{\log N}$ if all $N$ elements appear in a $O(1/N)$-fraction of the subsets.
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, Obata, and Trevisan (2002), which we enhance using the strong expanders of Alon and Capalbo (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.
Major revision
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, the parameters of the DPP need to be fitted to match the data; typically, we seek a set of parameters that maximize the likelihood of the data. 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.
It is natural to ask if there exist efficient algorithms for finding a maximum likelihood DPP model for a given data set. In seminal work on DPPs in Machine Learning, Kulesza conjectured in his PhD Thesis (2011) that the problem is NP-complete.
The lack of a formal proof prompted Brunel, Moitra, Rigollet and Urschel (COLT 2017) to conjecture that,
in opposition to Kulesza's conjecture, there exists a polynomial-time algorithm for computing a maximum-likelihood DPP. They also presented some preliminary evidence supporting their conjecture.
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. At the same time, we also obtain the first polynomial-time algorithm that achieves a nontrivial worst-case approximation
to the optimal log-likelihood: the approximation factor is $\frac{1}{(1+o(1))\log{m}}$ unconditionally (for data sets that consist of $m$ subsets), and can be improved to $1-\frac{1+o(1)}{\log N}$ if all $N$ elements appear in a $O(1/N)$-fraction of the subsets.
In terms of techniques, we reduce approximating the maximum log-likelihood of DPPs on a data set to
solving a gap instance of a ``vector coloring" problem on a hypergraph. Such a hypergraph is built on a bounded-degree graph construction of Bogdanov, Obata and Trevisan (FOCS 2002), and is further enhanced by the strong expanders of Alon and Capalbo (FOCS 2007) to serve our purposes.
