Learning the Parameters of Determinantal Point Process Kernels

Determinantal point processes (DPPs) are well-suited for modeling repulsion and have proven useful in many applications where diversity is desired. While DPPs have many appealing properties, such as efficient sampling, learning the parameters of a DPP is still considered a difficult problem due to the non-convex nature of the likelihood function. In this paper, we propose using Bayesian methods to learn the DPP kernel parameters. These methods are applicable in large-scale and continuous DPP settings even when the exact form of the eigendecomposition is unknown. We demonstrate the utility of our DPP learning methods in studying the progression of diabetic neuropathy based on spatial distribution of nerve fibers, and in studying human perception of diversity in images.

Approximate Inference in Continuous Determinantal Point Processes

Determinantal point processes (DPPs) are random point processes well-suited for modeling repulsion. In machine learning, the focus of DPP-based models has been on diverse subset selection from a discrete and finite base set. This discrete setting admits an efficient sampling algorithm based on the eigendecomposition of the defining kernel matrix. Recently, there has been growing interest in using DPPs defined on continuous spaces. While the discrete-DPP sampler extends formally to the continuous case, computationally, the steps required are not tractable in general. In this paper, we present two efficient DPP sampling schemes that apply to a wide range of kernel functions: one based on low rank approximations via Nystrom and random Fourier feature techniques and another based on Gibbs sampling. We demonstrate the utility of continuous DPPs in repulsive mixture modeling and synthesizing human poses spanning activity spaces.

Markov Determinantal Point Processes

A determinantal point process (DPP) is a random process useful for modeling the combinatorial problem of subset selection. In particular, DPPs encourage a random subset Y to contain a diverse set of items selected from a base set Y. For example, we might use a DPP to display a set of news headlines that are relevant to a user's interests while covering a variety of topics. Suppose, however, that we are asked to sequentially select multiple diverse sets of items, for example, displaying new headlines day-by-day. We might want these sets to be diverse not just individually but also through time, offering headlines today that are unlike the ones shown yesterday. In this paper, we construct a Markov DPP (M-DPP) that models a sequence of random sets {Yt}. The proposed M-DPP defines a stationary process that maintains DPP margins. Crucially, the induced union process Zt = Yt u Yt-1 is also marginally DPP-distributed. Jointly, these properties imply that the sequence of random sets are encouraged to be diverse both at a given time step as well as across time steps. We describe an exact, efficient sampling procedure, and a method for incrementally learning a quality measure over items in the base set Y based on external preferences. We apply the M-DPP to the task of sequentially displaying diverse and relevant news articles to a user with topic preferences.