Fast determinantal point processes via distortion-free intermediate sampling

by   Michal Derezinski, et al.
berkeley college

Given a fixed n× d matrix X, where n≫ d, we study the complexity of sampling from a distribution over all subsets of rows where the probability of a subset is proportional to the squared volume of the parallelopiped spanned by the rows (a.k.a. a determinantal point process). In this task, it is important to minimize the preprocessing cost of the procedure (performed once) as well as the sampling cost (performed repeatedly). To that end, we propose a new determinantal point process algorithm which has the following two properties, both of which are novel: (1) a preprocessing step which runs in time O(number-of-non-zeros(X)· n)+poly(d), and (2) a sampling step which runs in poly(d) time, independent of the number of rows n. We achieve this by introducing a new regularized determinantal point process (R-DPP), which serves as an intermediate distribution in the sampling procedure by reducing the number of rows from n to poly(d). Crucially, this intermediate distribution does not distort the probabilities of the target sample. Our key novelty in defining the R-DPP is the use of a Poisson random variable for controlling the probabilities of different subset sizes, leading to new determinantal formulas such as the normalization constant for this distribution. Our algorithm has applications in many diverse areas where determinantal point processes have been used, such as machine learning, stochastic optimization, data summarization and low-rank matrix reconstruction.


page 1

page 2

page 3

page 4


Exact sampling of determinantal point processes with sublinear time preprocessing

We study the complexity of sampling from a distribution over all index s...

Sampling from a k-DPP without looking at all items

Determinantal point processes (DPPs) are a useful probabilistic model fo...

Optimized Algorithms to Sample Determinantal Point Processes

In this technical report, we discuss several sampling algorithms for Det...

Matrix Completion from O(n) Samples in Linear Time

We consider the problem of reconstructing a rank-k n × n matrix M from a...

Amortized Edge Sampling

We present a sublinear time algorithm that allows one to sample multiple...

Sampling using Adaptive Regenerative Processes

Enriching Brownian Motion with regenerations from a fixed regeneration d...

A heuristic independent particle approximation to determinantal point processes

A determinantal point process is a stochastic point process that is comm...

Please sign up or login with your details

Forgot password? Click here to reset