Optimal Sublinear Sampling of Spanning Trees and Determinantal Point Processes via Average-Case Entropic Independence

We design fast algorithms for repeatedly sampling from strongly Rayleigh distributions, which include random spanning tree distributions and determinantal point processes. For a graph $G=(V, E)$, we show how to approximately sample uniformly random spanning trees from $G$ in $\widetilde{O}(\lvert V\rvert)$ time per sample after an initial $\widetilde{O}(\lvert E\rvert)$ time preprocessing. For a determinantal point process on subsets of size $k$ of a ground set of $n$ elements, we show how to approximately sample in $\widetilde{O}(k^\omega)$ time after an initial $\widetilde{O}(nk^{\omega-1})$ time preprocessing, where $\omega<2.372864$ is the matrix multiplication exponent. We even improve the state of the art for obtaining a single sample from determinantal point processes, from the prior runtime of $\widetilde{O}(\min\{nk^2, n^\omega\})$ to $\widetilde{O}(nk^{\omega-1})$. In our main technical result, we achieve the optimal limit on domain sparsification for strongly Rayleigh distributions. In domain sparsification, sampling from a distribution $\mu$ on $\binom{[n]}{k}$ is reduced to sampling from related distributions on $\binom{[t]}{k}$ for $t\ll n$. We show that for strongly Rayleigh distributions, we can can achieve the optimal $t=\widetilde{O}(k)$. Our reduction involves sampling from $\widetilde{O}(1)$ domain-sparsified distributions, all of which can be produced efficiently assuming convenient access to approximate overestimates for marginals of $\mu$. Having access to marginals is analogous to having access to the mean and covariance of a continuous distribution, or knowing "isotropy" for the distribution, the key assumption behind the Kannan-Lov\'asz-Simonovits (KLS) conjecture and optimal samplers based on it. We view our result as a moral analog of the KLS conjecture and its consequences for sampling, for discrete strongly Rayleigh measures

Improved Financial Forecasting via Quantum Machine Learning

Quantum algorithms have the potential to enhance machine learning across a variety of domains and applications. In this work, we show how quantum machine learning can be used to improve financial forecasting. First, we use classical and quantum Determinantal Point Processes to enhance Random Forest models for churn prediction, improving precision by almost 6%. Second, we design quantum neural network architectures with orthogonal and compound layers for credit risk assessment, which match classical performance with significantly fewer parameters. Our results demonstrate that leveraging quantum ideas can effectively enhance the performance of machine learning, both today as quantum-inspired classical ML solutions, and even more in the future, with the advent of better quantum hardware

A Unified Algorithm Framework for Unsupervised Discovery of Skills based on Determinantal Point Process

Learning rich skills through temporal abstractions without supervision of external rewards is at the frontier of Reinforcement Learning research. Existing works mainly fall into two distinctive categories: variational and Laplacian-based skill (a.k.a., option) discovery. The former maximizes the diversity of the discovered options through a mutual information loss but overlooks coverage of the state space, while the latter focuses on improving the coverage of options by increasing connectivity during exploration, but does not consider diversity. In this paper, we propose a unified framework that quantifies diversity and coverage through a novel use of the Determinantal Point Process (DPP) and enables unsupervised option discovery explicitly optimizing both objectives. Specifically, we define the DPP kernel matrix with the Laplacian spectrum of the state transition graph and use the expected mode number in the trajectories as the objective to capture and enhance both diversity and coverage of the learned options. The proposed option discovery algorithm is extensively evaluated using challenging tasks built with Mujoco and Atari, demonstrating that our proposed algorithm substantially outperforms SOTA baselines from both diversity- and coverage-driven categories. The codes are available at https://github.com/LucasCJYSDL/ODPP

On simulation of continuous determinantal point processes

We review how to simulate continuous determinantal point processes (DPPs) and improve the current simulation algorithms in several important special cases as well as detail how certain types of conditional simulation can be carried out. Importantly we show how to speed up the simulation of the widely used Fourier based projection DPPs, which arise as approximations of more general DPPs. The algorithms are implemented and published as open source software