[Paper Review] Monotonic Gaussian Process Flow

Ivan Ustyuzhaninov, Ieva Kazlauskaite+2 more|arXiv (Cornell University)|May 30, 2019

Gaussian Processes and Bayesian Inference43 references4 citations

TL;DR

This paper introduces Monotonic Gaussian Process Flow, a novel nonparametric Bayesian model that ensures monotonicity in stochastic processes via stochastic differential equations (SDEs). By leveraging SDE solutions with monotonic drift, the method guarantees monotonic sample paths, outperforms existing probabilistic monotonic models on benchmarks, and enables uncertainty-aware temporal alignment in hierarchical models.

ABSTRACT

We propose a new framework for imposing monotonicity constraints in a Bayesian nonparametric setting based on numerical solutions of stochastic differential equations. We derive a nonparametric model of monotonic functions that allows for interpretable priors and principled quantification of hierarchical uncertainty. We demonstrate the efficacy of the proposed model by providing competitive results to other probabilistic monotonic models on a number of benchmark functions. In addition, we consider the utility of a monotonic random process as a part of a hierarchical probabilistic model; we examine the task of temporal alignment of time-series data where it is beneficial to use a monotonic random process in order to preserve the uncertainty in the temporal warpings.

Motivation & Objective

• To address the challenge of enforcing global monotonicity in nonparametric Bayesian models, particularly in Gaussian processes.
• To develop a framework that guarantees monotonicity in sample paths while maintaining flexible, interpretable priors and principled uncertainty quantification.
• To enable uncertainty-aware temporal warping in hierarchical models, especially for time-series alignment where monotonic transformations are required.
• To improve upon existing monotonic models by ensuring exact monotonicity in samples and capturing compositional uncertainty in multi-layered probabilistic structures.

Proposed method

• Formulates a stochastic process via the solution of a stochastic differential equation (SDE) with a drift term that ensures almost surely monotonic sample paths.
• Uses a Gaussian process to model the drift function in the SDE, enabling nonparametric, flexible, and smooth monotonic functions.
• Applies the uniqueness theorem for SDE solutions to guarantee that all sample paths of the resulting process are monotonic by construction.
• Integrates the monotonic flow into a hierarchical two-layer model: a monotonic warping function $ g(ullet) $ followed by a GP observation model $ f(ullet) $, with shared variational inference.
• Employs variational inference with cross-layer correlations between the warping and observation functions to preserve compositional uncertainty.
• Leverages recent advances in variational inference for deep GPs to scale the model and handle complex, non-stationary data distributions.

Experimental results

Research questions

• RQ1 Can a nonparametric Bayesian model be constructed such that all sample paths are guaranteed to be monotonic, without relying on posterior constraints?
• RQ2 How can monotonicity be enforced in a way that supports interpretable priors and principled uncertainty quantification in hierarchical models?
• RQ3 What is the performance of the proposed monotonic flow compared to existing probabilistic monotonic models on standard regression benchmarks?
• RQ4 Can the inclusion of uncertainty in monotonic warping functions improve the interpretability and robustness of time-series alignment models?

Key findings

• The proposed Monotonic Gaussian Process Flow guarantees monotonicity in all sample paths due to the theoretical properties of SDE solutions with monotonic drift.
• The model achieves competitive regression performance on benchmark functions, outperforming or matching state-of-the-art probabilistic monotonic models.
• In hierarchical time-series alignment, the model captures a range of plausible warping functions consistent with the prior, revealing multiple plausible data alignments that point estimates miss.
• The inclusion of cross-layer correlations in variational inference preserves compositional uncertainty, leading to more informative and robust model estimates than point-estimate baselines.
• The method successfully models non-stationary, complex data distributions through monotonic transformations, demonstrating utility in real-world applications such as temporal alignment and trajectory estimation.
• The framework enables the use of monotonic functions as a general-purpose first layer in hierarchical models, especially when the underlying transformation is known to be non-stationary.