Series of univariate distributions indexed by equally spaced time points are ubiquitous in applications and their analysis constitutes one of the challenges of the emerging field of distributional data analysis. To quantify such distributional time series, we propose a class of intrinsic autoregressive models that operate in the space of optimal transport maps. The autoregressive transport models that we introduce here are based on regressing optimal transport maps on each other, where predictors can be transport maps from an overall barycenter to a current distribution or transport maps between past consecutive distributions of the distributional time series. Autoregressive transport models and their associated distributional regression models specify the link between predictor and response transport maps by moving along geodesics in Wasserstein space. These models emerge as natural extensions of the classical autoregressive models in Euclidean space. Unique stationary solutions of autoregressive transport models are shown to exist under a geometric moment contraction condition of Wu & Shao [(2004) Limit theorems for iterated random functions. Journal of Applied Probability 41, 425–436)], using properties of iterated random functions. We also discuss an extension to a varying coefficient model for first-order autoregressive transport models. In addition to simulations, the proposed models are illustrated with distributional time series of house prices across U.S. counties and annual summer temperature distributions.
Convex Neural Autoregressive Models: Towards Tractable, Expressive, and Theoretically-Backed Models for Sequential Forecasting and Generation
Three features are crucial for sequential forecasting and generation models: tractability, expressiveness, and theoretical backing. While neural autoregressive models are relatively tractable and offer powerful predictive and generative capabilities, they often have complex optimization landscapes, and their theoretical properties are not well understood. To address these issues, we present convex formulations of autoregressive models with one hidden layer. Specifically, we prove an exact equivalence between these models and constrained, regularized logistic regression by using semi-infinite duality to embed the data matrix onto a higher dimensional space and introducing inequality constraints. To make this formulation tractable, we approximate the constraints using a hinge loss or drop them altogether. Furthermore, we demonstrate faster training and competitive performance of these implementations compared to their neural network counterparts on a variety of data sets. Consequently, we introduce techniques to derive tractable, expressive, and theoretically-interpretable models that are nearly equivalent to neural autoregressive models.
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- Award ID(s):
- 2037304
- NSF-PAR ID:
- 10290937
- Date Published:
- Journal Name:
- ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
- Page Range / eLocation ID:
- 3890 to 3894
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/ICASSP39728.2021.9413662
