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Shape-constrained density estimation with Wasserstein projection

Takeru Matsuda, Ting-Kam Leonard Wong

TL;DR

This paper proves structural properties of the Wasserstein projection estimator, proposes a discretization which can be implemented by off-the-shelf solvers, and compares the projection estimator with the corresponding maximum likelihood estimator.

Abstract

Statistical inference based on optimal transport offers a different perspective from that of maximum likelihood, and has increasingly gained attention in recent years. In this paper, we study univariate nonparametric shape-constrained density estimation via projection with respect to the $p$-Wasserstein distance, with a focus on the quadratic case $p = 2$. By considering shape constraints given by displacement convex subsets of the Wasserstein space, Wasserstein projection estimation is a convex optimization problem. We focus on two fundamental examples, namely non-increasing densities on $\mathbb{R}_+ := [0, \infty)$ and log-concave densities on $\mathbb{R}$. In each case, we prove structural properties of the Wasserstein projection estimator, propose a discretization which can be implemented by off-the-shelf solvers, and compare the projection estimator with the corresponding maximum likelihood estimator.

Shape-constrained density estimation with Wasserstein projection

TL;DR

This paper proves structural properties of the Wasserstein projection estimator, proposes a discretization which can be implemented by off-the-shelf solvers, and compares the projection estimator with the corresponding maximum likelihood estimator.

Abstract

Statistical inference based on optimal transport offers a different perspective from that of maximum likelihood, and has increasingly gained attention in recent years. In this paper, we study univariate nonparametric shape-constrained density estimation via projection with respect to the -Wasserstein distance, with a focus on the quadratic case . By considering shape constraints given by displacement convex subsets of the Wasserstein space, Wasserstein projection estimation is a convex optimization problem. We focus on two fundamental examples, namely non-increasing densities on and log-concave densities on . In each case, we prove structural properties of the Wasserstein projection estimator, propose a discretization which can be implemented by off-the-shelf solvers, and compare the projection estimator with the corresponding maximum likelihood estimator.
Paper Structure (10 sections, 16 theorems, 136 equations, 4 figures)

This paper contains 10 sections, 16 theorems, 136 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Optimal transport and Wasserstein projection
  3. Wasserstein geometry of univariate distributions
  4. Wasserstein projection estimator
  5. Monotone density estimation
  6. Log-concave density estimation
  7. Implementation and examples
  8. Monotone density estimation
  9. Log-concave density estimation
  10. Discussion

Key Result

Proposition 2.2

Let $p \in [1, \infty)$. For $\mu, \nu \in \mathcal{P}_p(\mathbb{R})$, we have In particular, $\mathcal{P}_p(\mathbb{R})$ (equipped with the $p$-Wasserstein distance) and $\mathcal{Q}_p$ (equipped with the $L^p$ distance) are isometric via the mapping $\mu \mapsto Q_{\mu}$.

Figures (4)

  • Figure 1: Left: Estimated quantile functions from the Wasserstein projection estimator, for the mixture distributions given in \ref{['eqn:monotone1']} (Example \ref{['eg:monotone1']}). Right: Densities of the Wasserstein projection estimator (shaded) and Grenander's estimator (dashed), for two cases ($\lambda = 0, 0.8$) highlighted by thicker lines on the left panel. The support $\{0.2, 1\}$ of the data is shown by the crosses.
  • Figure 2: Left: Data and estimated quantile functions in the context of Example \ref{['eg:monotone3']}. Right: True and estimated densities.
  • Figure 3: Estimated densities for the two-point distribution in Example \ref{['eg:logconcave1']}. Left: $\lambda = 0.4$. Right: $\lambda = 0.2$.
  • Figure 4: Left: Empirical, true and estimated quantile functions in the context of Example \ref{['eg:logconcave2']}. Right: True and estimated densities. The data points are shown by the crosses.

Theorems & Definitions (49)

  • Definition 2.1: $p$-Wasserstein distance
  • Proposition 2.2: Isometry
  • proof
  • Remark 2.3: Convex costs
  • Definition 2.4: Displacement convexity
  • Remark 2.5
  • Example 2.6: Location-scale family
  • Example 2.7: bi-Lipschitz quantile functions
  • Theorem 2.8: Wasserstein projection
  • proof
  • ...and 39 more