Wasserstein Spatial Depth

1. Problem Statement

The paper addresses the challenge of performing statistical analysis on distribution-valued data (random probability measures) embedded within Wasserstein spaces ($\mathcal{P}_2(\mathbb{R}^d)$).

• Context: Modern data in fields like AI, climate science, and biology often consists of distributions (e.g., histograms, images, or batches of data) rather than simple vectors. Modeling these as random objects in a metric space captures their geometric structure and variability better than Euclidean models.
• The Gap: Conventional statistical tools, particularly statistical depth (which ranks data points by centrality, enabling outlier detection, ranking, and nonparametric testing), are designed for Euclidean spaces.
  • Wasserstein spaces are non-linear and possess a Riemannian manifold structure (geodesics are not straight lines).
  • Existing depth measures (e.g., Tukey depth, Monge-Kantorovich depth) either fail to generalize to non-linear spaces or are computationally intractable (exponential complexity) in high dimensions.
  • Simply embedding distributions into a Hilbert space (e.g., via RKHS) ignores the intrinsic geodesic geometry of the Wasserstein space, leading to distorted "ordering" and loss of information.

Goal: Develop a new notion of statistical depth specifically tailored for Wasserstein spaces that preserves the desirable properties of Euclidean depth (ranking, robustness, invariance) while respecting the underlying geometry.

2. Methodology: Wasserstein Spatial Depth (WSD)

The authors propose the Wasserstein Spatial Depth (WSD), an extension of the classical Euclidean spatial depth to the space of probability measures.

A. Geometric Foundation

The definition relies on the optimal transport (OT) framework:

• Let $P$ be a distribution of distributions (a measure on $\mathcal{P}_2(\mathbb{R}^d)$) and $Q$ be a specific distribution (a point in the space).
• In Euclidean space, spatial depth is defined by the expected direction of vectors from a point $x$ to random samples $X$.
• In Wasserstein space, the "direction" is replaced by the velocity field of the geodesic connecting $Q$ to a sample $P$.
• For absolutely continuous $Q$ and $P$, the constant-speed geodesic is generated by the optimal transport map $T_{Q,P}$. The velocity field at $t=0$ is $v = T_{Q,P} - I$.

B. Definition

The WSD of a distribution $Q$ with respect to a population of distributions $P$ is defined as:
$$\text{WSD}(Q; P) := 1 - \left( \int \left\| \mathbb{E}_{P \sim P} \left[ \frac{x - T_{Q,P}(x)}{W_2(P, Q)} \right] \right\|^2 dQ(x) \right)^{1/2}$$

• Intuition: It measures the "centrality" of $Q$ by calculating the average normalized velocity vector required to transport $Q$ to random samples $P$. If $Q$ is central, these vectors cancel out (low norm), yielding a depth near 1. If $Q$ is an outlier, the vectors align, yielding a depth near 0.
• Handling Discreteness: The paper provides a specific formulation for cases where $Q$ is not absolutely continuous, utilizing optimal transport plans rather than unique maps.

C. Estimation

The authors propose estimators for two common data scenarios:

1. One-stage sampling: Observing $n$ distributions $P_1, \dots, P_n$ directly.
2. Two-stage sampling: Observing an array of data points where $P_i$ are unknown and must be estimated from $m$ samples each ($X_{i,1}, \dots, X_{i,m}$).

• The estimator replaces the population measure $P$ with the empirical measure $P_n$ (or $P_{n,m}$) and the optimal transport maps with those computed from empirical distributions.
• Complexity: The computation is $O(nm^2)$, which is polynomial and feasible, unlike the exponential complexity of Tukey depth in high dimensions.

3. Key Contributions & Theoretical Results

A. Fundamental Properties

The paper proves that WSD satisfies the axiomatic properties required for a valid statistical depth:

1. Range: Values lie in $[0, 1]$.
2. Transformation Invariance: Invariant under isometries of the Wasserstein space (including trivial isometries and shape-preserving transformations).
3. Vanishing at Infinity: As $Q$ moves infinitely far from the support of $P$, $\text{WSD}(Q; P) \to 0$.
4. Maximality at the Center: The depth is maximized (value 1) at the geometric median (Fréchet mean) of the distribution $P$, provided the median is unique and absolutely continuous.
5. Geodesic Invariance: The depth remains constant along geodesics emanating from the median.
6. Continuity: The depth function is continuous with respect to both the target distribution $Q$ and the population distribution $P$.

B. Robustness

• Breakdown Point: The authors characterize the breakdown point of the empirical depth regions. They show that the breakdown point is approximately $\alpha/2$ for depth level $\alpha$, matching the robustness of Euclidean spatial depth.
• Influence Function: They derive the influence function of the WSD functional and prove it is bounded (specifically, the gross error sensitivity is $\le 2$), confirming the estimator's robustness against outliers.

C. Asymptotic Theory

• Consistency: The empirical WSD converges almost surely to the population WSD as sample sizes ($n$ and $m$) increase.
• Asymptotic Normality: Under the one-stage sampling model, the empirical WSD satisfies a Central Limit Theorem. For the two-stage model, convergence rates are established, depending on the dimension $d$ and sample size $m$ (e.g., $O(m^{-1/2})$ for $d=1$, $O(m^{-1/d})$ for $d>2$).

D. Statistical Inference

• Two-Sample Test: A nonparametric permutation test is constructed to test $H_0: P = Q$ vs. $H_1: P \neq Q$.
  • The test uses the Kolmogorov-Smirnov statistic on the WSD values of the two samples.
  • The authors prove the test controls the Type I error exactly (finite sample) and is consistent (power $\to 1$) as sample sizes grow.

4. Experimental Results

The paper validates the methodology through extensive simulations and a real-world application:

• Consistency: Empirical estimates closely match theoretical values across various distributions (Exponential, Weibull, Gaussian, Uniform).
• Comparison with Linear Embeddings: When distributions are embedded into a Hilbert space (RKHS) and standard functional depths (e.g., Modified Band Depth) are applied, the "ordering" is distorted. WSD correctly identifies outliers and centrality, whereas functional depths fail to distinguish them.
• Comparison with Other Metric Depths:
  • Tukey Depth: Computationally infeasible for Wasserstein spaces.
  • Lens Depth & Metric Spatial Depth: These general metric depths fail to detect outliers in Wasserstein space simulations and lack the property of maximizing at the geometric median.
• Outlier Detection: WSD successfully identifies "exotic" distributions (outliers) in high-dimensional settings ($d=10$) where other methods fail.
• Real-World Application (Climate Data):
  • Dataset: Daily temperatures from 40 European stations (1874–2023), modeled as distributions of monthly temperatures.
  • Finding: WSD identified specific "abnormal years" (e.g., 1879, 1929, 1942, 2018) with extremely low depth. These years correspond to historical records of extreme cold, heatwaves, or droughts, validating the method's ability to detect global climate anomalies.

5. Significance and Impact

• Theoretical Advancement: This work bridges the gap between optimal transport theory and nonparametric statistics, providing the first rigorous, computationally feasible depth measure for distributional data in Wasserstein spaces.
• Practical Utility: It enables standard statistical tasks (ranking, outlier detection, hypothesis testing) for complex data types that were previously difficult to analyze without losing geometric structure.
• Robustness: The method is robust to outliers and computationally efficient (polynomial time), making it suitable for modern high-dimensional data analysis.
• Future Directions: The paper opens avenues for depth-based clustering, regression, and classification specifically designed for the geometry of probability measures, moving beyond the limitations of linear approximations.

In summary, Wasserstein Spatial Depth is a foundational tool that allows statisticians to treat distributions as first-class citizens in a geometrically consistent manner, preserving the intrinsic structure of the data while enabling powerful inferential tools.