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Research · IIT Palakkad · Statistics

Gaussian Process Regression for Distribution-Valued Inputs

A replication of the simulation study in Bachoc et al. (2018), in which a Gaussian process takes probability distributions as inputs, and its extension to Wasserstein Tangent Gaussian Processes (WTGP): a GP prior on the tangent space of one-dimensional Wasserstein space.

InstitutionIIT Palakkad, Dept. of Statistics
SupervisorDr. Tanmay Sahoo
Period2026 – ongoing
StatusPaid research fellowship

Paper

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Open paper (PDF) ↗

Timeline

  • SelectionSelected as one of 122 fellows from more than 15,000 applications.
  • Week 1: reading and first proposalRead Bachoc et al. (2018) and presented an initial proposal built on spatial statistics: Moran’s I, Kriging and Wasserstein distance.
  • Replication of Table IImplemented data generation, the W₂ distance matrix, the power-exponential kernel, MLE fitting, Kriging prediction, and Legendre and PCA baselines.
  • Wasserstein Tangent GPIsometric quantile-function embedding of 1-D Wasserstein space into L²[0,1], a GP prior on the tangent space, kernel validity by a Kronecker-product spectral argument, closed-form posterior.
  • TheoryGP theory for monotonicity-constrained input spaces, high-dimensional additive GP models, and Bayesian inference in non-standard input geometries.

How the thinking developed

Why the similarity measure is the model

Two locations can share a mean and differ in the behaviour underneath it. A kernel on means cannot see that, so the project starts from the metric rather than the regression.

Why W₂ and not L²

Two densities differing by a small shift can be far apart in L² when their mass barely overlaps, yet close in W₂, which measures the cost of moving one into the other.

In one dimension, W₂ is sorting

The quantile function is the sorted sample, so W₂ is the L² distance between quantile functions. No density estimation is needed.

From replication to extension

Because the quantile map is an isometry into L²[0,1], a GP prior can be placed on the resulting tangent space. That is the step that turns the replication into WTGP.

Pipeline

  1. Distribution inputsSamples from μ₁,…,μₙ, generated with GP perturbations of a base family.
  2. Quantile functionsInverse CDFs on a common grid of levels.
  3. W₂ distance matrixd(μᵢ,μⱼ) = ‖F⁻¹ᵢ − F⁻¹ⱼ‖ in L²[0,1].
  4. KernelPower-exponential: K(μᵢ,μⱼ) = σ² exp(−(d/θ)^α), plus a nugget η on the diagonal.
  5. MLE fittingMaximise the log-likelihood over (σ², θ, α, η).
  6. Kriging predictionPosterior mean and variance at test distributions.
  7. BaselinesLegendre-projection and PCA-projection GPs on the same data.

Results

The replication reproduces the experiment behind Table I of Bachoc et al. (2018). The run shipped in the repository used 100 training and 100 test distributions; the paper uses 500 test distributions, so values approximate rather than match.

RMSE comparison
RMSE in the replication run: 0.143 for the W₂ GP against 1.018 for the Legendre order-5 baseline.
CIR comparison
Coverage of the 90% credible interval (CIR₀.₉) across the same models.

Reference values reported in the paper, as transcribed in the repository:

ModelRMSECIR₀.₉
Distribution (W₂)0.0940.92
Legendre, order 50.490.92
Legendre, order 100.340.89
PCA, order 50.630.82
PCA, order 100.520.87

WTGP. A 13% mean W₂ error reduction over a pointwise baseline, pooled over 10 seeds.

Simulated densities
Simulated input distributions.

Code

Simulation repository ↗

Reference: Bachoc, F. et al. (2018), A Gaussian Process Regression Model for Distribution Inputs.

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