Quiz: Lecture 12 — Wasserstein Barycenters

This quiz tests your understanding of Wasserstein barycenters — Fréchet means in Wasserstein space, including 1D quantile averaging, Gaussian barycenters, and computational approaches. There are 10 multiple-choice questions, each worth 1 mark. After submitting, your score is shown, and any questions you got wrong are highlighted — but the correct answer is not revealed. You can then redo only the questions you missed.

Q1. A Wasserstein barycenter \(\bar{\mu}\) of a distribution \(\Lambda\) on \(\mathcal{P}_2(\mathcal{X})\) is defined as:

A. The median of the random measure under the \(W_1\) distance B. The arithmetic mean of the supports of all measures in the population C. The measure with density equal to the average of the densities D. A minimizer of \(\Phi(\nu) = \int W_2^2(\nu, \mu) \, d\Lambda(\mu)\) — a Fréchet mean in \((\mathcal{P}_2(\mathcal{X}), W_2)\)

Q2. In one dimension, the Wasserstein barycenter of measures \(\mu_1, \ldots, \mu_n\) with weights \(\lambda_i\) has a closed form:

A. Its mean is the weighted average of the means, with no further constraints B. Its quantile function is the weighted average of the quantile functions: \(F_{\bar{\mu}}^{-1}(u) = \sum_{i=1}^n \lambda_i F_{\mu_i}^{-1}(u)\) C. Its density is the weighted average of the densities: \(f_{\bar{\mu}}(x) = \sum \lambda_i f_{\mu_i}(x)\) D. Its CDF is the weighted average of the CDFs: \(F_{\bar{\mu}}(x) = \sum \lambda_i F_{\mu_i}(x)\)

Q3. For Gaussian measures \(\mu_i = \mathcal{N}(m_i, \Sigma_i)\), the Wasserstein barycenter \(\bar{\mu} = \mathcal{N}(\bar{m}, \bar{\Sigma})\) satisfies:

A. \(\bar{m} = \sum \lambda_i m_i\) and \(\bar{\Sigma}\) solves the Bures–Wasserstein fixed-point equation \(\bar{\Sigma} = \sum_i \lambda_i (\bar{\Sigma}^{1/2} \Sigma_i \bar{\Sigma}^{1/2})^{1/2}\) B. \(\bar{m} = \text{median}(m_i)\) and \(\bar{\Sigma} = \text{median}(\Sigma_i)\) C. \(\bar{m} = \sum \lambda_i m_i\) and \(\bar{\Sigma} = \sum \lambda_i \Sigma_i\) (the arithmetic mean of covariances) D. \(\bar{\mu}\) cannot be Gaussian in general

Q4. Existence of Wasserstein barycenters requires:

A. That the number of measures \(n\) be less than the dimension \(d\) B. No conditions are needed — barycenters always exist C. That all measures be Gaussian D. Moment and compactness conditions — the population must have bounded second moment and the measures must be supported on a compact or boundedly compact space

Q5. Uniqueness of Wasserstein barycenters typically requires:

A. Uniqueness always holds — barycenters are unique by definition B. That all measures have the same variance C. That at least one of the measures be absolutely continuous (have a density), or that the data be 1D D. That the measures be defined on a Euclidean space

Q6. The convergence rate of the sample Wasserstein barycenter in one dimension is:

A. The rate is unknown in one dimension B. \(O_P(n^{-1/2})\) — the parametric rate, thanks to the 1D quantile representation C. \(O_P(n^{-1})\) — faster than parametric D. \(O_P(n^{-1/4})\) — slower than parametric due to the infinite-dimensional nature

Q7. Which of the following is NOT a computational approach for Wasserstein barycenters?

A. Fixed-point iteration (for Gaussian or regularized barycenters) B. Linear programming (exact, for discrete measures) C. Principal component analysis of the density functions D. Entropic regularization with Sinkhorn iterations

Q8. How does the Wasserstein barycenter differ from the Euclidean barycenter (arithmetic mean of densities)?

A. The Wasserstein barycenter is always a Gaussian B. The Euclidean barycenter is always more accurate C. The Euclidean barycenter averages densities pointwise and suffers from dilation/blurring, while the Wasserstein barycenter respects the geometry of the support — it translates, compresses, and shapes the measures without artificial smearing D. They are always identical

Q9. Sample Wasserstein barycenters (computed from empirical measures) are:

A. Strongly consistent — they converge almost surely to the population barycenter under appropriate conditions B. Only consistent in Euclidean spaces C. Biased but consistent — the bias vanishes only as \(n \to 0\) D. Inconsistent — they converge to a random limit that depends on the sample

Q10. The entropic regularization approach (Sinkhorn barycenters) trades off:

A. Memory usage for numerical precision only B. Statistical consistency for computational speed — stronger regularization gives faster convergence but a biased (blurred) barycenter C. Nothing — entropic barycenters are identical to exact barycenters D. Interpretability for flexibility only