Quiz: Lecture 11 — The Wasserstein Distance and Its Geometry – Lectures on Non-Euclidean Data Analysis

Q1. The \(p\)-Wasserstein distance \(W_p(\mu, \nu)\) between two probability measures is defined as:

A. The Kullback–Leibler divergence raised to the power \(p\) B. The total variation distance between \(\mu\) and \(\nu\) C. The infimum over all couplings \(\pi\) of \((\int d(x,y)^p \, d\pi(x,y))^{1/p}\), where the marginals of \(\pi\) are \(\mu\) and \(\nu\) D. The maximum distance between any point in the support of \(\mu\) and any point in the support of \(\nu\)

Q2. Brenier’s theorem states that for absolutely continuous \(\mu\) on \(\mathbb{R}^d\), the optimal transport map for \(W_2\) is:

A. Any measurable function that pushes \(\mu\) forward to \(\nu\) B. The gradient \(\nabla\varphi\) of a convex potential \(\varphi\), i.e., \(T = \nabla\varphi\) with \(T_\#\mu = \nu\) C. A linear transformation \(T(x) = Ax + b\) with \(A\) positive definite D. The identity map \(T(x) = x\)

Q3. In one dimension (\(\mathcal{X} = \mathbb{R}\)), the squared 2-Wasserstein distance has a closed-form quantile representation:

A. \(W_2^2(\mu,\nu) = \int_0^1 (F_\mu^{-1}(u) - F_\nu^{-1}(u))^2 \, du\) B. \(W_2^2(\mu,\nu) = \int (f_\mu(x) - f_\nu(x))^2 \, dx\) C. \(W_2^2(\mu,\nu) = (\mathbb{E}[X] - \mathbb{E}[Y])^2\) D. \(W_2^2(\mu,\nu) = \sup_{x} |F_\mu(x) - F_\nu(x)|\)

Q4. What is the Alexandrov curvature of the 2-Wasserstein space \(\mathcal{W}_2(\mathbb{R})\) for distributions on the real line?

A. Flat — \(\mathcal{W}_2(\mathbb{R})\) is isometric to a convex subset of \(L^2(0,1)\) B. Strictly positive (like a sphere) C. The curvature is undefined for Wasserstein spaces D. Strictly negative (like hyperbolic space)

Q5. For \(\mathcal{W}_2(\mathbb{R}^d)\) with \(d \ge 2\), the curvature is:

A. Flat, like \(\mathcal{W}_2(\mathbb{R})\) B. CAT(0) — nonpositive curvature everywhere C. Strictly negative everywhere D. CBB(0) (curvature bounded below by 0) but NOT CAT(0)

Q6. Displacement interpolation (McCann) provides:

A. An algorithm for computing Fréchet means B. The geodesic between two measures \(\mu\) and \(\nu\) in Wasserstein space, given by \(\mu_t = ((1-t)\mathrm{id} + tT)_\#\mu\) where \(T\) is the optimal map C. A technique for kernel density estimation D. A method to interpolate between two Euclidean points using splines

Q7. Convergence in \(W_p\) is equivalent to:

A. Pointwise convergence of density functions B. Convergence in total variation only C. Convergence in Kolmogorov–Smirnov distance D. Weak convergence (convergence in distribution) plus convergence of \(p\)-th moments

Q8. The Monge optimal transport problem seeks a deterministic map \(T\) pushing \(\mu\) forward to \(\nu\). The Kantorovich formulation generalizes this by:

A. Restricting the search to linear transport maps only B. Requiring the cost function to be convex in both arguments C. Adding a penalty term to enforce continuity of the transport map D. Allowing mass splitting via couplings (transport plans) \(\pi\) — a joint distribution with marginals \(\mu\) and \(\nu\) — instead of requiring a deterministic map

Q9. The Sinkhorn algorithm for computing Wasserstein distances introduces:

A. An entropic regularization term \(-\varepsilon H(\pi)\) that makes the problem strictly convex and solvable via iterative matrix scaling B. A neural network approximation to the transport plan C. A Monte Carlo sampling scheme for high-dimensional integrals D. A dimensionality reduction technique for the ground space

Q10. In 1D, the optimal transport plan between two empirical measures can be computed by:

A. Solving a large linear program with \(n \times m\) variables B. Using gradient descent on the space of couplings C. Sorting the observations and matching order statistics — the sort-and-match principle D. Applying the kernel trick and solving a quadratic program