Quiz: Lecture 13 — Probability Distributions from Samples – Lectures on Non-Euclidean Data Analysis

Q1. The empirical measure \(\hat{\mu}_n = \frac{1}{n}\sum_{i=1}^n \delta_{X_i}\) is:

A. A parametric estimator assuming normality B. A smoothed kernel density estimate of the true distribution C. The most direct nonparametric estimator — it places mass \(1/n\) at each observed point, with no smoothing or modeling assumptions D. Only defined for one-dimensional data

Q2. The plug-in estimator of \(W_p(\mu, \nu)\) uses \(W_p(\hat{\mu}_n, \hat{\nu}_m)\). Its consistency follows from:

A. The central limit theorem alone B. Varadarajan’s theorem (almost-sure weak convergence of \(\hat{\mu}_n\) to \(\mu\)) together with the strong law of large numbers for moments C. Only the Glivenko–Cantelli theorem D. The law of the iterated logarithm

Q3. The Fournier–Guillin bound reveals the Wasserstein curse of dimensionality. For a measure on \(\mathbb{R}^d\), the expected \(W_p^p\) distance between the empirical measure and the true measure scales as:

A. \(n^{-1/2}\) for all dimensions B. \(n^{-1}\) regardless of dimension C. \(n^{-1/2}\) in low dimensions (\(d < 2p\)), but slows to \(n^{-p/d}\) for \(d > 2p\) D. Exponentially fast in \(d\)

Q4. The Weed–Bach upper and lower Wasserstein dimensions \(\dim_{W,p}^*(\mu)\):

A. Provide an intrinsic notion of dimensionality — they characterize the optimal rate \(n^{-1/s}\) for \(s > \dim_{W,p}^*(\mu)\), which can be much smaller than the ambient dimension \(d\) B. Always equal the ambient dimension \(d\) C. Are always equal to 1 D. Are only defined for Gaussian measures

Q5. In one dimension, the asymptotic distribution of \(n W_2^2(\hat{\mu}_n, \mu)\) (Bobkov–Ledoux) is:

A. A chi-squared distribution with \(n-1\) degrees of freedom B. A normal distribution with mean 0 C. An exponential distribution D. A weighted Brownian bridge functional — \(\int_0^1 \frac{B(t)^2}{f(F^{-1}(t))^2} \, dt\), where \(B(t)\) is the standard Brownian bridge

Q6. For a density of smoothness \(s\), the minimax rate for estimating the density under \(W_p\) loss is:

A. Slower than the empirical measure rate in all cases B. \(n^{-(s+1)/(2s+d)}\) — which can be much faster than the empirical measure rate when \(s\) is large and \(d\) is small C. Independent of the smoothness \(s\) D. \(n^{-1/2}\) — the parametric rate

Q7. Why does the empirical measure suffer from the curse of dimensionality in Wasserstein space?

A. Because the Sinkhorn algorithm fails in high dimensions B. Because the number of atoms grows exponentially with \(d\) C. The empirical measure does not suffer from any curse of dimensionality D. Because in high dimensions, a finite sample cannot cover the space — the typical distance from a sample point to the nearest other sample point grows, and the transport cost must move mass over increasingly large distances

Q8. Two-sample Wasserstein distance estimation reduces to the one-sample case via:

A. The Berry–Esseen theorem B. The triangle inequality: \(|W_p(\hat{\mu}_n, \hat{\nu}_m) - W_p(\mu, \nu)| \le W_p(\hat{\mu}_n, \mu) + W_p(\hat{\nu}_m, \nu)\) C. The Cauchy–Schwarz inequality D. The law of large numbers

Q9. A measure that is supported on a low-dimensional manifold embedded in \(\mathbb{R}^d\) has:

A. Weed–Bach dimension close to the intrinsic manifold dimension, not \(d\) — so convergence can be much faster than the ambient-dimension bound suggests B. Weed–Bach dimension equal to the ambient dimension \(d\) C. No well-defined convergence rate D. The same rate as a measure with full-dimensional support

Q10. Smooth-density recovery (e.g., kernel density estimation) differs from the empirical measure in that:

A. It always performs worse than the empirical measure B. It does not use any smoothing — it is identical to the empirical measure C. It explicitly models smoothness, achieving the minimax rate \(n^{-(s+1)/(2s+d)}\), which exploits regularity to beat the empirical measure rate when \(s\) is large relative to \(d\) D. It is only defined in one dimension