Quiz: Lecture 3 — Fréchet Means: Consistency and Convergence Rate

This quiz tests your understanding of the core concepts from Lecture 3. 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. Strong consistency of sample Fréchet means means that:

A. \(\hat{\mu}_n \to \mu\) in probability B. \(\hat{\mu}_n \to \mu\) almost surely C. \(\hat{\mu}_n \to \mu\) in distribution D. \(\mathbb{E}[\hat{\mu}_n] = \mu\) for all \(n\)

Q2. An \(\varepsilon\)-cover of a set \(A\) in a metric space is:

A. A collection of disjoint balls of radius \(\varepsilon\) whose union is \(A\) B. A finite set of points \(\{x_i\}\) such that \(A \subset \bigcup_i B(x_i, \varepsilon)\) C. A subset of \(A\) where all pairwise distances are \(> \varepsilon\) D. A triangulation of \(A\) with all edges of length \(\le \varepsilon\)

Q3. The metric entropy \(H(\varepsilon, A, d)\) of the sphere \(\mathbb{S}^{p-1}\) grows as \(\varepsilon \to 0\) like:

A. \(\varepsilon^{-(p-1)}\) (polynomial in \(1/\varepsilon\)) B. \((p-1)\log(1/\varepsilon)\) (logarithmic in \(1/\varepsilon\)) C. \(\varepsilon^{-1}\) (inverse linear) D. A constant independent of \(\varepsilon\)

Q4. For finite-dimensional spaces like SPD matrices and BHV tree space, the entropy exponent \(\beta\) is:

A. \(\beta < 1\) B. \(\beta = 1\) C. \(\beta = 2\) D. \(\beta = 1/2\)

Q5. When the entropy exponent satisfies \(\beta < 1\), the convergence rate of the sample Fréchet mean is:

A. Slower than the parametric rate \(n^{-1/2}\) B. Parametric: \(n^{-1/2}\) C. Exponential: \(e^{-n}\) D. Logarithmic: \(\log n / n\)

Q6. Stickiness of sample Fréchet means refers to:

A. The algorithm for computing Fréchet means converging too slowly B. Sample Fréchet means getting trapped on lower-dimensional singular strata C. The population Fréchet mean being at the boundary of the space D. All data points being concentrated at a single location

Q7. Smeariness occurs primarily in:

A. Flat Euclidean spaces B. Smooth positively curved manifolds (like spheres) C. Hadamard (nonpositively curved) spaces D. Discrete metric spaces with finitely many points

Q8. The \(\varepsilon\)-net argument in the Bhattacharya–Patrangenaru consistency proof relies on:

A. The classical central limit theorem B. Uniform convergence of \(F_n\) to \(F\) on a compact set containing the sample minimizers C. The law of the iterated logarithm D. The Berry–Esseen bound on normal approximation error

Q9. In the 1D Wasserstein space \((\mathcal{P}_2([0,1]), W_2)\), the metric entropy grows as:

A. Logarithmically in \(1/\varepsilon\) (like finite-dimensional spaces) B. Polynomially: \(\varepsilon^{-1}\) C. Exponentially in \(1/\varepsilon\) D. It is constant (bounded independently of \(\varepsilon\))

Q10. For \(\beta > 1\), the convergence rate \(\eta_{\beta,n}\) is:

A. \(n^{-1/2}\) (same as parametric) B. \(n^{-1/(2\beta)}\), which is slower than \(n^{-1/2}\) C. \(e^{-n}\) (exponentially fast) D. \(\log n\) (diverges as \(n \to \infty\))