Quiz: Lecture 2 — Fréchet Means: Existence and Uniqueness

This quiz tests your understanding of the core concepts from Lecture 2. 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. The Fréchet mean generalizes the Euclidean mean by:

A. Minimizing expected distance (not squared) B. Minimizing expected squared distance C. Maximizing the likelihood function D. Using the component-wise definition of expectation

Q2. If \(\mathbb{E}\{d^2(X, x_0)\}< \infty\) for some \(x_0 \in \mathcal{M}\), then:

A. \(\mathbb{E}\{d^2(X, x)\}\) is finite only for \(x\) near \(x_0\) B. \(\mathbb{E}\{d^2(X, x)\}\) is finite for all \(x \in \mathcal{M}\) C. The Fréchet mean is guaranteed to exist D. The space must be Euclidean

Q3. In the example \(\mathcal{M} = \mathbb{R} \setminus \{0\}\) with \(X = \pm 1\) equally likely, the population Fréchet mean:

A. Is at \(x = 0\) B. Is at \(x = 1\) C. Is at \(x = -1\) D. Does not exist (the set is empty)

Q4. A geodesic space is one where:

A. Every pair of points can be joined by at least one geodesic B. Geodesics are always unique between any two points C. The space is a smooth manifold D. All distances are Euclidean

Q5. Which of the following is NOT a Hadamard space?

A. SPD matrices with the affine-invariant metric B. BHV tree space C. The sphere \(\mathbb{S}^2\) with geodesic distance D. \(\mathbb{R}^p\) with Euclidean distance

Q6. The \(\mathrm{CAT}(0)\) inequality compares:

A. Areas of geodesic triangles vs. comparison triangles B. Distances from a vertex to a point on the opposite side vs. the corresponding distance in the comparison triangle C. Perimeters of geodesic triangles D. Angles at corresponding vertices

Q7. In a Hadamard space, the squared distance function \(f(x) = d^2(z, x)\) for fixed \(z\) is:

A. Linear along geodesics B. Concave along geodesics C. Strongly convex along geodesics D. Neither convex nor concave along geodesics

Q8. The Heine–Borel property (every closed bounded subset is compact) guarantees:

A. Uniqueness of the Fréchet mean B. Existence of the Fréchet mean C. The convergence rate is \(n^{-1/2}\) D. The space is a Hadamard space

Q9. On the sphere \(\mathbb{S}^p\), the Fréchet mean of a uniform distribution is:

A. Unique and located at the north pole B. Every point on the sphere (completely non-unique) C. Always at the equator D. Does not exist

Q10. The key geometric property of Hadamard spaces that guarantees uniqueness of the Fréchet mean is:

A. Compactness of closed balls B. Positive sectional curvature C. Strong convexity of squared distance along geodesics D. Infinite dimensionality allowing more degrees of freedom