Quiz: Lecture 1 — Introduction to Non-Euclidean Data Analysis

This quiz tests your understanding of the core concepts from Lecture 1. 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. Which of the following is NOT one of the algebraic and geometric features of \(\mathbb{R}^p\) that classical statistics relies on?

A. Vector space structure B. Inner product and orthogonality C. Curved (non-zero curvature) geometry D. Convexity and projections

Q2. The parallelogram law \(\|u+v\|^2 + \|u-v\|^2 = 2\|u\|^2 + 2\|v\|^2\) is used to test whether:

A. A metric satisfies the triangle inequality B. A norm is induced by an inner product C. A space is finite-dimensional D. A distance is symmetric

Q3. The sphere \(\mathbb{S}^2\) with geodesic distance \(d(x,y) = \arccos(\langle x, y \rangle)\) fails to be Euclidean at which structural level?

A. Not a metric space B. Metric space, but not a vector space C. Normed space, but the norm is not inner-product induced D. Inner-product space, but infinite-dimensional

Q4. The truncated metric \(d(x,y) = \min\{1, |x-y|\}\) on \(\mathbb{R}\) is not induced by any norm. Which property fails?

A. Symmetry B. The triangle inequality C. Positive-definiteness D. Homogeneity: \(d(ax, ay) = |a|\,d(x,y)\)

Q5. The BHV tree space for rooted \(m\)-trees is not a vector space because:

A. Trees cannot be added or rescaled in a globally meaningful way B. The space has infinitely many dimensions C. The BHV distance does not satisfy the triangle inequality D. Trees are always binary

Q6. The “swelling effect” observed when using Euclidean distance on SPD matrices refers to:

A. The matrix exponential causing numerical overflow B. The Euclidean mean having a larger determinant than the original matrices C. The Frobenius norm growing without bound D. The dimension of the SPD cone increasing with sample size

Q7. The affine-invariant distance on \(\mathcal{S}_{++}^{m}\) is defined using:

A. The Cholesky decomposition of the difference matrix B. The matrix logarithm of \(\Sigma_0^{-1/2}\,\Sigma_1\,\Sigma_0^{-1/2}\) C. The element-wise absolute difference of the two matrices D. The trace of the product \(\Sigma_0 \Sigma_1\)

Q8. The Wasserstein space \(\mathcal{P}_2(\mathbb{R}^p)\) is not Euclidean. Which of the following is a reason?

A. Arbitrary linear combinations of probability measures may not yield probability measures B. The space is infinite-dimensional in general C. For \(p \ge 2\), the geometry is genuinely curved D. All of the above

Q9. According to the lecture, which approach provides more computational tractability once the geometric machinery is in place?

A. General metric spaces (only a distance function) B. Riemannian manifolds (charts, exp/log maps, curvature) C. Bare sets with no metric structure D. Both approaches are equally tractable

Q10. On \(\mathbb{R}\), define \(d(x,y) = (x-y)^2\). Which property of a metric does \(d\) fail to satisfy?

A. Symmetry: \(d(x,y) = d(y,x)\) for all \(x,y\) B. Positive-definiteness: \(d(x,y) = 0 \iff x = y\) C. Triangle inequality: \(d(x,z) \le d(x,y) + d(y,z)\) for all \(x,y,z\) D. None — \(d\) is a valid metric on \(\mathbb{R}\)