Quiz: Lecture 18 — Riemannian Manifolds: Geodesic Regression

This quiz tests your understanding of geodesic regression — the manifold analogue of linear regression, symmetric spaces, intrinsic \(R^2\), and comparison with metric-space Fréchet regression. 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. Geodesic regression on a Riemannian manifold is the direct analogue of:

A. Kernel regression B. Logistic regression C. Principal component analysis D. Simple linear regression — it fits a geodesic curve \(\exp_p(x v)\) to manifold-valued responses \(Y \in \mathcal{M}\), where \(p\) is the intercept, \(v\) is the slope (tangent vector), and \(x\) is the scalar predictor

Q2. The geodesic least-squares objective for data \((x_i, Y_i)\) is:

A. \(\sum_i (x_i - \bar{x}) d(Y_i, p)\) B. \(\sum_i d^2(Y_i, \exp_p(x_i v))\) — the sum of squared geodesic distances between the observed responses and the fitted geodesic C. \(\sum_i \|Y_i - (p + x_i v)\|^2\) — the Euclidean residual sum of squares D. \(\sum_i \|\log_{Y_i}(p)\|^2\)

Q3. A symmetric space is a Riemannian manifold where:

A. All points are equidistant from each other B. The dimension equals the rank C. For every point \(p\), there exists an isometry (the geodesic symmetry) that reverses all geodesics through \(p\) — geometrically, the manifold looks the same from every point D. The curvature is zero everywhere

Q4. Which of the following is NOT a symmetric space?

A. The SPD cone \(\mathcal{S}_{++}^m\) (with affine-invariant metric) B. A general Riemannian manifold without any special symmetry — symmetric spaces are a restricted class with strong geometric structure C. Euclidean space \(\mathbb{R}^m\) D. The sphere \(\mathbb{S}^m\)

Q5. On symmetric spaces, the geodesic least-squares problem has:

A. The same solution as in Euclidean space B. Infinitely many solutions in all cases C. Under Fletcher’s curvature and concentration bounds, a unique minimizer within the stated class of \(r\)-close geodesics; this is a local, qualified result rather than unconditional global uniqueness D. No guarantee of a solution

Q6. The intrinsic \(R^2\) for geodesic regression is defined as:

A. The trace of the Hessian of the least-squares objective B. \(R^2 = 1 - \frac{\sum_i d^2(Y_i, \hat{Y}_i)}{\sum_i d^2(Y_i, \hat{\mu})}\), where \(\hat{Y}_i\) are fitted values on the geodesic and \(\hat{\mu}\) is the Fréchet mean of \(\{Y_i\}\) — it generalizes the Euclidean \(R^2\) using geodesic distances C. The same formula as Euclidean \(R^2\) with no modifications D. The squared correlation between \(x\) and \(Y\)

Q7. Permutation tests for geodesic regression assess significance by:

A. Randomly permuting the predictor values \(x_i\), refitting the geodesic regression, and comparing the observed \(R^2\) to the permutation distribution — this tests \(H_0\): no association between \(x\) and \(Y\) B. Comparing the intrinsic \(R^2\) to a chi-squared distribution C. Computing the F-statistic from the ANOVA table D. Using the t-distribution with \(n-2\) degrees of freedom

Q8. Under Fletcher’s intrinsic Gaussian error model and the associated regularity conditions, geodesic least-squares estimation is equivalent to:

A. Bayesian estimation with a flat prior B. Nonparametric kernel regression C. Method of moments estimation D. Maximum likelihood for the geodesic parameters \((p,v)\) at fixed scale, because homogeneity makes the density’s normalizing constant independent of its center

Q9. How does geodesic regression differ from Fréchet regression (Lectures 6–9)?

A. Geodesic regression restricts the fitted response trajectory to one parametric geodesic \(\exp_p(xv)\), whereas Fréchet regression defines each target value through a weighted squared-distance minimization without imposing one geodesic across predictor values B. Fréchet regression only works for Euclidean responses C. Geodesic regression does not use the Riemannian metric D. They are mathematically identical

Q10. The gradient of the geodesic least-squares objective is computed using:

A. Differentials of the exponential map — specifically, the derivative \(d\exp_p(x v)\) with respect to both the base point \(p\) and the tangent vector \(v\), which can be computed via Jacobi fields or explicit formulas on symmetric spaces B. Finite differences only — the objective has no analytic gradient C. Automatic differentiation — the only viable approach D. Numerical integration over the manifold