Quiz: Lecture 19 — Riemannian Manifolds: Local Polynomial Regression

This quiz tests your understanding of intrinsic local polynomial regression — manifold-valued responses, tangent-space Taylor expansions, order-\(\ell\) polynomial curves, and comparison with metric-space local 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. The key difference between local Fréchet regression (metric-only) and intrinsic local polynomial regression (manifold) is:

A. Local Fréchet regression is always more accurate B. They use different kernel functions C. Local Fréchet regression only needs distances, while intrinsic local polynomial regression uses the manifold’s smooth structure (exp/log maps) to fit polynomials directly in tangent spaces, enabling explicit slope estimation D. There is no difference — they are the same method

Q2. The intrinsic mean-zero error model for manifold-valued responses is:

A. \(Y = \mu(X) + \varepsilon\) with \(\mathbb{E}[\varepsilon] = 0\) — the additive Euclidean error model B. \(Y = \exp_{\mu(X)}(\varepsilon)\) with \(\mathbb{E}[\varepsilon \mid X] = 0\) — within a normal neighborhood, \(\varepsilon=\log_{\mu(X)}Y\) lives in the tangent space C. \(Y = \log_{\mu(X)}(\varepsilon)\) — the error mapped via the logarithmic map D. \(Y = \mu(X) \cdot \varepsilon\) — a multiplicative error model

Q3. At target \(x_0\), an order-\(\ell\) local polynomial curve based at \(p\) is:

A. A single constant \(v_0\) only B. A Fourier series expansion C. \(m(z)=\exp_p\!\left\{\sum_{j=1}^{\ell}(z-x_0)^jv_j/j!\right\}\), where \(v_j\in T_p\mathcal M\) and \(m(x_0)=p\) D. A spline basis expansion

Q4. The order \(\ell = 0\) in intrinsic local polynomial regression recovers:

A. Local-constant (kernel Fréchet) regression — the estimator is the weighted Fréchet mean of nearby responses B. Linear regression C. Logistic regression D. Quadratic regression

Q5. Order \(\ell = 1\) gives local-linear regression. The tangent-space slope \(\hat{v}_1(x)\) provides:

A. The bandwidth for kernel smoothing B. Only the intercept of the regression C. The same information as \(\ell = 0\) D. A directly fitted first-order sensitivity in the tangent space; a basic local-constant Fréchet fit has no slope parameter

Q6. On the SPD manifold \(\mathcal{S}_{++}^m\) with the log-Euclidean metric, local-linear regression simplifies to:

A. A nonlinear programming problem with manifold constraints B. Weighted least squares in \(\mathbb{R}^{m(m+1)/2}\) — because the LE metric makes the manifold isometric to a Euclidean space, the log transform linearizes the problem C. A discrete optimization over permutations D. An intractable optimization problem requiring MCMC

Q7. On the SPD manifold with the affine-invariant metric, computation of local polynomial regression typically uses:

A. The closed-form quantile formula B. Ordinary least squares C. Simple matrix averaging D. AESAMC (annealing evolutionary stochastic approximation Monte Carlo) or a carefully derived Riemannian optimizer; the joint criterion is generally nonlinear

Q8. Boundary correction in intrinsic local polynomial regression is:

A. Requires a completely different statistical model B. Under standard smoothness and design conditions, adding the local slope removes the leading boundary-bias term, just as in Euclidean local polynomial regression C. Only possible with \(\ell = 0\) D. Impossible — the method fails at boundaries

Q9. The population local polynomial criterion minimizes:

A. \(\mathbb{E}[K_h(X-x)d^2\{Y,\exp_\omega(\sum_{j=1}^{\ell}(X-x)^jv_j/j!)\}]\) — kernel-weighted expected squared geodesic loss B. Only the sum of squared residuals C. The likelihood under a Gaussian model D. The cross-validated prediction error

Q10. Comparing local-constant (\(\ell=0\)) and local-linear (\(\ell=1\)) estimators on simulated data typically shows:

A. They always give identical results B. Local-linear has higher MSE everywhere due to increased variance C. Local-linear often reduces leading boundary bias for a smooth regression curve, but its extra slope parameter can increase variance, so finite-sample MSE (especially when the sample size is small) may favor either method D. Local-constant always outperforms local-linear