Quiz: Lecture 7 — Kernel Fréchet Regression – Lectures on Non-Euclidean Data Analysis

Q1. Why might global Fréchet regression (Lecture 6) be misspecified?

A. It cannot handle negative weights B. It only works for Euclidean response spaces C. It assumes a linear relationship between predictor and response, which may not hold D. It requires the predictor to be one-dimensional

Q2. The kernel Fréchet regression target \(\mu_h^{\mathrm{NW}}(x)\) is the minimizer of:

A. \(\frac{1}{n}\sum_{i=1}^n d^2(Y_i, \omega)\) (the unweighted Fréchet mean) B. \(\mathbb{E}\{K_h(X - x) d^2(Y, \omega)\}\) C. \(\mathbb{E}\{d^2(Y, \omega) \mid X = x\}\) (the exact conditional Fréchet mean) D. \(\mathbb{E}\{s(X, x) d^2(Y, \omega)\}\) where \(s\) is the global weight function

Q3. The sample kernel Fréchet regression estimator uses normalized weights. Why is normalization necessary?

A. To guarantee convergence at the parametric rate B. To remove the dependence on the bandwidth \(h\) C. To make all weights nonnegative D. To ensure the weights sum to 1, making the estimator a proper weighted Fréchet mean

Q4. How do kernel regression weights differ from global Fréchet regression weights?

A. Kernel weights are always nonnegative and decay with distance from \(x\); global weights can be negative and are nonlocal B. Kernel weights require matrix inversion; global weights do not C. Kernel weights are always negative; global weights are always positive D. They are identical — the only difference is the bandwidth

Q5. What happens to the kernel Fréchet regression estimator as the bandwidth \(h \to \infty\)?

A. It diverges to infinity B. It interpolates each data point exactly C. It approaches the unconditional Fréchet mean of all responses D. It becomes the global Fréchet regression estimator

Q6. What happens to the kernel Fréchet regression estimator as \(h \to 0\)?

A. It converges to the global linear fit B. All weights become equal C. It becomes undefined D. It approaches a nearest-neighbor fit — \(\hat{\mu}(X_i) \approx Y_i\) for each observation

Q7. In the bias–variance tradeoff for kernel Fréchet regression, a larger bandwidth \(h\) leads to:

A. Higher bias, higher variance B. Higher bias, lower variance C. Lower bias, higher variance D. Lower bias, lower variance

Q8. The kernel Fréchet regression target \(\mu_h^{\mathrm{NW}}(x)\) is not exactly the conditional Fréchet mean \(\mu(x)\). This is because:

A. The Fréchet mean does not exist for kernel-weighted objectives B. Kernel regression works only in Euclidean spaces C. Kernel smoothing induces a bias — the target is a smoothed version of the true conditional mean D. The kernel weights are always uniformly distributed

Q9. In the split proximal-point algorithm for computing \(\hat{\mu}_h^{\mathrm{NW}}(x)\), each step moves the current iterate toward a single response \(Y_i\) along a geodesic. The fraction \(t_i\) of the geodesic traversed is:

A. \(t_i = w_{in}^{\mathrm{NW}}(x, h)\) (independent of the step-size parameter \(\lambda\)) B. \(t_i = 1\) (move all the way to the data point \(Y_i\)) C. \(t_i = \dfrac{2\lambda\, w_{in}^{\mathrm{NW}}(x, h)}{1 + 2\lambda\, w_{in}^{\mathrm{NW}}(x, h)}\) D. \(t_i = \dfrac{\lambda}{1 + \lambda}\) (independent of the kernel weight)

Q10. In kernel Fréchet regression, how does the bandwidth \(h\) affect which observations contribute to the estimator at a given evaluation point \(x\)?

A. All \(n\) observations always contribute equally, regardless of \(h\) B. Smaller \(h\) concentrates weight on fewer observations near \(x\), reducing the effective local sample size C. Larger \(h\) reduces the effective sample size by shrinking all kernel weights toward zero D. The effective sample size depends only on the response space dimension, not on \(h\)