Quiz: Lecture 8 — Local Fréchet Regression

This quiz tests your understanding of local-linear Fréchet regression — extending local polynomial smoothing to metric-space responses. 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 primary motivation for local-linear Fréchet regression over local-constant (kernel) regression is:

A. Faster computation — local-linear requires fewer iterations B. The ability to handle categorical predictors C. Correction of \(O(h)\) boundary bias to \(O(h^2)\) everywhere D. Guaranteed nonnegative weights at all evaluation points

Q2. The local-linear weight function \(s(z, x, h)\) satisfies two key properties that enable the metric-space generalization. They are:

A. \(\mathbb{E}[s(X, x, h)] = 1\) and \(\mathbb{E}[s(X, x, h)(X - x)] = 0\) B. \(s(z, x, h) \ge 0\) and \(\int s(z, x, h) dz = 1\) C. \(s(z, x, h) = 1\) and \(s(z, x, h)(z - x) = 0\) for all \(z\) D. \(\mathbb{E}[s(X, x, h)] = 0\) and \(\operatorname{var}(s(X, x, h)) = 1\)

Q3. The property \(\mathbb{E}[s(X, x, h)(X - x)] = 0\) means that the local-linear weights are:

A. Always positive for data near \(x\) B. Identical to the global Fréchet regression weights C. Independent of the predictor distribution D. Orthogonal to linear trends — the estimator is exactly unbiased for linear conditional mean functions

Q4. Why can local-linear weights \(s_{in}(x, h)\) be negative?

A. This is a bug — weights should always be normalized to be positive B. Negative weights indicate that the Fréchet mean does not exist C. Near boundaries, \(\hat{\nu}_2 - \hat{\nu}_1(X_i - x)\) can become negative for faraway points, extrapolating the local slope D. Negative weights only occur when the bandwidth is chosen incorrectly

Q5. For the quadratic-curvature case \(\beta_1 = \beta_2 = 2\), the optimal bandwidth and convergence rate are:

A. \(h \asymp n^{-1/5}\) with rate \(n^{-2/5}\) B. \(h \asymp n^{-1/3}\) with rate \(n^{-1/3}\) C. \(h \asymp n^{-1/2}\) with rate \(n^{-1/2}\) D. \(h \asymp n^{-1}\) with rate \(n^{-1}\)

Q6. The local moments \(\nu_j(x, h) = \mathbb{E}[K_h(X - x)(X - x)^j]\) for \(j = 0, 1, 2\) are used to:

A. Estimate the curvature of the metric space B. Determine the optimal number of iterations for gradient descent C. Compute the geodesic distance between data points D. Construct the local-linear weight function from the normal equations of local polynomial regression

Q7. In the stress-testing application, why is local-linear preferred over kernel regression at extreme stress levels (\(x\) near boundaries)?

A. Kernel regression extrapolates linearly beyond the boundary B. Local-linear regression is always faster computationally C. Kernel regression at boundaries averages only one-sided data, pulling estimates inward — local-linear corrects this via slope extrapolation D. Kernel regression cannot be applied to SPD matrices

Q8. The convergence rate \(d(\mu(x), \hat{\mu}(x)) = O_p(h^{2/(\beta_1-1)} + (nh)^{-1/[2(\beta_2-1)]})\) generalizes the classical bias–variance decomposition. What do \(\beta_1\) and \(\beta_2\) represent?

A. \(\beta_1\) is the curvature exponent of the population Fréchet function (bias); \(\beta_2\) is the curvature exponent of the empirical objective (variance) B. \(\beta_1\) is the bandwidth exponent; \(\beta_2\) is the kernel order C. \(\beta_1\) is the predictor dimension; \(\beta_2\) is the response dimension D. \(\beta_1 = \beta_2 = 1\) always, by definition

Q9. How does the weight function \(s(z, x, h)\) generalize from Euclidean to metric-space local Fréchet regression?

A. It is recomputed using geodesic distances instead of Euclidean distances B. It remains unchanged — computed entirely from the scalar predictors, then used as weights in the metric-space Fréchet mean C. It must be recomputed using the exponential map at each evaluation point D. It is replaced by a uniform weight function

Q10. All three Fréchet regression methods (global, kernel, local-linear) share the same computational primitive under the log-Euclidean metric on SPD matrices. What is it?

A. Riemannian gradient descent on the SPD manifold B. A weighted average of matrix logarithms, \(\log \hat{\mu}(x) = \sum_i w_i(x) \log \Sigma_i\) C. The proximal-point algorithm with geodesic interpolation D. Newton’s method on the tangent space