Quiz: Lecture 6 — Global Fréchet Regression

This quiz tests your understanding of global Fréchet regression — extending linear regression 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 key insight that enables extending linear regression to metric-space responses is:

A. Linear regression can be rewritten as a weighted least-squares problem where the response enters only through squared distance B. Linear regression requires the response space to be a vector space C. The predictor must be transformed to match the metric of the response space D. The intercept term is unnecessary in metric-space regression

Q2. The global Fréchet regression weight function is \(s(z, x) = 1 + (z - \mathbb{E}X)^\top\Sigma^{-1}(x - \mathbb{E}X)\). What is \(\mathbb{E}[s(X, x)]\) for any fixed \(x\)?

A. 0 B. 1 C. \(x - \mathbb{E}X\) D. \(\operatorname{var}(X)\)

Q3. At \(x = \mathbb{E}X\), the global Fréchet regression curve must pass through:

A. The origin of the metric space B. The unconditional Fréchet mean of the responses C. The point farthest from all data D. The mode of the predictor distribution

Q4. The sample global Fréchet regression estimator \(\hat{\mu}(x)\) is computed as:

A. A weighted Fréchet mean of the responses with weights \(s_{in}(x)\) B. The unweighted Fréchet mean of all responses C. A kernel-weighted average of the predictor values D. The projection of \(\mathbb{E}Y\) onto the tangent space at \(x\)

Q5. The weight function \(s(z, x)\) can produce negative weights. What does this signify?

A. An error in the computation — weights must always be positive B. Extrapolation beyond the range of observed \(X\) values, reflecting the global linear model C. That the Fréchet mean does not exist at that \(x\) D. That the predictor distribution is non-Gaussian

Q6. The curvature condition (P2) in the convergence theory states that near the minimizer, \(M(\omega, x) - M(\mu(x), x) \ge C\, d(\omega, \mu(x))^{\beta}\). What does \(\beta = 2\) mean geometrically?

A. The Fréchet function has a sharper-than-quadratic minimum B. The squared distance grows quadratically near the minimum — the same as in Euclidean space C. The metric space is locally flat but globally curved D. The convergence rate is slower than \(n^{-1/2}\)

Q7. For SPD spaces with the log-Euclidean metric, the convergence rate of the global Fréchet regression estimator is:

A. \(O_p(n^{-1/4})\) B. \(O_p(n^{-1/3})\) C. \(O_p(n^{-1/2})\) (parametric rate) D. \(O_p(n^{-1})\) (super-efficient)

Q8. When \(\mathcal{M} = \mathbb{R}^q\) with the Euclidean metric, global Fréchet regression recovers:

A. The componentwise median of \(Y\) given \(X\) B. The usual multivariate linear regression of \(Y\) on \(X\), componentwise C. A nonparametric kernel regression of \(Y\) on \(X\) D. Logistic regression of \(Y\) on \(X\)

Q9. The entropy condition (P1) controls:

A. The curvature of the metric space at the Fréchet mean B. The local size/complexity of the parameter space near \(\mu(x)\), and therefore the stochastic oscillation of \(M_n - M\) C. The bandwidth selection for kernel regression D. The convergence rate of the proximal-point algorithm

Q10. Under the log-Euclidean metric, the global Fréchet regression estimator on SPD matrices has a closed form because:

A. SPD matrices under the log-Euclidean metric are isometric to a Euclidean space of symmetric matrices B. The Fréchet mean on SPD matrices is always the arithmetic mean C. The weight function \(s(z, x)\) is always 1 on SPD spaces D. SPD matrices form a compact space