Quiz: Lecture 10 — Logistic Regression with Metric-Space Covariates

This quiz tests your understanding of metric-space logistic regression — binary classification when predictors live in a nonlinear metric space. 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. How does metric-space logistic regression differ from Fréchet regression (Lectures 6–9) in terms of the roles of predictor and response?

A. Neither uses a metric structure — both are purely Euclidean methods B. Both place the metric structure on the response side C. Both place the metric structure on the predictor side D. Fréchet regression places the metric on the response; metric logistic regression places the metric on the covariate (predictor)

Q2. The metric-space logistic model uses the Fréchet mean of the covariates as an anchor point \(\mu^*\). Why is an anchor needed?

A. The anchor is required for computing geodesic distances between covariates B. The anchor is only used for visualization, not for estimation C. Arbitrary linear combinations are not defined in a general metric space, so there is no natural origin — the anchor provides a reference point D. The anchor determines the bandwidth for kernel smoothing

Q3. The Alexandrov inner product \(\langle q, r \rangle_p = d(p, q) d(p, r) \cos\angle_p(q, r)\) generalizes which Euclidean concept to metric spaces?

A. The ordinary Euclidean inner product \((q - p)^\top(r - p)\) B. The cross product \((q - p) \times (r - p)\) C. The Euclidean norm \(\|q - p\|\) D. The matrix determinant

Q4. In the metric-space logistic model, the geodesic from \(\mu^*\) to \(\beta^*\) represents:

A. The trajectory of the LIPO algorithm B. A discriminating direction — points whose geodesic from \(\mu^*\) projects positively onto this direction have higher log-odds of \(Y = 1\) C. The shortest path between any two data points D. The path along which the Fréchet mean is computed

Q5. Estimation in metric-space logistic regression proceeds in two stages. They are:

A. Stage 1: fit a linear model; Stage 2: apply a nonlinear transformation B. Stage 1: cluster the data; Stage 2: classify each cluster C. Stage 1: compute kernel weights; Stage 2: apply gradient descent D. Stage 1: estimate anchor \(\hat{\mu}\) via Fréchet mean; Stage 2: estimate coefficient \(\hat{\beta}\) via MLE

Q6. The generalized LIPO (Lipschitz global optimization) algorithm is used because:

A. The objective has a known closed-form maximizer B. Gradient descent fails in all metric spaces C. The log-likelihood \(L_n\) is typically neither convex nor differentiable in any linear sense on a general metric space D. Newton’s method is always faster but harder to implement

Q7. In the LIPO algorithm, a proposal \(\beta\) is accepted only if its Hölder upper envelope can still beat the current best value. What happens if the Hölder constant \(K\) is chosen too small?

A. The envelope is too tight — proposals that could improve on the current best are rejected, risking premature convergence B. The algorithm runs faster but finds the same optimum C. Almost every proposal is accepted — the algorithm becomes uniform random search D. The algorithm requires more memory to store additional proposals

Q8. In the fMRI application, the fitted geodesic reveals that the positive direction (toward language task) is associated with increased connectivity between which brain regions?

A. All brain regions equally B. The method cannot identify specific regional connections C. Primary motor cortex and visual cortex D. Superior temporal gyrus (auditory/language) and inferior frontal gyrus (speech/language)

Q9. On the sphere \(\mathbb{S}^2\), the decision boundary of the metric-space logistic classifier is:

A. A great circle orthogonal to the discriminating direction at the anchor B. A single point (the anchor) C. The entire sphere — no discrimination is possible D. A small circle parallel to the equator

Q10. How does the portfolio regime classification application complement the Fréchet regression framework from earlier lectures?

A. It only works for Euclidean data, unlike Fréchet regression B. It reverses the roles: instead of predicting covariance from a scalar stress indicator, it classifies the regime from the covariance matrix itself C. It uses the same methods with no changes D. It replaces all previous methods with a neural network