Quiz: Lecture 9 — Total-Variation Regularized Fréchet Regression

This quiz tests your understanding of TV-regularized Fréchet regression — fused-lasso and change-point detection for metric-space trajectories. 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 total variation of a metric-space-valued curve \(\gamma : [a, b] \to \mathcal{M}\) is defined as:

A. \(\mathbb{E}[d(\gamma(X), \gamma(Y))]\) for \(X, Y \sim \text{Uniform}(a, b)\) B. \(\int_a^b \|\gamma'(t)\|\, dt\) (the integral of the derivative) C. \(\sup_{a = s_0 < \cdots < s_m = b} \sum_{j=1}^m d(\gamma(s_j), \gamma(s_{j-1}))\) D. \(\max_{t \in [a,b]} d(\gamma(t), \gamma(a))\)

Q2. The TV-regularized Fréchet estimator minimizes \(L_{\lambda}(\gamma) = \frac{1}{n}\sum_i d^2(\gamma(t_i), Y_i) + \lambda \operatorname{TV}(\gamma)\). What role does \(\lambda\) play?

A. \(\lambda\) is the step size in the proximal-point algorithm B. \(\lambda\) trades off data fidelity against piecewise-constant structure; larger \(\lambda\) encourages fewer jumps C. \(\lambda\) determines the number of data points to include D. \(\lambda\) controls the bandwidth of kernel smoothing

Q3. The step-function representation theorem states that any minimizer of \(L_{\lambda}\) can be replaced by a step function with jumps only at the design points. What does this imply computationally?

A. The infinite-dimensional problem reduces to optimization over \(n\) points in \(\mathcal{M}\) B. The problem cannot be solved computationally C. The estimator requires smoothing at every data point D. The estimator is always constant (no jumps allowed)

Q4. The cyclic proximal-point algorithm for TV-regularized Fréchet regression alternates between two types of updates. They are:

A. Forward and backward passes through the ordered data B. Kernel smoothing and local-linear correction C. Data-fitting steps (moving toward observations along geodesics) and fusion steps (shrinking adjacent fitted values together) D. Gradient descent and Newton steps

Q5. In the fusion step, the parameter \(\theta_j = \min\{\alpha_r n\lambda / (2 d(p_j, p_{j+1})), 1/2\}\) is capped at \(1/2\). Why?

A. To ensure the step size satisfies the Robbins–Monro conditions B. To guarantee convergence within exactly 100 iterations C. Because the geodesic distance cannot exceed \(1/2\) in Hadamard spaces D. To prevent the points from overshooting each other (crossing) and to preserve the firmly nonexpansive property

Q6. The asymptotic convergence rate for TV-regularized Fréchet regression with \(\lambda \asymp n^{-2/3}\) is:

A. \(O_p(n^{-1/3})\) (canonical bounded-variation rate) B. \(O_p(n^{-1/2})\) (parametric rate) C. \(O_p(n^{-1})\) (super-efficient) D. \(O_p(n^{-2/5})\) (nonparametric smoothing rate)

Q7. For the TV-regularized Fréchet estimator \(L_{\lambda}(\gamma) = \frac{1}{n}\sum_i d^2(\gamma(t_i), Y_i) + \lambda \operatorname{TV}(\gamma)\), what happens as \(\lambda \to \infty\)?

A. The fitted curve converges to the original noisy data points (no regularization) B. The fitted curve converges to the Fréchet mean of all observations — a single constant, since total variation is driven to zero C. The fitted curve converges to a piecewise-linear interpolation of the data D. The fitted curve coincides with the local-linear Fréchet estimator

Q8. In the fMRI dynamic functional connectivity application, how is \(\lambda\) chosen?

A. \(\lambda\) is fixed at 1 for all applications B. By cross-validation on held-out subjects C. By Akaike Information Criterion (AIC) D. By selecting the smallest \(\lambda\) that yields the expected number of jumps (matching known stimulus transitions)

Q9. Under the log-Euclidean metric on SPD matrices, TV-regularized Fréchet regression simplifies to:

A. Entrywise classical fused lasso (1D total-variation denoising) on the matrix logarithms B. A geodesic convex optimization problem requiring Riemannian gradient descent C. A non-convex problem that cannot be solved exactly D. A kernel regression with a special choice of bandwidth

Q10. When would you prefer TV-regularized Fréchet regression over local-linear Fréchet regression for portfolio risk analysis?

A. When the predictor is high-dimensional B. When market dynamics exhibit abrupt regime changes (jumps) — TV preserves sharp transitions that smoothing would blur C. When computational efficiency is the primary concern D. When the covariance structure evolves smoothly with market conditions