Numerical differentiation of noisy data: A unifying multi-objective optimization framework

The paper introduces the loss L = RMSE(trapz(ẋ̂(Φ)) + μ, y) + γ TV(ẋ̂(Φ)) for parameter selection without ground truth and specifies μ via Eq. (6) and γ via an empirical heuristic (Eq. (8)) while advocating TVRJ solved in sliding windows and practical optimizers such as CVXPY/MOSEK . The candidate solution formalizes and proves key steps: (i) derives the unique optimal μ as the sample mean residual; (ii) gives mild conditions guaranteeing existence of a minimizer over Φ; (iii) writes an explicit convex TVRJ program with KKT conditions and existence; and (iv) interprets the paper’s heuristic log γ = −1.6 log f − 0.71 log dt − 5.1 (Eq. (8)) . There is a small sign/normalization typo in the paper’s μ definition (Eq. (6)) that the model corrects; otherwise, the paper’s framework and the model’s derivations agree in substance.