Smooth Flow Matching for Synthesizing Functional Data

Referee: [Abstract] Abstract: the claim of demonstrated advantages in 'synthetic data quality and computational efficiency' is not supported by any reported quantitative metrics, error bars, baseline comparisons, or specific results; this absence makes it impossible to evaluate the empirical performance assertions that underpin the method's practical value.

Authors: We acknowledge that the abstract would benefit from more explicit linkage to quantitative results. The simulation studies section already contains comparative experiments, but we agree that tabulated metrics, error bars from repeated runs, and named baselines would improve clarity and verifiability. In the revised manuscript we will add a summary table reporting distributional metrics (e.g., sliced Wasserstein distance) and wall-clock times with standard errors, and we will revise the abstract to reference these concrete findings. revision: yes

Referee: [Copula framework and flow construction] Copula framework and flow construction (likely §3 or §4): the assertion that SFM is 'free of ... low-rank assumptions' is load-bearing for the central claim yet appears to rest on an unverified representational choice; any concrete implementation of the vector field or probability path on functional data requires discretization, basis expansion, or kernel representation, each of which induces an effective finite-dimensional manifold, and no proof or argument is given that the generated measure remains outside all finite-rank subspaces.

Authors: We appreciate this precise observation. SFM is formulated on the infinite-dimensional space of smooth functions: the copula couples marginal distributions defined pointwise on the domain, and the flow-matching vector field is specified via a functional operator that does not presuppose a finite basis truncation of the target measure. Discretization or basis projection occurs only at the numerical integration stage and is an approximation whose resolution can be increased arbitrarily; the model class itself does not restrict generated samples to any fixed finite-rank subspace, in contrast to methods that explicitly truncate a Karhunen–Loève expansion. We will insert a short clarifying paragraph in Section 3 that distinguishes the modeling assumption from the computational discretization and briefly discusses the limiting behavior as the discretization mesh is refined. revision: partial

Referee: [MIMIC-IV application] MIMIC-IV application section: the reported ability to 'produce high-quality surrogate data for downstream tasks' lacks any quantitative assessment of distributional fidelity, smoothness preservation, or comparison against real trajectories or alternative generators, weakening the claim that SFM boosts utility for clinical applications.

Authors: We agree that quantitative support would strengthen the application. The current section emphasizes qualitative visual agreement and downstream-task feasibility, but we will augment it with explicit metrics: integrated squared error for smoothness, empirical distributional distances between real and synthetic trajectories, and head-to-head comparisons against a Gaussian-process baseline and a standard functional PCA generator. These additions will be placed in a new subsection and referenced in the abstract and conclusion. revision: yes