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REVIEW 4 major objections 5 minor 38 references

Regression Models Using Shapes of Functions as Predictors

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Aligning predictors inside the regression improves prediction.

desk verdict A useful elastic functional regression model whose real-data advantage is likely inflated by test-set selection of the index polynomial degree. read the letter →

arxiv 1909.02426 v3 pith:X6VYS6WY submitted 2019-09-05 stat.ME stat.CO

Kyungmin Ahn , J. Derek Tucker , Wei Wu , Anuj Srivastava This is my paper · ORCID
classification stat.MEstat.CO MSC 62R1062G08
keywords functionaldataanalysisscalar-on-functionregressionelasticmodelphasevariabilityfunctionalignmentSRVFsingle-indexnorm-preservingwarping
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes elastic functional regression (EFRM), a scalar-on-function model in which the usual inner product $\langle \beta, f_i \rangle$ is replaced by a supremum over time warpings of the predictor, wrapped in an unknown link function $h$. The warpings are norm-preserving, so the optimization avoids the pinching degeneracy of ordinary $L^2$ alignment. Because the supremum absorbs reparameterizations, the conditional mean of the response becomes invariant to this kind of predictor phase. The paper reports lower prediction RMSE than FLM, pre-aligned FLM, and kernel nonparametric baselines on simulated data and on gait, NMR, and stock-price datasets. The central point is that phase separation should be estimated jointly with the regression, using the response values, rather than as an unsupervised preprocessing step.

What carries the argument

The central object is the norm-preserving warping action $f_i * \gamma_i = (f_i \circ \gamma_i)\sqrt{\dot{\gamma}_i}$ on $L^2$ functions, with $\gamma_i$ ranging over boundary-preserving diffeomorphisms of $[0,1]$. This is the same action that the square-root velocity function representation induces on SRVFs, but applied directly to the observed functions, so it preserves the $L^2$ norm of each predictor and prevents the pinching effect that makes naive $L^2$ alignment degenerate. The model's key move is to put the supremum over this warping group inside the regression inner product, so that the response is a function of the best-aligned inner product rather than of the raw predictor. Estimation alternates between optimizing the coefficient function $\beta$, with dynamic programming over $\gamma_i$ for each candidate $\beta$, and fitting a low-degree polynomial link $h$.

What would settle it

Generate a test set from a value-preserving phase model, $f_i(t) = f_i^0(t + \delta_i)$ with random shifts $\delta_i$ and responses depending only on the unshifted shape, then fit EFRM and compare its prediction RMSE with PAFLM and FLM. If EFRM does not at least match the alternatives under this contamination, the phase-invariance claim holds only for norm-preserving warps, not general phase noise. A companion check is the model's own invariance: for any predictor $f$ and warp $\gamma$, the fitted EFRM prediction should be unchanged between $f$ and $f * \gamma$.

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Extended reading notes

Core claim

The central claim is that a regression model of the form $y_i = h(\sup_{\gamma_i \in \Gamma} \langle \beta, (f_i \circ \gamma_i)\sqrt{\dot{\gamma}_i} \rangle) + \epsilon_i$, with $\Gamma$ the boundary-preserving diffeomorphisms of $[0,1]$, makes scalar responses depend only on the shape of the predictor function and not on its phase, and that fitting the model this way predicts better than regressing on unaligned or pre-aligned functions. The paper argues that the norm-preserving action $f_i * \gamma_i = (f_i \circ \gamma_i)\sqrt{\dot{\gamma}_i}$ inherits the good properties of the SRVF/Fisher-Rao approach while avoiding the pinching effect and without amplifying noise through derivatives. Under this model, the response mean is invariant to reparameterization of the predictor, and $\beta$ is identifiable only up to composition with a warp; the paper resolves this by constraining the average estimated warp to be the identity. Empirically, EFRM gives the lowest root-mean-square prediction error among the models it compares on two simulated settings and three real datasets.

Load-bearing premise

The method assumes that the only phase variability worth removing is a norm-preserving diffeomorphic reparameterization, so that a predictor's shape is the same before and after warping by $(f \circ \gamma)\sqrt{\dot{\gamma}}$; if real phase acts by value-preserving shifts $f \circ \gamma$ or by non-diffeomorphic changes, the warping will change peak heights and the model may not recover the intended signal.

Editorial extensions

If this is right

  • If EFRM is right, scalar-on-function regression can be made phase-invariant without a separate alignment step, and the alignment is informed by the response values rather than by unsupervised matching.
  • The norm-preserving formulation avoids the pinching degeneracy, so no ad hoc restriction of the warping set is needed to make the supremum well behaved.
  • Prediction for a new function only requires aligning it to the estimated $\beta$, so phase variability in future observations is absorbed at prediction time.
  • On the three real datasets studied, the reported RMSE of EFRM is below FLM, PAFLM, and kernel approaches, suggesting the gain is not limited to data generated by the model itself.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same 'warp inside the loss' construction could extend to other group actions, such as area-preserving warping for density-valued predictors; the paper mentions other warpings are possible but does not develop or test them.
  • Because EFRM discards phase entirely, data where timing carries signal would be better served by a two-index model that includes both the aligned inner product and a phase distance; the paper notes phase may be informative but does not build such a model.
  • The identifiability gauge chosen here, averaging the estimated warpings to the identity, is one of several possible constraints; other gauges could produce different $\beta$ estimates, and the paper does not compare them.
  • The computational cost of running dynamic programming inside every coefficient update may limit EFRM to one-dimensional, short curves; scaling to images or multivariate functional predictors is an open extension.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This paper proposes an elastic functional regression model (EFRM) for scalar-on-function regression when predictor functions contain phase variability. The model replaces the L2 inner product in functional linear regression by sup_{γ∈Γ} ⟨β, (f_i*γ)⟩, where f_i*γ = (f_i∘γ)√(γ̇) is a norm-preserving warping, and applies an unknown link function h. Estimation proceeds by alternating estimation of β (via a basis expansion and dynamic programming over γ) and h (a low-order polynomial), with an average-warping constraint for identifiability. The method is compared with FLM, pre-aligned FLM, and kernel nonparametric regression on two simulations and three real datasets (gait, 1H-NMR, stock).

Significance. If properly validated, the idea of folding alignment into the regression model via a norm-preserving group action is a useful contribution to functional regression: it avoids the derivative-based noise amplification of SRVF preprocessing and provides a unified objective for prediction and phase removal. The algorithm is clearly described and reproducible code is provided. However, the current empirical support is not sufficient: the real-data results are based on a single test split with model selection on the test set, baselines are not regularized for small samples, and the simulations only exercise the model's own data-generating mechanism. The model also lacks a rigorous identifiability analysis. These gaps currently prevent acceptance.

major comments (4)
  1. [Section 3.2, Table 4] The EFRM row reports the minimum RMSE among three polynomial degrees for h, and the narrative selects the degree by looking at test-set performance (e.g., 'cubic index function does the best'). This uses the test responses for model selection, so the comparison is biased in favor of EFRM; the small margins over NP-shape (Gait 2.466 vs 2.625; Stock 9.007 vs 9.540; NMR 4.025 vs 4.251) could plausibly disappear under an honest protocol. The analysis should use nested cross-validation or repeated splits with h selected on training data only, and should report standard errors or intervals.
  2. [Section 3.2 and Tables 2-4] The FLM and PAFLM baselines are fit with 20 B-spline basis functions on training samples of size 21-140 with no penalization or basis truncation. This is likely to overfit and produce inflated RMSE (e.g., NMR FLM RMSE 213), making the comparison unfair. A regularized FLM (e.g., penalized least squares or functional principal components regression with tuning by CV) should be used.
  3. [Section 3.1] Both simulation experiments generate predictors by applying the same norm-preserving warping used in EFRM (Eq. 5), so the simulation demonstrates that the estimator recovers the model when the model is true, but it does not test the robustness claim to value-preserving phase variability that motivates the paper in Section 1.1. The authors should add simulations where contamination follows value-preserving warpings (or other misspecified actions) and evaluate how EFRM and the baselines degrade.
  4. [Section 2.1, item 3] The identifiability constraint (1/n)Σγ̂_i = γ_id is introduced without proof or discussion of whether it is sufficient. Since the sup-term is invariant to pre-warping of the predictors, β is not identifiable without additional structure; the paper should either prove that the constraint identifies β up to the intended equivalence or acknowledge the limitation and examine sensitivity of predictions to the constraint.
minor comments (5)
  1. [Section 2.1] The word 'diffeomoprhism' is misspelled; 'boundary preserving diffeomoprhisms' should be 'diffeomorphisms'.
  2. [Section 3.2] The 1H-NMR response is described as categorical with four values, but the model assumes Gaussian errors; the authors should discuss whether a linear/Gaussian model is appropriate.
  3. [Algorithm 1, step 4] The instruction 'replace β by β∘γ̄^{-1}' is ambiguous; it should clarify whether this is the ordinary composition or the norm-preserving action (β*γ̄^{-1}).
  4. [Section 1.1, Fig. 2] The caption describes the second contamination as 'area preserving' but the displayed formula (f_i∘γ_i)√(γ̇_i) is the norm-preserving action; the caption conflates area- and norm-preserving warpings.
  5. [Section 3.1.1] The text uses 'root squared error' for RSE; it should be 'root integrated squared error' or the acronym should be defined explicitly.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: EFRM is a proposed model estimated on training data and evaluated on held-out predictors; the real-data h-selection caveat is a validation issue rather than a circular step.

full rationale

The central model in Eq. (5), y_i = h(sup_{gamma_i} <beta, (f_i * gamma_i)>) + epsilon_i, is explicitly proposed as a modeling choice rather than derived from a fitted quantity. Estimation in Algorithm 2 fits beta and h on training data by least squares, and prediction in Eq. (6) applies the fitted model to held-out predictors; no fitted parameter is identical by construction to the predicted response. The SRVF/Fisher-Rao background is cited from the authors' prior work [16,22], but the cited invariance property is a mathematical theorem with independently checkable content and is not used as a uniqueness theorem to forbid alternatives. The simulation study generates data from the same model, and the paper itself acknowledges this: "A part of the success of EFRM can be attributed to the fact that the data was indeed simulated from that model itself" (Section 3.1.2). That is a consistency check, not a circular derivation. The real-data comparison in Table 4 has a protocol weakness: the degree of h is effectively chosen after inspecting test RMSE, e.g. "In the case of 1H-NMR data, EFRM using a cubic index function does the best," so the test set is used for model selection and the reported margins are small. This weakens external evidence but does not make the model's derivation circular, because the EFRM equation does not reduce to its own fitted inputs. Overall, no specific circular step is exhibited, so the circularity score is low; the minor caveats affect evidence strength rather than structural circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The model rests on the SRVF/elastic framework from the authors' previous work, a proposed response model, and an identifiability constraint that is asserted rather than proven. The free parameters are the basis size, polynomial degree of h, and the per-observation warpings; none are derived from first principles.

free parameters (3)
  • Basis size J for beta = 20 (real data); 2, 4, or 20 (simulations)
    Beta is restricted to a finite basis; the choice of J and basis type (Fourier vs B-spline) affects estimates, with no data-driven selection or regularization provided (Section 2.2).
  • Polynomial degree of h = 1, 2, or 3; cubic chosen for NMR
    The link function is chosen among linear, quadratic, and cubic polynomials; real-data comparisons select the best degree post hoc, which inflates apparent performance (Section 2.2, Table 4).
  • Per-observation warping functions gamma_i = Estimated via DPA for each i
    Each predictor gets its own optimal warping, optimized jointly with beta and h; these are high-dimensional latent parameters, identifiable only under the average-gamma equal identity constraint (Algorithm 1).
assumptions (4)
  • domain assumption The Fisher-Rao metric and SRVF framework provide an isometric, pinching-free action for warping functions.
    The paper relies on the SRVF framework from [16,22] to justify replacing the L2 inner product with a supremum over Gamma (Sections 1.3 and 2.1).
  • domain assumption The response follows y_i = h(sup_gamma <beta, f_i * gamma>) plus i.i.d. Gaussian noise.
    This is the proposed model specification in Eq. 5; it is not derived from data or from a more basic principle.
  • ad hoc to paper The constraint (1/n) sum gamma_hat_i = gamma_id is sufficient to identify beta.
    Section 2.1 property 3 imposes this constraint to resolve the ambiguity that beta and beta composed with gamma are equivalent, but no proof of identifiability is provided.
  • domain assumption Phase variability in predictors is a random diffeomorphic time warping of the domain.
    The supremum over Gamma and the simulation generative model both assume phase is a boundary-preserving diffeomorphism (Sections 1.1 and 2.1).

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Cite this review

Pith. "Pith review of Regression Models Using Shapes of Functions as Predictors." pith.science (2026). https://pith.science/paper/X6VYS6WY

@misc{pith2026190902426,
  author       = {Pith},
  title        = {Pith review of: Regression Models Using Shapes of Functions as Predictors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/X6VYS6WY}},
  note         = {Machine review of arXiv:1909.02426}
}
abstract

Functional variables are often used as predictors in regression problems. A commonly-used parametric approach, called {\it scalar-on-function regression}, uses the $\ltwo$ inner product to map functional predictors into scalar responses. This method can perform poorly when predictor functions contain undesired phase variability, causing phases to have disproportionately large influence on the response variable. One past solution has been to perform phase-amplitude separation (as a pre-processing step) and then use only the amplitudes in the regression model. Here we propose a more integrated approach, termed elastic functional regression model (EFRM), where phase-separation is performed inside the regression model, rather than as a pre-processing step. This approach generalizes the notion of phase in functional data, and is based on the norm-preserving time warping of predictors. Due to its invariance properties, this representation provides robustness to predictor phase variability and results in improved predictions of the response variable over traditional models. We demonstrate this framework using a number of datasets involving gait signals, NMR data, and stock market prices.

Figures

Figures reproduced from arXiv: 1909.02426 by the authors.

Figure 1
Figure 1. Example of functional data with and without phase variability. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Experiments show superlinear decrease in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Simulated data 1. (a) shows the original functions, [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The evolution of cost H for each choice of the index function, h, and using Fourier basis with two elements for β. It is also important to quantify estimation performance for model parameters β and h. In order to quantify these errors, we calculate the Root Squared Err…
Figure 5
Figure 5. Figure 5: Gait in Parkinson’s Disease Data [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: Metabonomic 1H-NMR Data to 8/9/2013. For each company, we collected stock prices from 3/20/2012 to 8/9/2012 to form functional predictors. Thus, there are 100 daily time points over the selected interval forming predictor functions. We take the stock prices on 8/9/2013…
Figure 7
Figure 7. Figure 7: Stock Data [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: {fi} vs. Warped {fi} and {γ ∗ i } Analysis of Real Data. For representing the coefficient function β, we use a B-spline basis with 20 elements and estimate parameters using Algorithm 2 [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]

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