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REVIEW 3 major objections 6 minor 37 references

Scalar-on-Shape Regression Models for Functional Data Analysis

T0 review · 3 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Scalar responses can be predicted from functional shape alone—invariant to time warping—via a Fisher-Rao inner product maximized over warpings; the fit defines regression phases and a regression mean.

desk verdict A useful phase-invariant functional regression framework, but the estimation algorithm's final centering step uses ordinary composition instead of the SRVF action, so as written the reported β is not the fitted β. read the letter →

arxiv 2411.15326 v1 pith:VIDR6YTP submitted 2024-11-22 stat.ME

Sayan Bhadra , Anuj Srivastava This is my paper · ORCID
classification stat.ME MSC 62R1062G08
keywords functionaldataanalysisshaperegressionscalar-on-shapemodelFisher-Raometricsquare-rootvelocityfunctionphase-amplitudeseparationsingle-indexmean
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

Functional data such as daily COVID curves mixes two components: shape, the timing-normalized pattern of peaks and valleys, and phase, the time warping that moves those peaks left or right. This paper tries to establish that scalar responses can be predicted from shape alone by building time warping into the regression model instead of removing it beforehand. The key move is to replace the ordinary inner product with the Fisher-Rao inner product on square-root velocity functions, so the alignment term $\sup_{\gamma_i} \langle \beta, q_i \star \gamma_i \rangle$ is provably invariant to replacing a predictor by any time-warped version of itself. When that works, the fitted coefficient $\beta$ acts as a regression mean—a shape whose alignment with each predictor best explains the response—and the optimal warpings are regression phases. The method delivers high predictive accuracy on simulated and real data when phases are uninformative, and the paper is explicit that discarding informative phases hurts.

What carries the argument

The machinery is the phase-invariant supremum of a Fisher-Rao inner product, made computable by the square-root velocity function $q = \text{sign}(f')\sqrt{|f'|}$. Because the Fisher-Rao metric satisfies $\langle q_1, q_2 \rangle = \langle q_1 \star \gamma, q_2 \star \gamma \rangle$ for every warping $\gamma$, the value $\sup_{\gamma \in \Gamma} \langle \beta, q_i \star \gamma \rangle$ depends only on the orbit of $q_i$ under $\Gamma$—that is, only on the shape of $f_i$. The supremum itself is the registration step: it searches over warpings to align each predictor to the current $\beta$, and dynamic programming makes that search feasible. The index function $h$ then wraps the aligned inner product into a nonlinear predictor-response map, and the offset $g(f_i(0))$ absorbs baseline differences.

What would settle it

Generate responses from two shape indices, $y_i = h_1(\sup_{\gamma} \langle \beta_1, q_i \star \gamma \rangle) + h_2(\sup_{\gamma} \langle \beta_2, q_i \star \gamma \rangle)$ with $\beta_1, \beta_2$ orthogonal, and fit the single-index SI-ScoSh model; if out-of-sample $R^2$ falls well below a two-index fit, the single-index structure is falsified. A quicker check already appears in the paper: data generated with informative phases give ScoSh test $R^2$ below zero, confirming the model is only sound when phases are genuine nuisances.

Watch

Extended reading notes

Core claim

The paper's central object is the single-index scalar-on-shape model $y_i = g(f_i(0)) + h(\sup_{\gamma_i \in \Gamma} \langle \beta, q_i \star \gamma_i \rangle) + \epsilon_i$, with $q_i$ the square-root velocity function of the predictor $f_i$ and $\Gamma$ the group of boundary-preserving warpings. The Fisher-Rao invariance $\langle q_1, q_2 \rangle = \langle q_1 \star \gamma, q_2 \star \gamma \rangle$ makes the supremum independent of the phase of $f_i$, so the response is a function of the shape class $[f_i]$ alone. Identifiability is restored by constraining $h$ and $g$ and by centering $\beta$ so the average of the optimal warpings is the identity. Estimation alternates dynamic-programming alignment of each $q_i$ to the current $\beta$ with least-squares updates of $\beta$, $h$, and $g$, and bootstrap experiments show the true parameters fall inside 95% intervals. The authors call the solution $\beta$ the regression mean and the optimizing warpings the regression phases, and the multiple-index version fits residuals iteratively.

Load-bearing premise

The load-bearing premise is that every relevant shape feature of a predictor enters the response through one scalar index, $h(\sup_{\gamma} \langle \beta, q_i \star \gamma \rangle)$, so shape information orthogonal to a single aligned coefficient $\beta$ is assumed irrelevant; the paper's own experiments show the model loses predictive power when phase carries real signal.

Editorial extensions

If this is right

  • Registration moves inside the regression: estimated warpings are not a preprocessing choice but part of the fit, so results are invariant to how the input curves were initially parameterized.
  • The estimated $\beta$ can be displayed as a curve, giving practitioners an interpretable regression mean shape that shows which timing-independent features drive the response.
  • In applications where phase is noise, such as pandemic waves with country-specific timings, shape-only prediction should outperform raw-function regression; the paper reports test $R^2$ above 0.92 on COVID hospitalization data and 0.89 on infection data.
  • With a nonlinear index $h$, the model captures curvature in the shape-response relation; simpler linear versions lose accuracy as the true $h$ becomes more complex.
  • Iterating the single-index fit on residuals yields a multiple-index ScoSh model, improving rainfall prediction from $R^2 = 0.39$ to $0.74$, so a small number of shape indices may suffice.

Reading between the lines

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

  • A natural extension the paper leaves implicit: the same supremum machinery could feed the fitted regression phases back as predictors, turning phase information into a covariate when it is informative rather than discarding it.
  • The single-index restriction is a strong structural assumption; comparing SI-ScoSh against a nonparametric shape regression would test whether one aligned index is sufficient or merely convenient.
  • The centering condition on the average of optimal warpings suggests a connection to means on the warping group; one could test estimator stability by replacing the Fourier basis with data-adaptive principal components and checking whether the shape estimate of $\beta$ changes.
  • Since the supremum is computed by dynamic programming at each iteration, the method's scalability hinges on that step; an obvious test is to benchmark runtime and accuracy on long curves against a pre-registered scalar-on-function baseline.
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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

3 major / 6 minor

Summary. The paper proposes a scalar-on-shape (ScoSh) regression model in which a scalar response is regressed on the shape of a functional predictor, with phase variability removed inside the regression model. The model replaces the usual L2 inner product with the phase-invariant supremum term sup_γ <β, q_i ⋆ γ>, where q_i is the square-root velocity function (SRVF) of the predictor and ⋆ is the SRVF warping action. A single-index extension (SI-ScoSh) introduces a nonlinear link h and an offset g, and the paper introduces the concepts of regression phase and regression mean, defined through the optimizing warpings and the estimated coefficient β. Estimation is performed by a coordinate-descent algorithm with dynamic programming for the warpings, and the paper reports simulation studies and several real-data applications (Spanish weather, COVID hospitalization, COVID infection, Australian rainfall). The paper also sketches a multiple-index extension and proposes bootstrap-based uncertainty evaluation.

Significance. If the method is technically sound, the paper makes a useful contribution to functional data analysis: it offers a principled way to regress scalars on function shapes without a separate registration step, and the phase-invariant Fisher-Rao/SRVF construction is appropriate. The paper contains substantial experimental work, including comparisons with functional linear models and single-index models, and it explicitly studies the important distinction between informative and uninformative phases. The bootstrap analysis in Section 2.4 is a good empirical check, and the real-data examples address practically relevant problems. However, the paper's central inferential claims depend on identifiability and estimation details that are not fully established, and at least one step of the proposed algorithm appears inconsistent with the model's own invariance argument.

major comments (3)
  1. [Section 2.2, Items 3-5] The centering step in Algorithm 1, Step 9 uses ordinary composition, bβ = bβ ∘ γ̄^{-1}, but the invariance property used to define the equivalence class of β is the SRVF action: sup_γ <β ⋆ γ0, q_i ⋆ γ> = sup_γ <β, q_i ⋆ γ>. These two transformations differ in general, and ordinary composition does not preserve the supremum scores used in the fitted model. As written, after Step 9 the reported β is not the function whose sup scores were used to fit h and to run the coordinate descent, so the final model and the claimed 'regression mean' are not well-defined. This is load-bearing because the regression mean and regression phase are stated as central contributions. Please correct Step 9 to use the SRVF action (or state explicitly that β is an SRVF and that centering is performed by β ⋆ γ̄^{-1}), and verify that all reported estimates, bootstrap intervals, and real-data results use the corrected definition.
  2. [Section 2.2, Items 3-5] The identifiability constraints are imposed but not justified. The condition (1/n) Σ_i γ_i = γ_id is a nonlinear equation in β because each γ_i is defined as argmax_γ <β, q_i ⋆ γ>; no existence or uniqueness proof is given, and it is not shown that every orbit under ⋆ contains exactly one β satisfying this condition. Similarly, the normalization of h by fixing its highest coefficient is only meaningful for a polynomial parameterization, and the paper does not state how the general continuous h is identified, nor does it discuss the joint additive ambiguity between h and g except for the g(0)=0 constraint. These are not merely technicalities: Section 2.2 claims that 'with these constraints, the model is fully specified, and the parameters are well-defined.' Please either provide a rigorous identifiability argument or explicitly weaken the claim to say that the algorithm selects a representative of an equivalence class.
  3. [Algorithms 1-2 and Section 2.4] The estimation procedure is heuristic: Algorithm 1 steps from 3 to 7 use dynamic programming, continuous optimization, and a stopping rule based only on whether H is 'large,' with no proof that the alternating updates decrease the objective or converge to a stationary point. The bootstrap analysis in Section 2.4 is purely empirical and covers only one simulation setting; it does not establish consistency or asymptotic validity of the bootstrap. The real-data sections report point predictions (R2 values) without error bars or intervals, so it is unclear whether the reported advantages over competing models are stable. This is a load-bearing issue because the paper's practical claims rest on the estimated parameters and their uncertainty. Please add convergence diagnostics, a clearly specified stopping criterion, and uncertainty quantification for the real-data results, or explicitly state that the method is heuristic and the real-data comparisons are exploratory.
minor comments (6)
  1. [Algorithm 1, Step 2] The initialization line reads 'Compute initial bβ(t) = Σ_{j=1}^J c_j,' which omits the basis functions b_j(t); it should be corrected to include them.
  2. [Section 2.4] There is a typo: 'haevn' should be 'have' in the first sentence of Section 2.4.
  3. [Equation (5)] The inner product notation in the displayed objective is broken: the expression should read 〈Σ_{j=1}^J c_j b_j, q_i ⋆ γ_i〉 with closing parentheses in the correct places.
  4. [Section 3.2 and Tables 5-6] The text states that the shape metric d_s, rather than RMSE, is used to evaluate estimation error for β, but Tables 5 and 6 label the error as RMSE and do not define the metric. Please make the table labels and captions consistent with the text.
  5. [Section 3.3] The text contains the typo 'ScoSch' instead of 'ScoSh' in the sentence 'The high performance of the ScoSch model underscores its invariance.'
  6. [Section 4] The real-data sections refer to detailed tables in the Supplementary Material, but this preprint does not include the supplementary material; without those tables, the real-data results cannot be fully checked. Please include the supplementary tables or summarize the additional results in the main text.

Circularity Check

1 steps flagged · score 1.0 of 10

No material circularity; the only mild issue is that 'regression mean/phase' are definitional labels for the fitted argmin/argmax variables.

  1. self definitional [Section 2.5, Eq. (7)]
    "Similarly, we define optimal time-warping in the ScoSh model using bγi = arg maxγ∈Γ ⟨bβ, qi ⋆ γ⟩, where the estimator of β is: bβ = arg inf β∈L2 Σ i ( yi − g(fi(0)) − h(sup γi∈Γ ⟨β, qi ⋆ γi⟩) )² = ... assuming h(x) = x, g = 0 . ... This motivates naming bβ as the regression mean of the shapes of {fi} w.r.t responses {yi}."

    The 'regression mean' is introduced as the name of the argmin of the same squared-error objective used for estimation (Eq. 5), and the 'regression phase' is the argmax used inside that objective. Thus the new concepts are the fitted optimization variables themselves; their content is fixed by the defining optimization, not derived from an independent principle. This is a definitional labeling rather than a load-bearing circular prediction: the paper's held-out R2 comparisons do not depend on what the quantity is called.

full rationale

Apart from the naming redundancy above, the derivation chain is self-contained. The SI-ScoSh model (Eq. 4) is defined directly with the phase-invariant Fisher-Rao/SRVF inner product; invariance is established by the SRVF action property (Eq. 3), not by fitting. Estimation (Algorithms 1-2) alternates coordinate descent on β, h, and g with dynamic-programming alignment; prediction is evaluated on held-out test folds, so no fitted training value is relabeled as a test prediction. Citations to Srivastava and Klassen (2016) supply standard SRVF/Fisher-Rao facts, not an unverified uniqueness theorem, and are not load-bearing in a circular way. The one technical concern flagged by the skeptic -- Algorithm 1 Step 9 centers β by ordinary composition bβ ∘ γbar^{-1} rather than the SRVF action bβ ⋆ γbar^{-1} -- is a correctness/implementation issue, not a circularity of the derivation; it does not raise the circularity score.

Assumptions & free parameters 5 free parameters · 5 assumptions · 2 invented entities

The model relies on standard SRVF geometry, an explicit domain assumption that phase is nuisance, and several ad hoc choices: the quadratic restriction on g (with missing justification), the scale normalization, and the centering constraint. The fitted quantities (basis dimension, polynomial degrees, number of layers) are free parameters selected per dataset.

free parameters (5)
  • Basis dimension J for beta = 6 in COVID real data; 4 or 6 in simulations
    Number of Fourier basis functions for the coefficient function beta; chosen by the user and reported as selected to give best performance.
  • Degree of index polynomial h = cubic in COVID real data; up to quartic in simulations
    The degree of the polynomial used for the link function h, or SVM hyperparameters; selected per dataset.
  • Degree of offset polynomial g = quadratic
    Paper restricts g to a quadratic polynomial with justification deferred to Appendix 6.1, which is not included.
  • Number of index layers K = 4 in rainfall/humidity example
    For the multiple index extension, K is chosen by stopping when prediction improvement becomes small.
  • Scale identifiability constraint = highest coefficient of h fixed to 1
    An arbitrary normalization to resolve the scaling ambiguity between beta and h.
assumptions (5)
  • standard math SRVF transformation Q is a bijection between AC0 and L2 and the Fisher-Rao inner product equals L2 on SRVFs
    Invoked in Section 2.1 as background from Srivastava and Klassen (2016).
  • domain assumption The response yi depends on the predictor fi only through its shape, i.e., phase is a pure nuisance
    Central modeling assumption stated in Sections 1 and 2.2; the paper's real-data justifications are that phases are arbitrary parameterizations.
  • ad hoc to paper The index function h and offset g are continuous functions; g is restricted to a quadratic polynomial
    The continuous assumption is standard; the quadratic restriction is justified only by an appendix reference (6.1) that is missing.
  • ad hoc to paper Identifiability constraints: average of argmax warpings equals identity, highest coefficient of h equals 1, g(0)=0
    Imposed in Section 2.2 to remove degrees of freedom; no proof that these constraints fully identify the parameters.
  • domain assumption The optimization algorithm (coordinate descent with dynamic programming) converges to a good solution
    No convergence guarantee is provided; Section 2.4 uses repeated random splits to argue empirically.
invented entities (2)
  • Regression phase
    purpose: Name for the optimal time warping gamma_i = argmax <beta, q_i*gamma> in the regression model
    A definition rather than a new physical entity; it is fully determined by the fitted beta, so it has no independent falsifiable handle.
  • Regression mean
    purpose: Name for the estimated coefficient function beta that minimizes prediction error
    Same as regression phase: it is the fitted parameter, not an independently measurable quantity.

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Cite this review

Pith. "Pith review of Scalar-on-Shape Regression Models for Functional Data Analysis." pith.science (2026). https://pith.science/paper/VIDR6YTP

@misc{pith2026241115326,
  author       = {Pith},
  title        = {Pith review of: Scalar-on-Shape Regression Models for Functional Data Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VIDR6YTP}},
  note         = {Machine review of arXiv:2411.15326}
}
read the original abstract

Functional data contains two components: shape (or amplitude) and phase. This paper focuses on a branch of functional data analysis (FDA), namely Shape-Based FDA, that isolates and focuses on shapes of functions. Specifically, this paper focuses on Scalar-on-Shape (ScoSh) regression models that incorporate the shapes of predictor functions and discard their phases. This aspect sets ScoSh models apart from the traditional Scalar-on-Function (ScoF) regression models that incorporate full predictor functions. ScoSh is motivated by object data analysis, {\it, e.g.}, for neuro-anatomical objects, where object morphologies are relevant and their parameterizations are arbitrary. ScoSh also differs from methods that arbitrarily pre-register data and uses it in subsequent analysis. In contrast, ScoSh models perform registration during regression, using the (non-parametric) Fisher-Rao inner product and nonlinear index functions to capture complex predictor-response relationships. This formulation results in novel concepts of {\it regression phase} and {\it regression mean} of functions. Regression phases are time-warpings of predictor functions that optimize prediction errors, and regression means are optimal regression coefficients. We demonstrate practical applications of the ScoSh model using extensive simulated and real-data examples, including predicting COVID outcomes when daily rate curves are predictors.

Figures

Figures reproduced from arXiv: 2411.15326 by the authors.

Figure 1
Figure 1. Top: The three panels plot 95% bootstrap confidence intervals of the estimated b b [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Left: Alignment and shape averaging of functions using SRVFs; Middle: Align [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Two examples contrasting amplitude-phase and regression phase. In each row, [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Simulating experimental data. Top: From left to right, some initial predictor [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: The estimate bh for htrue = x 2 − 4x + 4 (left) and htrue = (x − 1 2 )(x − 3)(x − 4.5) (middle) under the SI-ScoSh model. The right panel shows estimates of gb with a quadratic gtrue for SI-ScoSh and SI-ScoF(FR)models. 2. Estimation Regression Coefficient β: Here we st…
Figure 6
Figure 6. Figure 6: βb’s when βtrue has J = 4 (left) and J = 6 (middle). The numbers beside the colored lines in the legend show J used in estimating β. The orange diamonds show the estimated β for the SI-ScoF(FR) model. The right panel shows estimates of gb with a cubic gtrue for SI-ScoS…
Figure 7
Figure 7. Figure 7: Examples of fis (left), noisy measurements of yis (second), the original test set in solid lines and their perturbed version in hashed lines (third panel), and SI-ScoSh predicted ybi plotted versus true yis (last panel). Top: Responses yi are the max amplitude of predi…
Figure 8
Figure 8. Figure 8: Spanish weather results – Top: Predictor functions [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: Covid hospitalization results – Top: the daily hospitalization curves (left), cor [PITH_FULL_IMAGE:figures/full_fig_p025_9.png]
Figure 10
Figure 10. Figure 10: Covid infection results – Top: the daily infection curves (left), correspond [PITH_FULL_IMAGE:figures/full_fig_p026_10.png]
Figure 11
Figure 11. Figure 11: Top: the morning humidity curves for different counties (left) and the total [PITH_FULL_IMAGE:figures/full_fig_p028_11.png]

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