Lifting methods for manifold-valued variational problems

Daniel Cremers; Evgeny Strekalovskiy; Jan Lellmann; Thomas Vogt

REVIEW 4 major objections 5 minor 73 references

Lifting methods for manifold-valued variational problems

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

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Pith's one-line read The paper claims that first-order finite-element lifting makes manifold-valued variational problems solvable to a global optimum with far fewer range labels than earlier methods.

desk verdict A useful survey-and-extension of functional lifting for manifold-valued problems, with the key exactness claim formally unproven but honestly flagged. read the letter →

arxiv 1908.03776 v1 pith:QCGUMD5X submitted 2019-08-10 math.NA cs.NA

Thomas Vogt , Evgeny Strekalovskiy , Daniel Cremers , Jan Lellmann This is my paper · ORCID

classification math.NAcs.NA MSC 65K1065N3068U10

keywords functionalliftingmanifold-valuedimageprocessingconvexrelaxationsublabel-accuratediscretizationfiniteelementmethodstotalvariationHuberregularizationprimal-dualhybridgradient

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

This paper tries to establish that functional lifting, the technique of turning a hard variational problem into a convex one by embedding the value range into a higher-dimensional space, works for images whose values live on a curved manifold when the range is discretized with first-order finite elements. The authors argue that this finite-element discretization gives sublabel-accurate lifting on manifolds: the data term is represented more faithfully between the chosen sample points, so far fewer sample points are needed than in earlier manifold lifting based on total variation. The paper also drops the total-variation-only restriction, so regularizers such as Tikhonov and Huber fit the same convex saddle-point formulation. If the construction is right, one convex optimization routine can find approximate global minimizers for manifold-valued denoising, inpainting, and related problems, and the back-projected solutions stay close to the true optimum even on coarse range triangulations.

What carries the argument

The central object is the first-order finite element space $S_h$ on a triangulated manifold $M_h\subset\mathbb{R}^N$, with nodal basis $\chi_1,\ldots,\chi_L$. The space of probability measures $P(M)$ is replaced by the convex set of nonnegative measures $\mu_h$ on the vertices satisfying $\sum_k \langle\mu_h,\chi_k\rangle=1$, i.e. the probability simplex over the $L$ labels. The integrand is split as $\rho(x,z)+\eta(P_z\xi)$, where $P_z$ maps ambient Jacobians into the tangent subspace; on each simplex $T$ this becomes two epigraphical constraints involving the convex conjugates $\eta^*$ and $\rho^*_T$, linked by a linear equation. These constraints are what make the optimization Euclidean despite the curved range, and the resulting convex-concave saddle-point problem is solved by a primal-dual hybrid gradient iteration. On $S^1$ the construction also yields a curvature correction factor $\alpha_T=d_{S^1}(Z_1^T,Z_2^T)/\|Z_1^T-Z_2^T\|_2$, the ratio of geodesic to Euclidean distance between the two vertices of a simplex, which rescales the finite element gradient.

What would settle it

Take a manifold-valued denoising problem with a known exact global minimizer, for instance the $S^1$ two-point mean from the paper's motivating example, solve the lifted saddle-point problem with the coarsest reported triangulation, and compare the projected solution's original energy with the exact global energy. If the projected solution lands on the local minimum that gradient-style methods find, or if the energy gap does not shrink toward zero as the triangulation is refined, the central claim of near-global, label-bias-free optimization fails for that instance.

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Extended reading notes

Core claim

On its own terms, the paper claims to unify and extend manifold-valued functional lifting through a finite element interpretation. For a variational problem of the form $\min_{u:\Omega\to M} \int_\Omega (\rho(x,u(x)) + \eta(P_{u(x)}Du(x)))\,dx$ with convex regularizer $\eta$, the range $M$ is approximated by a triangulated manifold $M_h$, and each solution is lifted to a probability measure over the vertices of the triangulation. The lifted energy is written in saddle-point form: the convex conjugate of $\eta$ enters through the constraint $\eta^*(P_T\nabla_T p(x)) \le a_T(x)$, and the data term through $\rho^*_T(q_{T,1}(x)) \le b_T(x)$, with the linear coupling $a_T(x)+b_T(x)=-q_{T,2}(x)$. Because $p$ is piecewise linear on each simplex, the sublabel-accurate data approximation of Euclidean lifting carries over to the manifold. Numerically, the paper reports that TV, Huber, and Tikhonov regularizers all run in this framework on orientable and non-orientable manifolds, and that good results need only 60 vertices for $SO(3)$ instead of 720, 12 vertices for $S^2$ instead of 162, and a $5\times 5$ grid on the Klein bottle.

Load-bearing premise

The load-bearing assumption is that the formal lifting step, carried over from vector-valued problems without a complete proof, truly preserves the manifold problem's global minimizers; if the convex relaxation is not tight, the global solution of the lifted problem need not correspond to a global or near-global minimizer of the original problem.

Editorial extensions

If this is right

The same lifted saddle-point formulation handles total variation, Huber, and quadratic (Tikhonov) regularization for manifold-valued images, generalizing earlier methods that were limited to total variation.
Because the data term is approximated convexly between labels rather than only at labels, coarse range triangulations suffice: 60 vertices on $SO(3)$ and 12 on $S^2$ are reported to match earlier results that needed hundreds of labels.
The method applies to non-orientable manifolds such as the Klein bottle, because the tangent-basis map $P_z$ is not required to be continuous.
After the lifted problem is solved to its globally optimal saddle point, the solution is projected back to the manifold by a Riemannian center-of-mass computation, yielding a concrete denoising and inpainting pipeline.

Reading between the lines

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

The finite-element viewpoint suggests trying higher-order elements or adaptive refinement on the range manifold to push the required number of labels below the counts reported here; the paper itself only demonstrates first-order elements.
If the equivalence between the lifted convex problem and the original nonconvex problem can be proven rather than assumed, it would provide a priori optimality bounds for manifold-valued regularizers, which the paper notes are currently missing for the manifold case.
The per-simplex correction factor on $S^1$ hints that on more curved manifolds a per-simplex linear transformation between the geometric and the finite-element gradient could systematically correct coarse-triangulation bias; the paper finds the effect negligible in its experiments, so this is a testable prediction rather than a claim.
Because the lifted problem is a Euclidean convex-concave saddle-point problem, it inherits the fast GPU-friendly solvers of scalar lifting, which could make global manifold optimization practical at the resolution of real InSAR and elevation data.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This paper is a survey with a real extension: it takes functional lifting from TV-only on manifolds to general convex regularizers, and it reframes the discretization as finite elements on the range manifold, which reduces label bias. The Klein bottle and SO(3) experiments, plus the released code, make the claims concrete. I'd use it as a reference for sublabel-accurate lifting for manifold-valued problems.

The paper is also honest: it says in Section 2.1 that the derivations are formal and that function spaces and well-posedness are left for future work. That is exactly where the main risk sits. The exactness of the lift for vectorial problems, which this generalizes, is justified by calibration/tightness results. No analogue is proved here. Without a tightness result, the relaxed problem's global minimizer need not project to a near-minimizer of the original manifold-valued problem. The paper's own projection via Riemannian center of mass is non-convex and lacks optimality bounds, so the gap is real.

The second soft spot is the piecewise-flat approximation M_h. Replacing the manifold by a simplicial complex changes the geometry; the authors give only an S^1 curvature correction and rely on empirical negligibility. No Gamma-convergence or error analysis is provided. If the paper is read as a survey with heuristic improvements, that is fine; but the abstract promises approximate global minimizers, which needs more support.

The numerics are qualitative and compare against the authors' earlier method at different mesh resolutions. That does not close the theoretical gaps, though it does demonstrate practical benefit: 12 vertices on the icosahedron versus 162 previously.

I would send it to peer review. It deserves referee time, but a careful referee should ask for either a rigorous tightness/convergence analysis or a clear statement that the method is heuristic, plus some quantitative error evaluation on a problem with a known solution. The paper is worth reading for the survey alone; the extension is plausible but unproven.

Referee Report

4 major / 5 minor

Summary. This manuscript is a survey-style chapter on functional lifting for first-order variational problems u: Ω → M, where M ⊂ R^N is a Riemannian submanifold. The authors embed M into the space of probability measures via Dirac measures, define a lifted functional (11) through the convex conjugate of the integrand, and discretize the range by a triangulated manifold M_h with a first-order finite element space (15)-(16). After replacing the tangent-space projection P_z by per-simplex projections P_T in (17)-(19), they arrive at the saddle-point formulation (24)-(27). The paper claims that this framework extends sublabel-accurate lifting to manifolds and generalizes the previous TV-only lifting [47] to arbitrary convex regularizers such as Huber and Tikhonov. Several qualitative experiments are reported (Klein bottle, SO(3), S^2 normals, InSAR S^1 data). The paper explicitly declares the derivations in Section 2 formal and leaves function-space and well-posedness questions for future work.

Significance. If the central claims are correct, the paper offers a useful unifying presentation and a practical algorithmic extension: first-order finite elements on the range reduce label bias, the framework handles non-orientable manifolds and general convex regularizers, and the released code and real-data experiments make the approach reproducible. The treatment of non-orientable manifolds and the extension beyond TV are genuine contributions. However, because no tightness theorem, Γ-convergence result, or error estimate is proved, the significance is conditional: the paper does not establish that the global minimizer of the lifted problem yields a near-global minimizer of the original manifold-valued problem. The numerical evidence is visual and compares mainly with the authors' earlier methods, so the 'sublabel accuracy' advantage is not yet quantitatively validated. The paper is honest about these limitations, which is a strength.

major comments (4)

[Section 2.1, Eqs. (10)-(12)] The central promise that the lifted convex problem can be solved globally and that its projection approximates a global minimizer of (10) rests on an unproved transfer of exactness from the vectorial theory [51]. The text states that the construction of [51] can be generalized 'by replacing the range Γ with M' and immediately adds that 'most derivations will be formal' and that well-posedness is left for future work. No analogue of the calibration or exactness arguments available for scalar/vectorial ranges is supplied for manifold ranges, and Section 1.1 explicitly concedes that no projection guarantee is known in the manifold case. Since the relaxation from M to P(M) is not tight in general, the global minimizer of (11) may project to a value substantially above the infimum of (10). This gap is load-bearing for the headline claim and should be addressed, at minimum by formulating the exactness claim as a conjecture with a precise statement and by providing numerical evidence of the energy gap on a simple test problem.
[Section 2.2, Eqs. (17)-(19)] The substitution of the pointwise tangent-space projection P_z by per-simplex flat projections P_T changes the model to a piecewise-flat surrogate manifold. The paper derives a scalar correction only for S^1 in Section 2.3 and otherwise asserts empirical negligibility. No Γ-convergence, interpolation error, or mesh-refinement analysis is given for M_h → M. Without such a consistency result, the claim that sublabel accuracy allows much coarser range discretizations than [47] is not quantitatively established; the authors should either add a convergence analysis or report a numerical mesh-refinement study showing that the solution stabilizes as the triangulation is refined.
[Section 2.4, Eq. (32)] The final projection step to M via the Riemannian center of mass (32) is itself a non-convex problem with no optimality bound, as the paper acknowledges. Because this projection is an essential part of the pipeline, the lack of any guarantee for it compounds the tightness gap in the first major comment and directly affects the claim that the method finds approximate global minimizers of (10). Even an assumption-dependent bound, for example for data terms supported in a geodesically convex ball, would clarify the status of the claim.
[Section 3] The numerical evaluation is qualitative and compares mainly with the authors' own earlier methods at different mesh resolutions (12 vs 162 vertices on S^2 in Section 3.3; 60 vs 720 on SO(3) in Section 3.2). No energy values, ground-truth errors, or comparisons with local optimization baselines are reported. Since sublabel accuracy and label-bias reduction are central claims, a quantitative experiment, such as projected energy versus the true minimum on a problem with known solution, or error versus number of labels, is needed to substantiate the visual improvement.

minor comments (5)

[Section 3.1] The sentence 'The resolution of the signal (250 one-dimensional data points) is far below the resolution of the triangulation' appears to say the opposite of what is meant, since 250 data points are many more than the 25 vertices of the 5×5 triangulation; please rephrase.
[Equation (24)] The range of the dual variable p is not consistently stated: below (17) the text says p∈ S_h^d, while (24) writes p∈ C^1(Ω, S_h^{d+1}); please align the notation and state the dimension of p explicitly.
[Section 2.4] The stopping criterion 'relative gap between primal and dual objective fell below 10^{-5}' is not defined; please specify how the primal and dual objectives are evaluated for the discretized saddle-point problem.
[Figure 5 caption] The caption contains an apparent typesetting artifact ('/tieaccentlowercase'); please correct it.
[Section 1.2] The sentence 'This is very desirable for discrete Γ' appears right after a discussion of label bias; it would be clearer to state explicitly that label bias is desirable only in the discrete-label setting, not in the continuous-range setting.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the manifold lifting model is an explicit substitution into prior vectorial lifting ([51]) and finite-element discretization ([38], [50]), not a refitted prediction or self-citation chain; the admitted lack of a tightness proof is a correctness gap, not a circular step.

full rationale

The central construction is admittedly an extension by substitution: Section 2.1 states 'Formally, the lifting strategy for vectorial problems proposed in [51] can be generalized to this setting by replacing the range Γ with M,' and equations (11)-(12) are exactly that substitution. The finite-element discretization in Section 2.2, equations (17)-(27), similarly adapts [50] and [38]. This is normal use of prior work as a starting point, not circularity: no parameter is fitted to data and then reported as a prediction, no uniqueness theorem from the authors' own prior work is invoked to force the choice, and the cited results have assumptions (vectorial ranges) that do not include the target manifold result. The paper's explicit caveats—'Most derivations will be formal; we leave a rigorous choice of function spaces as well as an analysis of well-posedness for future work' (Section 2.1) and its admission in Section 1.1 that 'For the manifold-valued case considered here, we are not aware of a similar result yet' regarding projection optimality bounds—are genuine limitations and correctness risks, but they are gaps in justification, not equivalences between inputs and outputs. The numerical experiments compare against the authors' own earlier methods, but they are qualitative demonstrations of a proposed model, not a derived law masquerading as external validation. No specific reduction of a conclusion to its own premises by construction or by self-citation could be exhibited, so the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper relies on a small set of standard and domain-specific assumptions. The main burden is the unproven tightness of the convex relaxation for manifold ranges, inherited from prior work, and the adequacy of the simplicial approximation of the manifold. No new entities are postulated; the lifting space P(M) of probability measures is taken from prior work. The only hand-chosen numbers are the experimental regularization parameters.

free parameters (2)

Regularization weight λ (per experiment) = TV: 0.4 (DEM), 0.6 (InSAR); Huber: 0.75 (both); Tikhonov: 3.0 (DEM), 1.0 (InSAR)
Chosen by hand for each experiment to produce visually pleasing results; not fitted by the method and not part of the central claim, but affects all reported outputs.
Huber parameter α = 0.1 (DEM and InSAR)
Fixed manually as the transition point of the Huber regularizer; influences the smoothness/edge trade-off in the Huber experiments.

assumptions (5)

standard math Whitney embedding theorem allows restriction to submanifolds of R^N without loss of generality.
Invoked in Section 2 to justify formulating the model for submanifolds of R^N.
domain assumption The integrand decomposes as f(x,z,ξ) = ρ(x,z) + η(P_z ξ) with P_z a surjective linear map with kernel N_zM.
Equation (13). This restricts the class of first-order variational problems; not all manifold-valued energies fit this form, e.g., those depending on curvature or on the full ambient Jacobian without such decomposition.
domain assumption The convex lifting construction from [51] remains valid for manifold ranges without modification.
Section 2.1 states the derivations are formal and rigorous well-posedness is left for future work; the tightness of the relaxation is assumed, not proved.
domain assumption The triangulated manifold M_h with linear finite elements faithfully approximates the geometry of M, so that piecewise linear interpolation and facet gradients are adequate.
Sections 2.2 and 2.4. The paper notes a curvature correction factor for S^1 (α_T) but observes only negligible differences in experiments; for general manifolds no corresponding correction is derived.
domain assumption Projecting the lifted solution via the Riemannian center of mass (32)-(35) yields a reasonable approximation of the original solution.
Section 2.4. The projection problem itself is non-convex and solved by local Karcher iteration; the paper states it 'yields reasonable results' empirically but provides no guarantee.

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Cite this review

Pith. "Pith review of Lifting methods for manifold-valued variational problems." pith.science (2026). https://pith.science/paper/QCGUMD5X

@misc{pith2026190803776,
  author       = {Pith},
  title        = {Pith review of: Lifting methods for manifold-valued variational problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QCGUMD5X}},
  note         = {Machine review of arXiv:1908.03776}
}

read the original abstract

Lifting methods allow to transform hard variational problems such as segmentation and optical flow estimation into convex problems in a suitable higher-dimensional space. The lifted models can then be efficiently solved to a global optimum, which allows to find approximate global minimizers of the original problem. Recently, these techniques have also been applied to problems with values in a manifold. We provide a review of such methods in a refined framework based on a finite element discretization of the range, which extends the concept of sublabel-accurate lifting to manifolds. We also generalize existing methods for total variation regularization to support general convex regularization.

Discussion (0). Continue with ORCID to comment.

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