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Measuring dissimilarity between convex cones by means of max-min angles

T0 review · 2 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Two one-sided max-min angles between convex cones assemble into a true distance, equal to the Pompeiu-Hausdorff distance on their unit-sphere sections.

desk verdict The distance construction and closed forms are solid, but Theorem 5.3's local optimality claim is false as written—the proof conflates a master-program subgradient with the projection returned by the algorithm. read the letter →

arxiv 2511.10483 v2 pith:ZQ3IEUFH submitted 2025-11-13 math.OC

Welington de Oliveira , Valentina Sessa , David Sossa This is my paper · ORCID
classification math.OC MSC 52A4090C2690C4690C4749M37
keywords convexconesmax-minanglePompeiu-Hausdorffdistancecutting-planemethodpolyhedralimage-setclassificationfew-shotlearningprincipalangles
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 introduces a new way to quantify how different two convex cones are, built from the max-min angle: for each direction in one cone, take the smallest angle to the other cone, then take the largest of those. The authors prove that the two one-sided max-min angles combine into a genuine distance, Dis_r(P,Q) = 2||(sin(Θ(P,Q)/2), sin(Θ(Q,P)/2))||_r, which coincides with the Pompeiu-Hausdorff distance between the cones intersected with the unit sphere. For linear subspaces and revolution cones the measure has closed forms, and for polyhedral cones a cutting-plane method computes it, with a local-solution variant that still yields points satisfying necessary optimality conditions. Applied to few-shot image-set classification on the ETH-80 dataset, the measure achieves about 90% average accuracy even when training cones are built from only one to five objects. The work establishes that angle-based comparison of convex cones can be an exact metric, not merely a heuristic similarity.

What carries the argument

The engine is the max-min angle Θ(P,Q), a pair of directional quantities defined by optimizing over unit vectors in each cone; the metric structure rests on the identity Dis_r(P,Q)=2||(sin(Θ(P,Q)/2), sin(Θ(Q,P)/2))||_r, which links these angles to the Pompeiu-Hausdorff distance between the spherical sections P∩S^n and Q∩S^n through the chord-length relation ||u−v||=2 sin(arccos⟨u,v⟩/2). Computationally, Θ(P,Q) is obtained by minimizing the convex support function F_Q(u)=max_{v∈Q∩S^n}⟨u,v⟩ over the nonconvex set P∩S^n. For polyhedral cones, the paper builds a cutting-plane model of F_Q, leading to a nonconvex quadratically constrained master program; subgradients come from projecting onto Q∩S

What would settle it

Take two polyhedral cones in R^3 with known generator matrices, compute the max-min angle by exhaustive angular sampling on the unit sphere to high precision, then run the local variant of the cutting-plane algorithm; if the solver's final iterate violates the KKT system (3.5) — a direct algebraic check — the local convergence claim fails.

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Extended reading notes

Core claim

The central claim is Theorem 2.5: for closed convex cones P,Q in R^n, the max-min angle Θ(P,Q)=max_{u∈P∩S^n} min_{v∈Q∩S^n} arccos⟨u,v⟩, together with its mirror Θ(Q,P), defines a distance Dis_r(P,Q)=2||(sin(Θ(P,Q)/2), sin(Θ(Q,P)/2))||_r on the space of all such cones, and this distance equals the Pompeiu-Hausdorff-based distance between the truncated sets P∩S^n and Q∩S^n. Consequently, two cones coincide exactly when both one-sided angles vanish. For linear subspaces of equal dimension, Dis_r reduces to a multiple of the spectral distance; for revolution cones an explicit formula is derived. For polyhedral cones, an adapted cutting-plane algorithm computes the angles: solving the master prog

Load-bearing premise

The local-solver guarantee rests on the untested premise that the nonlinear solver reaches a stationary point of the nonconvex master problem at every iteration; if it fails, the returned angle may not satisfy the necessary optimality conditions.

Editorial extensions

If this is right

  • Dis_r is an angle-based dissimilarity on all closed convex cones that satisfies the metric axioms, so cone-comparison tasks can rely on it without ad hoc heuristics.
  • For linear subspaces of equal dimension, Dis_r coincides with a multiple of the spectral distance, computable directly from singular values.
  • For polyhedral cones, the global cutting-plane variant converges to the exact angle, while the local variant still returns a point satisfying necessary optimality conditions, justifying the use of fast local solvers on large instances.
  • The two one-sided angles Θ(P,Q) and Θ(Q,P) carry different information, so Dis_r captures a directed structure that a single scalar cannot encode.
  • In few-shot image-set classification on ETH-80, the distance yields about 90% accuracy even when training cones are built from one to five objects per class.

Reading between the lines

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

  • The ordered pair (Θ(P,Q), Θ(Q,P)) could itself serve as a two-dimensional interpretable representation of cone dissimilarity, preserving the asymmetry that Dis_r symmetrizes.
  • Because Dis_r equals the Pompeiu-Hausdorff distance between spherical sections, any future algorithm that approximates Hausdorff distance on spheres would immediately yield a way to compute Dis_r, connecting this work to computational geometry on spheres.
  • A testable extension would replace the chord-length factor sin(θ/2) by other monotone functions of angle; the proof structure of Theorem 2.5 suggests which functions preserve the metric axioms.
  • The cutting-plane framework may extend to non-polyhedral cones whenever projecting onto the cone's spherical section is tractable, for instance for certain spectral cones.
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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

2 major / 3 minor

Summary. The paper introduces a dissimilarity measure between closed convex cones based on a max-min angle Θ(P,Q), defines Dis_r(P,Q) as the ℓ_r norm of the two sine half-angle terms, and proves (Theorem 2.5) that it is a distance on C^n, coinciding with the Pompeiu-Hausdorff distance between the cones' intersections with the unit sphere. It derives closed forms for linear subspaces (Theorem 4.1) and revolution cones (Theorem 4.3). For polyhedral cones, it proposes a specialized cutting-plane method (Algorithm 5.1), proves global convergence when the master problem is solved globally (Theorem 5.2), and claims that when the master is solved only to stationarity the method still produces points satisfying necessary optimality conditions (Theorem 5.3). Numerical experiments on random cones and ETH-80 image-set classification are reported.

Significance. The geometric construction in Theorem 2.5 is elegant and genuinely useful: it connects a natural angle-based cone dissimilarity to the classical Pompeiu-Hausdorff distance, and the subspace and revolution-cone formulas are clean, parameter-free consequences of the framework. The global cutting-plane convergence theorem (Theorem 5.2) is also valuable and the algorithmic design is sensible. The paper is self-contained and the main distance result does not rely on fitted parameters or circular arguments. The image-set application is illustrative rather than a benchmarked contribution. The main weakness is the local-convergence guarantee: Theorem 5.3, as stated and proved, is not correct under the arbitrary projection tie-breaking used in Algorithm 5.1, and the numerical section does not certify that fmincon returns stationary points. These issues are fixable but are load-bearing for the paper's advertised local-solution guarantee.

major comments (2)
  1. [Section 5.1, Theorem 5.3] The proof of Theorem 5.3 assumes that the projection vector v_{k+1} computed in line 4 of Algorithm 5.1 belongs to ∂\hat F^k_Q(u_{k+1}). Stationarity of the master problem (5.2) only yields some subgradient w∈∂\hat F^k_Q(u_{k+1}), and there is no reason that w equals the projection of u_{k+1} onto Q∩S. This is not a solver artifact: it fails when the projection is non-unique. Example in R^3: P=span{e1,e2}, Q=cone{e1,e2,e3}; for u=-e1, J(u)={2,3}, so Proj_{Q∩S}(u)=cone{e2,e3}∩S. If the tie-break returns e2, then v-⟨u,v⟩u=e2∉P*=span{e3}, so system (3.5) fails, although -e1 is a global minimizer of the true problem and a stationary point of the master model max(u_x,u_y,0). Thus Theorem 5.3 as written is false for arbitrary tie-breaking. The proof requires v_{k+1} to be chosen as the multiplier/subgradient returned by the master solve, or a projection-selection rule that guarantees the corre
  2. [Section 6] The local variant is presented as satisfying necessary optimality conditions by Theorem 5.3, but the implementation uses fmincon with default parameters on the nonconvex equality-constrained master problem (5.2), and the paper never verifies stationarity of the returned points. For a nonconvex problem with the constraint ||u||=1, default fmincon can terminate at nonstationary points, and the associated multipliers are not used to define v_{k+1}. The reported angles and the classification accuracies in Figs. 8–9 therefore rest on an unverified solver guarantee. Additionally, for the 'global' variant, Gurobi reaches its time limit in about 20% of instances, so those labels are not certified global solutions either. Please report KKT residuals or exit-flag distributions for the local solves, and clearly separate certified-global, certified-stationary, and heuristic results.
minor comments (3)
  1. [Section 2] The definitions in C^n appear to include the zero cone, for which P∩S^n is empty and Θ(P,Q) is undefined. The paper should either state that all cones are assumed nonzero (or at least not the zero cone) or handle the zero cone separately.
  2. [Section 6.1.1 and Introduction] The text says the approach 'could correctly classify 7 classes out of 8 on average', while the introduction states 'an average 90% accuracy'. These statements should be reconciled; the aggregate accuracy depends on the norm and on k, so the precise claim should be stated with the corresponding configuration. Also, 'fmiconto' should be 'fmincon'.
  3. [References] Reference [7] is not the original source of the ETH-80 dataset. Please cite the primary dataset reference (e.g., Leibe and Schiele, 2003) in addition to, or instead of, the image-set-classification paper.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular dependency found; central distance theorem follows from definitions and standard tools.

full rationale

The claimed central result, Theorem 2.5, is derived by substituting the definition of the max-min angle into the expression for the one-sided Pompeiu-Hausdorff term and applying the elementary identity 1-cos(alpha)=2sin^2(alpha/2). No fitted parameter is later renamed as a prediction, and no quantity in the distance formula is defined in terms of the distance being proved. The self-citations to prior work by the same authors concern background material on critical angles and the complexity of related problems; they are not load-bearing for the new distance construction, which is self-contained apart from standard convex-analysis tools (e.g., [5] for cone-sphere projections). Algorithm 5.1 is a specialized Kelley cutting-plane scheme whose convergence statements are independent of the paper's new definitions. The proof of Theorem 5.3 does contain a step in which the projection used in Algorithm 5.1 is assumed to be the subgradient required by the master-problem stationarity conditions; this is a mathematical correctness issue, not a circularity, because it does not reduce a claimed result to an input definition or fitted value. Under the specified review rules, no quoted equation or citation chain makes a 'prediction' equal to its own construction, so the appropriate circularity score is 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central theory relies on standard convex analysis and two borrowed projection results. The only genuinely new object is the max-min angle itself, which is a mathematical definition rather than a postulated physical entity. The application adds an unverified empirical modeling assumption about CNN feature cones.

free parameters (2)
  • r (norm order in Dis_r) = 1, 2, ∞ in experiments
    Parameterizes the family of distances; the theorems hold for all 1≤r≤∞. Not fitted to data, but chosen by hand in the numerical section.
  • stopping tolerance tol = 10^-3
    Algorithm 5.1 termination tolerance. A standard algorithmic choice, not fitted to classification accuracy.
assumptions (6)
  • domain assumption The cones considered have nonzero intersection with the unit sphere, so P∩S^n is nonempty and compact.
    Used in Proposition 2.1 and Theorem 2.5. The zero cone is not discussed.
  • domain assumption Projection and subdifferential characterizations of Proj_{Q∩S^n} and ∂F_Q from Bauschke-Bui-Wang [5] are correct and applicable.
    Borrowed external theorem used in Proposition 3.1 and Proposition 5.1; F_Q evaluation and the cutting-plane algorithm depend on it.
  • standard math Classical cutting-plane convergence theory for convex objectives over compact feasible sets [1,3] applies to the nonconvex feasible set P∩S^n.
    Used in Theorem 5.2 to assert global convergence of Algorithm 5.1.
  • standard math Principal angles between linear subspaces are obtained from singular values of V^T U.
    Used in Theorem 4.1 to compute the distance between linear subspaces.
  • ad hoc to paper ETH-80 image classes are separable by angular distance between cones generated by ResNet-18 pool5 features.
    Empirical modeling assumption in Section 6.1.1; no independent geometric justification is provided.
  • domain assumption Image feature vectors used as cone generators are nonnegative or otherwise normalized so that the columns of G and H can be taken unit norm.
    The polyhedral cone model in Section 5 assumes unit-norm generators; this is not verified for the CNN features.

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Pith. "Pith review of Measuring dissimilarity between convex cones by means of max-min angles." pith.science (2026). https://pith.science/paper/ZQ3IEUFH

@misc{pith2026251110483,
  author       = {Pith},
  title        = {Pith review of: Measuring dissimilarity between convex cones by means of max-min angles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZQ3IEUFH}},
  note         = {Machine review of arXiv:2511.10483}
}
read the original abstract

This work introduces a novel dissimilarity measure between two convex cones, based on the max-min angle between them. We demonstrate that this measure is closely related to the Pompeiu-Hausdorff distance, a well-established metric for comparing compact sets. Furthermore, we examine cone configurations where the measure admits simplified or analytic forms. For the specific case of polyhedral cones, a nonconvex cutting-plane method is deployed to compute, at least approximately, the measure between them. Our approach builds on a tailored version of Kelley's cutting-plane algorithm, which involves solving a challenging master program per iteration. When this master program is solved locally, our method yields an angle that satisfies certain necessary optimality conditions of the underlying nonconvex optimization problem yielding the dissimilarity measure between the cones. As an application of the proposed mathematical and algorithmic framework, we address the image-set classification task under limited data conditions, a task that falls within the scope of the \emph{Few-Shot Learning} paradigm. In this context, image sets belonging to the same class are modeled as polyhedral cones, and our dissimilarity measure proves useful for understanding whether two image sets belong to the same class.

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