On the existence and non-existence of centres of mass on Hilbert spheres

Shahin Tavakoli

REVIEW 2 major objections 4 minor 16 references

On the existence and non-existence of centres of mass on Hilbert spheres

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper shows that on the infinite-dimensional Hilbert sphere, L^p centres of mass do not always exist, and gives a simple condition under which they do.

desk verdict Settles an open existence question on Hilbert spheres with a clean counterexample and a useful finite-dimensional reduction; one minor closure issue in Theorem 1 is easily fixed. read the letter →

arxiv 2607.13667 v1 pith:WNUIFZHF submitted 2026-07-15 math.ST stat.TH

Shahin Tavakoli This is my paper

classification math.STstat.TH MSC 62R3060D0546C05

keywords FréchetmeanL^pcentreofmassHilbertsphereinfinite-dimensionalexistencemeansdimensionreductionsamplefunctionaldata

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 asks whether an average — a centre of mass — always exists for probability distributions on the infinite-dimensional sphere, the unit sphere in a Hilbert space. The answer is no: for every p>1, a symmetric mixture of antipodal pairs spread over a complete orthonormal basis has no L^p centre of mass, because any minimiser would have to be orthogonal to every basis vector, which is impossible for a unit vector. The paper pairs this counterexample with a positive criterion: if the distribution's support spans a subspace with at least one orthogonal direction to spare, then a centre of mass exists for any continuous cost, including L^p for all p>0. For finite samples, the centre always exists and can be computed by an optimisation problem on a subsphere of dimension at most n, no matter how large the ambient sphere is. This makes sample Fréchet means on Hilbert spheres — natural for density and functional data — computable by finite-dimensional methods.

What carries the argument

The mechanism is the identity d_S(s,x) = arccos⟨s,x⟩, extended to the closed unit ball. For the counterexample, F_p decomposes into independent terms (π−α)^p + α^p whose minimum over α is at α = π/2; making s perpendicular to every basis direction lowers the cost toward its infimum, but completeness forbids a unit vector orthogonal to all basis vectors. For existence, extension to the ball maps minimisation on the sphere to minimisation over the closed unit ball of U, which is weakly compact; when an escape direction u⊥ ∈ U^⊥ ∩ S exists, any u ∈ B lifts to a sphere point s = u + √(1−‖u‖²)u⊥, transferring the infimum to an attained value.

What would settle it

For the basis-symmetric measure, minimise F_p over the full sphere of a large finite-dimensional truncation and watch the minimiser's component along the basis coordinates shrink as the dimension grows; the limiting infimum is (π/2)^p but the only attaining vector is zero. If a unit attaining vector existed in the limit, Proposition 1 would be wrong.

Watch

Extended reading notes

Core claim

The paper establishes a sharp pair of facts about centres of mass on the Hilbert sphere S = {x : ||x||=1}. Proposition 1: for any p > 1, a symmetric mixture of antipodal pairs on a complete orthonormal basis — ν = Σ w_n(δ_{u_n} + δ_{−u_n}) — has no L^p centre: F_p(s) = Σ w_n((π−α_n)^p + α_n^p) ≥ (π/2)^p, with equality only if every α_n = ⟨s,u_n⟩ = 0, which forces s = 0, outside S. Theorem 1: if U = span(supp ν) has a nontrivial orthogonal complement, a minimiser of F_φ exists for any continuous φ and, if unique, lies in U ∩ S. Theorem 2: for any finite sample x_1,…,x_n, a minimiser exists in span(u, x_1,…,x_n) ∩ S for any unit u orthogonal to the data, so the sample problem reduces to a mani

Load-bearing premise

The existence and dimension-reduction theorems hinge on the support of the distribution leaving at least one unit direction orthogonal to every data point; when no such free direction exists — and the span is understood in the closed sense required for weak compactness — the compactness argument fails and the paper's own counterexample shows existence can fail.

Editorial extensions

If this is right

Sample L^p centres of mass on the Hilbert sphere always exist and are computable by a smooth optimisation over a manifold of dimension at most n, independent of the infinite-dimensional ambient space.
The same reduction works for any continuous φ, so Fréchet means, Huber means, and other φ-centres for functional data are finite-dimensional optimisation problems.
Non-uniqueness of the sample centre can be detected from the computed solution: a nonzero coefficient on the added orthogonal direction u signals at least two minimisers.
Population centres of mass exist whenever the support leaves a free dimension; the only way to lose existence is to spread the data so densely (over a complete basis) that no orthogonal direction remains.
The result extends Yokota's existence theorem from samples inside a spherical cap of radius < π/2 to arbitrary samples, without cap assumptions.

Reading between the lines

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

The escape-to-zero mechanism in Proposition 1 probably applies to any distribution whose support is 'balanced' around a complete basis with symmetric antipodal weights, including continuous analogues on pairwise-orthogonal subspheres; the paper sketches this, and a unified non-existence criterion could be proved.
Theorem 2's dimension reduction relies only on the existence of an orthogonal operator fixing the data — a property shared by other homogeneous spaces. Similar reductions may hold for pre-shape spaces and other quotient spheres, making infinite-dimensional shape analysis finite-dimensional per sample.
Since for any finite sample the free dimension always exists, non-existence is a purely infinite-population phenomenon; one might expect sample centres to drift slowly along the escaping direction as the sample size grows, suggesting a new form of inconsistency without cap assumptions.
The uniqueness condition in Theorem 1 (s* ∈ U∩S) suggests a way to test whether a population centre is unique: if numerical solutions on the subsphere land with nonzero component along u, uniqueness fails and the centre set contains a continuum.

Share X Bluesky LinkedIn Reddit HN

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

Punchline: the paper settles an open existence question on Hilbert spheres with a simple, explicit counterexample, and proves a finite-dimensional reduction for sample centres of mass that removes the old cap assumptions. Both main claims are correct. The one soft spot is a missing closure convention in Theorem 1, and it is a one-line fix.

What's new: Proposition 1 gives a concrete probability measure on the infinite-dimensional sphere with no L^p centre for any p>1 — a symmetric two-point mixture over an orthonormal basis. The infimum is (pi/2)^p and the only candidate minimizer is zero, which is not on the sphere. This is clean and convincing. Theorem 1 says existence holds whenever the supporting subspace has a non-trivial orthogonal complement. The proof reduces the problem to the closed unit ball of the support's span, uses weak compactness, then lifts back to the sphere using the perpendicular direction. Theorem 2 extends Yokota's span reduction to all finite samples, without radius conditions and without uniqueness: a minimizer always exists in the (n+1)-dimensional subsphere spanned by the data plus one extra orthogonal vector. The algorithm in Section 3 is a reasonable implementation of that reduction.

Where it gets soft: the proof of Theorem 1 silently treats U = span{supp(nu)} as closed. It applies orthogonal projection onto U and uses weak compactness of U's unit ball, which are only valid for a closed subspace. For finite samples this is automatic, but for general nu the statement should say U = cl(span(supp nu)). The stress-test note has it right: the repair is one line and none of the main conclusions change. The condition dim U^perp >= 1 is genuinely load-bearing, since the proof needs a perpendicular vector to lift the minimizer from the ball to the sphere. I would not call this more than a minor rigor flaw.

The rest is solid. The proofs are short, self-contained, and start from definitions; there is no circularity or fitted-parameter issue. The literature is cited appropriately, including Yokota, Dai, and Jaffe. The discussion of statistical implications is honest about what is and isn't proven — consistency and asymptotics are left out.

Who it's for: anyone working on Frechet means on spheres, directional statistics, or functional data on the Hilbert sphere. It's a subfield-level result, not a big breakthrough, but it clears up a known open question and gives a practical dimension reduction.

Recommendation: send it to review. With a note to fix the closed-span definition, I'd expect acceptance after minor revision.

Referee Report

2 major / 4 minor

Summary. The paper studies the existence and computation of φ-centres of mass (including L^p Fréchet means) on the unit sphere of an infinite-dimensional separable Hilbert space. It presents three main results. Proposition 1 constructs, for every p > 1, a probability distribution on the sphere with no L^p centre of mass: a two-point mixture on a complete orthonormal basis, for which the infimum of F_p is (π/2)^p but is attained only at the zero vector. Theorem 1 gives a sufficient condition for existence: if the closed linear span of the support has a non-trivial orthogonal complement, then for any continuous φ a φ-centre exists in the sphere. Theorem 2 shows that for finite samples, a sample φ-centre always exists and can be found in the finite-dimensional subsphere spanned by the data plus one additional orthogonal unit vector, reducing the problem to an optimization on a manifold of dimension at most n. An algorithm for this reduction is provided and justified, and statistical implications are discussed.

Significance. The paper resolves a genuine open question: whether centres of mass always exist on infinite-dimensional Hilbert spheres. The non-existence example is explicit and elementary, and the sufficient condition in Theorem 1 is simple and new. The finite-dimensional reduction for sample centres in Theorem 2 is practically valuable for functional data analysis, where data are often infinite-dimensional but the sample size is moderate. The proofs are self-contained, rely only on standard Hilbert-space facts, and are clearly presented. The paper is likely to interest statisticians and probabilists working with non-Euclidean data, shape analysis, and Fréchet means in infinite-dimensional spaces.

major comments (2)

[Theorem 1, Section 2 (proof in Appendix)] The subspace U is defined as span{supp(ν)}. The proof uses the orthogonal projection onto U, the decomposition H = U ⊕ U^⊥, and weak compactness of the closed unit ball in U. These are valid only when U is closed. If 'span' means the algebraic span (finite linear combinations), the proof is incomplete: the projection need not be bounded or well-defined on H, and the closed unit ball in U need not be weakly compact. Moreover, the statement 'if s* is unique, then s*∈U∩S' may fail, because a unique minimizer could lie in the closure of U but not in the algebraic span. Please redefine U as the closed linear span of the support (or state explicitly that span is taken in the closed sense). This repair does not affect Proposition 1 or Theorem 2, which only involve finite-dimensional spans.
[Theorem 2, last sentence, Section 2] The statement 'The results of Theorem 2 also hold for the finite-dimensional sphere S^k, k≥1' is too broad. The proof requires choosing u ∈ span{x_1,...,x_n}^⊥ ∩ S, which is possible only if the data do not span the ambient R^{k+1}. When k is small or the data span all directions, no such u exists. Since the finite-dimensional case already has existence by compactness, the reduction statement needs a qualifier (e.g., 'whenever such a u exists'). As written, it is inaccurate for finite-dimensional spheres.

minor comments (4)

[Proof of Theorem 2, Appendix] The existence of the orthogonal operator R such that R x_i = x_i and R s ∈ U∩S is asserted without proof. A brief justification—constructing R as the identity on span{x_i} and a rotation in the orthogonal complement that maps the residual component of s onto ±u—would make the argument fully transparent.
[Proof of Proposition 1, Appendix] The claim that F_p(s) = (π/2)^p if and only if ⟨s,u_n⟩ = 0 for all n relies on the strict convexity of t ↦ (π−t)^p + t^p for p > 1, which has a unique minimum at t = π/2. This is correct but could be stated explicitly for readers.
[Algorithm 1, Section 3] The notation x_i^T x_j is used for the inner product in a general Hilbert space. Replacing this with ⟨x_i, x_j⟩ would be more consistent with the rest of the paper and avoid confusion in the L^2 setting.
[References, Section 4] The reference to Jaffe [2026] is listed with a date of May 2026; please ensure this is consistent with the publication status of the cited work.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the paper's results are derived from definitions and standard Hilbert-space/optimization facts; the only flagged issue is a one-line rigor gap (closed span), not circularity.

full rationale

The paper is a pure existence/reduction theorem paper. Proposition 1 derives its non-existence result by an explicit computation of F_p on the symmetric two-point mixture over a complete orthonormal basis, showing the infimum (π/2)^p is attained only at the zero vector, which is not on the sphere. Theorem 1 uses standard Hilbert-space arguments: orthogonal projection, weak compactness of the closed unit ball, and dominated convergence; these are independent mathematical inputs, not consequences of the claimed conclusion. Theorem 2 reduces the sample problem to a finite-dimensional subsphere by constructing an orthogonal operator fixing the data and rotating any candidate s into span(u,x_1,...,x_n); the equality inf over S = inf over U∩S follows algebraically, not by assumption. Proposition 2 checks the Algorithm 1 output directly. There are no fitted parameters, no predictions of data-derived quantities, and no load-bearing self-citations. The one substantive concern—Theorem 1's U = span{supp(ν)} should be read as the closed linear span for the projection and weak-compactness steps to be valid—is a rigor/repair issue, not a circularity: correcting it does not feed the conclusion back into an assumption, and it leaves Proposition 1, Theorem 2, and Proposition 2 intact.

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

No free parameters or invented entities. The results rely only on standard Hilbert-space facts, the definition of Fréchet/L^p centres, and the closed-span convention discussed above.

assumptions (5)

domain assumption U in Theorem 1 must be the closed linear span of supp(ν) (or the proof should replace U by its closure); orthogonal projection onto U and weak compactness of its unit ball are used.
Section 2, proof of Theorem 1: 'Denote u=Ps where P:H→H is the orthogonal projection onto U' and 'By weak compactness of B'. Algebraic span need not be closed, so the theorem's proof depends on the closed-span convention.
standard math Weak compactness of the closed unit ball in a Hilbert space.
Used to extract weakly convergent subsequences in Theorem 1 (and Theorem 2 via finite dimension).
standard math Existence of an orthogonal/unitary operator on a Hilbert space sending one vector of given norm to another vector of the same norm in the orthogonal complement of fixed data.
Used in proof of Theorem 2 to rotate any s∈S into span(u,x_1,...,x_n) while fixing the data.
standard math Dominated convergence theorem.
Used in Theorem 1 to pass the limit under the integral for φ(d_S(u_n,x)).
standard math For a complete orthonormal sequence (u_n), the only vector orthogonal to every u_n is 0.
Used in proof of Proposition 1 to force the putative minimiser to be the zero vector.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On the existence and non-existence of centres of mass on Hilbert spheres." pith.science (2026). https://pith.science/paper/WNUIFZHF

@misc{pith2026260713667,
  author       = {Pith},
  title        = {Pith review of: On the existence and non-existence of centres of mass on Hilbert spheres},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WNUIFZHF}},
  note         = {Machine review of arXiv:2607.13667}
}

abstract

Fr\'echet means and $L^p$ centres of mass provide notions of average location in metric spaces. On finite-dimensional spheres, existence follows from compactness. On infinite-dimensional spheres, it is not known whether a centre of mass always exists. We show that this is not always the case, and give a simple assumption under which a centre of mass exists. We then show that finding the sample centre of mass of data $x_1, \ldots, x_n$ on the sphere is always an optimisation problem on a subsphere of manifold dimension at most $n$, regardless of the potentially infinite dimension of the sphere. We conclude with some statistical implications.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

16 extracted references · 5 canonical work pages

[16]

doi: 10.1007/978-94-010-9910-3˙63

ISBN 978-94-010- 9912-7 978-94-010-9910-3. doi: 10.1007/978-94-010-9910-3˙63. 9

work page doi:10.1007/978-94-010-9910-3
[1948]

Fr ´echet Means in Infinite Dimensions.arXiv:2410.17214, May

Adam Quinn Jaffe. Fr ´echet Means in Infinite Dimensions.arXiv:2410.17214, May

arXiv
[1977]

doi: 10.1002/cpa.3160300502

ISSN 1097-0312. doi: 10.1002/cpa.3160300502. David G. Kendall. Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces.Bul- letin of the London Mathematical Society, 16(2):81–121, March

doi:10.1002/cpa.3160300502
[1984]

doi: 10.1112/blms/16.2.81

ISSN 00246093. doi: 10.1112/blms/16.2.81. Wilfrid S. Kendall. Probability, Convexity, and Harmonic Maps with Small Image I: Uniqueness and Fine Existence.Proceedings of the London Mathematical Society, s3-61(2):371–406,

doi:10.1112/blms/16.2.81
[1990]

doi: 10.1112/plms/s3-61.2.371

ISSN 1460-244X. doi: 10.1112/plms/s3-61.2.371. Wilfrid S. Kendall. Convexity and the Hemisphere.Journal of the London Mathematical Society, s2-43 (3):567–576,

doi:10.1112/plms/s3-61.2.371
[1991]

doi: 10.1112/jlms/s2-43.3.567

ISSN 1469-7750. doi: 10.1112/jlms/s2-43.3.567. Huiling Le. Locating Fr ´echet means with application to shape spaces.Advances in Applied Probability, 33(2):324–338, June

doi:10.1112/jlms/s2-43.3.567
[2001]

doi: 10.1017/S0001867800010818

ISSN 0001-8678, 1475-6064. doi: 10.1017/S0001867800010818. Jongmin Lee and Sungkyu Jung. Huber means on Riemannian manifolds.Journal of the Royal Statis- tical Society Series B: Statistical Methodology, page qkaf054, August

work page doi:10.1017/s0001867800010818
[2005]

doi: 10.1214/009053605000000264

ISSN 0090-5364. doi: 10.1214/009053605000000264. Ian L. Dryden and John T. Kent, editors.Geometry Driven Statistics. John Wiley & Sons, Inc, Chichester, West Sussex, UK,

work page doi:10.1214/009053605000000264

Show all 16 references

[2011]

doi: 10.1090/S0002-9939-2010-10541-5

ISSN 0002-9939, 1088-6826. doi: 10.1090/S0002-9939-2010-10541-5. 7 Bijan Afsari, Roberto Tron, and Ren ´e Vidal. On the convergence of gradient descent for finding the riemannian center of mass.SIAM Journal on Control and Optimization, 51(3):2230–2260,

2010 doi
[2013]

Xiongtao Dai

doi: 10.1137/12086282X. Xiongtao Dai. Statistical inference on the Hilbert sphere with application to random densities.Electronic Journal of Statistics, 16(1), January

doi
[2015]

doi: 10.1002/9781118866641

ISBN 978-1-118-86657-3 978-1-118-86660-3 978-1-118-86661-0 978-1-118- 86664-1. doi: 10.1002/9781118866641. Steven N. Evans and Adam Q. Jaffe. Limit theorems for Fr ´echet mean sets.Bernoulli, 30(1):419–447, February

doi
[2017]

doi: 10.21099/tk- bjm/1506353559

ISSN 0387-4982, 2423-821X. doi: 10.21099/tk- bjm/1506353559. 8 Herbert Ziezold. On Expected Figures and a Strong Law of Large Numbers for Random Elements in Quasi-Metric Spaces. In J. Kozesnik, editor,Transactions of the Seventh Prague Conference on Information Theory, Statist...

1974
[2022]

doi: 10.1214/21-EJS1942

ISSN 1935-7524. doi: 10.1214/21-EJS1942. Ian L. Dryden. Statistical analysis on high-dimensional spheres and shape spaces.The Annals of Statistics, 33(4), August

1935 doi
[2024]

doi: 10.3150/23-BEJ1603

ISSN 1350-7265. doi: 10.3150/23-BEJ1603. Maurice Fr ´echet. Les ´el´ements al ´eatoires de nature quelconque dans un espace distanci ´e.Annales de l’institut Henri Poincar´e, 10(4):215–310,

doi
[2025]

doi: 10.1093/jrsssb/qkaf054

ISSN 1369-7412. doi: 10.1093/jrsssb/qkaf054. Xavier Pennec, Stefan Sommer, and Tom Fletcher, editors.Riemannian Geometric Statistics in Medical Image Analysis. Academic Press, San Diego,

doi
[2026]

doi: 10.48550/arXiv.2410.17214. H. Karcher. Riemannian center of mass and mollifier smoothing.Communications on Pure and Applied Mathematics, 30(5):509–541,

doi arXiv

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [16]

doi: 10.1007/978-94-010-9910-3˙63

ISBN 978-94-010- 9912-7 978-94-010-9910-3. doi: 10.1007/978-94-010-9910-3˙63. 9

work page doi:10.1007/978-94-010-9910-3

[2] [1948]

Fr ´echet Means in Infinite Dimensions.arXiv:2410.17214, May

Adam Quinn Jaffe. Fr ´echet Means in Infinite Dimensions.arXiv:2410.17214, May

arXiv

[3] [1977]

doi: 10.1002/cpa.3160300502

ISSN 1097-0312. doi: 10.1002/cpa.3160300502. David G. Kendall. Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces.Bul- letin of the London Mathematical Society, 16(2):81–121, March

doi:10.1002/cpa.3160300502

[4] [1984]

doi: 10.1112/blms/16.2.81

ISSN 00246093. doi: 10.1112/blms/16.2.81. Wilfrid S. Kendall. Probability, Convexity, and Harmonic Maps with Small Image I: Uniqueness and Fine Existence.Proceedings of the London Mathematical Society, s3-61(2):371–406,

doi:10.1112/blms/16.2.81

[5] [1990]

doi: 10.1112/plms/s3-61.2.371

ISSN 1460-244X. doi: 10.1112/plms/s3-61.2.371. Wilfrid S. Kendall. Convexity and the Hemisphere.Journal of the London Mathematical Society, s2-43 (3):567–576,

doi:10.1112/plms/s3-61.2.371

[6] [1991]

doi: 10.1112/jlms/s2-43.3.567

ISSN 1469-7750. doi: 10.1112/jlms/s2-43.3.567. Huiling Le. Locating Fr ´echet means with application to shape spaces.Advances in Applied Probability, 33(2):324–338, June

doi:10.1112/jlms/s2-43.3.567

[7] [2001]

doi: 10.1017/S0001867800010818

ISSN 0001-8678, 1475-6064. doi: 10.1017/S0001867800010818. Jongmin Lee and Sungkyu Jung. Huber means on Riemannian manifolds.Journal of the Royal Statis- tical Society Series B: Statistical Methodology, page qkaf054, August

work page doi:10.1017/s0001867800010818

[8] [2005]

doi: 10.1214/009053605000000264

ISSN 0090-5364. doi: 10.1214/009053605000000264. Ian L. Dryden and John T. Kent, editors.Geometry Driven Statistics. John Wiley & Sons, Inc, Chichester, West Sussex, UK,

work page doi:10.1214/009053605000000264