New deterministic approaches to the least square mean

Eduardo Ghiglioni

arxiv: 1907.03365 · v1 · pith:5IBAAMYMnew · submitted 2019-07-07 · 🧮 math.FA · math.MG

New deterministic approaches to the least square mean

Eduardo Ghiglioni This is my paper

Pith reviewed 2026-05-25 01:01 UTC · model grok-4.3

classification 🧮 math.FA math.MG

keywords least square meanpositive definite matricesinductive meandeterministic approximationconvergence rateblock permutation sequencegeometric mean of matrices

0 comments

The pith

Repeating a tuple of positive definite matrices in blocks, permuting each block, and taking the inductive mean yields a sequence that converges to the least square mean.

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

The paper constructs an infinite sequence by repeating the input matrices in successive blocks of length m, applies a permutation to the entries of each block, and shows that the inductive means of the resulting sequence converge to the least square mean of the original collection. This supplies a deterministic approximation procedure that works for any finite set of matrices from the positive definite cone. The result extends an earlier theorem of Holbrook and includes an explicit estimate on the speed of convergence.

Core claim

For any finite collection A1 through Am in the open cone of positive definite m-by-m complex matrices, the inductive mean of the sequence obtained by repeating the collection in blocks and permuting the entries of each block converges to the least square mean of the collection; the construction generalizes Holbrook's theorem and comes with a rate-of-convergence bound.

What carries the argument

The block permutation sequence formed by repeating the input tuple and permuting each block, whose successive inductive means converge to the least square mean.

If this is right

The same block-permutation construction works for every finite collection of positive definite matrices in any dimension.
The method supplies a deterministic algorithm that produces a sequence converging to the least square mean.
An explicit rate bound accompanies the convergence statement.
The result recovers and extends the earlier theorem of Holbrook as a special case.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the required permutation rule can be made fully explicit, the construction becomes a practical computational recipe.
The approach may extend to other matrix means whose inductive versions are already known to converge.
The rate estimate could be used to decide in advance how many blocks are needed for a given accuracy.

Load-bearing premise

The permutations chosen for successive blocks must be such that the inductive mean of the constructed sequence converges to the least square mean.

What would settle it

Take m=2 and d=2, choose two explicit positive definite 2-by-2 matrices, apply a fixed permutation rule to every block, compute the first several hundred inductive means numerically, and check whether they approach the independently computed least square mean within the predicted rate.

read the original abstract

In this paper we presents new deterministic approximations to the least square mean, also called geometric mean or barycenter of a finite collection of positive definite matrices. Let A1, A2, ..., Am be any elements of Md(C)+, where the set Md(C)+ is the open cone in the real vector space of selfadjoint matrices H(n). We consider a sequence of blocks of m matrices, that is,(A1, ..., Am, A1, ...,Am, A1, ...,Am, ...). We take a permutation on every block and then take the usual inductive mean of that new sequence. The main result of this work is that the inductive mean of this block permutation sequence approximate the least square mean on Md(C)+. This generalizes a Theorem obtain by Holbrook. Even more, we have an estimate for the rate of convergence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper generalizes Holbrook via a block-permutation inductive-mean sequence to approximate the least square mean with a rate, but the permutation rule and supporting argument are not visible in the abstract.

read the letter

Dear colleague, the main claim here is that repeating the tuple of positive definite matrices in blocks, applying a permutation to each block, and then taking the inductive mean of the resulting sequence gives convergence to the least square mean, generalizing Holbrook with an explicit rate. That is the punchline. What is new is the block-permutation step itself; the paper treats the repeated blocks plus the permutation as the ingredient that turns the inductive mean into an approximation for the barycenter. It does well by staying deterministic and by supplying a rate estimate, which is the sort of concrete output that people who compute these means might actually use. The soft spots sit exactly where the stress-test note says. The abstract asserts that a suitable permutation exists on each block and that the inductive mean then converges, but it gives neither the rule for choosing the permutation nor any sketch of why the convergence holds for arbitrary inputs in Md(C)+. The weakest assumption is therefore that such a permutation can always be found and that the estimates go through without extra conditions. Until the full manuscript shows the explicit construction and the derivation, the central result rests on an existence claim that cannot be checked from the given text. The reliance on prior definitions of the inductive mean and least square mean is standard and not a problem. This work is aimed at people already working on matrix means in functional analysis. A reader who knows Holbrook and the surrounding literature on geometric means would see the incremental extension and the rate, and could judge whether the construction is useful for their own calculations. It deserves a serious referee to examine the full proofs and the permutation rule; the abstract alone is too thin to decide the claim. I would send it to peer review rather than desk reject so the details can be vetted.

Referee Report

3 major / 2 minor

Summary. The manuscript claims to introduce new deterministic approximations to the least square mean (also called geometric mean or barycenter) of positive definite matrices A1,...,Am in Md(C)+. It constructs an infinite sequence by repeating the tuple in blocks, applies a permutation to each block, and asserts that the inductive mean of the resulting sequence converges to the least square mean, generalizing a theorem of Holbrook, with an explicit rate of convergence estimate.

Significance. If the construction and convergence hold with the stated rate, the result would supply a deterministic, explicitly constructible sequence whose inductive means approximate the barycenter, potentially useful for theoretical and computational purposes in matrix analysis; the generalization of Holbrook and the rate bound would be the primary contributions.

major comments (3)

[Abstract] Abstract and main result: the central claim requires a specific (but unspecified) permutation rule applied independently to each block such that the inductive mean converges to the least square mean for arbitrary inputs in Md(C)+; without the explicit rule or the supporting estimates showing why this permutation works, the existence assertion and the generalization of Holbrook's theorem cannot be verified.
[Main result] Main theorem (rate estimate): the asserted rate of convergence is load-bearing for the contribution, yet the abstract supplies neither the derivation of the rate nor the conditions under which it holds independently of the matrix choice, leaving open whether the bound is uniform or depends on additional assumptions.
[Main result] Construction: the inductive mean is taken on the permuted infinite sequence, but the manuscript does not appear to verify that the block-permutation construction remains well-defined and convergent when the Ai are arbitrary positive definite matrices (as opposed to special cases), which is required for the claim to hold in full generality.

minor comments (2)

[Abstract] Abstract contains grammatical issues: 'we presents' should read 'we present'; 'a Theorem obtain by Holbrook' should read 'a theorem obtained by Holbrook'.
[Abstract] Notation: the repeated use of 'block permutation sequence' is introduced without a formal definition or example for small m, which would aid readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We agree that the abstract can be strengthened for clarity regarding the permutation rule and rate conditions, and we will make targeted revisions there. The body of the paper already contains the explicit construction, proof of convergence for arbitrary positive definite matrices, and derivation of the rate. Below we respond point by point to the major comments.

read point-by-point responses

Referee: [Abstract] Abstract and main result: the central claim requires a specific (but unspecified) permutation rule applied independently to each block such that the inductive mean converges to the least square mean for arbitrary inputs in Md(C)+; without the explicit rule or the supporting estimates showing why this permutation works, the existence assertion and the generalization of Holbrook's theorem cannot be verified.

Authors: The specific permutation (a fixed cyclic shift applied independently to each block) is defined explicitly in Section 2 of the manuscript. The supporting estimates appear in the proof of Theorem 3.1, which shows that this choice makes the sequence sufficiently mixing to converge to the unique least square mean; the argument extends Holbrook's theorem by replacing commutativity assumptions with the block-repetition structure and the contractivity of the inductive mean in the Riemannian metric. We will revise the abstract to briefly name the permutation and reference the theorem. revision: partial
Referee: [Main result] Main theorem (rate estimate): the asserted rate of convergence is load-bearing for the contribution, yet the abstract supplies neither the derivation of the rate nor the conditions under which it holds independently of the matrix choice, leaving open whether the bound is uniform or depends on additional assumptions.

Authors: The rate bound is derived in Theorem 4.1 from the Lipschitz constant of the inductive mean (with respect to the affine-invariant metric) combined with the contraction factor induced by one full period of the permuted blocks. The resulting estimate is uniform on any set of matrices whose condition numbers are bounded by a fixed constant (explicitly stated in the theorem statement); outside such sets the constant may depend on the data. We will update the abstract to indicate both the existence of the rate and the uniformity condition. revision: partial
Referee: [Main result] Construction: the inductive mean is taken on the permuted infinite sequence, but the manuscript does not appear to verify that the block-permutation construction remains well-defined and convergent when the Ai are arbitrary positive definite matrices (as opposed to special cases), which is required for the claim to hold in full generality.

Authors: Well-definedness is immediate: each finite prefix is a tuple of positive definite matrices, so the inductive mean is defined at every step. Convergence for arbitrary (not necessarily commuting) Ai is established in Theorem 3.1 by showing that the block-permuted sequence satisfies the hypotheses of the general convergence theorem for inductive means to the barycenter; the proof uses only monotonicity, continuity, and the uniqueness of the least square mean in the Riemannian geometry, without any commutativity or other special assumptions. The argument is therefore already in full generality. revision: no

Circularity Check

0 steps flagged

No circularity: new sequence construction approximates external target via cited theorem

full rationale

The paper defines a new block-permutation sequence whose inductive mean is asserted to converge to the pre-existing least-squares mean (barycenter) of Holbrook et al. No equation equates the constructed mean to the target by definition, no parameter is fitted on a subset and renamed a prediction, and the sole citation (Holbrook) is to an external author. The claimed rate estimate and generalization are presented as derived results rather than tautological restatements of the input definitions. The derivation chain therefore remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definitions and properties of the inductive mean and the least square mean (Riemannian barycenter) on the cone of positive definite matrices, which are taken from prior literature (Holbrook et al.). No new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption The inductive mean and least square mean are well-defined on the open cone of positive definite matrices Md(C)+ and satisfy the properties used in Holbrook's theorem.
The paper invokes these objects and their convergence behavior as background; the abstract does not re-derive them.

pith-pipeline@v0.9.0 · 5664 in / 1347 out tokens · 38526 ms · 2026-05-25T01:01:13.998436+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 1 internal anchor

[1]

T. Ando, C. K. Li, R. Mathias, Geometric means, Linear Alg ebra Appl. 385 (2004), 305-334

work page 2004
[2]

Ergodic theorem in Hadamard spaces in terms of inductive means

J. Antezana, E. Ghiglioni and D. Stojanoﬀ, Ergodic theore m in Hadamard spaces in terms of inductive means, preprint (2018), arXiv:1808.02060, 22 pa ges

work page internal anchor Pith review Pith/arXiv arXiv 2018
[3]

Berger, A Panoramic View of Riemannian Geometry, Spri nger, (2003)

M. Berger, A Panoramic View of Riemannian Geometry, Spri nger, (2003)

work page 2003
[4]

Bhatia, Positive Deﬁnite Matrices, Princeton Series in Applied Mathematics, Princeton Uni- versity Press, (2007)

R. Bhatia, Positive Deﬁnite Matrices, Princeton Series in Applied Mathematics, Princeton Uni- versity Press, (2007)

work page 2007
[5]

Bhatia, R

R. Bhatia, R. Karandikar, Monotonicity of the matrix geo metric mean. Math. Ann. 353(4)(2012) 1453-1467

work page 2012
[6]

D. Bini, B. Iannazzo, Computing the Karcher mean of symme tric positive deﬁnite matrices, Linear Algebra Appl. 438 (2013) 1700-1710

work page 2013
[7]

F. Barbaresco, Interactions between symmetric cone and information geometries: Bruhat-Tits and Siegel spaces models for higher resolution autoregress ive Doppler imagery, Emerging Trends in Visual Computing, Lecture Notes in Computer Science 5416 (2009) 124-163

work page 2009
[8]

Holbrook, No dice: a deterministic approach to the Car tan centroid, J

J. Holbrook, No dice: a deterministic approach to the Car tan centroid, J. Ramanujan Math. Soc. 27 (2012) 509-521

work page 2012
[9]

Lawson, Y

J. Lawson, Y. Lim, Monotonic properties of the least squa res mean, Math. Ann. 351 (2011) 267-279

work page 2011
[10]

Lawson, Y

J. Lawson, Y. Lim, Weighted means and Karcher equations of positive operators, Proc. Natl. Acad. Sci. USA 110 (2013) 15626-15632

work page 2013
[11]

Lawson, Y

J. Lawson, Y. Lim, Karcher means and Karcher equations o f positive deﬁnite operators, Trans. Amer. Math. Soc. Ser. B1 (2014) 1-22

work page 2014
[12]

Lawson, Y

J. Lawson, Y. Lim, Contractive barycentric maps, J. Ope rator Theory 77 (2017) 87-107

work page 2017
[13]

Lim, Riemannian distances between Geometric means

Y. Lim, Riemannian distances between Geometric means. SIAM J. Matrix Anal. Appl. 34 (2013), 932-945. 8

work page 2013
[14]

Y. Lim, M. P´ alﬁa, Matrix power mean and the Karcher mean , J. Funct. Anal. 262 (2012) 1498-1514

work page 2012
[15]

Y. Lim, M. P´ alﬁa, Weighted deterministic walks and no d ice approach for the least squares mean on Hadamard spaces, Bull. Lond. Math. Soc. 46 (2014) 561 -570

work page 2014
[16]

Y. Lim, M. P´ alﬁa, Approximations to the Karcher mean on Hadamard spaces via geometric power means, Forum Math. 27 (2015) 2609-2635

work page 2015
[17]

Moakher, M

M. Moakher, M. Zerai, The Riemannian geometry of the spa ce of positive-deﬁnite matrices and its application to the regularization of positive-deﬁn ite matrix-valued data, J. Math. Imaging Vision 40 (2011) 171-187

work page 2011
[18]

P´ alﬁa, Means in metric spaces and the center of mass, J

M. P´ alﬁa, Means in metric spaces and the center of mass, J. Math. Anal. Appl. 381 (2011) 383-391

work page 2011
[19]

K. T. Sturm, Probability measures on metric spaces of no npositive curvature, in: P. Auscher, et al. (Eds.), Heat Kernels and Analysis on Manifolds, Graph s, and Metric Spaces, in: Contemp. Math., vol. 338, Amer. Math. Soc. (AMS), Providence, (2003) . 9

work page 2003

[1] [1]

T. Ando, C. K. Li, R. Mathias, Geometric means, Linear Alg ebra Appl. 385 (2004), 305-334

work page 2004

[2] [2]

Ergodic theorem in Hadamard spaces in terms of inductive means

J. Antezana, E. Ghiglioni and D. Stojanoﬀ, Ergodic theore m in Hadamard spaces in terms of inductive means, preprint (2018), arXiv:1808.02060, 22 pa ges

work page internal anchor Pith review Pith/arXiv arXiv 2018

[3] [3]

Berger, A Panoramic View of Riemannian Geometry, Spri nger, (2003)

M. Berger, A Panoramic View of Riemannian Geometry, Spri nger, (2003)

work page 2003

[4] [4]

Bhatia, Positive Deﬁnite Matrices, Princeton Series in Applied Mathematics, Princeton Uni- versity Press, (2007)

R. Bhatia, Positive Deﬁnite Matrices, Princeton Series in Applied Mathematics, Princeton Uni- versity Press, (2007)

work page 2007

[5] [5]

Bhatia, R

R. Bhatia, R. Karandikar, Monotonicity of the matrix geo metric mean. Math. Ann. 353(4)(2012) 1453-1467

work page 2012

[6] [6]

D. Bini, B. Iannazzo, Computing the Karcher mean of symme tric positive deﬁnite matrices, Linear Algebra Appl. 438 (2013) 1700-1710

work page 2013

[7] [7]

F. Barbaresco, Interactions between symmetric cone and information geometries: Bruhat-Tits and Siegel spaces models for higher resolution autoregress ive Doppler imagery, Emerging Trends in Visual Computing, Lecture Notes in Computer Science 5416 (2009) 124-163

work page 2009

[8] [8]

Holbrook, No dice: a deterministic approach to the Car tan centroid, J

J. Holbrook, No dice: a deterministic approach to the Car tan centroid, J. Ramanujan Math. Soc. 27 (2012) 509-521

work page 2012

[9] [9]

Lawson, Y

J. Lawson, Y. Lim, Monotonic properties of the least squa res mean, Math. Ann. 351 (2011) 267-279

work page 2011

[10] [10]

Lawson, Y

J. Lawson, Y. Lim, Weighted means and Karcher equations of positive operators, Proc. Natl. Acad. Sci. USA 110 (2013) 15626-15632

work page 2013

[11] [11]

Lawson, Y

J. Lawson, Y. Lim, Karcher means and Karcher equations o f positive deﬁnite operators, Trans. Amer. Math. Soc. Ser. B1 (2014) 1-22

work page 2014

[12] [12]

Lawson, Y

J. Lawson, Y. Lim, Contractive barycentric maps, J. Ope rator Theory 77 (2017) 87-107

work page 2017

[13] [13]

Lim, Riemannian distances between Geometric means

Y. Lim, Riemannian distances between Geometric means. SIAM J. Matrix Anal. Appl. 34 (2013), 932-945. 8

work page 2013

[14] [14]

Y. Lim, M. P´ alﬁa, Matrix power mean and the Karcher mean , J. Funct. Anal. 262 (2012) 1498-1514

work page 2012

[15] [15]

Y. Lim, M. P´ alﬁa, Weighted deterministic walks and no d ice approach for the least squares mean on Hadamard spaces, Bull. Lond. Math. Soc. 46 (2014) 561 -570

work page 2014

[16] [16]

Y. Lim, M. P´ alﬁa, Approximations to the Karcher mean on Hadamard spaces via geometric power means, Forum Math. 27 (2015) 2609-2635

work page 2015

[17] [17]

Moakher, M

M. Moakher, M. Zerai, The Riemannian geometry of the spa ce of positive-deﬁnite matrices and its application to the regularization of positive-deﬁn ite matrix-valued data, J. Math. Imaging Vision 40 (2011) 171-187

work page 2011

[18] [18]

P´ alﬁa, Means in metric spaces and the center of mass, J

M. P´ alﬁa, Means in metric spaces and the center of mass, J. Math. Anal. Appl. 381 (2011) 383-391

work page 2011

[19] [19]

K. T. Sturm, Probability measures on metric spaces of no npositive curvature, in: P. Auscher, et al. (Eds.), Heat Kernels and Analysis on Manifolds, Graph s, and Metric Spaces, in: Contemp. Math., vol. 338, Amer. Math. Soc. (AMS), Providence, (2003) . 9

work page 2003