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arxiv: 2606.19223 · v1 · pith:KSZWP6ZXnew · submitted 2026-06-17 · 🧮 math.DG

Diffeomorphic Logarithm of Special Orthogonal Matrices

Zhifeng Deng , P.-A. Absil , Kyle A. Gallivan , Wen Huang This is my paper

Pith reviewed 2026-06-26 19:43 UTC · model grok-4.3

classification 🧮 math.DG
keywords special orthogonal groupmatrix logarithmexponential mapdiffeomorphic regionsSchur decompositionKarcher meanskew-symmetric matrices
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The pith

Each special orthogonal matrix has at most two skew-symmetric preimages under the exponential map in the region containing the principal logarithm.

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

The paper characterizes the local diffeomorphism structure of the exponential map from skew-symmetric matrices to special orthogonal matrices in the set where the derivative is invertible. It organizes this set into diffeomorphic regions by means of a canonical alignment of Schur decompositions. The region that contains the principal logarithm is shown to have a multiplicity structure in which every matrix in SO_n admits at most two preimages. This structure is used to define a diffeomorphic logarithm together with an efficient and stable algorithm for its computation. When the logarithm is applied to the Karcher mean problem on SO_n, the resulting mean varies continuously under small perturbations of the input matrices.

Core claim

The set of skew-symmetric matrices where the derivative of the exponential map is invertible can be partitioned into diffeomorphic regions using a canonical alignment of Schur decompositions. The region containing the principal logarithm has the property that every matrix in SO_n has at most two preimages in this region. This geometric structure allows the introduction of the diffeomorphic logarithm together with an efficient algorithm for its computation.

What carries the argument

Canonical alignment of Schur decompositions, which partitions the set of skew-symmetric matrices with invertible exponential derivative into diffeomorphic regions.

If this is right

  • The diffeomorphic logarithm supplies a continuous single-valued branch of the matrix logarithm on SO_n.
  • An efficient and stable numerical algorithm exists for computing the diffeomorphic logarithm.
  • Karcher means computed with the diffeomorphic logarithm vary continuously when the input matrices are perturbed.
  • The principal logarithm fails to capture this continuous dependence in the same setting.

Where Pith is reading between the lines

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

  • The same Schur-based partitioning technique might be adapted to other compact matrix Lie groups to bound the number of logarithm branches.
  • Numerical optimization routines on rotation manifolds could switch to the diffeomorphic logarithm to avoid discontinuities that appear with the principal branch.
  • Global topology questions about the covering properties of the exponential map on SO_n become more accessible once the local regions are explicitly described.

Load-bearing premise

A canonical alignment of Schur decompositions can partition the skew-symmetric matrices where the derivative of the exponential is invertible into diffeomorphic regions without overlaps or gaps that would allow more than two preimages per special orthogonal matrix.

What would settle it

Exhibit a matrix in SO_n together with three distinct skew-symmetric preimages that all lie inside the single diffeomorphic region containing the principal logarithm and where the derivative of the exponential map remains invertible.

Figures

Figures reproduced from arXiv: 2606.19223 by Kyle A. Gallivan, P.-A. Absil, Wen Huang, Zhifeng Deng.

Figure 1
Figure 1. Figure 1: D-components of Skew5 \ S illustrated by the non-auxiliary angle. The vertical and horizontal lines in S are due to the presence of the auxiliary angle θ3 = 0 in view of Proposition 3.3. When n = 4, those lines disappear and the rest of the illustration is unchanged. (i) C∗ ∩ exp−1 (Q) has at most two preimages, as characterized in Proposition 3.12; (ii) Ce ∩exp−1 (Q) for Ce ̸= C∗ has at most one preimage,… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of angular trajectories on the ( [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Scatter patterns of (θi + 2πxi , θj + 2πxj ) in the (u,v)-chessboard for (xi ,xj )∈Z 2 . Red lines: Rank-deficient conditions u = 2πl and v = 2πl for l ∈ Z \ {0}. Blue crosses: the (u, v) representations of (θi + 2πxi , θj + 2πxj ) for (xi , xj ) ∈ Z 2 . If the continuous angles α(t) of S(t) satisfies (3.7), then (u α ij (t), vα ij (t)) connects (θi + θj + 2π(xi+xj ), θi+θj+2π(xi−xj )) and (θi+θj+2π(yi+yj … view at source ↗
Figure 4
Figure 4. Figure 4: Illustration for the proof of Proposition 4.11. strict D-preimages Y with the canonical shift ξ is D-connected with the strict D-preimages Xε(1) via (4.3), see illustration in [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Venn Diagram of a Preimage X of Q with S ∈ Skewn \ S Proof. Suppose X is a D-preimage that satisfies ∥S − X∥2 < π. Let R be a Schur basis of Q and X = Pm i=1 R[i] [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Compute time and error of different logarithms [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Converged minimizers of the moving data points [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
read the original abstract

The special orthogonal group $\mathbb{SO}_n$ is a Lie group whose geometry and local structure are encoded by the exponential map in its Lie algebra $\mathbf{Skew}_n$, the set of skew-symmetric matrices. The associated multi-valued inverse problem -- the matrix logarithm -- in $\mathbb{SO}_n$ exhibits a highly nontrivial local diffeomorphism structure, which differs from the matrix logarithm for invertible matrices. This work characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices where its derivative is invertible. We show that this set with an invertible derivative can be organized into diffeomorphic regions, using a canonical alignment of Schur decompositions. In particular, the region that contains the principal logarithm has a special multiplicity structure: each matrix in $\mathbb{SO}_n$ admits at most two skew-symmetric preimages in this region. Based on this geometric framework, we introduce the diffeomorphic logarithm of special orthogonal matrices together with an efficient and stable algorithm. Moreover, it is applied to the Karcher mean problem in $\mathbb{SO}_n$, demonstrating continuous behavior of the mean under perturbations of the data, which is not captured by the principal logarithm.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript characterizes the local diffeomorphism structure of the exponential map from skew-symmetric matrices (where its derivative is invertible) to SO(n). Using a canonical alignment of Schur decompositions, it organizes this set into diffeomorphic regions and shows that the distinguished region containing the principal logarithm has the property that every element of SO(n) has at most two preimages therein. It defines the diffeomorphic logarithm, supplies an efficient stable algorithm for its computation, and applies the construction to the Karcher mean on SO(n) to exhibit continuous dependence on data perturbations.

Significance. If the multiplicity bound and partition hold, the work supplies a geometrically motivated logarithm on SO(n) with controlled branching that avoids some discontinuities of the principal logarithm, which is relevant for manifold optimization and statistics. The explicit algorithm and its demonstration on the Karcher mean constitute practical strengths; the result would be of interest in Lie-group numerics.

major comments (2)
  1. [Sections describing the Schur alignment and the principal region] The central construction (canonical alignment of Schur decompositions to partition the locus where Dexp is invertible) is load-bearing for both the diffeomorphic-region claim and the 'at most two preimages' multiplicity bound in the principal region. The manuscript must supply an explicit, continuous rule for the alignment (ordering of 2x2 blocks, choice of signs/angles) together with a proof that the resulting regions are disjoint, cover the locus, and induce diffeomorphisms; absent this, the multiplicity assertion and the subsequent algorithm rest on an unverified assumption.
  2. [Algorithm section and numerical examples] The algorithm for the diffeomorphic logarithm is presented as efficient and stable, yet no complexity analysis, derivative formulas, or numerical verification against the claimed multiplicity bound appears. Without these, it is impossible to confirm that the output remains inside the asserted region for all inputs.
minor comments (2)
  1. [Introduction and preliminaries] Notation for the distinguished regions and the precise definition of 'canonical alignment' should be introduced earlier and used consistently.
  2. [Application section] The Karcher-mean application would benefit from a side-by-side comparison (principal vs. diffeomorphic log) on a concrete perturbation example with quantitative continuity measures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's detailed feedback on our manuscript. The comments highlight areas where the presentation of the Schur alignment and the algorithm can be strengthened with more explicit details and verifications. We will revise the manuscript to address these points fully.

read point-by-point responses

  1. Referee: The central construction (canonical alignment of Schur decompositions to partition the locus where Dexp is invertible) is load-bearing for both the diffeomorphic-region claim and the 'at most two preimages' multiplicity bound in the principal region. The manuscript must supply an explicit, continuous rule for the alignment (ordering of 2x2 blocks, choice of signs/angles) together with a proof that the resulting regions are disjoint, cover the locus, and induce diffeomorphisms; absent this, the multiplicity assertion and the subsequent algorithm rest on an unverified assumption.

    Authors: We acknowledge that the current description of the canonical alignment, while outlined in the sections on Schur decompositions, would benefit from a more explicit and self-contained statement of the ordering rule and the accompanying proof. In the revised version, we will introduce a new subsection that defines the alignment procedure step-by-step: the blocks are ordered by increasing angle magnitude with ties broken by a continuous sign convention derived from the real parts, and we will prove that this yields a partition into disjoint regions each diffeomorphic via the exponential map, covering the entire locus where the derivative is invertible. This will also confirm the multiplicity bound of at most two preimages in the principal region. revision: yes

  2. Referee: The algorithm for the diffeomorphic logarithm is presented as efficient and stable, yet no complexity analysis, derivative formulas, or numerical verification against the claimed multiplicity bound appears. Without these, it is impossible to confirm that the output remains inside the asserted region for all inputs.

    Authors: We agree that additional supporting material is needed for the algorithm. The revised manuscript will include: (i) a complexity analysis showing the dominant cost is the Schur decomposition at O(n^3), with subsequent steps linear in the number of blocks; (ii) explicit formulas for the derivative of the diffeomorphic logarithm using the inverse of the derivative of the exponential on each region; (iii) numerical experiments on random SO(n) matrices for n up to 10, verifying that the computed logarithm stays in the principal region and that no matrix has more than two preimages therein. These additions will substantiate the claims of efficiency, stability, and correctness. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on local invertibility of Dexp and Schur alignment without self-referential reduction

full rationale

The paper's central construction organizes the set where Dexp is invertible into diffeomorphic regions via canonical Schur alignment and then asserts a multiplicity bound of two preimages in the principal region. This is presented as a theorem derived from the Lie group structure and the local diffeomorphism property of exp, not as a definition or a fit to data. No equations reduce a claimed prediction to an input by construction, no load-bearing self-citation is invoked to justify uniqueness, and no ansatz is smuggled via prior work. The framework is self-contained against the external benchmark of the matrix exponential on skew-symmetric matrices.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review limits visibility into explicit parameters or entities; the work rests on standard Lie-group facts plus the new Schur-alignment construction.

axioms (1)
  • domain assumption The exponential map from Skew_n to SO_n is a local diffeomorphism precisely where its derivative is invertible.
    This invertibility condition is used to define the sets that are then organized into regions.

pith-pipeline@v0.9.1-grok · 5742 in / 1222 out tokens · 22333 ms · 2026-06-26T19:43:25.760397+00:00 · methodology

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Reference graph

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