Conjugation, loop and closure invariants of the iterated-integrals signature

Jeremy Reizenstein; Joscha Diehl; Rosa Prei{\ss}

REVIEW 3 major objections 6 minor 26 references

Conjugation, loop and closure invariants of the iterated-integrals signature

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves that the conjugation-invariant features of the iterated-integrals signature are exactly the cyclic rotations of words, characterizes loop invariants by a bracket annihilation condition, and constructs a closure projection…

desk verdict Conjugation invariants are completely nailed down by a short, clean theorem; the loop-invariant half is an attractive but under-proved sketch. read the letter →

arxiv 2412.19670 v1 pith:NIU7VBZY submitted 2024-12-27 math.RA math.ACmath.COmath.PR

Joscha Diehl , Rosa Prei{ss} , Jeremy Reizenstein This is my paper · ORCID

classification math.RAmath.ACmath.COmath.PR MSC 17B0116T3068R15

keywords iterated-integralssignatureconjugationinvariantsloopclosureshufflealgebrafreeLiecombinatorialnecklacessignedarea

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 gives algebraic characterizations of three natural invariance classes for the iterated-integrals signature of a path: features unchanged by path conjugation, features of closed loops unchanged by moving the starting point, and features unchanged by closing a path with a straight segment. It proves that the conjugation invariants are exactly the sums of cyclic rotations of words, $\operatorname{rot}(w)$, so they correspond level by level to combinatorial necklaces. For loops it proves that a feature $\varphi$ is invariant exactly when $\langle[v,i],\varphi\rangle=0$ for every zero-increment grouplike element $v\in V$ and every coordinate direction $i$, and it constructs a projection $\mathrm{rcl}$ whose image is the right-closure invariants and whose kernel is the shuffle ideal $S$. Loop-and-closure invariants then coincide, as a graded algebra, with the letter-reduced loop invariants, and explicit dimension formulas follow from the generating function $(1-q)^d/(1-dq)$.

What carries the argument

The central objects are the rotation operator $\operatorname{rot}(w)$, which sums the cyclic rotations of a word; the shuffle ideal $S$ generated by the letters; and the space $V$ spanned by grouplike elements whose signature has zero increment. A further load-bearing piece is the coordinates of the first kind indexed by Lyndon words, which freely shuffle-generate the tensor algebra and identify $V_n^\perp$ with $S_n$. The closure results are carried by the explicit operator $\mathrm{rcl} = \mathrm{sh}\circ(\mathrm{id}\otimes H)\circ\Delta_\bullet$, where $\mathrm{sh}$ is the shuffle product and $H$ is the signed shuffle-antipode map; its kernel is exactly $S$, making it a projection onto the right-closure invariants.

What would settle it

Compile the pairing matrix between the space $V_n$ (spanned by grouplike elements with zero increment) and the span of shuffle monomials in non-letter coordinates of the first kind, and check that its rank equals $\dim V_n$ predicted by the generating function $(1-q)^d/(1-dq)$. A rank mismatch gives a concrete counterexample to Lemma 3.5 and hence to the loop-invariant theorem; a concrete element satisfying the bracket condition but failing loop invariance would also falsify Theorem 3.10 directly.

Watch

Extended reading notes

Core claim

The paper's central discovery is that three a priori geometric invariance questions about paths reduce to finite algebraic conditions on words. Theorem 2.11 classifies conjugation invariants completely: $\mathrm{ConjInv} = \operatorname{im}\operatorname{rot}$, the image of the operator that maps a word to the sum of its cyclic rotations, with the repetition multiplicity counted. Theorem 3.10 characterizes loop invariants as $[V,\mathbb{R}^d]^\perp$, where $V$ is the span of grouplike elements with vanishing increment; equivalently, $\varphi$ is a loop invariant iff $\langle[v,i],\varphi\rangle=0$ for all $v\in V$ and all letters $i$. The paper further proves that $\varphi$ is a right-closure invariant iff it lies in the image of the explicit operator $\mathrm{rcl} = \mathrm{sh}\circ(\mathrm{id}\otimes H)\circ\Delta_\bullet$, and that the kernel of $\mathrm{rcl}$ is the shuffle ideal generated by the letters. It leaves as conjectures the precise structure of the loop invariants in low dimensions, in particular whether they are generated by signed areas and conjugation invariants.

Load-bearing premise

The loop-invariant theorem rests on the assumption that a certain set of non-letter coordinates freely generates the whole tensor algebra under the shuffle product; if that generation statement is false or incomplete, the key identification of the zero-increment grouplike span with the orthogonal complement of the shuffle ideal, and with it the characterization of loop invariants, collapses.

Editorial extensions

If this is right

At each word length $m$, the conjugation invariants form a vector space of dimension equal to the number of $m$-bead necklaces, with a canonical basis given by the rotations $\operatorname{rot}(w)$ over necklace representatives.
A signature element is a loop invariant exactly when it is orthogonal to every bracket $[v,i]$ with $v$ in the zero-increment grouplike span $V$; level by level this is a finite-dimensional linear condition.
Right-closure invariants are exactly the image of the explicit operator $\mathrm{rcl}$, and since $\ker\mathrm{rcl}=S$, the loop-and-closure invariants are isomorphic as graded algebras to the letter-reduced loop invariants.
Dimension formulas follow: the loop-and-closure invariants at level $n$ have dimension $\dim V_n - \dim[V_{n-1},\mathbb{R}^d]$, computed from the generating function $(1-q)^d/(1-dq)$.
For $d\ge 2$ there are infinitely many shuffle-algebraically independent conjugation invariants and infinitely many shuffle-independent loop-and-closure invariants, so the invariant feature sets are large enough for practical use.

Reading between the lines

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

A practical consequence the paper does not spell out: because $\operatorname{rot}$ is a linear operator on words, one can precompute a basis of conjugation invariants for each fixed degree once, then project any signature word vector onto this basis to obtain a canonical, start-point-independent feature set.
The closure projection $\mathrm{rcl}$ suggests a data-cleaning step: replacing a raw signature feature $\varphi$ by $\mathrm{rcl}(\varphi)$ removes exactly the component that changes when the path is closed by a straight segment, which is testable on piecewise-linear trajectory data.
The conjectured low-dimensional structure $\mathrm{LoopInv}=S+\mathrm{AreaConj}$ in dimensions 2 and 3 is a strong falsifiable prediction: a computer search through the letter-reduced loop-invariant tables should either find a first counterexample or support the conjecture, effectively settling Conjecture 3.12.

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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

Here is my honest take. The paper's headline result is Theorem 2.11: conjugation invariants of the iterated-integrals signature are exactly the image of the cyclic rotation operator rot. The proof is short and I could not find a gap: Proposition 2.6 reduces to a Lie-bracket condition via Chen-Chow, Lemma 2.7 handles the right-comb induction, and Proposition 2.10 plus the averaging argument gives the classification. That settles a natural question cleanly.

The other main contribution is the framework for closure invariants, with the rcl operator and the projection property. Lemma 4.7 and the isomorphisms in Section 4 are genuinely useful; I have not seen this way of separating the trivial loop invariants from the letter-reduced ones before.

The soft spots are in Section 3. Theorem 3.10, which characterizes loop invariants as [V, R^d]^{\\perp}, is only sketched. The proof says 'by derivation and linearization' and moves from conditions on linear paths to the Lie-bracket condition on all of V. That step is doing real work: one has to show that the letter-level conditions propagate to all Lie polynomials, via the Jacobi identity and Lemma 3.13. The induction is absent, and there is also a duplicate reference typo in the proof. I believe the theorem is likely true — I can see the induction one would write — but the paper as published does not actually prove it. Lemma 3.5, which identifies the orthogonal complement of V with the shuffle ideal S, also has an internally inconsistent definition of R_n: first it is 'shuffle monomials in non-letter words', then immediately it is 'shuffle monomials in non-letter coordinates of the first kind'. Those spaces differ (e.g., for d=2,n=2: span{12,21} vs span{12−21}). The lemma's statement is probably correct and the proof can be fixed, but as written it is not right.

The reliance on [DLPR21, Cor. 4.5] for free shuffle-generation of coordinates of the first kind is real, but that is an external published result, and not obviously suspect; the dependency is at least explicit. The dimension tables and the conjectures are honest and look consistent.

Net: this is a paper worth publishing after revision. The conjugation-invariant classification and the rcl machinery are solid, and the loop-invariant theorem is worth having even in its current sketched form, but it should not go to press without a complete proof and a repaired Lemma 3.5. I would send it to a serious referee, and I would cite the conjugation and closure results in my own work.

Who for: anyone working on signature invariants, rough paths, or combinatorial Hopf algebras. The reading group would enjoy the clean parts and could spend a session reconstructing the missing induction.

Referee Report

3 major / 6 minor

Summary. The paper studies three classes of signature invariants for paths in R^d: conjugation invariants, loop invariants, and closure invariants. Section 2 proves that the space of conjugation invariants is exactly the image of the cyclic rotation operator rot on words (Theorem 2.11), with a clean proof via Chen's identity and a characterization in terms of annihilation of Lie brackets with letters (Proposition 2.6). Section 3 characterizes loop invariants: Theorem 3.10 asserts that an element is a loop invariant if and only if it annihilates all brackets [v,i] with v in the span V of zero-increment grouplikes and i a letter. Section 4 introduces a right-closure operator rcl and proves that right-closure invariants form the image of rcl, with dimension counts given by the generating function of V; it also relates loop-and-closure invariants to letter-reduced loop invariants. Several conjectures are clearly stated as such, including the description LoopInv = S + AreaConj and related dimension conjectures.

Significance. If the main theorems are correct, the paper gives a complete and explicit algebraic classification of conjugation invariants as necklace words, and a tractable linear criterion for loop invariants, together with a concrete operator formulation for closure invariants. Theorem 2.11 is proved in full and is a satisfying result with direct computational consequences. The loop-invariant characterization in Theorem 3.10 is the central new result and is potentially very useful, but its proof is incomplete. The closure-invariant section is largely self-contained and contains useful explicit identities and dimension data. The paper is honest about its conjectural parts and provides reproducible-looking tables and OEIS links, which strengthens its value as a reference.

major comments (3)

[Section 3.1, Theorem 3.10] The proof of Theorem 3.10 is only a sketch and omits the load-bearing reduction from Lie brackets to letter brackets. After Proposition 3.3 one obtains <[L,g],phi>=0 for all Lie polynomials L and all zero-increment grouplikes g, hence <[L,v],phi>=0 for all v in V. To pass from this to condition (7) one must prove that [FL(R^d), V] is generated, as a linear span or closure, by brackets [i,v] with i a letter and v in V; this is not immediate and requires an induction using the Jacobi identity and Lemma 3.13. Conversely, to recover loop invariance from (7) one must show that the letter-level conditions imply <[L,v],phi>=0 for every Lie polynomial L; this reverse induction is also absent. The sentence 'by derivation and linearization' is therefore not a complete proof of the central characterization and must be expanded into a full argument.
[Section 3.1, Lemma 3.5] The definition of R_n inside the proof of Lemma 3.5 is internally inconsistent. The proof first defines R_n as the span of shuffle monomials in non-letter words and then identifies it with the span of shuffle monomials in non-letter coordinates of the first kind. These two spaces differ; for d=2 and n=2 they are respectively span{12,21} and span{12-21}. With the first choice the direct sum T_n = S_n + R_n fails because S_2 and span{12,21} intersect nontrivially. Consequently the argument for the dimension identity V_n = S_n^\perp is not established as written. Since Lemma 3.5 is used in Theorem 3.10, Proposition 3.6, Corollary 3.7, and the dimension counts in Section 4, this proof needs to be repaired or replaced with a consistent definition of R_n.
[Section 3.1, Proposition 3.6] The proof of Proposition 3.6 relies on [DLPR21, Corollary 4.5] for the assertion that the dual elements f_i freely shuffle-generate T(R^d). This same result is also used in Lemma 3.5 to construct R_n. Because the cited item is a self-cited preprint and is load-bearing for the loop-invariant characterization, the statement should be reproduced in the paper, or the proof should be given directly. As it stands, the reader cannot verify from the manuscript that the particular dual basis constructed here has the required free shuffle-generation property.

minor comments (6)

[Section 3.1, Theorem 3.10 proof] The proof says 'By Proposition 3.2 and Proposition 3.2'; the second reference should presumably be Proposition 3.3.
[Section 3.1, after Corollary 3.7] The unnumbered display contains 'dim(V_n/ dim[V_{n-1}, R^d])'; the stray 'dim' in the denominator should be removed so the expression reads 'dim(V_n/[V_{n-1},R^d])'.
[Abstract and title page] The abstract contains the typo 'interated integrals', and the title page has spacing errors such as 'TU Berli n' and 'a natura l'; these should be corrected.
[Table 3.1] The level-12 row is missing entries for [V_{n-1},R^2] and for the letter-reduced loop invariants; the missing values should be supplied or the row should be annotated as incomplete.
[References] The reference [DLPR21] should include the arXiv identifier or a journal/digital object identifier if one is available.
[Remark 2.14] The phrase 'For dimension 2 and up' should be 'For dimensions 2 and up'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main conjugation-invariant theorem is derived from Chen's identity and Chen-Chow, and the loop-invariant characterization has proof gaps but does not reduce to its own inputs.

full rationale

The paper's headline result ConjInv = im rot (Theorem 2.11) is not circular. Proposition 2.6 reduces conjugation invariance to annihilation by brackets with letters, using Chen's identity and the Chen-Chow theorem for the grouplike-to-path step; Proposition 2.10 and Theorem 2.11 then finish with linear algebra on the cyclic shift. No parameter is fitted and no target statement is used as an assumption. The loop-invariant section contains an abbreviated step in Theorem 3.10 ('by derivation and linearization') that is not fully written out; making it rigorous would require an induction over Lie polynomials using Lemma 3.13 and the Jacobi identity, so this is a completeness gap rather than a circular reduction. Lemma 3.5 and Proposition 3.6 rely on the self-cited shuffle-generation result [DLPR21, Cor. 4.5]. Under the reviewing rules this citation is parameter-free, does not assume the target invariant characterization, and is externally checkable; it therefore supplies real evidence rather than a circular justification. There is also an internal inconsistency in Lemma 3.5's definition of R_n as written (for d=2, n=2, shuffle monomials in non-letter words span {12,21}, whereas non-letter coordinates of the first kind span {12−21}), but this is a proof error that affects completeness, not a self-referential derivation. The paper also honestly labels its unresolved cases as conjectures. No circular step is exhibited.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is pure mathematics with no fitted parameters and no new postulated entities. The central derivations use standard results from signature theory (Chen's identity, Chen-Chow) and one non-trivial shuffle-generation result from the authors' own prior work (DLPR21). The latter is the main external dependency for the loop invariant part.

assumptions (4)

standard math Chen's identity: sig(A ⊔ B) = sig(A) sig(B) under the concatenation product.
Used in Prop 2.4 to prove that rot(w) is a conjugation invariant, and throughout the paper for signature computations.
standard math Chen-Chow theorem: every group-like element in the tensor algebra can be realized as the signature of a piecewise smooth path.
Used in Prop 2.6, 3.3, and 4.2 to pass from signature evaluations to arbitrary group-like elements.
domain assumption The signature is well defined on paths modulo tree-like equivalence.
Used in Prop 3.2, where A ⊔ B^{-1} ⊔ B is identified with A via tree-like equivalence.
domain assumption Coordinates of the first kind (a dual basis of the free Lie algebra) freely shuffle-generate the tensor algebra (DLPR21, Cor. 4.5).
Load-bearing for Lemma 3.5 and Prop 3.6, where it is used to prove V_n^⊥ = S_n and V_n = span of non-letter Lie monomials. This is a self-cited result and is not machine-checked.

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Cite this review

Pith. "Pith review of Conjugation, loop and closure invariants of the iterated-integrals signature." pith.science (2026). https://pith.science/paper/NIU7VBZY

@misc{pith2026241219670,
  author       = {Pith},
  title        = {Pith review of: Conjugation, loop and closure invariants of the iterated-integrals signature},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NIU7VBZY}},
  note         = {Machine review of arXiv:2412.19670}
}

read the original abstract

Given a feature set for the shape of a closed loop, it is natural to ask which features in that set do not change when the starting point of the path is moved. For example, in two dimensions, the area enclosed by the path does not depend on the starting point. In the present article, we characterize such loop invariants among all those features known as interated integrals of a given path. Furthermore, we relate these to conjugation invariants, which are a canonical object of study when treating (tree reduced) paths as a group with multiplication given by the concatenation. Finally, closure invariants are a third class in this context which is of particular relevance when studying piecewise linear trajectories, e.g. given by linear interpolation of time series. Keywords: invariant features; concatenation of paths; combinatorial necklaces; shuffle algebra; free Lie algebra; signed area; signed volume; tree-like equivalence.

Figures

Figures reproduced from arXiv: 2412.19670 by the authors.

**Figure 2.1.** Two conjugate paths. Their signatures will agre [PITH_FULL_IMAGE:figures/full_fig_p004_2_1.png] view at source ↗

**Figure 2.2.** The path γ which consists of four axis-aligned line segments. 2.2.2 Signed volume For paths in more than two dimensions, the first conjugation invariants which are not just shuffles of letters occur at level 3, in the homogeneity class(es) which contain one each of three distinct letters. Specifically, using the first three letters, rot(123) and rot(132) are distinct conjugation invariants, but only their sum rot(12… view at source ↗

**Figure 3.1.** Two paths A and B which are loops differing only by starting point. A is the concatenation of a path C from a to b and a path D from b to a, while B is the concatenation of D with C. Definition 3.1. A signature element φ ∈ T(R d ) is called a loop invariant if for any two closed paths A and B which only differ by their starting point, hsig(A), φi = hsig(B), φi. A signature element φ ∈ T(R d ) is called a conjugation… view at source ↗

Figures from the paper (1 more)

**Figure 3.2.** Figure 3.2: A closed path A conjugated by a path B. the graded subalgebra of loop invariants (which, by the previous statements is the same as the subalgebra of conjugation invariants for loops). Clearly ConjInv ⊂ LoopInv, and the inclusion is strict for d ≥ 2, since for example…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

The infinite brownian loop on a symmetric space

Jean-Philippe Anker, Philippe Bougerol, and Thierry Jeulin. The infinite brownian loop on a symmetric space. Revista Matemática Iberoamericana , 18(1):41--97, 2002

work page 2002

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work page 2019

[3] [3]

Cyclic quasi-symmetric functions

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work page 2021

[4] [4]

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Carlos Am \'e ndola, Darrick Lee, and Chiara Meroni. Convex hulls of curves: Volumes and signatures. In International Conference on Geometric Science of Information , pages 455--464. Springer, 2023

work page 2023

[5] [5]

The signature of a rough path: uniqueness

Horatio Boedihardjo, Xi Geng, Terry Lyons, and Danyu Yang. The signature of a rough path: uniqueness. Advances in Mathematics , 293:720--737, 2016

2016

[6] [6]

Gustavo E.A. P.A. Batista, Eamonn J. Keogh, Oben Moses Tataw, and Vinícius M.A. de Souza. CID: an efficient complexity-invariant distance for time series . Data Mining and Knowledge Discovery , 28(3):634--669, 2014

work page 2014

[7] [7]

Invariants and coinvariants of the symmetric group in noncommuting variables

Nantel Bergeron, Christophe Reutenauer, Mercedes Rosas, and Mike Zabrocki. Invariants and coinvariants of the symmetric group in noncommuting variables. Canadian Journal of Mathematics , 60(2):266--296, 2008

work page 2008

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Emmanuel Briand, Mercedes Rosas, and Mike Zabrocki. On the sn-module structure of the noncommutative harmonics. Journal of Combinatorial Theory, Series A , 115(6):1077--1085, 2008

work page 2008