REVIEW 2 major objections 4 minor 2 cited by
Computing Path Signature Varieties in Macaulay2
T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash
Pith's one-line read A Macaulay2 package computes path signature varieties as ideals, dimensions, and degrees.
desk verdict A solid software paper for Macaulay2 that fills a real niche; the math checks out on the worked examples, but the missing public release of the package is the main blocker. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery centers on the signature map $\sigma$ and three algebraic structures built around it. Chen's formula computes the signature of a concatenated path from the signatures of its pieces, which is what makes piecewise polynomial paths tractable. The shuffle product makes $\sigma$ an algebra homomorphism to the coefficient ring, and the shuffle algebra is free commutative over Lyndon words; bracketings of those Lyndon words provide a basis of the free Lie algebra, so elements of $\mathrm{Lie}_k(\mathbb{R}^d)$ can be exponentiated by `tensorExp` to parametrize the universal variety $U_{d,k}$. For families of paths, the package precomputes the core tensors of the canonical axis and monomial paths and transports them by the diagonal action of a linear map.
What would settle it
Compute the signature of a piecewise polynomial path with a zero-length segment or a repeated concatenation and compare against direct numerical integration of the iterated integrals; a disagreement would show the Chen-formula implementation misses an edge case. Alternatively, use the package to compute the dimension and degree of a signature variety not in the literature tables, such as $L_{4,3,3}$ or $P_{3,5,4}$, and check against a numerical irreducible decomposition.
Extended reading notes
Core claim
The paper claims that signature varieties of piecewise polynomial paths can be computed routinely in Macaulay2 by combining a few algebraic ingredients: Chen's formula for concatenating segment signatures, the shuffle product on the free associative algebra, Lyndon words as free commutative generators whose bracketings span the free Lie algebra, and the exponential map $\exp$ from Lie coordinates to signature tensors. With these ingredients, the package produces parametrizations of the universal variety $U_{d,k}$ and of the polynomial and piecewise linear signature varieties $P_{d,k,m}$ and $L_{d,k,m}$, and implicitizes them to obtain defining ideals, dimensions, and degrees. The authors demonstrate this on examples, including $U_{2,3}$ (dimension 5, degree 4, six quadrics), $L_{3,3,2}$ (projective dimension 5, degree 90, one linear and 162 quadratic generators), and $P_{2,4,3}$ and $L_{2,4,3}$ (projective dimension 5, degrees 192 and 64), matching tables in [AFS19]. The package also implements the adjoint homomorphism for polynomial transformations of paths, so the signature of a transformed path can be obtained by evaluating an algebra map instead of computing new iterated integrals.
Load-bearing premise
The claim depends on the implementation of Chen's formula and the Lyndon-word shuffle algebra being exact for every piecewise polynomial path; the paper verifies this on selected examples but does not prove correctness or specify behavior for degenerate segments, high-degree polynomials, or repeated concatenations.
Editorial extensions
If this is right
- For any piecewise linear or polynomial family of paths, the package yields a parametrization of the associated signature variety, and from it the defining ideal, dimension, degree, and minimal generators.
- When symbolic implicitization is infeasible, the package’s parametrizations plug directly into numerical implicitization routines, extending computations to larger $d$, $k$, and $m$.
- Signatures under polynomial transformations of paths reduce to evaluating the adjoint word map, so no iterated integrals need to be computed anew for each transformed path.
- The reproduced examples give users a checkable baseline: $U_{2,3}$, $L_{3,3,2}$, $P_{2,4,3}$, and $L_{2,4,3}$ all match published dimensions, degrees, and generator counts.
Reading between the lines
- If the implementation is exact beyond the tested cases, the package is a ready data generator for conjectures about signature varieties, such as degree formulas or generator structures for larger $d$ and $k$.
- The same Lyndon-word and shuffle-algebra pipeline could be ported to other computer algebra systems or to numerical libraries, because the underlying identities are independent of Macaulay2.
- A natural next test is to exercise degenerate cases; the current examples do not include zero-length segments, repeated concatenations, or high-degree polynomials, so those behaviors remain unverified.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript introduces a Macaulay2 package, PathSignatures, for computing signatures of piecewise polynomial paths and the algebraic varieties associated with signature tensors. It defines a Path type for parametrized piecewise polynomial paths, computes signature tensors via Chen's formula, implements shuffle and half-shuffle products, Lyndon-word bases, and methods for universal varieties and core signature tensors. The package is demonstrated on several examples, including the implicitization of the universal variety U_{2,3}, the piecewise linear signature variety L_{3,3,2}, and numerical computations for P_{2,4,3} and L_{2,4,3}, with results matching tables from AFS19. The paper also states an equivariance relation for polynomial transformations of paths and verifies an instance of it.
Significance. If the package is made publicly available and its signature engine is reliable, this is a useful software contribution to the algebraic statistics and nonlinear algebra community: it is the first dedicated Macaulay2 package for path signature varieties, and it combines exact symbolic implicitization with numerical fallback through NumericalImplicitization. The worked examples reproduce published dimensions, degrees, and ideals from AFS19, providing a credible sanity check for the computations. The mathematical content is largely computational rather than new theoretical results, but the package has the potential to lower the barrier for studying signature varieties. The examples are machine-checked transcripts, which is a strength: they are specific and reproducible in principle.
major comments (2)
- [Section 1] The central claim of the paper is the introduction of a Macaulay2 package, yet no code, repository, version identifier, or installation instructions are provided. The Macaulay2 session transcripts in the examples cannot be run or checked by the reader. For a software paper, the artifact is load-bearing: without it, the contribution is not independently verifiable. Please release the package with a persistent identifier and a version, and document its dependencies (NCAlgebra and NumericalImplicitization).
- [Section 2] The paper states that "the path signature is computed from the polynomial pieces through Chen's Formula" but gives no specification of the implementation's behavior for edge cases such as zero-length or degenerate segments, repeated concatenations, high-degree polynomials, or coefficient rings that are not fields. The examples in Sections 4 and 6 test only a few low-dimensional cases. Since the abstract and introduction claim support for all piecewise polynomial paths, this is a gap in evidence: a hidden implementation bug in one of these regimes would not be caught by the printed examples. Please provide a thorough test suite or a correctness argument for the signature engine, and specify the exact assumptions on the allowed inputs.
minor comments (4)
- [Section 4] In Example 4.1, the line defining the ring Q is hard to parse: `Q = QQ new Array from (apply(lwords,i->y_i) | {Degrees => apply(lwords, i->length(i))});` uses `i` for a Lyndon word, which is confusing. Please use a different index name, such as `w`.
- [Section 6] In the sentence 'each coordinate σ_{i1,...,ik} of the tensor σ(k)(X) is a homogeneous polynomial of degree k', the notation σ_{i1,...,ik} is undefined. Consider defining it as the coefficient of the word i1...ik in the signature.
- [Section 3] The output of lyndonShuffle in Example 3.3 is a nested HashTable, which is hard to read. A more human-readable display would help, but this is a presentation issue.
- [Throughout] The authors should ensure that all code blocks are syntactically valid Macaulay2 code as printed; for instance, in Example 2.1 the line `i4 : M = id_(QQ^3);` is missing a space between `id` and `_` in the Macaulay2 convention (the printed form is acceptable, but please verify consistency).
Circularity Check
No significant circularity: the package computations are self-contained and benchmarked against independently published tables.
full rationale
The paper's derivation chain is self-contained. Section 2 encodes piecewise polynomial paths by polynomial segments and computes signatures from those segments via Chen's formula, which is a standard theorem and is cited to [AFS19, Corollary 5.1] only as a reference, not as a fitted input. Section 3 constructs shuffle products and Lyndon-word representations directly from the free associative algebra, with no parameter fitting. Section 4 computes the universal variety by taking a Lyndon-word basis of the free Lie algebra and applying the exponential map, then obtains the prime ideal by implicitization; this is an independent algebraic computation whose output is compared with [AFS19, Table 2] as a benchmark, not used as an input. Section 5 verifies the equivariance and polynomial-transformation formulas by direct computation, with the half-shuffle homomorphism taken from [CP20], an external prior work not authored by the present paper's authors. Section 6 constructs signature varieties from explicit core tensors and linear images, and again compares dimensions and degrees to [AFS19, Table 3] only as consistency checks. No fitted parameter is renamed as a prediction, no uniqueness theorem from the authors' own prior work is invoked to force a choice, and no ansatz is smuggled in via citation. The shared author in [AFS19] does not make the comparison circular because the cited results are previously published, peer-reviewed, and externally checkable; the package's computations reproduce them rather than being defined by them. Thus the paper shows no significant circularity.
Assumptions & free parameters
assumptions (4)
- domain assumption Chen's formula for the signature of concatenated paths is valid and correctly implemented.
- standard math Lyndon words form a basis for the free Lie algebra and the shuffle algebra, so the package's Lyndon-word representation is sound.
- standard math The Chen-Chow theorem correctly identifies the image of the signature map with the exponential of the Lie algebra, justifying the universal variety parametrization.
- domain assumption Macaulay2's NCAlgebra and NumericalImplicitization packages are reliable for the computations performed.
Cite this review
Pith. "Pith review of Computing Path Signature Varieties in Macaulay2." pith.science (2026). https://pith.science/paper/3GZVNGX3
@misc{pith2026250601429,
author = {Pith},
title = {Pith review of: Computing Path Signature Varieties in Macaulay2},
year = {2026},
howpublished = {\url{https://pith.science/paper/3GZVNGX3}},
note = {Machine review of arXiv:2506.01429}
}
abstract
The signature of a path is a non-commutative power series whose coefficients are given by certain iterated integrals over the path coordinates. This series almost uniquely characterizes the path up to translation and reparameterization. Taking only fixed degree parts of these series yields signature tensors. We introduce the Macaulay2 package $\texttt{PathSignatures}$ to simplify the study of these interesting objects for piecewise polynomial paths. It allows for the creation and manipulation of parametrized families of paths and provides methods for computing their signature tensors and their associated algebraic varieties.
Forward citations
Cited by 2 Pith papers
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Learning Barycenters from Signature Matrices
For quadratic truncation, the minimal number of segments needed to recover a path from the barycenter of signature matrices is min(d, total segments) if all segment counts are even, and min(d, total segments minus the...
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SignatureTensors.jl: A Package for Signature Tensors in Julia
SignatureTensors.jl is a new Julia package that computes signature tensors of paths, supporting both exact symbolic and numerical computations via compatibility with the OSCAR computer algebra system.
Reference graph
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Reviewed August 7, 2026 · model on record in the stance chip above.
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