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arxiv: 1907.08423 · v1 · pith:6I7UZSRUnew · submitted 2019-07-19 · 🧮 math.PR · cs.NA· math.NA

Insertion algorithm for inverting the signature of a path

Jiawei Chang , Terry Lyons This is my paper

Pith reviewed 2026-05-24 19:27 UTC · model grok-4.3

classification 🧮 math.PR cs.NAmath.NA
keywords signature inversioninsertion methodnormalized signaturesmooth pathspiecewise linear pathspath reconstructionrough paths
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The pith

An insertion method reconstructs a path from its signature by bounding differences in normalized terms.

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

The paper presents an insertion algorithm designed to invert the signature and recover the original path. It establishes that for smooth paths the difference between each inserted n-th term and the subsequent (n+1)-th term of the normalized signature admits a converging upper bound. It further shows that for piecewise linear paths a subsequence of the normalized signature terms possesses a constant lower bound. These analytic results are illustrated through numerical examples that demonstrate the reconstruction procedure.

Core claim

We introduce the insertion method for reconstructing the path from its signature, i.e. inverting the signature of a path. For this purpose, we prove that a converging upper bound exists for the difference between the inserted n-th term and the (n+1)-th term of the normalised signature of a smooth path, and we also show that there exists a constant lower bound for a subsequence of the terms in the normalised signature of a piecewise linear path.

What carries the argument

The insertion method, which inserts successive terms into the signature sequence and uses the proven bounds to reconstruct the underlying path.

If this is right

  • The signature of a smooth path can be inverted by successively inserting terms whose differences converge.
  • A constant lower bound on a subsequence of normalized signature terms permits identification of the underlying piecewise linear path.
  • Numerical evaluation of the insertion procedure recovers concrete paths from their signatures in both the smooth and piecewise-linear cases.

Where Pith is reading between the lines

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

  • The same insertion bounds may supply error estimates when the method is applied to approximate or discretised signatures arising in applications.
  • If the lower-bound property extends beyond piecewise-linear paths to paths of bounded variation, the algorithm could apply to a wider class of time series.

Load-bearing premise

The paths must belong to the smooth class for the converging upper bound or to the piecewise-linear class for the constant lower bound.

What would settle it

Finding a smooth path for which the difference between the inserted n-th term and the (n+1)-th term of the normalized signature fails to admit a converging upper bound would disprove the first claim.

Figures

Figures reproduced from arXiv: 1907.08423 by Jiawei Chang, Terry Lyons.

Figure 1
Figure 1. Figure 1: Reconstruction of y as a function of x for the path in Example 2.1 We note that by solving the linear equation (2) for γu − γ0 and γT − γu, we are able to reconstruct exactly the underlying path. We can now computationally reconstruct a path consisting of two linear pieces. If γ : [0, t] → R d is a path consisting of linear pieces, Let the 2d × d n+1 matrix A represent the linear mapping · ⊗ S n 0,T (γ) + … view at source ↗
Figure 2
Figure 2. Figure 2 [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The Hilbert-Schmidt norm of the normalised signature of a monotone lattice [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Reconstruction of a semicircle under ` 2 norm, where n is the level of signature used 28 [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Reconstruction of a circle under ` 2 norm, where n is the level of signature used 29 [PITH_FULL_IMAGE:figures/full_fig_p029_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Reconstruction of the digit ‘8’ using the insertion method [PITH_FULL_IMAGE:figures/full_fig_p031_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Reconstruction of digits from the data set [10] using signature level 9 and 10 [PITH_FULL_IMAGE:figures/full_fig_p032_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Reconstruction of digits from the data set [10] using signature level 9 and 10 [PITH_FULL_IMAGE:figures/full_fig_p033_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Reconstruction of digits from the data set [10] using signature level 9 and 10 [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Reconstruction of digits from the data set [10] using signature level 9 and 10 [PITH_FULL_IMAGE:figures/full_fig_p035_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Reconstruction of digits from the data set [10] using signature level 9 and 10 [PITH_FULL_IMAGE:figures/full_fig_p038_13.png] view at source ↗
read the original abstract

In this article we introduce the insertion method for reconstructing the path from its signature, i.e. inverting the signature of a path. For this purpose, we prove that a converging upper bound exists for the difference between the inserted n-th term and the (n+1)-th term of the normalised signature of a smooth path, and we also show that there exists a constant lower bound for a subsequence of the terms in the normalised signature of a piecewise linear path. We demonstrate our results with numerical examples.

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

0 major / 3 minor

Summary. The paper introduces an insertion method for inverting the signature of a path (i.e., reconstructing the path from its signature). It proves the existence of a converging upper bound on the difference between the inserted n-th term and the (n+1)-th term of the normalised signature for smooth paths, and the existence of a constant lower bound for a subsequence of terms in the normalised signature for piecewise linear paths. The claims are supported by numerical examples.

Significance. Signature inversion is a core problem in rough path theory with applications in stochastic analysis and data science. If the stated bounds are rigorously established as claimed, the insertion method supplies a constructive approach with explicit convergence guarantees differentiated by path regularity class (smooth vs. piecewise linear). The provision of both an upper bound that converges and a constant lower bound on a subsequence constitutes a concrete theoretical contribution that could guide practical reconstruction algorithms.

minor comments (3)
  1. [Abstract] The abstract refers to the 'normalised signature' without a definition or citation; a one-sentence clarification or pointer to the relevant section would improve readability for readers outside the immediate subfield.
  2. Numerical examples are mentioned but the manuscript does not specify the path classes, truncation levels, or error metrics used in the demonstrations; adding these details would strengthen the connection between the proved bounds and the reported illustrations.
  3. Notation for the inserted terms and the difference being bounded should be introduced with an explicit equation reference in the main text to avoid ambiguity when the bounds are stated.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of its significance for signature inversion, and recommendation of minor revision. We appreciate the note that the insertion method could supply a constructive approach with explicit convergence guarantees. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces the insertion method and states two explicit theorems: a converging upper bound on the difference between inserted n-th and (n+1)-th terms for smooth paths, and a constant lower bound on a subsequence for piecewise-linear paths. Both results are presented as direct proofs from the definition of the signature transform on the stated path classes, with no reduction of any claim to a fitted parameter, self-referential definition, or load-bearing prior result by the same authors. The derivation chain therefore remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the work relies on standard properties of signatures for smooth and piecewise linear paths.

pith-pipeline@v0.9.0 · 5599 in / 1142 out tokens · 17823 ms · 2026-05-24T19:27:21.515157+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. SignatureTensors.jl: A Package for Signature Tensors in Julia

    cs.SC 2026-04 unverdicted novelty 4.0

    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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