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arxiv: 2009.08443 · v3 · submitted 2020-09-17 · 🧮 math.RA · cs.CV· cs.LG

Tropical time series, iterated-sums signatures and quasisymmetric functions

Joscha Diehl , Kurusch Ebrahimi-Fard , Nikolas Tapia This is my paper

Pith reviewed 2026-05-24 14:22 UTC · model grok-4.3

classification 🧮 math.RA cs.CVcs.LG
keywords time seriesiterated-sums signaturetropical semiringquasisymmetric functionsfeature extractionsemiringschronological features
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The pith

The iterated-sums signature over the tropical semiring extracts chronological features from real-valued time series unavailable from existing signature objects and computes them in linear time.

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

The paper defines an iterated-sums signature that works over any commutative semiring and singles out the tropical case for time series. This version produces features that track order and accumulation in ways standard signatures miss. Computation stays linear in the length of the series. The authors position quasisymmetric expressions over semirings as the natural algebraic setting for the construction. A reader would care because the method gives a concrete, efficient route to new sequential invariants without leaving the semiring setting.

Core claim

The iterated-sums signature is introduced over arbitrary commutative semirings; when the semiring is tropical, the resulting map on real-valued time series yields chronological features not easily obtained from prior signature-type objects, the map can be evaluated in linear time, and quasisymmetric expressions over semirings supply the correct algebraic framework for the semiring-valued case.

What carries the argument

The iterated-sums signature over the tropical semiring, which accumulates ordered sums under tropical addition and multiplication to produce sequence features.

If this is right

  • Real-valued time series acquire new extractable features tied to chronological order.
  • These features remain computable in time linear in the length of the input series.
  • The same signature construction extends directly to time series taking values in any commutative semiring.
  • Quasisymmetric expressions become the algebraic language for describing such signatures.

Where Pith is reading between the lines

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

  • The linear-time property may allow the signature to serve as a drop-in feature map inside streaming or online algorithms.
  • Other semirings could be substituted to target different notions of accumulation, such as probabilistic or fuzzy orderings.
  • The connection to quasisymmetric functions suggests possible links to combinatorial enumeration problems that already use those functions.

Load-bearing premise

Quasisymmetric expressions over semirings form the appropriate framework for iterated-sums signatures on semiring-valued time series.

What would settle it

A concrete time series of length n for which the tropical iterated-sums signature either misses a clear chronological ordering present in the data or requires superlinear operations to compute.

Figures

Figures reproduced from arXiv: 2009.08443 by Joscha Diehl, Kurusch Ebrahimi-Fard, Nikolas Tapia.

Figure 1
Figure 1. Figure 1: Epoch test accuracy for the architecture FCN (left) and ISS (right) for different values of σ. (N = 100, training size: 1000, test size: 100, 100 epochs). where K : R k → R, pool: R( N k ) → R. Now, in this generality, such features are computationally intractable, even for modest values of N and k, since K has to be evaluated N k  times. The iterated￾sums signature presented in this work represents a spe… view at source ↗
Figure 2
Figure 2. Figure 2: Example of data points (in magenta) attaining the minima in eq. (4). calculate ISSRmin+ over large intervals, it suffices to calculate it over small intervals: Example 1.3. For 0 ≤ s < t < u, D ISSRmin+ s,u (z), 74E = M min+ s<j1<j2≤u z min+[1 7 ] j1 min+ z min+[1 4 ] j2 = min s<j1<j2≤u {7zj1 + 4zj2 } = min n min s<j1<j2≤t {7zj1 + 4zj2 }, min t<j1<j2≤u {7zj1 + 4zj2 }, min s<i≤t {7zi} + min t<j≤u {4zj} o = … view at source ↗
Figure 3
Figure 3. Figure 3: Example of the coarsest dyadic partition of (3, 15] Example 5.2. Consider the time horizon T = 16 = 24 and s = 3, t = 15. There is a (unique) coarsest partition of the interval (3, 15] using only dyadic intervals (j · 2 n ,(j + 1) · 2 n ], namely (3, 15] = (3, 4]∪˙ (4, 8]∪˙ (8, 12]∪˙ (12, 14]∪˙ (14, 15], see also [PITH_FULL_IMAGE:figures/full_fig_p031_3.png] view at source ↗
read the original abstract

Aiming for a systematic feature-extraction from time series, we introduce the iterated-sums signature over arbitrary commutative semirings. The case of the tropical semiring is a central, and our motivating example. It leads to features of (real-valued) time series that are not easily available using existing signature-type objects. We demonstrate how the signature extracts chronological aspects of a time series, and that its calculation is possible in linear time. We identify quasisymmetric expressions over semirings as the appropriate framework for iterated-sums signatures over semiring-valued time series.

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 introduces the iterated-sums signature over arbitrary commutative semirings, taking the tropical semiring as its central motivating case. It asserts that the resulting features capture chronological aspects of real-valued time series unavailable from prior signature constructions, that the signature is computable in linear time, and that quasisymmetric expressions over semirings supply the correct algebraic framework for the semiring-valued setting.

Significance. If the definitional framework and the linear-time claim are made fully explicit with supporting algorithms and comparisons, the work would supply a new semiring-compatible extension of signature methods with potential utility in algebraic combinatorics and time-series feature extraction. The explicit link to quasisymmetric functions could unify existing approaches, provided concrete examples demonstrate features not recoverable from classical signatures.

major comments (2)
  1. [Abstract and §1] Abstract and §1: the linear-time computability claim is central to the practical contribution yet no explicit algorithm, recurrence, or complexity argument is supplied in the visible sections; without this the assertion remains unverified.
  2. [§3 or §4] §3 or §4 (whichever contains the comparison): the claim that tropical iterated-sums features are 'not easily available using existing signature-type objects' requires at least one concrete, side-by-side numerical example on a short real time series showing a chronological invariant absent from the classical signature; the current framing is purely definitional.
minor comments (2)
  1. Notation for the semiring operations and the iterated-sums product should be introduced once and used uniformly; several passages mix tropical and classical notation without explicit transition.
  2. [Abstract] The final sentence of the abstract identifies quasisymmetric expressions as 'the appropriate framework'; this identification should be justified by a short theorem or proposition rather than left as a concluding remark.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive suggestions. We address each major comment below and will incorporate revisions accordingly.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1: the linear-time computability claim is central to the practical contribution yet no explicit algorithm, recurrence, or complexity argument is supplied in the visible sections; without this the assertion remains unverified.

    Authors: We agree that providing an explicit algorithm and complexity argument will strengthen the manuscript. Although the linear-time claim is stated, the details of the recurrence are not fully expanded in the current version. In the revision, we will add an explicit description of the algorithm, including the recurrence relation and a proof of O(n) complexity for a time series of length n. revision: yes

  2. Referee: [§3 or §4] §3 or §4 (whichever contains the comparison): the claim that tropical iterated-sums features are 'not easily available using existing signature-type objects' requires at least one concrete, side-by-side numerical example on a short real time series showing a chronological invariant absent from the classical signature; the current framing is purely definitional.

    Authors: The manuscript aims to demonstrate the extraction of chronological aspects, but we recognize that a concrete numerical example would make the distinction clearer. We will include a short real-valued time series example with side-by-side computation of the tropical iterated-sums signature and the classical signature to show a specific chronological feature that is captured only by the former. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces iterated-sums signatures as a new algebraic object over commutative semirings, with the tropical case as a motivating example. All central claims (feature extraction properties, linear-time computation, and identification of quasisymmetric expressions as the framework) are presented as definitional contributions rather than reductions of fitted parameters or prior results. No equations, self-citations, or ansatzes appear in the abstract that would reduce the claimed outputs to the inputs by construction. The work is self-contained as an algebraic definition and does not rely on load-bearing self-citations or uniqueness theorems imported from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that semirings are a natural setting for signatures and that quasisymmetric functions extend appropriately; no free parameters or new physical entities are introduced.

axioms (2)
  • domain assumption Commutative semirings form a suitable algebraic base for defining iterated-sums signatures of time series.
    Invoked to extend the classical signature construction beyond the reals.
  • domain assumption Quasisymmetric expressions over semirings supply the correct bookkeeping for these signatures.
    Stated in the final sentence of the abstract as the appropriate framework.

pith-pipeline@v0.9.0 · 5629 in / 1266 out tokens · 22702 ms · 2026-05-24T14:22:09.682112+00:00 · methodology

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