Lag Operator SSMs: A Geometric Framework for Structured State Space Modeling

Kenji Doya; Noboru Murata; Sutashu Tomonaga

REVIEW 3 major objections 3 minor 3 references

Lag Operator SSMs: A Geometric Framework for Structured State Space Modeling

T0 review · 3 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read Discrete structured state-space models can be built directly from a lag operator, with the entire recurrence following from one inner product between warped basis functions.

desk verdict The lag-operator re-framing is useful and the continuous-limit algebra mostly works, but Eq. (23) silently extends the old basis beyond its defining interval, so the paper's central projection derivation is wrong as written and the numerical validation does not test the claimed mechanism. read the letter →

arxiv 2512.18965 v2 pith:SLZHS7PQ submitted 2025-12-22 cs.LG

Sutashu Tomonaga , Kenji Doya , Noboru Murata This is my paper

classification cs.LG

keywords structuredstatespacemodelslagoperatortimewarpingorthogonalprojectionLegendrepolynomialsmemorycompressiononlinerecurrencecontinuous-timelimit

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

The paper seeks to show that the discrete-time recurrence at the heart of structured state-space models can be derived geometrically, without first writing a continuous-time ODE and discretizing it. Its core move is to define a "lag operator" that compares the projection basis at two consecutive times; the next state's transition matrix is then just one inner product involving this operator. This would make SSM design modular: choose a basis and a time warp, and the recurrence follows. To justify the claim, the authors prove that a Legendre-basis instance with exponential time warping reproduces, in the continuous-time limit, the state and input matrices of a well-known polynomial-projection memory system, and they verify the match numerically.

What carries the argument

The central object is the backward lag operator, the self-map of the canonical interval obtained by pulling a coordinate back from the new time horizon through the inverse warp and pushing it forward through the old warp. This single composition carries all information about how the orthogonal projection basis changes between steps, so the transition matrix collapses to an inner product and the input vector is the projection of the input-extension shape onto the new basis. Everything else—the continuous-time generator, the exact finite-step transition, and the match with known memory systems—follows by expanding or exponentiating this inner product.

What would settle it

Take a time warp that is piecewise-smooth but not analytic, for example linear on the recent interval and exponential on the distant past. Compute the transition matrix two ways, once with the lag-operator inner product over the full canonical interval and once by explicitly zero-extending the previous approximation before projecting onto the new basis. If the two matrices differ at any finite timestep, the recurrence is not the claimed projection update.

Watch

Extended reading notes

Core claim

The central claim is that the projection update from time t to t+1 can be written directly as c_{t+1}=A_t c_t + B_t u_{t+1} with (A_t)_{nm}=<phi_n, phi_m composed with sigma_t composed with sigma_{t+1}^{-1}>. The lag operator sigma_t composed with sigma_{t+1}^{-1} expresses how the basis functions "slide" as the history interval extends, and the inner product gives the transition matrix at once, bypassing the ODE and discretization stages. The authors then show that for Legendre polynomials on (0,1] with the exponential warp s maps to e^{s-t}, the continuous-time generator obtained from this discrete construction matches the generator of the established Legendre-based polynomial projection m

Load-bearing premise

The recurrence assumes the previous step's basis functions can be evaluated across the next step's whole domain via the warp composition, even though those functions are only defined on the earlier time interval; if that extension is not valid, the single-inner-product transition is not the projection update it claims to be.

Editorial extensions

If this is right

Choosing a canonical basis and a time-warping map now fully determines the SSM recurrence; the state transition needs no separate discretization step.
Since the warp's derivative is the measure that defines memory weighting, memory profiles, including multi-resolution or adaptive time-dependent profiles, become direct design choices.
For stationary warps, the finite-step transition is the matrix exponential of the derived generator, giving a first-principles path to time-invariant recurrences of the kind used in practice.
The exponential-warp and Legendre instance recovers the continuous-time matrices of a known polynomial-projection memory system, so that system's efficiency can be understood as a consequence of projection geometry rather than of separate ODE analysis.
The exact finite-step transition for that instance is invertible and includes a scalar decay correction, which the paper verifies numerically against standard discretizations as the step size shrinks.

Reading between the lines

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

The paper's proof-of-concept only instantiates stationary analytic warps; a natural next test is whether non-analytic or piecewise time warps still satisfy the single-inner-product identity, since the basis functions may not extend cleanly outside their defining interval.
The transition derivation appears to rely on evaluating the previous basis over the extended domain through the warp composition; if that extension fails for some warp, the recurrence would coincide with the projection update only approximately, so checking this identity is the fastest way to stress-test the framework.
A practical extension the authors leave implicit: because the input hold scheme only changes finite-step corrections and not the continuous limit, downstream tasks with high-frequency inputs may be more sensitive to hold choice than the asymptotic matching suggests.
One could tune the warp parameters end-to-end in a learned model, using the induced measure as an interpretable memory knob; this is a testable consequence the paper does not yet benchmark.

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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

The paper's core idea is genuinely attractive: derive a discrete-time SSM recurrence by measuring how a time-warped basis changes from one step to the next, and write the state transition as a single inner product with a lag operator. The modular design space (basis choice × time warp) is a clean way to organize thinking about structured SSMs, and the generator formulas in Propositions 1 and 2 are not literally present in the cited HiPPO papers. For a reader who already knows HiPPO, this re-framing does offer a new geometric handle.

The soft spot is not minor. Equation (23) rewrites the inner product ⟨ψ_{t+1,n}, ψ_{t,m}⟩_{ω_{t+1}} as an integral over the full canonical interval Z. But ψ_{t,m} is defined only on T_t, and the prior approximation is zero outside T_t (Eq. 18). Under the change of variables, the support of ψ_{t,m} maps to (0, σ_{t+1}(t)], not all of Z. The paper silently analytic-continues the composition φ_m∘σ_t∘σ_{t+1}^{-1} beyond that interval. That is not the projection update. Computing the update honestly for the exponential warp gives A_Δ = e^{Δ A_HiPPO} directly—no need for the inverse-and-decay correction in Eq. (32). The "tilting" shift the paper adds by hand is exactly the boundary term that the analytic continuation introduces. So the validation in Section 4 compares the corrected matrix, not the actual A_Δ from Eq. (23), and Table 3 does not support the mechanism the abstract claims.

There are smaller issues: the identity A_Δ = e^{Δ A_gen} is asserted rather than proven, and the "exact recovery" framing is stronger than Table 3 supports (the difference at Δ=0.1 is 0.955). To the paper's credit, the appendices are detailed enough that the problem is findable, and the authors are honest that this is a proof of concept with no downstream benchmarks.

Who is this for? Researchers who care about the theoretical foundations of SSMs might find the lag-operator viewpoint stimulating once the domain issue is fixed—and the fix likely makes the derivation simpler, not more complex. As written, though, the central derivation needs major revision. I would send it to peer review rather than desk reject: the flaw is specific, reproducible, and fixable, and the underlying geometric idea deserves referee attention. A good referee should ask the authors to redo the projection update with the correct domain and show whether the framework still reproduces HiPPO in that honest computation.

Referee Report

3 major / 3 minor

Summary. The paper introduces a discrete-time SSM construction based on time-warped orthonormal bases. The state transition is claimed to be computable directly from a single inner product involving a backward lag operator ℓ_{t+1→t} = σ_t ∘ σ_{t+1}^{-1}, thereby bypassing the continuous-time ODE and discretization pipeline of HiPPO/S4. For the exponential warp and Legendre basis, the authors claim to recover the HiPPO-LegS system. Numerical experiments compare the resulting matrices to HiPPO-LegS and demonstrate reconstruction of a Lorenz63 signal.

Significance. If the construction were correct, it would provide a clean geometric unification of discrete SSM design, with a modular design space over the warp σ_t and the canonical basis Φ. The paper is careful in its continuous-limit algebra, and the numerical tables (Tables 1–2) show high precision for the objects they compute. The claimed recovery of HiPPO-LegS would be a valuable sanity check. However, the central projection derivation has a domain-truncation gap that affects the main claim, so the significance is contingent on a substantive revision.

major comments (3)

[Section 3.3, Eq. (23)] The derivation of (A_t)_{nm} = ⟨ψ_{t+1,n}, ψ_{t,m}⟩_{ω_{t+1}} = ⟨φ_n, φ_m ∘ σ_t ∘ σ_{t+1}^{-1}⟩ silently changes the integration domain. ψ_{t,m} is defined only on T_t = (−∞, t]; Eq. (18) explicitly states that the prior approximation is zero elsewhere. Under z = σ_{t+1}(s), the support of ψ_{t,m} maps to (0, σ_{t+1}(t)] = (0, e^{−Δ}], not to the full canonical interval Z = (0,1]. The second equality in Eq. (23) integrates over all of Z, implicitly using an analytic continuation of φ_m beyond Z and replacing the zero-extension with that continuation. The honest projection inner product is ∫_0^{e^{−Δ}} φ_n(z) φ_m(e^{Δ} z) dz, not Eq. (23). Thus Eq. (23) is not the projection update defined by Eqs. (18)–(19), and the 'single inner product' recurrence is not the geometric projection claimed.
[Sections 3.4–3.5, Eq. (32)] Because of the domain truncation, the continuous generator A_gen in Proposition 1 is the derivative of the extended full-interval inner product, not of the projection update. The 'tilting' relation A_HiPPO = −(A_gen + I)^T and the corrected discrete matrix A_corrected_Δ = (A_Δ)^{-1} e^{−Δ} are introduced to match HiPPO, but they are not derived from the projection geometry. The numerical validation in Section 4 (Table 3 and Fig. 4) uses A_corrected_Δ, not the A_Δ obtained from Eq. (31). The experiments therefore validate the patched matrix, not the claimed end-to-end recurrence of Eq. (23). The paper should either derive the honest projection matrix (including the boundary contribution) and show how the stable HiPPO matrix follows, or explicitly abandon the projection interpretation and define the lag-operator recurrence as a separate construction.
[Abstract and Section 3.4, 'exact recovery' claim] The paper claims that the exponential-warp/Legendre instance 'exactly recovers' the HiPPO recurrence. However, Propositions 1 and 2 establish only a match of continuous-time generators (A_gen, B_gen) under the extension convention, not the exact discrete-time recurrence. The discrete transition obtained directly from Eq. (23)/(31) does not match the discretized HiPPO system; the match requires the ad hoc inverse-and-decay correction in Eq. (32). The 'exact recovery' claim is therefore unsupported as stated. The authors should qualify the claim to refer to the continuous limit, or supply a derivation that produces the corrected matrix from the projection update itself.

minor comments (3)

[Section 3.1–3.3] The paper uses the same symbol ψ_{t,m} both as a function on T_t and as a function on T_{t+1} in Eq. (23). This notational ambiguity obscures the domain issue; the authors should explicitly define the extension (or zero-extension) used in each inner product.
[Section 4.2.1, Tables 1–2] The relative Frobenius differences are reported without confidence intervals or a discussion of numerical integration error. This is acceptable for a proof of concept, but a sentence on the integration rule used for A_Δ (e.g., Gauss–Legendre quadrature) would improve reproducibility.
[General] The manuscript contains several apparent rendering or formatting artifacts (e.g., 'F ramework', 'T omonaga', 'P ad´ e'), which should be cleaned up in a revision. Also, the phrase 'HiPPO's memory dynamics' in Section 4.2.3 is slightly imprecise: the comparison is to the S4/HiPPO matrix convention, not a dynamics benchmark.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lag-operator derivation is self-contained; the HiPPO match is an external consistency check.

full rationale

The load-bearing construction (Eq. 23) defines the transition matrix as an inner product of time-warped basis functions and rewrites it by change of variables; it does not fit parameters to data or define the target in terms of the result. The validation against HiPPO-LegS is an external benchmark, not a self-citation, and the choice of exponential warp plus Legendre basis is an explicit instance, not a hidden assumption. Propositions 1 and 2 derive A_gen and B_gen from the lag-operator generator and then note the algebraic identity A_HiPPO = -(A_gen+I)^T; importing the 'tilting' term from the cited HiPPO derivation weakens the claim of a fully independent derivation but does not make the derivation circular. The numerical section uses the corrected matrix A_corrected_Δ = (A_Δ)^{-1}e^{-Δ} (Eq. 32), which is deliberately constructed to match the stable HiPPO system; this reduces the evidential value of the numerical match but is not a fitted-input-called-prediction in the statistical sense. The domain-extension issue in Eq. (23) (ψ_{t,m} is zero-extended outside T_t while the lag-operator formula integrates over all of Z) is a correctness concern, not a circularity. Therefore no circular step is established.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The central derivation rests on standard inner-product/change-of-variables mathematics plus a zero-order-hold signal model, but the most load-bearing axiom is the implicit analytic extension of the old basis beyond its defining interval. The paper introduces no fitted numerical parameters; the exponential warp and Legendre basis are chosen to match HiPPO.

assumptions (5)

domain assumption The discrete signal is interpolated by zero-order hold (ZOH) into a continuous function before projection (Eq. 7).
This enables continuous inner products, and the input matrix B_Delta depends on the hold model; the paper shows all hold models converge to the same B_gen in the limit.
ad hoc to paper The old basis psi_{t,m} can be evaluated on the extended interval T_{t+1} through sigma_t o sigma_{t+1}^{-1} over the full canonical interval Z (Eq. 23 and Eq. 31).
Without this implicit analytic extension, the projection inner product in Eq. (20) is restricted to z in sigma_{t+1}(T_t), not all of Z, and the resulting A_Delta differs from the paper's formula.
domain assumption The discrete transition and continuous generator satisfy A_Delta = exp(Delta A_gen), used to obtain A_corrected in Eq. (32).
This matrix-exponential identity is asserted and supported numerically by Table 1, but not proven in the text.
standard math In the continuous limit, the input signal u(s) is smooth enough that u_{t+Delta} tends to u(t) and first-order Taylor expansions are valid (Appendix 8.2).
Standard continuity assumption used to pass from discrete recurrences to the continuous ODE generator.
standard math The warped basis is orthonormal under the induced inner product with weight omega_t=|sigma_t'| (Eq. 11).
Follows from a change of variables; unproved but straightforward.

invented entities (2)

Lag operator ell_{t+1->t} = sigma_t o sigma_{t+1}^{-1}
purpose: Measures the geometric shift of the projection basis between timesteps and defines the transition matrix A_t through a single inner product.
A new mathematical operator introduced in this paper. It has no falsifiable handle outside the framework; its validity rests entirely on the projection derivation and the implicit analytic-extension assumption.
Domain expansion
purpose: Conceptual device describing how basis functions and the signal history extend as the time window grows.
The paper's geometric narrative for the lag operator; not independently testable and is the source of the unflagged extension assumption.

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Cite this review

Pith. "Pith review of Lag Operator SSMs: A Geometric Framework for Structured State Space Modeling." pith.science (2026). https://pith.science/paper/SLZHS7PQ

@misc{pith2026251218965,
  author       = {Pith},
  title        = {Pith review of: Lag Operator SSMs: A Geometric Framework for Structured State Space Modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SLZHS7PQ}},
  note         = {Machine review of arXiv:2512.18965}
}

read the original abstract

Structured State Space Models (SSMs), which are at the heart of the recently popular Mamba architecture, are powerful tools for sequence modeling. However, their theoretical foundation relies on a complex, multistage process of continuous-time modeling and subsequent discretization, which can obscure intuition. We introduce a direct, first-principles framework for constructing discrete-time SSMs that is both flexible and modular. Our approach is based on a novel lag operator, which geometrically derives the discrete-time recurrence by measuring how the system's basis functions undergo what we call a domain expansion from one timestep to the next. The resulting state matrices are computed via a single inner product involving this operator, enabling a modular design space for creating novel SSMs by flexibly combining different basis functions and time-warping schemes. To validate our framework, we demonstrate that a specific instance exactly recovers the recurrence of the influential HiPPO model. Numerical simulations confirm our derivation, providing new theoretical tools for designing flexible and robust sequence models.

Figures

Figures reproduced from arXiv: 2512.18965 by the authors.

**Figure 1.** Illustration of a time warping function σt(s). The infinite time interval Tt = (−∞, t] (s-axis) is mapped to the finite canonical interval Z = (0, 1] (zaxis), concentrating recent history (near t) and compressing older history (towards −∞). Construct an orthonormal basis on Tt by warping ϕn: ψt,n(s) = ϕn(σt(s)) = ϕn ◦ σt(s), (9) denoted as Ψt: Ψt = {ψt,n(s) | n = 0, . . . , N − 1} = Φ ◦ σt. (10) 3.1.4 Induced Inne… view at source ↗

**Figure 2.** Conceptualization of the extended signal [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Example basis functions lag-shifted forward [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Visualization of the final compressed mem [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Works this paper leans on

3 extracted references · 1 linked inside Pith

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[Gu and Dao, 2023] Gu, A. and Dao, T. (2023). Mamba: Linear-time sequence modeling with selec- tive state spaces.arXiv [cs.LG]. [Gu et al., 2020] Gu, A., Dao, T., Ermon, S., Rudra, A., and R´ e, C. (2020). HiPPO: Recurrent memory with optimal polynomial projections.Neural Infor- mation Processing Systems, abs/2008.07669:1474–

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

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and Dao, T

[Gu and Dao, 2023] Gu, A. and Dao, T. (2023). Mamba: Linear-time sequence modeling with selec- tive state spaces.arXiv [cs.LG]. [Gu et al., 2020] Gu, A., Dao, T., Ermon, S., Rudra, A., and R´ e, C. (2020). HiPPO: Recurrent memory with optimal polynomial projections.Neural Infor- mation Processing Systems, abs/2008.07669:1474–

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[Voelker and Eliasmith, 2018] Voelker, A. R. and Eliasmith, C. (2018). Improving Spiking Dy- namical Networks: Accurate Delays, Higher-Order Synapses, and Time Cells.Neural computation, 30(3):569–609. Lag Operator SSMs: A Geometric F ramework for Structured State Space Modeling Appendix 6 Notation Summary for Discrete-to-Continuous Matrices Table 4: Summa...

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[Gu et al., 2022a] Gu, A., Goel, K., and Re, C. (2022a). Efficiently Modeling Long Sequences with Structured State Spaces. [Gu et al., 2022b] Gu, A., Johnson, I., Timalsina, A., Rudra, A., and R´ e, C. (2022b). How to Train Your HiPPO: State Space Models with Generalized Or- thogonal Basis Projections.arXiv [cs.LG]. [Somvanshi et al., 2025] Somvanshi, S.,...

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