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arxiv: 2606.10199 · v1 · pith:BIF3BRUFnew · submitted 2026-06-08 · 💻 cs.LG · cs.CL

A Continuous-Time Markov Chain Framework for Insertion Language Models

Dhruvesh Patel , Benjamin Rozonoyer , Soumitra Das , Tahira Naseem , Tim G.J. Rudner , Andrew McCallum This is my paper

Pith reviewed 2026-06-27 17:10 UTC · model grok-4.3

classification 💻 cs.LG cs.CL
keywords insertion language modelscontinuous-time Markov chainsdiffusion modelsdenoising objectivessequence generationvariable-length sequencesgenerative models
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The pith

Insertion language models arise as special cases of a continuous-time Markov chain denoising process on variable-length sequences.

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

The paper derives a diffusion-style denoising objective for insertion language models directly from first principles by modeling the corruption of sequences as a continuous-time Markov chain on the space of variable-length sequences. This formulation unifies previous ad-hoc insertion approaches as special cases inside the same framework. A sympathetic reader would care because insertion models allow flexible generation order unlike left-to-right or mask-based methods, and a first-principles derivation supplies a tractable training objective. Empirical checks on planning and language tasks confirm the resulting models keep the flexibility advantages while remaining competitive.

Core claim

By formulating the noising process as a continuous-time Markov chain on the space of variable-length sequences, the authors derive a tractable denoising objective that recovers the original insertion dynamics and shows that previous formulations of ILMs can be viewed as special cases of this general framework.

What carries the argument

The continuous-time Markov chain on variable-length sequences, whose transition rates are set to yield a tractable denoising objective that recovers insertion dynamics.

If this is right

  • The approach retains the benefits of insertion-based generation over left-to-right generation and masked diffusion models on a synthetic planning task.
  • In language modeling the diffusion-based approach is competitive with left-to-right generation and masked diffusion models.
  • The framework supplies additional flexibility in sampling compared to existing insertion language models.
  • All prior ILM formulations become recoverable as special cases inside the single CTMC denoising setup.

Where Pith is reading between the lines

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

  • Different choices of transition rates in the CTMC could generate new families of insertion dynamics beyond those previously studied.
  • The same first-principles construction might extend to other variable-length or tree-structured generation problems outside language.
  • Links to existing diffusion models on discrete spaces could allow transfer of variance-reduction or sampling techniques developed elsewhere.

Load-bearing premise

The noising process on variable-length sequences can be exactly represented by a continuous-time Markov chain whose transition rates permit a tractable denoising objective that recovers the original insertion dynamics.

What would settle it

If training with the derived objective on sequences produced by a known insertion model fails to recover the same distribution under sampling, or if no choice of CTMC rates exactly matches the variable-length noising steps of prior ILMs, the unification claim would not hold.

Figures

Figures reproduced from arXiv: 2606.10199 by Andrew McCallum, Benjamin Rozonoyer, Dhruvesh Patel, Soumitra Das, Tahira Naseem, Tim G.J. Rudner.

Figure 1
Figure 1. Figure 1: Diffusion-based Insertion Language Mod￾els (DILMs) retain the benefits of Insertion Lan￾guage Models (ILMs) while offering principled train￾ing and sampling procedures based on diffusion using a continuous-time Markov chain framework. Masked diffusion models can generate sequences in ar￾bitrary order while also generating multiple tokens per step (Lou et al., 2024; Sahoo et al., 2024; Shi et al., 2024). MD… view at source ↗
Figure 2
Figure 2. Figure 2: The noising process, shown on the left, is a continuous-time Markov chain that deletes one token at [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Generative perplexity vs. number of sampling steps for models trained on LM1B (left) and OpenWeb [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sample path of a right-continuous CTMC with states [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Empirical joint distributions of initial length [PITH_FULL_IMAGE:figures/full_fig_p031_6.png] view at source ↗
Figure 5
Figure 5. Figure 5: Log-linear noise schedule. We use the log-linear schedule for joint noising defined as σ(t) = σmin  σmax σmin t , where 0 < σmin < σmax are hyperparameters (in partic￾ular σ(0) = σmin and σ(1) = σmax). The corresponding integrated rate from proposition 2 is σ¯0,t = Z t 0 σ(u) du = σmin ln(σmax/σmin)  σmax σmin t − 1 ! . During training we sample t uniformly from [ε, 1], with ε small (e.g. ε = 10−4 in t… view at source ↗
Figure 7
Figure 7. Figure 7: Illustration of the insertion update. The top panel isolates the insertion slots: [PITH_FULL_IMAGE:figures/full_fig_p035_7.png] view at source ↗
read the original abstract

Insertion Language Models (ILMs) offer several advantages over left-to-right generation and mask-based generation. However, existing formulations of insertion-based generation have largely been ad-hoc. In this paper, we derive a diffusion-style denoising objective for ILMs from first principles by formulating the noising process as a continuous-time Markov chain on the space of variable-length sequences. We show that previous formulations of ILMs can be viewed as special cases of this denoising framework. Through empirical evaluation on a synthetic planning task, we show that the proposed approach retains the benefits of insertion-based generation over left-to-right generation and masked diffusion models. In language modeling, our diffusion-based approach is competitive with left-to-right generation and masked diffusion models, while offering additional flexibility in sampling compared to existing insertion language models.

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 / 0 minor

Summary. The paper claims to derive a diffusion-style denoising objective for Insertion Language Models (ILMs) from first principles by formulating the noising process as a continuous-time Markov chain (CTMC) on the space of variable-length sequences. It shows that previous ILM formulations are special cases of this framework and provides empirical results on a synthetic planning task and language modeling, where the approach is competitive with left-to-right generation and masked diffusion models while offering sampling flexibility.

Significance. If the CTMC rate construction, forward process, and reverse-process objective hold, the work supplies a principled unification of ad-hoc ILM methods under a first-principles denoising framework. The explicit demonstration that prior formulations are special cases, together with the retention of insertion-generation benefits in experiments, constitutes a clear advance over existing diffusion and autoregressive approaches for variable-length sequence modeling.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive review, accurate summary of our contributions, and recommendation to accept the manuscript. We are pleased that the referee recognizes the value of the first-principles CTMC derivation and the unification of prior ILM methods.

Circularity Check

0 steps flagged

No significant circularity; first-principles CTMC derivation is self-contained

full rationale

The paper formulates a continuous-time Markov chain on variable-length sequences to derive a denoising objective for insertion language models from first principles. It recovers prior ILM formulations as special cases without reducing any central claim to a fitted parameter, self-citation chain, or definitional equivalence. No load-bearing step equates the output to its inputs by construction; the rate construction and reverse-process objective stand independently of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated. The central modeling choice (CTMC on variable-length sequences) functions as an implicit domain assumption whose justification is not visible.

pith-pipeline@v0.9.1-grok · 5676 in / 1130 out tokens · 16750 ms · 2026-06-27T17:10:25.433181+00:00 · methodology

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Reference graph

Works this paper leans on

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    For all figures and tables that present empirical results, check if you include: (a) The code, data, and instructions needed to reproduce the main experimental re- sults (either in the supplemental material or as a URL). [Yes, the code used for this work is available at https://github.com/ dhruvdcoder/ctmc_dilm. Appendix E con- tains some key implementati...

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    If you are using existing assets (e.g., code, data, models) or curating/releasing new assets, check if you include: (a) Citations of the creator If your work uses ex- isting assets. [Yes] (b) The license information of the assets, if ap- plicable. [Not Applicable] (c) New assets either in the supplemental mate- rial or as a URL, if applicable. [Yes, we pr...

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    Bayes rule and the Markov property imply q(xk | xk+1, x0) = q(xk | x0) q(xk+1 | x0) q(xk+1 | xk) = r(k, k + 1)q(xk+1 | xk), (6) where we denote the ratio q(xk|x0) q(xk+1|x0) with r(k, k + 1). With slight abuse of notation, we will denote the probability law of the noising process withq and its time-reversal with p. To begin, recall that the discrete-time ...

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    log ˆKt(x0[b], x0[a ∨ ek]) CTMC F ramework for Insertion Language Models where ek ∈ Bn,1 is the vector of length n with all 0s except for the k-th position, Bn,m(x0, b) = {a ∈ Bn,m : x0[a] = x0[b]}, and si(a) is the set of indices of zeros in a that fall between the ( i − 1)-th and i-th 1s. The re- indexing in the third line above is justified by a biject...

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    (15) (*) Here, in the second step, we invoke the following approximation

    log ˆpθ ins(i, xk 0 | x0[b]) = σt ¯σ0,t E m,b (n − m) ˜Sn,m,σt X i∈[m+1], w∈V ptarget ins (i, w | x0, b) log ˆpθ ins(i, w | x0[b]) . (15) (*) Here, in the second step, we invoke the following approximation. We fix the alignment between the positions of the current sequence x0[b] and a predecessor state, and identify the predecessor statex0[b∨ek] by the in...

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    This is exact whenever the predecessor state determines a unique insertion location-token pair, and becomes approximate in the repeated-token case discussed below

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    can be seen as the target joint probability distribution over positions and tokens. To see this, note that Pm+1 i=1 si(b) = n − m. Putting eq. (14) and eq. (15) together, we get the final expression. Remark 4 (Rao-Blackwellization). Instead of using the unbiased estimator obtained by fixing a = b, we could marginalize over all a ∈ Bn,m(x0, b) explicitly. ...

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    for some k ∈ si(b), then Proposition 4 gives pt|0(y | x0) pt|0(x | x0) = ρm+1 t (1 − ρt)n−m−1 ρm t (1 − ρt)n−m = ρt 1 − ρt , and the forward deletion rate from y to x is σt. Hence every missing original index contributes the same coefficient γind t = σt ρt 1 − ρt . CTMC F ramework for Insertion Language Models As in Corollary 1, we now use the same approx...

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