On Achievable Rates Over Noisy Nanopore Channels

Nir Weinberger; V. Arvind Rameshwar

REVIEW 4 major objections 4 minor 54 references

On Achievable Rates Over Noisy Nanopore Channels

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proves that the noisy nanopore channel—a duplication channel with Markov inputs and memoryless noise—achieves rates arbitrarily close to the noiseless capacity in the long-memory erasure regime and in the high-sampling-rate…

desk verdict Two interesting achievability results for noisy nanopore channels, but the erasure/large-tau proof has a Fano factor-of-m error that needs a per-burst repair before the paper is publishable. read the letter →

arxiv 2501.02917 v5 pith:WT6IGYOG submitted 2025-01-06 cs.IT math.IT

V. Arvind Rameshwar , Nir Weinberger This is my paper · ORCID

classification cs.ITmath.IT MSC 94A1594A2494A40

keywords nanoporesequencingnoisychannelduplicationchannelscapacityerasurenoisechange-pointdetectiondeBruijnMarkovprocessDNAstorage

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 noisy nanopore channel (NNC) models a nanopore sequencer as a duplication channel: an input sequence of overlapping $\tau$-mers (length-$\tau$ DNA words that slide by one base) is repeated a random number of times, and then each repeated symbol passes through a memoryless noise channel. This paper's central claim is that the NNC is much more informative than its noisy appearance suggests. In the erasure-noise regime with long memory, the achievable rate is within $O(\tau\epsilon^\tau)$ of the no-self-loop noiseless capacity, which itself approaches $1$; in the high sampling-rate regime, a change-point detection decoder achieves rates up to that same noiseless capacity. The paper also supplies a tight lower bound for the noiseless channel and the first computable general lower and upper bounds for noisy nanopore channels. These results matter because nanopore sequencing is a leading candidate for DNA-based archival storage, and the regimes considered are the ones relevant to practical sequencers.

What carries the argument

The key structural fact is Lemma V.1 for a no-self-loop de Bruijn Markov input process—a first-order Markov chain whose states are length-$\tau$ words over the base alphabet, with each next state obtained by shifting one symbol and appending a new symbol. The lemma says that any two $\tau$-mers no more than $\tau$ positions apart completely determine the intervening sequence of $\tau$-mers. This makes the decoder $\mathcal{D}_{\mathrm{clean}}$ able to reconstruct the input from the output whenever no erasure burst has length $\tau$ or more, because each short burst is pinned down by its endpoints; the error probability is then bounded by the probability that some burst has length at least $\tau$, which is at most $m\,\mathbb{E}[K]\,\epsilon^\tau$. In the high sampling-rate regime the machinery is a two-stage decoder: quickest change-point detection estimates the run boundaries in the output, a trimming step discards the uncertain tail of each estimated run, and a MAP decoder reads the remaining samples as repeated noisy views of a single $\tau$-mer.

What would settle it

Take a small alphabet, enumerate the no-self-loop de Bruijn graph paths of length about $\tau$, and look for two distinct paths with the same first and last $\tau$-mers whose interiors fall inside an erasure burst of length less than $\tau$; if such a pair exists, Lemma V.1 fails and the decoder $\mathcal{D}_{\mathrm{clean}}$ will mis-reconstruct that output. Alternatively, simulate the decoder on all erasure patterns of length $\tau-1$ for a fixed no-self-loop codeword and check whether any pattern yields two valid reconstructions.

Watch

Extended reading notes

Core claim

For an NNC whose memoryless noise is an erasure channel over the $|\mathcal{X}|^\tau$-ary alphabet, the paper proves (Theorem V.1) that $C^{(\tau)}(\mathcal{W}_{\mathrm{nn,EC}}) \ge C_\tau^{\mathrm{no\text{-}noise,no\text{-}loop}} - O(\tau\epsilon^\tau)$, and since $C_\tau^{\mathrm{no\text{-}noise,no\text{-}loop}} \to 1$, the capacity of the erasure-noise nanopore channel tends to $1$ as the memory length $\tau$ grows. In the high sampling-rate regime, it proves (Theorem VI.2 and Corollary VI.1) that a decoder built from a quickest change-point detection routine and MAP decoding of the detected repeat blocks has vanishing error probability, so rates up to $C_\tau^{\mathrm{no\text{-}noise,no\text{-}loop}}$ are achievable. The paper therefore establishes that, in both regimes, the noisy nanopore channel can carry information at rates essentially equal to the noiseless channel capacity.

Load-bearing premise

The load-bearing premise is that in a no-self-loop de Bruijn Markov input, an erasure burst shorter than $\tau$ is always uniquely determined by the surviving $\tau$-mers on either side; if two different inputs could produce the same output after such a burst, the near-capacity achievability result for the erasure regime would have to be weakened.

Editorial extensions

If this is right

With erasure noise and sufficiently large $\tau$, information can be sent at rates arbitrarily close to $1$ bit per input symbol, using the standard no-self-loop de Bruijn Markov input distribution and the simple decoder $\mathcal{D}_{\mathrm{clean}}$.
At high sampling rates, the change-point detection decoder makes the block error probability go to $0$ as the input length grows, so the full no-self-loop noiseless rate $C_\tau^{\mathrm{no\text{-}noise,no\text{-}loop}}$ is achievable.
The noiseless capacity lower bound in Theorem III.1 is computable for duplication distributions such as the elementary i.i.d. duplication channel and the binomial duplication channel, and is tight according to the argument adapted from [28].
General lower bounds based on Bhattacharya parameters and upper bounds such as $\mathbb{E}[K]\,C(W)$ give explicit capacity estimates for arbitrary noisy nanopore channels, though they are most useful for short memory lengths.
In both regimes studied, the rate loss to noise vanishes asymptotically, so the noisy nanopore channel behaves like a noiseless constrained channel in the limits considered.

Reading between the lines

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

The finite-$\tau$ gap $O(\tau\epsilon^\tau)$ suggests a concrete design rule for DNA storage: choose the memory parameter $\tau$ so that $\tau\epsilon^\tau$ is below the target rate loss, and the erasure bursts the decoder must tolerate scale accordingly.
The same two-stage decoder could be tested experimentally on raw nanopore current traces by treating the change-point estimates as run boundaries and then applying MAP decoding over the trimmed samples; the paper does not run such an experiment.
Extending the erasure-region argument to substitution or insertion-deletion noise would require a replacement for Lemma V.1's endpoint determinism; if such a property holds for other noise classes, the same near-capacity conclusion may follow, but that is not shown here.
The high sampling-rate analysis leaves a tradeoff between sampling cost and rate; optimizing the block-length-dependent parameters rather than the paper's specific choices could sharpen the finite-length performance.

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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

Colleague,

My take: the paper has two genuinely interesting achievability results for noisy nanopore channels, but one of them—Theorem V.1 on erasure noise with large tau—rests on a proof step that is currently wrong in a concrete, correctable way. The stress-test note is accurate on this point.

The genuinely new content is in Sections V and VI. The change-point detection decoder is a nice idea, tied to a real practical subroutine, and the error analysis in Theorem VI.2 is coherent. The general bounds in Section IV are simple but useful as a first explicit handle on this channel. The noiseless lower bound is a clean exposition of Kirsch–Drinea, as the authors acknowledge. Credit where due: the results are plausible and the high-level intuition—long tau-mers and high sampling rates both mitigate noise—is appealing.

The soft spot is Proposition V.1. Equation (60) applies Fano's inequality to S^m, whose alphabet size is |X|^(τ m), not |X|^τ. The correct bound is H(S^m|Y) ≤ 1 + Pr(E)·τ m log|X|, so the per-symbol gap should be Pr(E)·τ log|X|, not (1/m)Pr(E)·τ log|X| as written. Even worse, E is the event that at least one erasure burst has length ≥ τ; for fixed τ and ε>0, Pr(E) → 1 as m→∞, so any block-error Fano bound is too weak. The repair is straightforward: count N, the number of long bursts, use E[N] ≤ m E[K] ε^τ, and bound the entropy contribution per burst by O(τ log|X|). That restores the claimed O(τ ε^τ) gap. But as written, Eqs. (60)–(61) are internally inconsistent. This is load-bearing for Theorem V.1.

Minor issues: Lemma V.3's exact reconstructibility ignores bursts touching the start or end of the output, which is patchable with an o(1) term. There is an undefined E0 in Eq. (72). Section VI's use of the authors' own prior result [46] in Eq. (79) is acceptable since that result is independently published, though it should be cited more carefully.

Who benefits: information theorists working on synchronization-error channels or DNA storage. The paper gives theoretical backing to oversampling and long t-mers as noise countermeasures, which is worth knowing.

Recommendation: send to peer review, but require a corrected proof of Proposition V.1 before acceptance. The flaw is mechanical, not fatal, and the contribution is solid enough to justify the referee time.

Referee Report

4 major / 4 minor

Summary. The paper studies the noisy nanopore channel (NNC), a duplication channel with structured de Bruijn Markov inputs followed by a memoryless channel. It gives a lower bound on the noiseless NNC capacity, general lower and upper bounds for noisy NNCs, and then analyzes two asymptotic regimes: long memory length τ with erasure noise, and high sampling rates. The main claimed results are Theorem V.1, stating that the erasure-NNC capacity is at least the no-self-loop noiseless capacity minus O(τ ε^τ) and hence tends to 1 as τ→∞, and Theorem VI.2/Corollary VI.1, stating that with a change-point-detection decoder and high sampling rates, error probability vanishes and rates up to the no-self-loop noiseless capacity are achievable. The proofs are built on entropy bounds, a reconstruction property of no-self-loop de Bruijn paths, and a Shiryaev change-point detector.

Significance. If the main claims hold, the paper gives substantial progress on a difficult channel model of practical relevance for nanopore DNA storage. The idea of using no-self-loop de Bruijn inputs to make erasure bursts of length less than τ uniquely reconstructible is elegant, and the change-point-detection decoder is a concrete, practically motivated algorithm. The paper also provides explicit computable bounds and clean examples for the noiseless and general noisy cases, which are useful contributions in themselves. However, the central proof of Section V currently contains a Fano-inequality error, and the Appendix proof of the key monotonicity lemma appears incorrect as written; these issues are load-bearing for Theorem V.1 and Theorem V.2. The high-sampling-rate section also needs a more careful formalization of the channel sequence. The results are plausible and likely repairable, but the current manuscript is not yet technically sound.

major comments (4)

[V-B, Eq. (60)] In the proof of Proposition V.1, Eq. (60) bounds H(S^m|Y^{T_m}) by 1 + Pr(E)·τ log|X|. Since S^m takes values in (X^τ)^m, Fano's inequality gives H(S^m|Y^{T_m}) ≤ 1 + Pr(E)·m τ log|X|. Using Pr(E) ≤ m E[K] ε^τ from Lemma V.3, the per-symbol bound becomes 1/m + m E[K] ε^τ τ log|X|, which diverges with m. Thus Proposition V.1 and Theorem V.1 do not follow as written. A repair is likely available by counting the number of ambiguous input symbols rather than using a block error event, but the current proof is internally inconsistent and needs to be rewritten.
[V-B, Lemma V.3] The assertion that absence of an output erasure burst of length at least τ makes S^m exactly reconstructible ignores bursts that touch the first or last position of Y^{T_m}: such a burst has only one endpoint, so Lemma V.1 cannot be applied. A fully erased prefix block of length r < τ occurs with probability on the order of (E[ε^K])^r, which for r=1 is O(ε), not O(ε^τ). This contributes only O(1) bits of block uncertainty and is patchable by an o(1) per-symbol term, but the statement 'exactly reconstructible' is false as written.
[VI, Eq. (66)] Eq. (66) requires P_K(ℓ_m ≤ K ≤ h_m) ≥ 1 − 1/m^{1+η} with ℓ_m = m^2 (ln m)^3, while P_K was fixed in Section II-B. For any fixed distribution on the positive integers, P_K(K ≥ ℓ_m) tends to 0 as m→∞ because the threshold tends to infinity, so (66) cannot hold for a fixed channel. The high-sampling-rate regime must be formalized as a sequence of channels W_nn^{(m)} whose duplication distribution depends on m, and the statements of Theorem VI.2 and Corollary VI.1 should be rephrased accordingly; as written, the limiting operation is not over a well-defined single channel.
[Appendix A, Lemma V.2] The proof of Lemma V.2 asserts A_Gno-loop_{τ−1} = Σ_i A_i and λ(A) ≥ λ(A_Gno-loop_{τ−1}). For q=2 and τ=2, the block-diagonal matrix A has spectral radius 0 because its diagonal blocks are nilpotent, whereas λ(A_Gno-loop_1) = 1. Thus the claimed inequality λ(A) ≥ λ(A_Gno-loop_{τ−1}) is false in this example. The lemma may still be true, but this proof does not establish it; a different comparison argument is needed for the convergence C_no-noise,no-loop_τ → 1 used in Theorem V.2.

minor comments (4)

[VI, proof of Theorem VI.2] In the line immediately after Eq. (72), the expression 'Pr[E^c_1 | E^c_0]' appears to be a typo; the event should be E^c_2 conditioned on E^c_1.
[V-B, Eqs. (54)-(55)] The letter E is used both for the block error event {S^m ≠ f(Y^{T_m})} and for the erasure-burst event {some burst of erasures has length at least τ}; these should be renamed to avoid confusion.
[IV, Theorem IV.3] The upper bound in Theorem IV.3 can exceed 1 when the subtracted term E[K]H(Z|Y) − H(K) is negative; it should be stated explicitly that the bound is combined with the trivial upper bound of 1.
[II-C, Theorem II.1] The remark after Theorem II.1 states a simplification for τ = o(m), but the capacity definition and all subsequent theorems fix τ as a constant; this asymptotic notation could confuse readers and should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central capacity bounds rest on external standards and an independently published prior theorem, not on a self-referential reduction.

full rationale

Walking the derivation chain: the capacity representation in Theorem II.1 is imported from [21] (external). The noiseless lower bound in Theorem III.1 is proved self-contained, with tightness credited to the external result [28]; no fitted quantity is renamed as a prediction. The general bounds in Theorems IV.1-IV.3 are derived from standard inequalities (conditioning reduces entropy, Wald's lemma, Fano, Bhattacharyya bounds). In the erasure/large-tau regime, the input P^* is a fixed no-self-loop max-entropy de Bruijn process, and the identity H(S2|S1)=C_no-noise,no-loop is the standard constrained-system result [50, Thm. 3.23], not a circular definition of the noisy capacity. The only self-citation in the derivation chain is [46, Thm. 3.1], used in Eq. (79) to exponentiate the Bhattacharyya bound for the k-view channel in the Stage-2 MAP decoding analysis. That theorem is a separately published, parameter-free result about multi-view channels; its assumptions do not include the present paper's achievability claim, and it is combined with the external bound [47, Prop. 4.8]. Corollary VI.1 then follows from Theorem VI.2 by a standard Fano argument, not by assuming the conclusion. I find no step in which a target quantity is defined in terms of the conclusion, no fitted parameter is relabeled as a prediction, and no load-bearing argument reduces to a self-citation chain. The apparent Fano-alphabet-size issue in Proposition V.1 (Eq. (60)) is a correctness concern about the proof as written, not a circularity, and is therefore not reflected in this score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claims rest on standard information-theoretic theorems (e.g., [21, Thm. 4], [28, Thm. 1], [51, Thm. 3.1], [47, Prop. 4.8]) and on modeling restrictions introduced by the authors, most notably the no-self-loop de Bruijn input process used in Section V. No free parameters are fitted to data; all numeric choices are channel parameters or asymptotic scaling factors that do not affect the conclusions.

assumptions (7)

standard math Capacity formula of [21, Thm. 4]: ergodic capacity equals the supremum of lim 1/m I(S^m; Y^{T_m}) over stationary ergodic de Bruijn Markov kernels.
Used in Theorem II.1 as the starting point for all bounds; taken from the cited paper without proof.
standard math Tightness of the noiseless lower bound per [28, Thm. 1].
The paper states the bound in Theorem III.1 is tight by [28], and uses this to claim the lower bound is the exact noiseless capacity expression.
ad hoc to paper No-self-loop de Bruijn Markov input process.
Introduced in Section V-A; ensures all input runs have length 1, so any tau consecutive tau-mers are endpoint-determined (Lemma V.1). This is a coding restriction, not part of the original NNC model.
domain assumption Regularity condition: W_{Y|z} and W_{Y|z'} are distinct distributions for all z != z'.
Stated before Theorem VI.2; required for the change-point detector to distinguish runs from different input states.
domain assumption High sampling rate concentration: P_K(ell_m <= K <= h_m) >= 1 - 1/m^{1+eta}.
Defines the high sampling rate regime in Eq. (66); the achievability theorem is conditional on this scaling.
standard math Shiryaev change-point detection delay bound (Theorem VI.1 from [51, Thm. 3.2] and [53]).
Used to bound false alarm and detection delay probabilities in the proof of Theorem VI.2.
standard math Bhattacharyya parameter bound for the k-view DMC from [47, Prop. 4.8] and [46, Thm. 3.1].
Used in Eq. (78)-(79) to show MAP decoding error over long blocks decays exponentially. Note that [46] is by the present authors.

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Pith. "Pith review of On Achievable Rates Over Noisy Nanopore Channels." pith.science (2026). https://pith.science/paper/WT6IGYOG

@misc{pith2026250102917,
  author       = {Pith},
  title        = {Pith review of: On Achievable Rates Over Noisy Nanopore Channels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WT6IGYOG}},
  note         = {Machine review of arXiv:2501.02917}
}

read the original abstract

In this paper, we consider a recent channel model of a nanopore sequencer proposed by McBain, Viterbo, and Saunderson (2024), termed the noisy nanopore channel (NNC). In essence, an NNC is a duplication channel with structured, Markov inputs, that is corrupted by memoryless noise. We first discuss a (tight) lower bound on the capacity of the NNC in the absence of random noise. Next, we present lower and upper bounds on the channel capacity of general noisy nanopore channels. We then consider two interesting regimes of operation of an NNC: first, where the memory of the input process is large and the random noise introduces erasures, and second, where the rate of measurements of the electric current (also called the sampling rate) is high. For these regimes, we show that it is possible to achieve information rates close to the noise-free capacity, using low-complexity encoding and decoding schemes. In particular, our decoder for the regime of high sampling rates makes use of a change-point detection procedure -- a subroutine of immediate relevance for practitioners.

Figures

Figures reproduced from arXiv: 2501.02917 by the authors.

**Figure 1.** The noisy nanopore channel 𝑊nn B. Channel Model The noisy nanopore channel (NNC), as mentioned earlier, is a noisy duplication channel with an input source that is constrained to have a specific first-order Markov structure. The NNC 𝑊nn = 𝑊nn (X, Y, 𝜏, 𝑃𝐾, 𝑊) that we describe here is that introduced in [17], with the difference that we assume that the noise arises from a general memoryless channel, and not specifica… view at source ↗

**Figure 2.** (a) Our lower bound for 𝐶(𝑊nn,EC), for an i.i.d. duplication channel with parameter 𝑝 = 0.999; (b) Our upper bound for 𝐶(𝑊nn,EC), for an i.i.d. duplication channel with parameter 𝑝 = 0.3. In both cases, we use |X| = 3 and 𝜏 = 2. our bounds hold generally for any stationary, ergodic Markov input process 𝑃𝑆|𝑆 − , which is not necessarily a de Bruijn Markov input process. Let 𝐶(𝑊) denote the capacity of the memoryless … view at source ↗

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