Bidirectional Mamba state-space model for anomalous diffusion

Juliette Lacherez (LOMA); Maxime Lavaud (LOMA); Thomas Salez (LOMA); Yacine Amarouchene (LOMA); Yosef Shokeeb (LOMA)

REVIEW 4 major objections 6 minor 35 references

Bidirectional Mamba state-space model for anomalous diffusion

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Bi-Mamba, a bidirectional state-space deep-learning model, recovers the effective diffusion coefficient and anomalous exponent from single short noisy trajectories, outperforming a bidirectional RNN baseline on the AnDi-2 challenge.

desk verdict Useful first Mamba-for-physics demo with honest AnDi-2 participation, but the headline comparison to the RNN baseline needs error bars before I'd trust it. read the letter →

arxiv 2412.07299 v1 pith:PVG5NVBM submitted 2024-12-10 cond-mat.soft physics.bio-phphysics.opticsstat.ML

Maxime Lavaud (LOMA) , Yosef Shokeeb (LOMA) , Juliette Lacherez (LOMA) , Yacine Amarouchene (LOMA) , Thomas Salez (LOMA) This is my paper · ORCID

classification cond-mat.softphysics.bio-phphysics.opticsstat.ML

keywords AnomalousdiffusionSingle-particletrackingState-spacemodelsMambaBidirectionalscanDeeplearningcoefficientexponent

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

Characterizing anomalous diffusion often comes down to estimating the effective diffusion coefficient $K$ and the anomalous exponent $\alpha$ from particle trajectories that are short, sparse, and noisy. This paper argues that Bi-Mamba, a deep state-space architecture with a bidirectional scan mechanism, can infer both parameters from a single two-dimensional trajectory while also segmenting the trajectory into different diffusion states. The model is evaluated on the AnDi-2 challenge datasets, which contain five bio-mimetic models of single-particle motion, and the authors report that it scores better overall and in every category than a bidirectional RNN baseline. If that benchmark result carries over to real experiments, biophysicists would gain a practical tool for single-particle tracking in environments where only a few short tracks are available.

What carries the argument

The central object is the Bi-Mamba block, defined by a Mamba layer — a selective state-space model that compresses a sequence into a hidden state while deciding, in an input-dependent way, what to remember and what to ignore — applied once to the trajectory features and once to their time-flipped counterparts, then concatenating the two outputs through a feedforward layer. The full model stacks three such blocks: one performs hot-encoded diffusion-state segmentation, one regresses $K$ under a mean-squared-logarithmic-error loss, and one regresses $\alpha$ under a mean-absolute-error loss. The features fed in are the per-axis displacements, the one-dimensional mean-squared displacements, the angle between consecutive displacements, and the distance from the origin. This bidirectional design is what allows the model to capture time dependencies across the whole trajectory while keeping training linear in sequence length.

What would settle it

Apply the trained Bi-Mamba model to experimental single-particle trajectories whose ground-truth $K$ and $\alpha$ are known from an independent measurement — for example, a calibrated optical trap in a well-characterized viscoelastic fluid — and compare inferred values with that ground truth; errors substantially larger than the benchmark's reported errors would falsify the transfer claim.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that Bi-Mamba — a selective state-space model equipped with a bidirectional scan — can take one short, noisy, two-dimensional trajectory and reliably infer the effective diffusion coefficient $K$ and anomalous exponent $\alpha$ (with $\mathrm{MSD}(t) = 4 K t^\alpha$), as well as identify changes in the diffusion state along the trajectory. Using the AnDi-2 challenge datasets, which represent five bio-mimetic models including single-state, multi-state, dimerization, transient confinement, and quenched-trap motion, the authors report that Bi-Mamba outperforms a bidirectional recurrent neural network baseline both overall and in each task category. They further report that the model trains for more epochs without overfitting and with lower loss variance, and they describe a three-block architecture in which segmentation, $K$ regression, and $\alpha$ regression are handled by dedicated blocks with a summed loss.

Load-bearing premise

The AnDi-2 challenge's synthetic datasets, with their five bio-mimetic models, noise levels, and short trajectories, are representative enough of real experimental single-particle tracking data that the reported benchmark gains transfer to practical use.

Editorial extensions

If this is right

Single-particle-tracking experiments with short, noisy trajectories could estimate $K$ and $\alpha$ from individual tracks without averaging over many trajectories.
The same model can return both global parameters and a per-time segmentation, so binding and unbinding events can be read directly from one network output.
Because the model trains stably for a larger number of epochs, fine-tuning specialized versions on simulations generated near the general model's predictions is a practical route to further accuracy.
The authors' stated expectation that Mamba-based U-Net variants will improve segmentation points to a concrete next step for the architecture.

Reading between the lines

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

The input-dependent selection mechanism may make Bi-Mamba robust to trajectories with more than two diffusion states, but the paper only tests the two-state scenarios of the challenge; this remains an open extension.
The hidden state at each time step could serve as a learned low-dimensional representation of trajectory dynamics, enabling clustering or anomaly detection beyond the two regression targets; the paper does not explore this.
The comparison against a single bidirectional RNN baseline does not by itself locate the source of the gain; a systematic comparison with Transformers or a unidirectional Mamba would separate the contribution of the bidirectional scan from the choice of state-space backbone.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 genuinely new thing here is the application of a bidirectional Mamba state-space architecture to anomalous diffusion inference from short, noisy trajectories. That specific transfer looks like a first, and the paper does something right: it evaluates on the external AnDi-2 benchmark, reports its official ranks, and provides a code link. That is real evidence and deserves credit.

The architecture is described clearly enough: a segmentation block followed by dedicated K and alpha regression blocks, with a bidirectional scan borrowed from Bi-Mamba+. The task-specific blocking is sensible, and the loss curves in Fig. 3 show that Mamba trains more stably than the bidirectional RNN baseline (lower variance, no overfitting up to 500 epochs). That stability result is probably the most useful observation in the paper.

The soft spots are real and load-bearing. The claim that Bi-Mamba scores "better overall and in each category" of the AnDi2 Challenge rests entirely on Fig. 3, which shows loss curves without error bars, confidence intervals, or any statement about the number of independent runs. The "smaller loss variance" is asserted but not quantified. The official ranks (7th alpha, 9th K, 3rd diffusion type, 10th change-point) are point estimates with no score distributions or participant counts, so we cannot tell if those ranks are close or far from the leaders. Hyperparameters (layers, state dimension, learning rate, epochs) are not reported, which makes the baseline comparison hard to reproduce. None of these are fatal; they are fixable with a few independent training runs and a small table.

The math and data handling look fine. The MSD model, the five bio-mimetic AnDi-2 models, and the feature set (displacements, 1D MSDs, angle, distance from origin) are standard. No circularity: training on AnDi-2 train and evaluating on the held-out test set is the right protocol.

Who gets value from this: anyone working on single-particle tracking inference who wants to know whether Mamba-style architectures are worth trying, and researchers thinking about transferring modern sequence models into physics. It is not a SOTA paper, but it is a legitimate incremental contribution on a timely topic. I would send it to peer review, with the request that the authors add error bars, hyperparameters, and a concrete comparison table to other AnDi-2 participants before acceptance. My own verdict is conditional, leaning positive.

Referee Report

4 major / 6 minor

Summary. The manuscript proposes Bi-Mamba, a bidirectional state-space deep-learning architecture, for characterizing anomalous diffusion from single, short trajectories. The model uses a three-block design that first segments the trajectory into diffusion states and then performs separate regressions of the effective diffusion coefficient K and the anomalous exponent alpha. The authors train and evaluate on the AnDi-2 challenge datasets, reporting loss curves against a bidirectional RNN baseline and listing their AnDi-2 challenge ranks. The central claim is that Bi-Mamba infers K and alpha efficiently and scores better overall and in each category than the RNN baseline.

Significance. If the comparative claim is substantiated, Bi-Mamba would be a useful addition to the toolbox for single-particle-tracking analysis, with the practical advantage of efficient, bidirectional sequence processing. Strengths of the work include public availability of the implementation on GitHub, use of the standardized AnDi-2 benchmark, and the inclusion of a bidirectional RNN as an internal baseline. However, the current evidence is not yet sufficient to support the headline superiority claim: the comparison rests on qualitative loss curves from what appears to be a single run, and the AnDi-2 ranks are reported without statistical context. The work is therefore promising but needs a substantially strengthened empirical evaluation.

major comments (4)

[Results paragraph and Fig. 3] The central claim that Bi-Mamba "scor[es] better overall and in each category" than the bidirectional RNN baseline is supported only by qualitative loss curves plotted from what appears to be a single training run. No error bars, confidence intervals, repeated-seed statistics, or paired significance tests are reported, so the reported advantage may lie within run-to-run variability. Because the abstract's "infers efficiently" and the conclusion's "notable efficiency" rest on this comparison, the authors should rerun the experiment with multiple seeds and report a quantitative comparison table with means and dispersions.
[AnDi-2 ranks paragraph] The listed AnDi-2 results (7th in alpha, 9th in K, 3rd in diffusion type, 10th in change-point) are point estimates without score distributions, participant counts, or the score gaps to neighboring teams. Without this context, a reader cannot tell whether these ranks indicate a meaningful improvement or a knife-edge placement, and the claim of competitiveness is therefore undersupported.
[Architecture description and Fig. 2] The manuscript does not report the hyperparameters of Bi-Mamba or of the bidirectional RNN baseline (number of layers, state dimension, learning rate, batch size, epochs, optimization details), despite the text asserting a "fair benchmarking" comparison. Reproducibility requires these details, and the GitHub link alone does not substitute for reporting them in the paper.
[Multitask loss paragraph] The total loss is described as the sum of the MAE, MSLE, and WCE losses, but no weighting or balancing scheme is given. Since these losses have different scales and units, an unweighted sum can be dominated by one task; the paper should specify the weighting and, ideally, show sensitivity of the results to it.

minor comments (6)

[Architecture paragraph (p. 3)] The text says diffusion modes are "hot-encoded"; this should read "one-hot encoded".
[Evaluation paragraph (p. 3)] There is a typo in "Mean-Squared Logarithic Error"; it should be "Mean-Squared Logarithmic Error".
[Fig. 3 caption] The figure legend uses "RNN" and "Mamba", while the text compares a bidirectional RNN with Bi-Mamba; the caption should define both labels.
[Data description paragraph (p. 3)] The trajectory count "10 4" in the Fig. 3 caption should be typeset as "10^4".
[Abstract and data availability] The phrase "tested on the AnDi-2 challenge datasets among others" is vague; the paper should either identify the "other" datasets or remove the phrase.
[Reference [15]] Reference [15] is the AnDi-2 registered-report protocol; if the challenge results have been published, that version should be cited instead of or in addition to the protocol.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claim is an empirical benchmark result on external AnDi-2 data against an RNN baseline, with no fitted parameter renamed as a prediction.

full rationale

The paper makes no first-principles derivation of anomalous-diffusion quantities; its central claim is that the Bi-Mamba architecture performs well at inference tasks on AnDi-2 challenge data. The model is trained on those datasets and evaluated on a separate test set of 10^4 trajectories, with a bidirectional RNN as an internal baseline. The quoted statement 'For fair benchmarking purposes, we compare our model to a bidirectional RNN' and the reported AnDi-2 ranks (7th in alpha, 9th in K, 3rd in diffusion type, 10th in change-point detection) are external benchmark results, not quantities derived from the model's own fitted constants. The only self-citation in the reference list ([8], prior work by some authors on particle tracking) is background material and is not load-bearing for the architecture or the benchmark claims. The architecture inspiration from Bi-Mamba+ [33] is an independent external citation, not a self-citation. There is no uniqueness theorem, no ansatz smuggled in via citation, and no renaming of a known result. The weaknesses of the paper—absence of error bars or significance tests on the RNN comparison, and the transferability of AnDi-2 synthetic data to real experiments—are concerns about empirical support and external validity, not circularity. The derivation chain is therefore self-contained with respect to circularity, and the appropriate score is 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper introduces no physical constants or ad hoc scaling factors; its central claim is empirical. The main dependencies are the choice of input features, the architecture hyperparameters (not reported), and a set of domain assumptions about AnDi-2 benchmark realism and zero-padding.

free parameters (1)

Architecture hyperparameters (layers, state dimension, learning rate, epochs, batch size) = not reported
Chosen by hand and not specified, these tuning values affect the reported AnDi-2 scores and are needed for reproduction.

assumptions (4)

domain assumption The AnDi-2 challenge datasets are a valid external benchmark for anomalous diffusion characterization.
The central evaluation uses these synthetic trajectories as ground truth; realism relative to experimental data is assumed without a transfer study.
domain assumption The five phenomenological models (single-state, multi-state, dimerization, transient confinement, quenched trap) cover relevant experimental scenarios.
The paper maps each model to biological observations but does not justify completeness or representativeness.
domain assumption Zero-padding trajectories shorter than 200 time steps does not bias the learned representations.
The data paragraph states short trajectories are zero-padded; the effect of padding on the bidirectional scan is not analyzed.
domain assumption Hand-crafted features (displacements, 1D MSDs, displacement angle, distance from origin) are sufficient for K and alpha inference and segmentation.
The feature set is selected a priori; no feature importance or ablation is reported.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Bidirectional Mamba state-space model for anomalous diffusion." pith.science (2026). https://pith.science/paper/PVG5NVBM

@misc{pith2026241207299,
  author       = {Pith},
  title        = {Pith review of: Bidirectional Mamba state-space model for anomalous diffusion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PVG5NVBM}},
  note         = {Machine review of arXiv:2412.07299}
}

read the original abstract

Characterizing anomalous diffusion is crucial in order to understand the evolution of complex stochastic systems, from molecular interactions to cellular dynamics. In this work, we characterize the performances regarding such a task of Bi-Mamba, a novel state-space deep-learning architecture articulated with a bidirectional scan mechanism. Our implementation is tested on the AnDi-2 challenge datasets among others. Designed for regression tasks, the Bi-Mamba architecture infers efficiently the effective diffusion coefficient and anomalous exponent from single, short trajectories. As such, our results indicate the potential practical use of the Bi-Mamba architecture for anomalousdiffusion characterization.

Figures

Figures reproduced from arXiv: 2412.07299 by the authors.

**Figure 2.** FIG. 2. a) Bi-Mamba block implementation. b) Deeper Bi [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. Loss values calculated on a test dataset consisting [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

35 extracted references · 30 canonical work pages

[1]

Metzler and J

R. Metzler and J. Klafter, Physics reports 339, 1 (2000)

2000
[2]

MacKintosh and C

F. MacKintosh and C. Schmidt, Current opinion in col- loid & interface science 4, 300 (1999)

work page 1999
[3]

Sharifian Gh., Molecular Pharmaceutics 18, 2122 (2021), pMID: 33914545

M. Sharifian Gh., Molecular Pharmaceutics 18, 2122 (2021), pMID: 33914545

work page 2021
[4]

N. J. Yang and M. J. Hinner, Methods Mol. Biol. 1266, 29 (2015)

work page 2015
[5]

Dahan, S

M. Dahan, S. L´ evi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, Science 302, 442 (2003)

work page 2003
[6]

Cognet, D

L. Cognet, D. A. Tsyboulski, J.-D. R. Rocha, C. D. Doyle, J. M. Tour, and R. B. Weisman, Science 316, 1465 (2007)

work page 2007
[7]

Matse, M

M. Matse, M. V. Chubynsky, and J. Bechhoefer, Physical Review E 96, 042604 (2017)

2017
[8]

Lavaud, T

M. Lavaud, T. Salez, Y. Louyer, and Y. Amarouchene, Physical Review Research 3, L032011 (2021)

work page 2021

Show all 35 references

[9]

Vilquin, V

A. Vilquin, V. Bertin, E. Rapha¨ el, D. S. Dean, T. Salez, and J. D. McGraw, Physical Review Letters 130, 038201 (2023)

2023
[10]

Michael Schurr and V

J. Michael Schurr and V. Bloomfield, CRC Crit. Rev. Biochem. 4, 371 (1977)

1977
[11]

Stetefeld, S

J. Stetefeld, S. A. McKenna, and T. R. Patel, Biophys. Rev. 8, 409 (2016)

2016
[12]

Cerbino and P

R. Cerbino and P. Cicuta, J. Chem. Phys. 147, 110901 (2017)

2017
[13]

Midtvedt, S

B. Midtvedt, S. Helgadottir, A. Argun, J. Pineda, D. Midtvedt, and G. Volpe, Applied Physics Reviews 8, 011310 (2021)

2021
[14]

Mu˜ noz-Gil, G

G. Mu˜ noz-Gil, G. Volpe, M. A. Garcia-March, E. Aghion, A. Argun, C. B. Hong, T. Bland, S. Bo, J. A. Conejero, N. Firbas, `O. Garibo I Orts, A. Gentili, Z. Huang, J.-H. Jeon, H. Kabbech, Y. Kim, P. Kowalek, D. Krapf, H. Loch-Olszewska, M. A. Lomholt, J.-B. Masson, P. G. Meyer...

2021
[15]

Mu˜ noz-Gil, H

G. Mu˜ noz-Gil, H. Bachimanchi, J. Pineda, B. Midtvedt, M. Lewenstein, R. Metzler, D. Krapf, G. Volpe, and C. Manzo, Quantitative evaluation of methods to analyze motion changes in single-particle experiments [registered report stage 1 protocol] (2023)

2023
[16]

Saxton, Biophysical journal 66 2 Pt 1, 394 (1994)

M. Saxton, Biophysical journal 66 2 Pt 1, 394 (1994)

1994
[17]

M. J. Saxton, Biophysical journal 70 3, 1250 (1996)

1996
[18]

Nicolau, J

D. Nicolau, J. Hancock, and K. Burrage, Biophysical journal 92 6, 1975 (2007)

2007
[19]

Geisel and S

T. Geisel and S. Thomae, Physical Review Letters 52, 1936 (1984)

1984
[20]

Eggeling, C

C. Eggeling, C. Ringemann, R. Medda, G. Schwarzmann, K. Sandhoff, S. Polyakova, V. N. Belov, B. Hein, C. von Middendorff, A. Sch¨ onle, and S. W. Hell, Nature 457, 1159 (2009)

2009
[21]

Manzo, T

C. Manzo, T. S. van Zanten, and M. F. Garcia-Parajo, Biophys. J. 100, L8 (2011)

2011
[22]

Honigmann, V

A. Honigmann, V. Mueller, H. Ta, A. Schoenle, E. Sez- gin, S. W. Hell, and C. Eggeling, Nat. Commun. 5, 5412 (2014)

2014
[23]

Mainali and E

D. Mainali and E. A. Smith, Eur. Biophys. J. 42, 281 (2013)

2013
[24]

Yanagawa, M

M. Yanagawa, M. Hiroshima, Y. Togashi, M. Abe, T. Ya- mashita, Y. Shichida, M. Murata, M. Ueda, and Y. Sako, Sci. Signal. 11, eaao1917 (2018)

2018
[25]

da Rocha-Azevedo, S

B. da Rocha-Azevedo, S. Lee, A. Dasgupta, A. R. Vega, L. R. de Oliveira, T. Kim, M. Kittisopikul, Z. A. Malik, and K. Jaqaman, Cell Rep. 32, 108187 (2020)

2020
[26]

A. M. Achimovich, T. Yan, and A. Gahlmann, Biophys. J. 122, 3254 (2023)

2023
[27]

S. T. Low-Nam, K. A. Lidke, P. J. Cutler, R. C. Roovers, P. M. P. van Bergen en Henegouwen, B. S. Wilson, and D. S. Lidke, Nat. Struct. Mol. Biol. 18, 1244 (2011)

2011
[28]

Grimes, Z

J. Grimes, Z. Koszegi, Y. Lanoisel´ ee, T. Miljus, S. L. O’Brien, T. M. Stepniewski, B. Medel-Lacruz, M. Baidya, M. Makarova, D. M. Owen, A. K. Shukla, J. Selent, S. J. Hill, and D. Calebiro (2022)

2022
[29]

A. V. Weigel, M. M. Tamkun, and D. Krapf, Proc. Natl. Acad. Sci. U. S. A. 110, E4591 (2013)

2013
[30]

K. M. Spillane, J. Ortega-Arroyo, G. de Wit, C. Eggeling, H. Ewers, M. I. Wallace, and P. Kukura, Nano Lett. 14, 5390 (2014)

2014
[31]

Rossier, V

O. Rossier, V. Octeau, J.-B. Sibarita, C. Leduc, B. Tessier, D. Nair, V. Gatterdam, O. Destaing, C. Al- big` es-Rizo, R. Tamp´ e, L. Cognet, D. Choquet, B. Lounis, and G. Giannone, Nat. Cell Biol. 14, 1057 (2012)

2012
[32]

Gu and T

A. Gu and T. Dao, ArXiv abs/2312.00752 (2023)

2023 arXiv
[33]

Liang, X

A. Liang, X. Jiang, Y. Sun, X. Shi, and K. Li, Bi- mamba+: Bidirectional mamba for time series forecast- ing (2024), arXiv:2404.15772 [cs.LG]

2024 arXiv
[34]

Kabbech and I

H. Kabbech and I. Smal, Journal of Open Source Soft- ware 9, 6157 (2024)

2024
[35]

J. Ma, F. Li, and B. Wang, arXiv preprint arXiv:2401.04722 (2024)

2024 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Metzler and J

R. Metzler and J. Klafter, Physics reports 339, 1 (2000)

2000

[2] [2]

MacKintosh and C

F. MacKintosh and C. Schmidt, Current opinion in col- loid & interface science 4, 300 (1999)

work page 1999

[3] [3]

Sharifian Gh., Molecular Pharmaceutics 18, 2122 (2021), pMID: 33914545

M. Sharifian Gh., Molecular Pharmaceutics 18, 2122 (2021), pMID: 33914545

work page 2021

[4] [4]

N. J. Yang and M. J. Hinner, Methods Mol. Biol. 1266, 29 (2015)

work page 2015

[5] [5]

Dahan, S

M. Dahan, S. L´ evi, C. Luccardini, P. Rostaing, B. Riveau, and A. Triller, Science 302, 442 (2003)

work page 2003

[6] [6]

Cognet, D

L. Cognet, D. A. Tsyboulski, J.-D. R. Rocha, C. D. Doyle, J. M. Tour, and R. B. Weisman, Science 316, 1465 (2007)

work page 2007

[7] [7]

Matse, M

M. Matse, M. V. Chubynsky, and J. Bechhoefer, Physical Review E 96, 042604 (2017)

2017

[8] [8]

Lavaud, T

M. Lavaud, T. Salez, Y. Louyer, and Y. Amarouchene, Physical Review Research 3, L032011 (2021)

work page 2021