Pith. sign in

REVIEW 3 major objections 4 minor 55 references

The data geometry of masking diffusion: Certified-optimal schedules via unmasking growth complexity

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read A path-resolved measure of data geometry, the unmasking growth complexity, controls KL discretization error and yields certified-optimal masking diffusion schedules.

desk verdict A real theoretical advance in masking diffusion with a certificate that is narrower than advertised; the main theorems hold up but the data-dependent guarantee needs an extra layer for B_alpha. read the letter →

arxiv 2608.13520 v1 pith:VDPKUXV7 submitted 2026-08-13 cs.LG cs.AIcs.ITmath.ITmath.STstat.MLstat.TH

Martin J. Wainwright This is my paper · ORCID
classification cs.LGcs.AIcs.ITmath.ITmath.STstat.MLstat.TH MSC 62B1094A1760J22
keywords maskingdiffusionunmaskinggrowthcomplexityKLdiscretizationerrorcertified-optimalscheduleslog-reveal-oddscoordinatesBernoullifixed-cardinalitydata-dependentsampling
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

Masking diffusion samplers generate discrete data by progressively revealing masked coordinates, and their accuracy is set by how finely the reveal path is discretized. This paper claims that for both Bernoulli-subset and fixed-cardinality unmasking, the Kullback–Leibler (KL) discretization error is controlled by the local increments of one path-resolved quantity: the unmasking growth complexity (UGC), defined as a weighted integral of the curvature of mutual information along the reveal process. The increments are additive, and they can be estimated directly from clean samples through KL increments along coupled reveal trajectories, which lets a user build schedules that certify a prescribed KL error with high probability and run within a constant factor of the oracle-optimal iteration count. In log-reveal-odds coordinates the UGC density becomes the intrinsic local geometry, and its square-root integral gives the sharp leading-order optimal Euler discretization error in the fine-partition limit. The point of the paper is that schedule design for masking diffusion can be reduced to estimating and then allocating effort according to a measurable, data-dependent complexity path.

What carries the argument

The load-bearing object is the unmasking growth complexity (UGC), an additive interval measure $H(p,q)=\int_p^q t(1-t)h'(t)\,dt$ built from the Bernoulli unmasking gain $h(t)=\sum_i \mathrm{Info}(Z_i;X_t\mid i\in M(X_t))$; its additivity $H(p,r)=H(p,q)+H(q,r)$ is what lets global schedules be decomposed into blocks. Its log-reveal-odds density $q(\lambda)=r^2(1-r)^2h'(r)$, with $r=e^{\lambda}/(1+e^{\lambda})$, carries the local geometry, and the partition complexity $C(P)=\left(\sum_k \sqrt{S_k H_k}\right)^2$ quantifies the cost of a $K$-block geometric schedule. The mechanism that makes the theory data-driven is the sandwich of $H(p,q)$ between the KL unmasking increment $D(p,q)$ and twice it, estimated by running forced-mask reveal trajectories on clean samples and applying an empirical-Bernstein tail bound, which converts unknown UGC masses into certifiable confidence intervals.

What would settle it

Run the certified $K$-block procedure on the noisy repeated-bit ensemble with known UGC density, compute the true $H(p,q)$ exactly, and compare it with the truncated estimator $\hat{H}_m(p,q)$ over many sample sizes; the factor-two sandwich of Proposition 2 would be falsified if the estimate plus confidence radius fails to contain $H(p,q)$ at a rate exceeding the chosen $\eta$ for some distribution satisfying the moment condition (33a).

Watch

Extended reading notes

Core claim

The central claim is that the unmasking growth complexity $H(p,q)=\int_p^q t(1-t)h'(t)\,dt$, where $h(t)$ sums the conditional mutual information between each still-masked coordinate and the revealed ones, is the correct local currency for masking-diffusion error: the one-step KL defect of a Bernoulli unmasking step is at most $(\psi(q)/\psi(p)-1)H(p,q)$, with $\psi(t)=t/(1-t)$ the reveal odds, and an analogous bound holds for fixed-cardinality samplers with $H_{\mathrm{card}}$. The paper further claims that $H(p,q)$ is sandwiched to within a factor two by the KL increment $D(p,q)$ between forced-mask reveal trajectories, so a truncated Monte Carlo estimator of those increments yields high-probability upper confidence bounds on every block. Plugging these bounds into geometric schedules gives Theorem 2: for any $K$-block partition and any target accuracy, the sampler reaches $D_{\mathrm{KL}}(P_Z\|\hat{P}_{\hat{Z}})\le\varepsilon$ with probability at least $1-\eta$ using at most $N\approx 8\hat{C}_{\mathrm{UGC}}(P)/\varepsilon$ unmasking rounds. Theorem 3 sharpens the picture in the fine-partition limit: the infimum of the partition complexity is $(\int\sqrt{q}\,d\lambda)^2$, and the optimal $N$-step Euler discretization error is $\left(\int\sqrt{q}\,d\lambda\right)^2/(2N)+o(1/N)$.

Load-bearing premise

The certificate in Theorem 2 requires a known finite moment bound $B_\alpha$ on the aggregate KL changes of the exact one-coordinate Bayes denoisers over each reveal-odds dyadic interval, together with access to those exact denoisers; if the denoisers are learned, an extra approximation term enters the KL bound and is not certified.

Editorial extensions

If this is right

  • Single-block unmasking complexity is governed by the aggregate UGC mass, so existing fixed-cardinality guarantees are sharpened and Bernoulli unmasking matches CTMC-based $\tau$-leaping guarantees.
  • A $K$-block geometric schedule with explicit multipliers is within a factor 4 of the optimal dynamic-program allocation, so the main computation shifts from solving an integer program to estimating UGC increments.
  • Choosing $N\ge 8\hat{C}_{\mathrm{UGC}}(P)/\varepsilon$ rounds gives an end-to-end certificate $D_{\mathrm{KL}}\le\varepsilon$ with probability at least $1-\eta$, replacing oracle knowledge of the target geometry by sample estimates.
  • In the fine-partition limit, the optimal Euler error is determined by $\left(\int\sqrt{q}\,d\lambda\right)^2$, so regions of large $q$ require smaller reveal-odds steps and the square-root UGC density is the fundamental schedule-optimality object.
  • Geometry-aware block boundaries can yield substantial dimension-dependent gains, including $\widetilde{\Omega}(\sqrt{d})$ improvements with a constant number of adaptively placed blocks in the random XORSAT example.

Reading between the lines

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

  • The factor-two sandwich suggests a practical convergence diagnostic: per-block estimates of UGC mass from held-out samples could flag reveal times at which a trained denoiser has not captured the dominant dependencies, since those blocks will show persistently large confidence radii.
  • Because only the multiplicative reveal-odds ratio $\psi(q)/\psi(p)$ enters the one-step defect, the same additive machinery should extend to non-uniform or learned masking probabilities, although the paper does not analyze those variants.
  • The modulus-of-continuity bound for uniform partitions implies that for densities with sharp multi-peaked UGC, adaptive boundary selection should need far fewer blocks than uniform partitions; quantifying that gap for hierarchical mixture models is a testable extension of the paper's Lemma 3.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper introduces the unmasking growth complexity (UGC), a path-resolved functional of the masking reveal process, and uses it to bound the KL discretization error of Bernoulli-subset and fixed-cardinality unmasking samplers. The main technical results are: Theorem 1, bounding the KL error of both samplers by additive sums of local UGC increments; Proposition 2, a tail-robust estimator that sandwiches UGC increments from samples using KL increments along coupled reveal trajectories; Theorem 2, a data-dependent multi-block sampler with a high-probability KL certificate under stated moment and exact-denoiser assumptions; and Theorem 3, identifying the fine-partition limit with the squared integral of the square-root UGC density and the sharp leading-order optimal Euler error. The paper also connects aggregate UGC to classical dependence measures and exhibits examples with exponential-in-sqrt(d) gains in the ratio between coarse and fine partition complexities.

Significance. If the results hold as stated, the UGC framework is a valuable unification: it gives explicit additive KL bounds for two unmasking schemes, connects Bernoulli and fixed-cardinality analyses through exact information-profile representations, and provides a data-dependent estimation procedure with finite-sample Bernstein-type guarantees. The paper is refreshing in that the main Bernoulli proof is self-contained and the fixed-cardinality part is explicitly built on the exact representation of Chen et al. The fine-partition/Euler result (Theorem 3) is a clean sharp leading-order statement. The central weakness, already acknowledged in Section 7, is that the headline certified-optimal claim in Theorem 2 requires a known moment bound B_alpha and exact Bayes denoisers; for learned denoisers the additional approximation term (18) is not certified. This is not a mathematical inconsistency, but it is a material gap between the abstract's promise of certifiable samplers and the theorem's hypotheses.

major comments (3)
  1. [4.4 (Theorem 2, Eq. (37b))] The certified-optimal guarantee depends on condition (33a), which presumes a known finite moment bound B_alpha on aggregate denoiser KL increments, and on access to exact Bayes denoisers. No procedure is given for obtaining a valid high-probability upper bound on B_alpha from samples while preserving the simultaneous certificate. Estimating B_alpha from the same data would introduce an additional error term that is not accounted for in Proposition 2 or Theorem 2. Since equation (35) promises data-dependent parameter choices that guarantee D_KL <= epsilon with high probability, the claim as stated is only established under an oracle-like moment condition. Section 7 acknowledges the learned-denoiser issue, but the abstract and the certified-optimal terminology in Section 4.4 overstate what the theorem proves.
  2. [4.2 (Proposition 1)] The proof of Proposition 1 is delegated to the companion paper [Wai26] with only a 'mutatis mutandis' explanation. Because [Wai26] is an unpublished preprint and Proposition 1 is load-bearing for the near-optimality claim of the oracle schedule, this leaves a proof gap in the present manuscript. A self-contained proof, or a version-of-record reference with the specific lemmas adapted to the log-reveal-odds clock, is needed.
  3. [4.4.2 (Certified-optimal boundary selection)] The dynamic-programming boundary selection is described conceptually, but no theorem is stated or proved that the data-dependent selected partition satisfies a KL certificate after optimizing the block boundaries. Theorem 2 applies only to a fixed K-block partition; when the partition itself is chosen from data using estimated edge costs, the selection procedure introduces additional uncertainty that is not covered by the union bound in the proof of Theorem 2. As written, the claim of certified boundary selection is not supported by a formal guarantee.
minor comments (4)
  1. [2.2.1] The notation eOmega(sqrt(d)) in the abstract and Section 2.2.1 is never defined; please define it or use standard asymptotic notation.
  2. [3.2, Eq. (30a)] The display X_{t|i} := (M_{t|i}, Z_{(M_{t|i})^c}) appears to conflate the masked set with the masked vector; the first component should presumably be the vector of mask symbols on M_{t|i}. This is a notation issue but it makes the forced-mask process harder to read.
  3. [4.2, Proposition 1] The proposition refers to 'score evaluations' while the rest of the paper speaks of 'unmasking rounds' or 'iterations'; please use consistent terminology and clarify whether the score budget counts one denoiser evaluation per coordinate per round.
  4. [5.2.2, Lemma 3] The uniform-partition convergence bound (46) is stated for f=sqrt(q) continuous on the closed interval, but the discussion after it mentions a Lipschitz constant Lip(f) without specifying how the Lipschitz constant scales with dimension; please make the dependence explicit.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction: the UGC bound is derived from mutual-information increments, and the certified guarantee composes confidence intervals rather than fitting the target KL error.

full rationale

The central derivation chain is self-contained. The unmasking growth complexity H(p,q) is defined through the mutual-information derivative h' in equation (2c), not through the target KL divergence. Theorem 1's proof via Lemma 4 derives the exact one-step representation Gamma_umask(p,q) = integral_p^q (q-u) h'(u) du and then bounds it by (psi(q)/psi(p)-1) H(p,q); this is a genuine inequality, not an identity by construction. Lemma 1 derives D(p,q)/(q-p) = h(q)-h(p) from the Markov property of the forced-mask coupling, so the estimator in Proposition 2 estimates the UGC increment H, not the final sampling KL error. Proposition 2 and Theorem 2 then compose a high-probability upper-confidence sandwich; the bound contains the estimated complexity, but that is standard certified optimization, not a fitted prediction. The possible blockers to the headline guarantee are explicit assumptions rather than circular definitions: condition (33a) requires a known finite moment bound B_alpha on denoiser KL increments, and equation (18), together with Section 7, states that learned denoisers add an unestimated approximation term Eden. Those are correctness and assumption gaps, not circular self-reference. The proofs of Proposition 1 and Lemma 3 refer to the companion Gaussian paper [Wai26] for analogous arguments; since those are parameter-free proofs of an analogous but different result and do not assume the masking conclusion, they are self-citations but not load-bearing circularity. Overall, no step of the derivation reduces to its own input.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper introduces UGC as a mathematical functional, not as a physical entity; the main extra assumptions are the known moment bound B_alpha, access to exact Bayes denoisers, and regularity of q for the limit theorems. No new particles, forces, or dimensions are postulated.

free parameters (1)
  • B_alpha (moment bound) = unknown or assumed
    The tail-robust estimator (33b) and confidence radius (34b) require a known finite bound B_alpha on the alpha-th moment of aggregate KL increments; B_alpha is not estimated from data, yet the certified guarantee depends on it.
assumptions (5)
  • standard math Standard information-theoretic and probability tools: chain rules for KL and mutual information, Fubini, empirical Bernstein inequality (Maurer-Pontil).
    Invoked throughout the proofs of Theorems 1, 2, 3 and Proposition 2.
  • domain assumption The target distribution P_Z over a finite alphabet and the unmasking process driven by i.i.d. Uniform[0,1] variables independent of Z.
    Defines the problem setting in Section 2.1, equations (1a) and (1b).
  • ad hoc to paper Exact Bayes single-site posterior denoisers mu_i are available, and their KL increments satisfy the moment condition (33a) with a known B_alpha.
    Required for Proposition 2 and Theorem 2; the paper acknowledges in Section 7 that learned denoisers add an uncertified term (18).
  • domain assumption The UGC density q is continuous and strictly positive on [-l_d, l_d], and f = sqrt(q) is continuous for Lemma 3.
    Assumed in Theorem 3 and Lemma 3 to obtain the fine-partition limit and uniform convergence bounds.
  • standard math Existence of a good random matrix A in the XORSAT construction via the probabilistic method.
    Used in Appendix D, Lemma 2 proof, with Hoeffding bounds and union bounds over random subspaces.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The data geometry of masking diffusion: Certified-optimal schedules via unmasking growth complexity." pith.science (2026). https://pith.science/paper/VDPKUXV7

@misc{pith2026260813520,
  author       = {Pith},
  title        = {Pith review of: The data geometry of masking diffusion: Certified-optimal schedules via unmasking growth complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VDPKUXV7}},
  note         = {Machine review of arXiv:2608.13520}
}
abstract

We study masking diffusion for discrete sampling and introduce a path-resolved measure of data geometry called the \emph{unmasking growth complexity} ({\textsf{UGC}\xspace}). Its local increments directly control Kullback--Leibler (KL) discretization error, yielding a unified analysis of Bernoulli-subset and fixed-cardinality unmasking schemes. In log-reveal-odds coordinates, this structure yields optimized single-block and multi-block schedules, and quantifies the gains from adapting computational effort to data geometry. Crucially, we show how {\textsf{UGC}\xspace} increments can be estimated from samples via KL increments along coupled reveal trajectories. This leads to \emph{certified-optimal} samplers that achieve a prescribed KL error with high probability and iteration complexity within a constant factor of the corresponding oracle procedure. Collapsing the \ugc path yields the aggregate {\textsf{UGC}\xspace} mass, which connects to classical multivariate dependence measures and complexity measures from previous analyses of discrete diffusion. In the fine-partition limit, the squared integral of the square-root {\textsf{UGC}\xspace} density determines the sharp leading-order optimal Euler discretization error. Examples exhibit substantial dimension-dependent gains over coarse schedules, including $\widetilde{\Omega}(\sqrt{d})$ improvements achievable with a constant number of adaptively placed blocks.

Figures

Figures reproduced from arXiv: 2608.13520 by the authors.

Figure 1
Figure 1. Plots of the log-reveal-odds UGC density q for the noisy repeated-bit ensemble for η ∈ {0.01, 0.30, 0.45}. The density has a sharp peak for η = 0.01, and then flattens out and shifts to the right as η → 0.5. Up to two digits of accuracy, we have Ratio(PZ ) ∈ {4.51, 2.16, 1.65} in panels (a), (b), and (c), respectively. For this example, we have Ratio(PZ ) ≍ log(d) as the dimension grows [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figure 2
Figure 2. Plots of the log-reveal-odds UGC density q for the discrete mixture model. For each dimension d, the latent vector Z ∈ {0, 1} d is generated by selecting uniformly from M = 2d/4 cluster centers, and then flipping each coordinate independently with probability η = 0.02. Panels (a)–(c) correspond to d ∈ {32, 48, 64}, and hence M ∈ {256, 4096, 65536}, respectively. Increasing the dimension produces a progressively shar… view at source ↗
Figure 3
Figure 3. Hierarchical binary-prototype mixtures and their log-reveal-odds UGC densities. The rows corre￾spond to hierarchy depths L = 2 and L = 4. Left column panels (a) and (c) show the leaf prototypes, while right column panels (b) and (d) show the corresponding UGC densities q. 2.2.2 From coarse to fine geometry Let us describe how we exploit finer-grained UGC geometry for algorithmic purposes. A single-block scheme uses … view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Block geometry for the d = 64 random discrete mixture with M = 2d/4 and η = 0.02. Both panels show the square-root UGC density √q versus the log-reveal-odds λ. Panel (a) uses a single geometric multiplier, yielding CUGC ≈ 65.0. Panel (b) uses the illustrated K = 3 bloc…
Figure 5
Figure 5. Figure 5: Log-reveal-odds UGC densities for the repeated-common-bit model in panel (a) and the global￾parity model in panel (b), both with d = 64. The two densities are reflections of one another about λ = 0 in log-reveal-odds coordinates: their information is concentrated near …
Figure 6
Figure 6. Figure 6: plots the UGC density for d = 64 and α ∈ {0.2, 0.5, 0.8}. The density is unimodal, with its peak moving from right to left as α increases. To quantify the resulting gain, we focus on the symmetric case α = 0.5. 2 0 2 Log reveal odds 0 5 10 15 U G C d ensity q( ) 2 0 2 …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

55 extracted references · 43 canonical work pages

  1. [1]

    Spencer , edition =

    Noga Alon and Joel H. Spencer , edition =. The Probabilistic Method , year =

    work page

  2. [2]

    Chen and K

    S. Chen and K. Cong and J. Li , eprint =. arXiv preprint arXiv:2511.04647 , title =

    arXiv

  3. [3]

    Dmitriev and Z

    D. Dmitriev and Z. Huang and Y. Wei , eprint =. arXiv preprint arXiv:2602.15008 , title =

    arXiv

  4. [4]

    arXiv , arxivid =:2506.02318 , title =

    Yuchen Liang and Renxiang Huang and Lifeng Lai and Ness Shroff and Yingbin Liang , booktitle =. arXiv , arxivid =:2506.02318 , title =

    arXiv

  5. [5]

    arXiv , arxivid =:2509.16756 , title =

    Yuchen Liang and Yingbin Liang and Lifeng Lai and Ness Shroff , booktitle =. arXiv , arxivid =:2509.16756 , title =

    arXiv

  6. [6]

    Rotskoff and Molei Tao and Lexing Ying , booktitle =

    Yinuo Ren and Haoxuan Chen and Yuchen Zhu and Wei Guo and Yongxin Chen and Grant M. Rotskoff and Molei Tao and Lexing Ying , booktitle =. arXiv , arxivid =:2502.00234 , title =

    arXiv

  7. [7]

    Rotskoff and Lexing Ying , booktitle =

    Yinuo Ren and Haoxuan Chen and Grant M. Rotskoff and Lexing Ying , booktitle =. arXiv , arxivid =:2410.03601 , title =

    arXiv

  8. [8]

    Gillespie , doi =

    Daniel T. Gillespie , doi =. Approximate Accelerated Stochastic Simulation of Chemically Reacting Systems , volume =. The Journal of Chemical Physics , number =

    work page

Show all 55 references
  1. [9]

    Petzold and Yang Cao and Daniel T

    Muruhan Rathinam and Linda R. Petzold and Yang Cao and Daniel T. Gillespie , doi =. Consistency and Stability of Tau-Leaping Schemes for Chemical Reaction Systems , volume =. Multiscale Modeling & Simulation , number =

  2. [10]

    Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems , volume =

    Tiejun Li , doi =. Analysis of Explicit Tau-Leaping Schemes for Simulating Chemically Reacting Systems , volume =. Multiscale Modeling & Simulation , number =

  3. [11]

    Anderson and Arnab Ganguly and Thomas G

    David F. Anderson and Arnab Ganguly and Thomas G. Kurtz , doi =. Error Analysis of Tau-Leap Simulation Methods , volume =. The Annals of Applied Probability , number =

  4. [12]

    Linear Dependence Structure of the Entropy Space , volume =

    Te Sun Han , doi =. Linear Dependence Structure of the Entropy Space , volume =. Information and Control , number =

  5. [13]

    Nonnegative Entropy Measures of Multivariate Symmetric Correlations , volume =

    Te Sun Han , doi =. Nonnegative Entropy Measures of Multivariate Symmetric Correlations , volume =. Information and Control , number =

  6. [14]

    Information Theoretical Analysis of Multivariate Correlation , volume =

    Satosi Watanabe , doi =. Information Theoretical Analysis of Multivariate Correlation , volume =. IBM Journal of Research and Development , number =

  7. [15]

    Proceedings of the 22nd Conference on Learning Theory , eprint =

    Empirical. Proceedings of the 22nd Conference on Learning Theory , eprint =

  8. [16]

    Wainwright , eprint =

    Martin J. Wainwright , eprint =. Denoising Growth Complexity:. arXiv preprint arXiv:2607.26285 , number =. 2026 , institution =

    2026 arXiv

  9. [17]

    Approximate Maximizers of Intricacy Functionals , volume =

    J. Approximate Maximizers of Intricacy Functionals , volume =. doi:10.1007/s00440-011-0350-y , journal =

    doi

  10. [18]

    Mean Mutual Information and Symmetry Breaking for Finite Random Fields , volume =

    J. Mean Mutual Information and Symmetry Breaking for Finite Random Fields , volume =. doi:10.1214/11-AIHP416 , journal =

    doi

  11. [19]

    Edelman , doi =

    Giulio Tononi and Olaf Sporns and Gerald M. Edelman , doi =. A Measure for Brain Complexity: Relating Functional Segregation and Integration in the Nervous System , volume =. Proceedings of the National Academy of Sciences of the United States of America , number =

  12. [20]

    How Should Complexity Scale with System Size? , volume =

    Eckehard Olbrich and Nils Bertschinger and Nihat Ay and J. How Should Complexity Scale with System Size? , volume =. doi:10.1140/epjb/e2008-00134-9 , journal =

    doi

  13. [21]

    Buckley and Seth Bullock , doi =

    Lionel Barnett and Christopher L. Buckley and Seth Bullock , doi =. Neural Complexity and Structural Connectivity , volume =. Physical Review E , number =

  14. [22]

    Rosas and Pedro A

    Fernando E. Rosas and Pedro A. M. Mediano and Michael Gastpar and Henrik J. Jensen , doi =. Quantifying High-Order Interdependencies via Multivariate Extensions of the Mutual Information , volume =. Physical Review E , number =

  15. [23]

    Varley and Maria Pope and Joshua Faskowitz and Olaf Sporns , doi =

    Thomas F. Varley and Maria Pope and Joshua Faskowitz and Olaf Sporns , doi =. Multivariate Information Theory Uncovers Synergistic Subsystems of the Human Cerebral Cortex , volume =. Communications Biology , pages =

  16. [24]

    G. G. Lorentz , edition =

  17. [25]

    G. M. Phillips , publisher =

  18. [26]

    doi:10.18653/v1/D19-1633 , pages =

    Marjan Ghazvininejad and Omer Levy and Yinhan Liu and Luke Zettlemoyer , booktitle =. doi:10.18653/v1/D19-1633 , pages =

    doi

  19. [27]

    Freeman , booktitle =

    Huiwen Chang and Han Zhang and Lu Jiang and Ce Liu and William T. Freeman , booktitle =

  20. [28]

    Freeman and Michael Rubinstein and Yuanzhen Li and Dilip Krishnan , booktitle =

    Huiwen Chang and Han Zhang and Jarred Barber and Aaron Maschinot and Jose Lezama and Lu Jiang and Ming-Hsuan Yang and Kevin Patrick Murphy and William T. Freeman and Michael Rubinstein and Yuanzhen Li and Dilip Krishnan , booktitle =

  21. [29]

    Hauptmann and Ming-Hsuan Yang and Yuan Hao and Irfan Essa and Lu Jiang , booktitle =

    Lijun Yu and Yong Cheng and Kihyuk Sohn and Jose Lezama and Han Zhang and Huiwen Chang and Alexander G. Hauptmann and Ming-Hsuan Yang and Yuan Hao and Irfan Essa and Lu Jiang , booktitle =

  22. [30]

    Chiu and Alexander Rush and Volodymyr Kuleshov , booktitle =

    Subham Sekhar Sahoo and Marianne Arriola and Yair Schiff and Aaron Gokaslan and Edgar Marroquin and Justin T. Chiu and Alexander Rush and Volodymyr Kuleshov , booktitle =. Simple and Effective Masked Diffusion Language Models , volume =

  23. [31]

    Diffusion Language Models Are Versatile Protein Learners , volume =

    Xinyou Wang and Zaixiang Zheng and Fei Ye and Dongyu Xue and Shujian Huang and Quanquan Gu , booktitle =. Diffusion Language Models Are Versatile Protein Learners , volume =

  24. [32]

    Yuancheng Wang and Haoyue Zhan and Liwei Liu and Ruihong Zeng and Haotian Guo and Jiachen Zheng and Qiang Zhang and Xueyao Zhang and Shunsi Zhang and Zhizheng Wu , booktitle =

  25. [33]

    Large Language Diffusion Models , volume =

    Shen Nie and Fengqi Zhu and Zebin You and Xiaolu Zhang and Jingyang Ou and Jun Hu and Jun Zhou and Yankai Lin and Ji-Rong Wen and Chongxuan Li , booktitle =. Large Language Diffusion Models , volume =

  26. [34]

    and De Bortoli, V

    Benton, J. and De Bortoli, V. and Doucet, A. and Deligiannidis, G. , booktitle =. Nearly d -Linear Convergence Bounds for Diffusion Models via Stochastic Localization , url =

  27. [35]

    Brooks and A

    S. Brooks and A. Gelman and G. L. Jones and X. L. Meng , publisher =. Handbook of

  28. [36]

    Chen and H

    H. Chen and H. Lee and J. Lu , booktitle =. 2211.01916 , title =

    arXiv

  29. [37]

    Chen and S

    S. Chen and S. Chewi and J. Li and Y. Li and A. Salim and A. R. Zhang , booktitle =. arXiv , arxivid =:2209.11215 , title =

    arXiv

  30. [38]

    and Mei, S

    Chen, M. and Mei, S. and Fan, J. and Wang, M. , journal =. An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimization , url =

  31. [39]

    Croitoru and V

    F.-A. Croitoru and V. Hondru and R. T. Ionescu and M. Shah , doi =. Diffusion Models in Vision: A Survey , volume =. IEEE Transactions on Pattern Analysis and Machine Intelligence , number =

  32. [40]

    Gelman and J

    A. Gelman and J. Carlin and H. S. Stern and D. B. Dunson and A. Vehtari and D. K. Salomon , journal =

  33. [41]

    Ho and A

    J. Ho and A. Jain and P. Abbeel , journal =. Denoising diffusion probabilistic models , volume =

  34. [42]

    and Casella, George , title =

    Robert, Christian P. and Casella, George , title =. 2004 , doi =

    2004

  35. [43]

    Rombach and E

    R. Rombach and E. Blattmann and S. L. Dhariwal and A. M. D. M. L. and P. E. S. , journal =. High-Resolution Image Synthesis with Latent Diffusion Models , year =

  36. [44]

    R. Y. Rubinstein and D. P. Kroese , edition =. Simulation and the Monte Carlo Method , year =

  37. [45]

    Sohl‑Dickstein and E

    J. Sohl‑Dickstein and E. Weiss and N. Maheswaranathan and S. Ganguli , booktitle =. Deep Unsupervised Learning using Nonequilibrium Thermodynamics , url =

  38. [46]

    Song and S

    Y. Song and S. Ermon , booktitle =. Generative Modeling by Estimating Gradients of the Data Distribution , year =

  39. [47]

    Song and J

    Y. Song and J. Sohl-Dickstein and D. P. Kingma and A. Kumar and S. Ermon and B. Poole , booktitle =. Score-Based Generative Modeling through Stochastic Differential Equations , year =

  40. [48]

    Yang and Z

    L. Yang and Z. Zhang and Y. Song and et al. , journal =. Diffusion Models: A Comprehensive Survey of Methods and Applications , year =

  41. [49]

    Advances in Neural Information Processing Systems , title =

    Emiel Hoogeboom and Didrik Nielsen and Priyank Jaini and Patrick Forr. Advances in Neural Information Processing Systems , title =

  42. [50]

    Johnson and Jonathan Ho and Daniel Tarlow and Rianne van den Berg , booktitle =

    Jacob Austin and Daniel D. Johnson and Jonathan Ho and Daniel Tarlow and Rianne van den Berg , booktitle =. Structured Denoising Diffusion Models in Discrete State-Spaces , volume =

  43. [51]

    doi:10.52202/068431-2049 , title =

    Andrew Campbell and Joe Benton and Valentin De Bortoli and Thomas Rainforth and George Deligiannidis and Arnaud Doucet , booktitle =. doi:10.52202/068431-2049 , title =

    doi

  44. [52]

    Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution , url =

    Aaron Lou and Chenlin Meng and Stefano Ermon , booktitle =. Discrete Diffusion Modeling by Estimating the Ratios of the Data Distribution , url =

  45. [53]

    Titsias , booktitle =

    Jiaxin Shi and Kehang Han and Zhe Wang and Arnaud Doucet and Michalis K. Titsias , booktitle =. Simplified and Generalized Masked Diffusion for Discrete Data , volume =

  46. [54]

    arXiv , arxivid =:2505.21400 , title =

    Gen Li and Changxiao Cai , booktitle =. arXiv , arxivid =:2505.21400 , title =

  47. [55]

    arXiv preprint arXiv:2510.25544 , title =

    Hugo Lavenant and Giacomo Zanella , eprint =. arXiv preprint arXiv:2510.25544 , title =

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.