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REVIEW 2 major objections 6 minor 16 references

Stochastic-interpolant generative modeling can shed neural-network training: the drift comes from a P×P linear system, and with the Girsanov-optimal diffusion, sampling reduces to integrating an SDE.

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2026-08-02 21:25 UTC pith:MCD27ZBZ

load-bearing objection The kernelized interpolant idea is neat, but the advertised KL guarantee is vacuous at t=1 for finite features and the experiments are too thin to fully support the claims. the 2 major comments →

arxiv 2602.20070 v3 pith:MCD27ZBZ submitted 2026-02-23 cs.LG

Generative Modeling via Kernelized Stochastic Interpolants

Florentin Coeurdoux , Etienne Lempereur , Nathana\"el Cuvelle-Magar , St\'ephane Mallat , Eric Vanden-Eijnden This is my paper
classification cs.LG
keywords generative modelingstochastic interpolantskernel methodslinear systemsoptimal diffusionscattering transformmodel combinationtraining-free generation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to show that the velocity field of a stochastic interpolant — the process that transports Gaussian noise into data — can be estimated by solving a P×P linear system at each time step, using gradients of a feature map evaluated on paired noise–data samples. This replaces the neural-network regression that normally learns the drift, and because P is independent of the data dimension d, the precomputation is cheap. The authors derive the diffusion coefficient that minimizes a path-KL bound on generation error; it diverges at t=0 and vanishes at t=1, and they supply an integrator that handles the singular endpoint without clamping. They demonstrate the approach on financial log-returns, turbulent and cosmological fields, and images, using scattering transforms and pretrained velocity fields as feature maps, and show that the linear system can combine weak or cross-domain generative models into a stronger sampler. If correct, generative modeling becomes a linear-algebra preconditioning step.

Core claim

The central result is Proposition 2.1: for the ansatz b̂_t(x) = ∇φ(x)^⊤ η_t, the minimizer of the velocity regression loss over η_t is obtained by solving K_t η_t = E[∇φ(I_t)·İ_t], where K_t is the Gram matrix of feature gradients under the interpolant measure. The corresponding score estimate satisfies the exact drift–score relation, so no separate score regression is needed. Proposition 2.2 identifies the diffusion coefficient D_t^* = α_t γ_t / β_t as the pointwise minimizer of the path-KL integrand, and the paper proves that with this choice the generative SDE takes the form of Eq. (13), which integrates stably despite D_0^* = ∞. The KL between generated and target distributions is bounde

What carries the argument

The feature-gradient ansatz b̂_t(x) = ∇φ(x)^⊤ η_t is the engine: it reduces drift estimation to a dense P×P linear system solvable from samples of the interpolant. The Gram matrix K_t = E[∇φ(I_t)∇φ(I_t)^⊤] and cross-moment r_t = E[∇φ(I_t)·İ_t] are both accessible from paired noise–data draws, so all coefficients η_{t_k} can be precomputed without any optimization loop. The Girsanov-optimal diffusion D_t^* = α_t γ_t / β_t is the second key object; it simplifies the SDE to Eq. (13) and makes the time-reversed dynamics score-free (Appendix B), so the backward process depends only on the interpolant schedule. The integrator (Eq. 16), derived from d(β_t X_t) with a trapezoidal rule, handles the d

Load-bearing premise

The whole construction stands or falls on whether the true interpolant velocity b_t(x) is well approximated by the span of the feature-map gradients {∇φ_i(x)} under the interpolant measure, and whether the empirical Gram matrix K̂_t is invertible and not nearly singular at every time step; if either fails, the bound (Eq. 11) is vacuous and the generated samples silently miss the target.

What would settle it

Choose a target distribution whose exact velocity field can be computed analytically but is provably not in the span of a given feature map (e.g., a Gaussian mixture under linear features), run Algorithm 1 with a fine grid and large N, and compare the empirical terminal KL against the bound (Eq. 10). If the bound is violated or the error does not track the drift residual, the central reduction is unsound; in the S&P experiment, also check that N>P and that K̂_t is full-rank at all K time steps, since a rank-deficient system at any step invalidates the Proposition 2.1 guarantee.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any number of samples can be drawn after one precomputation: solve Eq. (7) on the time grid, store {η_{t_k}}, then integrate Eq. (16) with no further optimization or backpropagation.
  • Pretrained or even weakly-trained velocity fields serve as feature gradients; Eq. (17) turns model combination into a linear system, and the experiments show the ensemble outperforms its weakest members and generalizes across training domains.
  • Single-realization generation becomes feasible: a single long time series (e.g., the S&P 500 log-returns) provides enough paired samples for the linear system, a regime in which neural-network training is impossible.
  • The bound (Eq. 11) makes generation error a function of the drift residual, so the feature map's expressivity is the only hyperparameter (beyond discretization) that needs to be tuned; richer features reduce the bound.
  • With a characteristic kernel in the infinite-dimensional setting, the linear-system estimate recovers the exact velocity, meaning the training-free paradigm is not limited to finite feature maps.

Where Pith is reading between the lines

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

  • Because the coefficients are solved from paired noise–data samples, the method could be applied in an online or streaming setting, updating η_t incrementally as new data arrive — a straightforward extension the paper does not explore.
  • As P grows, the Gram matrix K̂_t becomes higher-dimensional and its condition number will degrade; a ridge-regularized version of Eq. (7) would be a natural robustness fix, and the paper does not address this.
  • The cross-domain results suggest that any sufficiently diverse set of velocity fields acts as a universal feature library; a testable implication is that randomly initialized networks with frozen weights might already provide a usable feature set, eliminating the pretraining requirement entirely.
  • The divergence D_0^* = ∞ implies the first integration step is essentially a Gaussian resampling; one could exploit this by using an adaptive time grid that is coarse near t=0 and refined near t=1, which the fixed-grid experiments do not.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The paper proposes a kernelized reformulation of stochastic interpolant generative models. The drift is restricted to the span of gradients of a finite feature map, so its estimate reduces to solving a P×P linear system at each time step; the optimal diffusion coefficient from previous Girsanov-based analyses is applied, and a custom integrator handles the singular endpoints of the optimal schedule. The method is demonstrated on financial time series, physical fields, and ensembles of weak generative models, including a cross-domain composition experiment.

Significance. If the theoretical claims held, this would be a valuable contribution: it would reduce generative-model training to linear algebra, enable training-free model combination, and make the dependence on data dimension explicit. The linear-system construction (Prop. 2.1) is clean and correctly derived, and the integrator (Eq. 16) is a thoughtful treatment of the singular D_t* endpoints. The empirical demonstrations, especially the cross-domain ensemble, are creative and suggestive. However, the advertised finite-KL guarantee via Girsanov is not established for generic finite feature maps, and one key empirical setup is under-specified. The core algorithmic idea may still be useful, but the paper's central theoretical claim needs substantial repair.

major comments (2)
  1. [§2.3, Eq. (10)–(12), Remark 2.3] The claimed KL bound is infinite for generic finite feature maps. For D_t* = α_t γ_t / β_t, the integrand in Eq. (10) becomes (β_t/(α_t γ_t)) r_t, where r_t = E|b_t(I_t) − ∇φ(I_t)^T η_t|². For the linear interpolant α_t=1−t, β_t=t, γ_t=1, b_1(x)=x. If the feature-gradient span does not contain the identity map, which is the generic case for scattering features or pretrained networks, then lim_{t→1} r_t > 0 and the integrand behaves like r_1/(1−t), making the integral in Eq. (10) diverge logarithmically. Consequently Eq. (11) is vacuous. The truncation at t=1−δ suggested in Remark 2.3 does not fix the issue because the truncated integral diverges as δ→0. This is not a conditioning problem but an internal inconsistency in the statement that D_t* ``controls generation error.'' The pointwise minimization itself is correct, but the Girsanov bound must either be replaced by a finite-horizon st
  2. [§3.1, Algorithm 1] The single-realization S&P experiment is missing the central empirical ingredient: how the N pairs (z^n, a^n) used in Eq. (7) are constructed from one observed sequence of length d=6064. With a single observed realization, a^n cannot be i.i.d. draws from the target measure on R^d unless the sequence is reinterpreted as a collection of blocks or windows, which changes the meaning of d. The paper reports K=1200 and P=217 but never states N, nor the conditioning of the Gram matrix, nor any regularization. Without this information the flagship ``generation from a single realization'' claim cannot be evaluated, and it is also unclear whether the algorithm is solving the stated problem or a different degenerate problem. Please specify the exact construction of the data pairs and report the resulting Gram-matrix condition numbers on the time grid.
minor comments (6)
  1. [Eq. (15)–(16), Algorithm 1] The symbol σ_t appears in the stochastic integral variance where γ_t is meant; the diffusion coefficient is sqrt(2 α_t β_t γ_t). This typo should be corrected.
  2. [Abstract and §2.2] The phrase ``P independent of the data dimension d'' is only true in the sense of the linear-system size. For the scattering features used in the experiments, P = O(log^3 d) (Appendix D), which depends on d. Please qualify the statement.
  3. [Appendix A, Theorem A.1] The proof of exact recovery uses a sketch: it does not state assumptions for interchanging derivative and expectation, existence/uniqueness of the η* equation, or the precise definition of the operator on the Hilbert space. If this theorem is meant to be more than a heuristic, it needs rigorous hypotheses; otherwise it should be labeled as a formal argument.
  4. [§3.2] The physical-field results are only qualitative. Quantitative comparisons (e.g., power spectra, scattering spectra, or MMD) would materially strengthen the claim that generated fields match the target statistics.
  5. [§3.3] The ``oracle log-likelihood'' evaluation is not defined precisely: how the oracle U-Net computes a likelihood for 28×28 images, whether it is a discrete or continuous likelihood, and how the average is computed. This is needed to interpret the monotone improvement in the ensemble-size plot.
  6. [Figure 2] The caption labels the third column ``2D turbulence'', while the text (§3.2) refers to ``3D magnetic turbulence (vorticity)''. The labels should be made consistent.

Circularity Check

0 steps flagged

No significant circularity: the linear-system drift estimate, Girsanov KL bound, and optimal-diffusion integrand minimizer are derived in-paper; self-citations are attribution/background, not load-bearing.

full rationale

The claimed derivation chain is self-contained. Proposition 2.1 (Eq. 6) is the normal equation of the quadratic regression loss (Eq. 3): minimizing E|∇φ(I_t)^T η_t − İ_t|² over η_t gives K_t η_t = E[∇φ(I_t)·İ_t]; the target distribution enters only through the data-dependent right-hand side, and the proposition is a projection identity rather than a restatement of the target. Proposition 2.2 gives D*_t = α_tγ_t/β_t as the pointwise minimizer of the nonnegative Girsanov integrand in Eq. 10, with the derivative shown in its proof; the citations to Chen et al. (2024) and Ma et al. (2024) are attribution, not the argument. The KL bound (Eq. 11) is an inequality bounding terminal KL by the fitted drift residual, not an equality that identifies the prediction with the fit. The empirical systems (Eq. 7, Eq. 17) are least-squares normal equations; the numerical sections compare generated samples to target statistics and to an oracle log-likelihood, rather than reporting fitted quantities as predictions. Self-citations to stochastic interpolants, scattering spectra, and Moment-Guided Diffusion are background or contrast. Remark 2.3 itself flags the divergence of the t→1 weight in Eq. 10; that is a mathematical-validity concern about the finiteness of the bound for finite feature maps, not a circularity, so it does not change this verdict.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 0 invented entities

The ledger items are mostly inherited constructions (interpolants, scattering spectra, optimal diffusion) rather than newly invented entities. No new particles, forces, or abstract quantities are postulated. The genuine premises the method adds are: (i) the finite feature-gradient span approximates the true velocity — a domain assumption that all error control (Eqs. 10-12) rests on; (ii) well-posedness of the empirical Gram system, effectively requiring N ≥ P; (iii) a density-positivity assumption that the experiments implicitly violate (empirical measures). The schedule, K, and scattering depth are hand-chosen hyperparameters affecting the bound but not the theory's validity.

free parameters (4)
  • Interpolant schedule (α_t, β_t) = trigonometric (cos/sin) for S&P and physical fields; linear (1-t, t) for MNIST/CelebA
    Chosen by hand; the schedule fixes γ_t = α_t ˙β_t − ˙α_t β_t and through D_t* = α_t γ_t/β_t governs the singular behavior at t=0 and t=1. Results are presented for two of many possible C^1 schedules.
  • Scattering hyperparameters (J scales, L orientations) = J=8 for S&P (P=217); J=6, L=4 for 2D fields (P=6803)
    Follows Morel et al. 2024 and Cheng et al. 2024; these numbers control expressivity of the feature-gradient span, hence the magnitude of the drift projection error in Proposition 2.2's bound.
  • Number of integration steps K = K=1200 (S&P), 5000 (fields), 1000 (MNIST/CelebA)
    Chosen per experiment; discretization error of the integrator (Eq. 16) is not part of the path-KL bound, so the net error includes unquantified discretization error.
  • Number of data pairs N for the Gram matrix = N=10,000 for MNIST/CelebA; not stated for S&P and physical fields
    Controls conditioning and rank of the empirical Gram matrix K̂_t (rank ≤ min(N,P)); with P=6803 the field experiments need N ≥ 6803 samples, which is not reported.
axioms (6)
  • domain assumption Target µ has a differentiable, everywhere positive density
    Stated at the start of §2.1; needed for the score s_t and the Girsanov KL to be well-defined. Fails for empirical/atomic distributions, the actual object in all experiments (finite sample sets).
  • domain assumption K_t = E[∇φ(I_t)·∇φ(I_t)⊤] is positive-definite for every t
    Assumed in Proposition 2.1 to make η_t unique; for the empirical version K̂_t this requires N ≥ P and non-collinear feature gradients, neither of which is verified in §3.
  • domain assumption Stochastic interpolant property: SDE (2) with exact b_t, s_t has marginals ρ_t = law(I_t)
    Imported from Albergo et al. 2023 (cited); used in Eqs. 2 and 10-11 to identify the path KL and bound the terminal KL by data processing inequality.
  • standard math Girsanov theorem and data processing inequality for diffusion path measures
    Used to derive Eq. 10 (path KL between exact and approximate SDEs with common diffusion √(2D_t)) and Eq. 11; requires absolute continuity of path measures.
  • standard math Score-drift identity s_t(x) = (β_t b_t(x) − ˙β_t x)/(α_t γ_t) and Tweedie identity s_t = −α_t^{-1} E[z|I_t=x]
    Equation 4 and following text; exact identities conditional on the interpolant construction, used to convert drift error into score error (Eq. 8).
  • standard math Characteristic kernels imply equality of laws (Appendix A)
    Used only in the Hilbert-space appendix to argue exact recovery in infinite dimension; not needed for the finite-dimensional main text.

pith-pipeline@v1.3.0-alltime-deepseek · 12410 in / 25629 out tokens · 220983 ms · 2026-08-02T21:25:38.575091+00:00 · methodology

0 comments
read the original abstract

We develop a kernel method for generative modeling within the stochastic interpolant framework, replacing neural network training with linear systems. The drift of the generative SDE is $\hat b_t(x) = \nabla\phi(x)^\top\eta_t$, where $\eta_t \in \mathbb{R}^P$ solves a $P\times P$ system computable from data, with $P$ independent of the data dimension $d$. Since estimates are inexact, the diffusion coefficient $D_t$ affects sample quality; the optimal $D_t^*$ from Girsanov diverges at $t=0$, but this poses no difficulty and we develop an integrator that handles it seamlessly. The framework accommodates diverse feature maps: scattering transforms, pretrained generative models, etc, enabling generation and model combination without neural network training. We demonstrate the approach on financial time series, turbulence, and image generation.

Figures

Figures reproduced from arXiv: 2602.20070 by Eric Vanden-Eijnden, Etienne Lempereur, Florentin Coeurdoux, Nathana\"el Cuvelle-Magar, St\'ephane Mallat.

Figure 1
Figure 1. Figure 1: Generation from a single S&P 500 daily log-returns realization ( [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Generation of two-dimensional physical fields using scattering-transform features. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Left: MNIST samples. Rows 1–2: individual models (50 and 100 SGD steps). Row 3: kernelized ensemble (P = 20, 100-step cohort). Right: Oracle log-likelihood vs. ensemble size P. Blue: 50-step; orange: 100-step. Error bars: ±1 std over 5 subsets [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: CelebA generation (128 × 128). Top: samples from an individual weak model (5 epochs of training). Bottom: samples from the kernelized ensemble of P = 25 weak models via Algorithm 1. The ensemble produces coherent face images despite each constituent model being severely under-trained. commodating diverse feature maps—scattering transforms and pretrained generative models—the framework enables training-free… view at source ↗
Figure 5
Figure 5. Figure 5: Cross-domain model composition for MNIST generation. Rows 1–4: samples from individual weak models [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

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Reference graph

Works this paper leans on

16 extracted references · 10 linked inside Pith

  1. [5]

    Arthur Gretton, Karsten M Borgwardt, Malte J Rasch, Bernhard Sch ¨olkopf, and Alexander Smola

    URLhttps://arxiv.org/abs/2405.06780. Arthur Gretton, Karsten M Borgwardt, Malte J Rasch, Bernhard Sch ¨olkopf, and Alexander Smola. A kernel two- sample test.The Journal of Machine Learning Research, 13(1):723–773,

    Pith/arXiv arXiv

  2. [6]

    Geoffrey Hinton, Oriol Vinyals, and Jeff Dean

    URLhttps://arxiv.org/abs/2504.11713. Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network.arXiv preprint arXiv:1503.02531,

    Pith/arXiv arXiv

  3. [9]

    Albergo, Nicholas M

    Nanye Ma, Mark Goldstein, Michael S. Albergo, Nicholas M. Boffi, Eric Vanden-Eijnden, and Saining Xie. SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers.arXiv:2401.08740,

    Pith/arXiv arXiv

  4. [10]

    Rudy Morel, St´ephane Mallat, and Jean-Philippe Bouchaud

    URLhttps:// arxiv.org/abs/2401.03892. Rudy Morel, St´ephane Mallat, and Jean-Philippe Bouchaud. Path shadowing Monte Carlo.Quantitative Finance, 24 (9):1199–1225,

    Pith/arXiv arXiv

  5. [11]

    Jean Morlet, Georges Arens, Eliane Fourgeau, and Dominique Glard

    doi: 10.1080/14697688.2024.2399285. Jean Morlet, Georges Arens, Eliane Fourgeau, and Dominique Glard. Wave propagation and sampling theory—part i: Complex signal and scattering in multilayered media.Geophysics, 47(2):203–221, 1982a. Jean Morlet, Georges Arens, Eliane Fourgeau, and Dominique Glard. Wave propagation and sampling theory—part ii: Sampling the...

    arXiv 2024

  6. [14]

    Score-based generative modeling through stochastic differential equations.arXiv:2011.13456,

    Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations.arXiv:2011.13456,

    Pith/arXiv arXiv 2011

  7. [15]

    URLhttps://arxiv.org/abs/2501.00212. 10 Training-Free Generative Modeling via Kernelized Stochastic InterpolantsA PREPRINT A General Hilbert Space Formulation The finite-dimensional framework of Section 2 extends to general Hilbert spaces, providing the theoretical foundation and connecting to characteristic kernel theory. Letϕ:R d → Fbe a feature map int...

    Pith/arXiv arXiv

  8. [16]

    Proof.Consider the ODE ˙X ∗ t =⟨∇ϕ(X ∗ t ), η∗ t ⟩F withX ∗ 0 =I 0 ∼N(0,Id d), whereη ∗ t solvesE X ∗ t [∇ϕ(X ∗ t )· ∇ϕ(X ∗ t )]η ∗ t =E It [∇ϕ(It)· ˙It]

    The key result specific to this setting is: Theorem A.1(Exact recovery under characteristic kernels).Ifkis characteristic andK t is positive-definite for all t∈[0,1], then ˆbt(x) =⟨∇ϕ(x), ηt⟩F recovers the true velocity fieldb t(x) =E[ ˙It|It =x]exactly for allt. Proof.Consider the ODE ˙X ∗ t =⟨∇ϕ(X ∗ t ), η∗ t ⟩F withX ∗ 0 =I 0 ∼N(0,Id d), whereη ∗ t sol...

    2020

  9. [2007]

    ISBN 9781595937643

    Association for Computing Machinery. ISBN 9781595937643. doi: 10.1145/1362622.1362654. URLhttps: //doi.org/10.1145/1362622.1362654. Viraj Shah, Nataniel Ruiz, Forrester Cole, Erika Lu, Svetlana Lazebnik, Yuanzhen Li, and Varun Jampani. Ziplora: Any subject in any style by effectively merging loras. InEuropean Conference on Computer Vision, pp. 422–438. Springer,

    arXiv

  10. [2015]

    Lora: Low-rank adaptation of large language models.arXiv preprint arXiv:2106.09685,

    Edward J Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen. Lora: Low-rank adaptation of large language models.arXiv preprint arXiv:2106.09685,

    Pith/arXiv arXiv

  11. [2017]

    Data exploration of turbulence simulations using a database cluster

    9 Training-Free Generative Modeling via Kernelized Stochastic InterpolantsA PREPRINT Eric Perlman, Randal Burns, Yi Li, and Charles Meneveau. Data exploration of turbulence simulations using a database cluster. InProceedings of the 2007 ACM/IEEE Conference on Supercomputing, SC ’07, New York, NY , USA,

    2007

  12. [2019]

    Diffusion soup: Model merging for text-to-image diffusion models

    8 Training-Free Generative Modeling via Kernelized Stochastic InterpolantsA PREPRINT Benjamin Biggs, Arjun Seshadri, Yang Zou, Achin Jain, Aditya Golatkar, Yusheng Xie, Alessandro Achille, Ashwin Swaminathan, and Stefano Soatto. Diffusion soup: Model merging for text-to-image diffusion models. InComputer Vision–ECCV 2024: 18th European Conference, pp. 256...

    2024

  13. [2021]

    MGD: Moment guided diffusion for maximum entropy generation.arXiv preprint arXiv:2602.17211,

    Etienne Lempereur, Nathana ¨el Cuvelle-Magar, Florentin Coeurdoux, St ´ephane Mallat, and Eric Vanden-Eijnden. MGD: Moment guided diffusion for maximum entropy generation.arXiv preprint arXiv:2602.17211,

    arXiv

  14. [2022]

    Stochastic interpolants: A unifying framework for flows and diffusions.arXiv preprint arXiv:2303.08797,

    Michael S Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions.arXiv preprint arXiv:2303.08797,

    Pith/arXiv arXiv

  15. [2024]

    Sihao Cheng, Rudy Morel, Erwan Allys, Brice M ´enard, and St´ephane Mallat

    URLhttps://arxiv.org/abs/ 2403.13724. Sihao Cheng, Rudy Morel, Erwan Allys, Brice M ´enard, and St´ephane Mallat. Scattering spectra models for physics. PNAS nexus, 3(4):pgae103,

    Pith/arXiv arXiv

  16. [2025]

    Joan Bruna and St´ephane Mallat

    URLhttps://arxiv.org/abs/ 2502.02483. Joan Bruna and St´ephane Mallat. Multiscale sparse microcanonical models.Mathematical Statistics and Learning, 1 (3):257–315,

    Pith/arXiv arXiv