Minimax Theory of Likelihood-Based Deep Learning for Speckle Regression

Soham Jana

REVIEW 2 major objections 4 minor 13 references

Minimax Theory of Likelihood-Based Deep Learning for Speckle Regression

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper shows that likelihood-based deep neural networks estimate signals corrupted by multiplicative speckle noise at the minimax-optimal rate, matching additive-noise regression up to logarithmic factors.

desk verdict The paper's central rate claims are probably right, but Theorems 1–2 are only proved for σ_τ = 0; the asserted transfer to σ_τ > 0 is a genuine gap that a referee should insist on closing. read the letter →

arxiv 2607.14064 v1 pith:C52K3IAX submitted 2026-07-15 math.ST stat.MLstat.TH

Soham Jana This is my paper

classification math.STstat.MLstat.TH MSC 62G0562G2062C20

keywords specklenoisenonparametricregressiondeepneuralnetworksminimaxoptimalitylikelihood-basedestimationsparsehigh-dimensionalReLU

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

Speckle noise multiplies the signal, so the regression function is invisible to the usual least-squares loss: E[y|x]=0. This paper proposes a likelihood-based deep neural network estimator, using the negative log-likelihood with f^2+σ_τ² in the denominator, and proves that its mean squared error is, up to logarithmic factors, n^{-2γ*/(2γ*+1)} for smooth hierarchical functions in both low-dimensional and sparse high-dimensional settings. A matching lower bound shows the estimator is minimax optimal up to logs. The rates coincide with additive-noise nonparametric regression, so multiplicative speckle does not add statistical difficulty—only the estimation strategy must change.

What carries the argument

The load-bearing object is the negative log-likelihood ℓ(f) = (1/n) Σ_i [ y_i²/(f(x_i)²+σ_τ²) + log(f(x_i)²+σ_τ²) ]. Its expectation difference ℓ̄(f)−ℓ̄(f*) equals (1/n) Σ_i [(f*/f)² − 1 − log((f*/f)²)], which is lower-bounded by a constant times Σ_i (f*−f)² via the inequality x−1−log x ≥ a(x−1)² for bounded x. This converts the MLE's likelihood inequality into a squared-error bound. The random fluctuation is controlled with the Hanson-Wright inequality on the quadratic form y^T(Σ* − Σ_f)y, where Σ_f = diag(1/f(x_i)²); a covering-number bound for the ReLU network class (width N ≈ (n/log n)^{1/(4γ*+2)}) lets the argument go through without sample splitting. For sparse high-dimensional feature

What would settle it

Run the likelihood estimator on data generated from y = f*(x)ξ + τ with a fixed σ_τ>0 and a smooth f*, and measure the empirical MSE as n grows. If the error does not decay as (log n)^{2γ*/(2γ*+1)} n^{−2γ*/(2γ*+1)}—e.g., it plateaus or decays slower—the σ_τ-transfer claim fails. Alternatively, a direct mathematical check of whether the proof's lower bound on ℓ̄(f)−ℓ̄(f*) holds with (f*)²+σ_τ² in the denominator would settle it.

Watch

Extended reading notes

Core claim

The paper's central claim is that estimating a signal f* from observations y_i = f*(x_i)ξ_i + τ_i — speckle noise (Gaussian multiplier) plus additive Gaussian noise — via maximum likelihood with a deep ReLU network is minimax optimal. For f* in a hierarchical smooth function class whose hardest composition has dimension-adjusted smoothness γ* = β*/d* > 1/2, the likelihood-based estimator satisfies, with high probability, ||f̂−f*||_n² ≤ c (log n)^{2γ*/(2γ*+1)} n^{−2γ*/(2γ*+1)} (Theorem 1). The same rate holds in sparse high-dimensional settings through an ℓ1-penalized projection layer (Theorem 2), and a matching lower bound Ω(n^{−2γ*/(2γ*+1)}) (Theorem 3) shows the estimator is within logarit

Load-bearing premise

The paper depends on knowing the additive noise level σ_τ and on the assertion—stated but not fully proved in Section IV—that the theory for pure speckle (σ_τ=0) carries over to σ_τ>0 by estimating f* as the square root of f̂²−σ_τ².

Editorial extensions

If this is right

Despeckling with deep networks is provably near-optimal: the DNN estimator matches the minimax lower bound up to log factors, so practitioners can use likelihood-based DNNs without paying a statistical penalty.
In sparse high-dimensional settings, the same rate is achieved automatically: the ℓ1-projection recovers active coordinates, so the curse of dimensionality is avoided when the signal depends on few features.
Multiplicative speckle noise does not make function recovery fundamentally harder than additive Gaussian noise: the rates agree, so only the loss function needs to change, not the sample complexity.
The logarithmic slack is linked to the DNN analysis; the author suggests classical nonparametric estimators may remove the logs, which would pin down the exact minimax rate.

Reading between the lines

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

If σ_τ is unknown, the model identifies only f*²+σ_τ²; one could add a smoothness prior on f* or a bounded-variance assumption to estimate σ_τ and then recover f* by the same square-root step.
The proof's σ_τ=0-to-σ_τ>0 transfer is asserted rather than demonstrated; a dedicated numerical or theoretical check with σ_τ>0 would either confirm the rate or reveal where the argument needs strengthening.
Because the lower bound uses bump functions with disjoint supports, the result likely extends to any fixed design satisfying a Riemann-sum condition, not just grids; this could be verified directly with random designs via the paper's Proposition 1.

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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

You should know the main thing before anything else: this paper has a real result, but its headline theorems outrun the proof. The minimax rate for likelihood-based DNN despeckling—matching additive-noise rates up to log factors—is a legitimate contribution, and the multidimensional lower bound is genuinely new. But Theorem 1 and Theorem 2 are stated for the full model with multiplicative plus additive noise, and the proof is carried out only for the pure speckle case σ_τ = 0. The transfer to σ_τ > 0 is asserted in Section IV, not proved. I checked the stress-test note about this and it holds up: the full likelihood (2) and the simplified loss (8) are different objectives, and the argument would need h = sqrt(f*^2 + σ_τ^2) to inherit the same smoothness class and the same DNN approximation rate, plus a square-root stability result for the empirical L2 norm. None of that is shown. As written, the upper bounds are unsupported for the main model.

What is actually new and good: first multidimensional minimax lower bound for Gaussian speckle regression, first DNN likelihood upper bounds for this model, and a sparse high-dimensional extension with an l1-penalized projection layer. The machinery is substantial and mostly credible—the likelihood expansion, the Hanson-Wright based empirical process control, the covering number bounds, and the peeling argument for the sparse case are all laid out in detail. The lower bound uses the standard hypercube construction and Berry-Esseen; it looks sound, and the σ_τ dependence is handled inside the likelihood ratio. The Fan–Gu approximation lemma is imported cleanly, and the self-citations are background, not load-bearing.

Soft spots, in proportion: the σ_τ transfer is the one that matters. I think it is fixable—if f* is bounded away from zero, h should be at least as smooth as f*^2 and the square-root step is likely a Lipschitz perturbation—but it has to be written down, and the likelihood lower bound (39) must be re-derived with h. Minor points: the numerical experiments show consistency but no baselines, no code, and no comparison to existing despeckling methods; the paper's own Remarks 5 and 8 admit the σ_τ restriction, which is honest. The log factors in the upper bounds are standard DNN artifacts.

Bottom line: this is a solid contribution to the minimax theory of deep learning under multiplicative noise, and the rate identity—speckle is no harder than additive noise up to logs—is well supported for σ_τ = 0. It deserves a serious referee. I would send it to peer review with the expectation that the σ_τ transfer be made rigorous or the theorems explicitly restricted.

Referee Report

2 major / 4 minor

Summary. The paper studies nonparametric regression under a model with multiplicative Gaussian speckle noise and additive Gaussian noise: y_i = f*(x_i) ξ_i + τ_i, with ξ_i, τ_i independent zero-mean Gaussian noises. Because E[y_i|x_i]=0, ordinary least-squares regression is inapplicable; the authors instead minimize a negative log-likelihood over deep ReLU networks, in both low-dimensional and sparse high-dimensional settings. The main claims are finite-sample upper bounds of order Õ(n^{-2γ*/(2γ*+1)}) on the squared estimation error, where γ* is a dimension-adjusted smoothness parameter of a hierarchical composition class H(d,l,P), and a matching lower bound of order Ω(n^{-2γ*/(2γ*+1)}), so that the method is near-minimax and the difficulty is comparable to additive-noise nonparametric regression. Numerical experiments illustrate the behavior of the estimator and the ability of the sparse version to detect active coordinates.

Significance. If correct, this is the first minimax theoretical analysis of likelihood-based DNN estimators for speckle regression in multivariate and sparse settings, extending the one-dimensional results of Malekian et al. (2025). The paper contains a substantial σ_τ=0 upper-bound proof based on empirical process theory and Hanson–Wright inequalities, and a detailed multidimensional lower-bound construction in Appendix D that appears carefully executed. It is also transparent in Remark 5 that the general-σ_τ case is not fully treated. However, the central theorems as stated cover the general model (1), and the proofs do not; the asserted transfer from σ_τ=0 to σ_τ>0 is a load-bearing gap rather than a cosmetic omission.

major comments (2)

[Section IV and Appendices A–B] Theorems 1–2 are stated for model (1) with known σ_τ=O(1) and for the full likelihood (2), but all displayed proofs are for the pure speckle model (9) with loss (8). Section IV asserts that for general σ_τ the proof 'remains essentially the same' because the optimizer of (8) estimates h=sqrt(f*^2+σ_τ^2), from which f* is recovered by sqrt. This transfer is not proved and is nontrivial: (i) the theorem's estimator minimizes (2), not (8), and the relation between the two optimizers is not established; (ii) h is not shown to lie in H(d,l,P) or to admit the same N^{-2γ*} DNN approximation rate; (iii) the expected-likelihood lower bound (39) is derived for (8)/(9), not for (2)/(1); (iv) the square-root map is not shown to preserve the ∥·∥_n rate, and pointwise feasibility f̂^2≥σ_τ^2 is not guaranteed. Since Theorems 1–2 are the main upper-bound claims for the general model, this gap is load-b
[Appendix B, Eqs. (63)–(64)] The proof of Theorem 2 drops the term λ|J| without a condition on |J|. From (63), the bound contains λ|J|; with λ≍log(ndN)/n, this is of order |J|logn/n. The displayed rate (64) requires |J| ≲ n^{1/(2γ*+1)} up to logarithms. The theorem statement only says |J| << d, which does not ensure this when d≤n^{c0}. The statement should either assume |J| is bounded (or specify the required growth condition) or include a |J|-dependent term in the rate. As written, the proof does not close for diverging |J|.

minor comments (4)

[Theorem 2 statement] The notation ψ_τ(x)=|x|^τ ∧1 reuses τ, which also denotes the additive-noise variance σ_τ, and the displayed formula for λ appears to be missing parentheses: it should presumably read λ=c_2(log(ndN)+Llog(BN))/n. Please clarify.
[Appendix D, start of proof] The reduction to the class of (β*,C)-smooth d*-variate functions uses, without proof, the inclusion of this class in H(|J|,l,P) for l≥2. The inclusion is true by representing coordinate projections as elements of H(|J|,1,P), but it should be stated and proved.
[Definition 4, p. 6] The phrase 'satisfying (3)' in Definition 4 appears to refer to Definition 3, not to equation (3). The cross-reference should be corrected.
[Section III-B, Fig. 1] The figure captions use 'Test MSE' while the text describes the mean squared error relative to the true generating function. Please align the terminology so that it is clear whether the plotted quantity is an oracle MSE or a held-out-data MSE.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: rates are derived from external approximation and empirical-process bounds; the σ_τ>0 transfer is a completeness gap, not a circular step.

full rationale

The paper derives the upper bounds from a standard DNN approximation lemma (Fan and Gu 2024) and empirical process/Hanson-Wright concentration (Appendix A), explicitly balancing N^{-2γ*} against N^2 log n/n (Eqs. (34)-(42)). The lower bound (Appendix D) is an explicit hypothesis-testing construction using Varshamov-Gilbert, Berry-Esseen and likelihood ratios; no fitted numerical parameter is recycled as a prediction. Self-citations (Chen et al. 2025, Xing et al. 2025, Fan et al. 2025) appear for background or for omitted secondary remarks and do not supply the main theorem. The principal weakness is that Section IV reduces the claimed general-σ_τ result to the σ_τ=0 case: 'under the assumption of a known σ_τ = O(1), the proof remains essentially the same when we consider the special case σ_τ = 0... Hence, for the rest of the paper we present the error guarantees for deep learning optimizer of (8), under the model without additive noise (9).' Appendices A and B prove the rates only for model (9)/loss (8), while Theorems 1-2 are stated for model (1)/loss (2). The asserted transfer via sqrt(f̂²-σ_τ²) is not demonstrated, and Remark 5 candidly treats the restriction as technical. This is an unproved bridge in the proof as stated, not a definitional/fitted-input circularity, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim rests mainly on modeling assumptions (boundedness, known σ_τ) and on one unproved transfer step from σ_τ=0 to σ_τ>0. The approximation and concentration theorems are standard or cited. No data-fitted constants enter the theoretical rate.

assumptions (8)

domain assumption f*(x) ∈ [f_min, f_max] ⊂ (M^{-1}, M) and network outputs are truncated to [1/M, M]
Assumption 1(a); prevents the likelihood denominator from vanishing and is central to all bounds.
domain assumption σ_τ is known and O(1)
Assumption 1(a); the paper acknowledges non-identifiability if σ_τ is unknown.
domain assumption Speckle ξ_i ~ N(0,1) and additive τ_i ~ N(0,σ_τ²), independent across i
Model (1); fully developed speckle model.
domain assumption γ* > 1/2 and f* ∈ H(d,l,P), a hierarchical composition class with bounded P
Assumption 1(a); ensures approximability and non-vanishing rates.
domain assumption Deterministic design points satisfy Assumption 2 (uniform Riemann approximation)
Needed for the lower bound (Theorem 3); excludes pathological designs.
standard math Deep ReLU approximation rates for H(d,l,P) (Lemma 1, Fan and Gu 2024)
Borrowed theorem used to choose f_φ* with sup error N^{-2γ*}.
standard math Standard concentration and hypothesis-testing tools (Hanson-Wright, Bernstein, Berry-Esseen, Gilbert-Varshamov)
Used throughout the proofs.
ad hoc to paper σ_τ=0 proof transfers to σ_τ>0 by estimating f*²+σ_τ² and taking sqrt
Section IV; asserted without proof; all appendix proofs use σ_τ=0.

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Cite this review

Pith. "Pith review of Minimax Theory of Likelihood-Based Deep Learning for Speckle Regression." pith.science (2026). https://pith.science/paper/C52K3IAX

@misc{pith2026260714064,
  author       = {Pith},
  title        = {Pith review of: Minimax Theory of Likelihood-Based Deep Learning for Speckle Regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C52K3IAX}},
  note         = {Machine review of arXiv:2607.14064}
}

read the original abstract

Speckle noise is a multiplicative noise commonly encountered in coherent imaging modalities such as synthetic aperture radar, optical coherence tomography, and digital holography. Although deep learning methods, in practice, have achieved state-of-the-art performance for speckle denoising, their fundamental statistical limits remain largely unexplored. Unlike additive noise models, multiplicative speckle noise makes the regression function unidentifiable from the conditional mean, rendering conventional least-squares-based deep learning approaches inapplicable. We study the minimax estimation of smooth nonparametric regression functions using likelihood-based deep neural network (DNN) estimators under a model with both multiplicative speckle noise and additive Gaussian noise. Our framework accommodates both low-dimensional and sparse high-dimensional features. We establish finite-sample upper bounds on the estimation error of the proposed DNN estimators and derive minimax lower bounds for nonparametric function recovery under our model, showing that they match up to logarithmic factors in the sample size. Moreover, these minimax rates coincide, up to logarithmic factors, with those for nonparametric regression under additive Gaussian noise alone, demonstrating that the intrinsic difficulty of estimation remains essentially unchanged despite the challenges posed by multiplicative speckle noise. Numerical experiments further supports consistency of our DNN-based despeckling methods and demonstrate their effectiveness.

Figures

Figures reproduced from arXiv: 2607.14064 by the authors.

**Figure 2.** Visualization of the estimated feature selection matrix [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

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Reference graph

Works this paper leans on

13 extracted references · 1 linked inside Pith

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Achim, A., Bezerianos, A., and Tsakalides, P. (2001). Novel bayesian multiscale method for speckle removal in medical ultrasound images.IEEE transactions on medical imaging, 20(8):772–783. Argenti, F., Lapini, A., Bianchi, T., and Alparone, L. (2013). A tutorial on speckle reduction in synthetic aperture radar images.IEEE Geoscience and remote sensing mag...

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We specify the data generating regression function class based on the above construction

R g2(x)dx=C 2 R ¯g2(x)dxand R ¯g2(x)dx>0; 3)gis a(β ∗, C2s−1)smooth function. We specify the data generating regression function class based on the above construction. Define the regression functionν (ωl)(x), indexed by the vectorω l = (ωl,1, . . . , ωl,M d∗ n )∈ {0,1}M d∗ n , as ν(ωl)(x) = 1− M d∗ nX j=1 ωl,jgn,j(x), g n,j(x) =M −β∗ n g(Mn(x−a n,j)). 29 ...

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, l|J |},Θ ∗ ij =1{1≤i≤ |J |, j=l i}

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L+1δ 1 n nX i=1 |ℓ′ i( efi)| ! .(36) To this end, we note the Bernstein’s inequality for independent sub-exponential variables. Lemma 5.(Vershynin, 2018, Theorem 2.9.1) Suppose thatZ 1, . . . , Zn are subexponential random vari- ables, with subexponential parameterσ max. Then we have P 1 n nX i=1 (Zi −E[Z i])≥t ! ≤exp −c·nmin t2 σ2 max , t σmax .(37) 21 S...

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We provide the details here for completeness

The proof closely follows the proof of the special case ofd= 1provided in (Malekian et al., 2025, Lemma 3.12), extended here to the multidimensional scenario. We provide the details here for completeness. Proof of Lemma 14.Our goal is to prove that fory i :=ν l(xi)ξ i +τ i,i= 1, . . . , n, Pνl X i y2 i 1 +σ 2 τ σ2 τ + 1 σ2 τ +ν 2 l (xi) −1 + X i log σ2 τ ...

2025

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

Achim, A., Bezerianos, A., and Tsakalides, P. (2001). Novel bayesian multiscale method for speckle removal in medical ultrasound images.IEEE transactions on medical imaging, 20(8):772–783. Argenti, F., Lapini, A., Bianchi, T., and Alparone, L. (2013). A tutorial on speckle reduction in synthetic aperture radar images.IEEE Geoscience and remote sensing mag...

2001

[2] [2]

We specify the data generating regression function class based on the above construction

R g2(x)dx=C 2 R ¯g2(x)dxand R ¯g2(x)dx>0; 3)gis a(β ∗, C2s−1)smooth function. We specify the data generating regression function class based on the above construction. Define the regression functionν (ωl)(x), indexed by the vectorω l = (ωl,1, . . . , ωl,M d∗ n )∈ {0,1}M d∗ n , as ν(ωl)(x) = 1− M d∗ nX j=1 ωl,jgn,j(x), g n,j(x) =M −β∗ n g(Mn(x−a n,j)). 29 ...

2012

[3] [4]

, l|J |},Θ ∗ ij =1{1≤i≤ |J |, j=l i}

Let J={l 1, . . . , l|J |},Θ ∗ ij =1{1≤i≤ |J |, j=l i}. Given the minimizer bϕ, bΘof the optimization (43), let bf SPN(x) =f SPN(x; bϕ, bΘ) =f bϕ(TrM ( bΘ⊤x)).(46) Using the parametersϕ ∗,Θ ∗, and noting thatΘ ∗ has|J |-many nonzero entries andf ϕ∗(TrM (Θ∗⊤x)) = fϕ∗(x), P i,j ψτ ( bΘ∗ i,j) =|J |, we have ℓ( bf SPN) +λ X i,j ψτ ( bΘi,j)≤ℓ(f ϕ∗) +λ|J |.(47)...

2024

[4] [6]

Gilbert, E

Cambridge university press. Gilbert, E. N. (1952). A comparison of signalling alphabets.The Bell System Technical Journal, 31(3):504–

1952

[5] [7]

A., Maleki, A., and Jalali, S

Chen, X., Jana, S., Metzler, C. A., Maleki, A., and Jalali, S. (2025). Multilook coherent imaging: Theoretical guarantees and algorithms.arXiv preprint arXiv:2505.23594. Chen, X., Maleki, A., and Jalali, S. (2026). Maximum likelihood reconstruction for multi-look digital holography with markov-modeled speckle correlation.arXiv preprint arXiv:2604.20154. C...

arXiv 2025

[6] [8]

Laurent, B

Springer Science & Business Media. Laurent, B. and Massart, P. (2000). Adaptive estimation of a quadratic functional by model selection. Annals of statistics, pages 1302–1338. LeCun, Y ., Bengio, Y ., and Hinton, G. (2015). Deep learning.nature, 521(7553):436–444. Lopez-Martinez, C. and Fabregas, X. (2003). Polarimetric sar speckle noise model.IEEE Transa...

2000

[7] [10]

Lemma 5.(Vershynin, 2018, Theorem 2.9.1) Suppose thatZ 1,

L+1δ 1 n nX i=1 |ℓ′ i( efi)| ! .(36) To this end, we note the Bernstein’s inequality for independent sub-exponential variables. Lemma 5.(Vershynin, 2018, Theorem 2.9.1) Suppose thatZ 1, . . . , Zn are subexponential random vari- ables, with subexponential parameterσ max. Then we have P 1 n nX i=1 (Zi −E[Z i])≥t ! ≤exp −c·nmin t2 σ2 max , t σmax .(37) 21 S...

2018

[8] [14]

We provide the details here for completeness

The proof closely follows the proof of the special case ofd= 1provided in (Malekian et al., 2025, Lemma 3.12), extended here to the multidimensional scenario. We provide the details here for completeness. Proof of Lemma 14.Our goal is to prove that fory i :=ν l(xi)ξ i +τ i,i= 1, . . . , n, Pνl X i y2 i 1 +σ 2 τ σ2 τ + 1 σ2 τ +ν 2 l (xi) −1 + X i log σ2 τ ...

2025