Non-Convex Tensor Recovery from Local Measurements

Jicong Fan; Tongle Wu; Ying Sun

REVIEW 3 major objections 6 minor 63 references

Non-Convex Tensor Recovery from Local Measurements

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A low-tubal-rank tensor can be recovered from independent compressive measurements of each lateral slice, and a preconditioned alternating-minimization method reaches epsilon-accurate recovery in O(log 1/epsilon) iterations independent of…

desk verdict Useful new local tensor sensing model, but the headline κ-independent convergence is not proven as stated because the Theorem 3 warm start is too weak for the contraction lemma unless r κ³ is bounded. read the letter →

arxiv 2412.17281 v1 pith:WXXUQWAM submitted 2024-12-23 cs.LG

Tongle Wu , Ying Sun , Jicong Fan This is my paper · ORCID

classification cs.LG MSC 15A6990C2694A12

keywords tensorcompressedsensinglowtubalranklocalmeasurementsnon-convexoptimizationalternatingminimizationpreconditionedgradientdescentsamplecomplexityt-SVD

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to prove that a low-tubal-rank tensor can be recovered from local measurements, meaning each lateral slice is sensed by its own independent random matrix rather than the whole tensor being projected as one unit. It formulates recovery as a nonconvex factored problem $\mathcal{X}^\star = \mathcal{U} * \mathcal{V}$ and gives two alternating algorithms: Alt-PGD-Min and a preconditioned version Alt-ScalePGD-Min. The headline result is that Alt-ScalePGD-Min reaches $\epsilon$-accurate recovery in $O(\log 1/\epsilon)$ iterations regardless of the tensor condition number $\kappa$, with quantified per-slice sample complexity. This matters for distributed, streaming, video, and MRI settings, where the data arrive slice by slice and whole-tensor sensing is infeasible.

What carries the argument

The load-bearing machinery is the low-tubal-rank factorization $X = U * V$ with $U^c * U = I_r$ under the t-product, in which tensor multiplication is block-circulant convolution and becomes elementwise matrix products in the Fourier domain. The algorithm alternates an exact least-squares solve for each $V$ slice (a closed form via $H_{t,i}H_{t,i}^c$ inversion) with a projected gradient step on $U$ followed by QR projection onto the orthogonal-tensor constraint. Alt-ScalePGD-Min changes only the $U$ update to $\hat U_{t+1}=U_t-\eta T_t*V_t^c*(V_t*V_t^c)^{-1}$; the extra inverse is an $r\times r\times n_3$ tensor and is cheap to compute. The proof tracks the principal-angle distance $\mathrm{Dis}(U_t,U_\star)$ through a Wedin-type sin $\theta$ bound, and uses sample splitting so each iteration sees fresh independent measurements.

What would settle it

Take a synthetic tensor with $r=4$ and $\kappa=10$ and initialize by the proved spectral bound: $\mathrm{Dis}(U_0,U_\star)\le 0.016\sqrt{4}\,10^2=3.2$, which is larger than the small constant required by the contraction step (67); running Alt-ScalePGD-Min on such a tensor and checking whether the $O(\log 1/\epsilon)$ iteration bound actually holds would settle whether the condition-number-free claim is valid as stated.

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Extended reading notes

Core claim

The paper's central claim is Theorem 5 with Corollaries 4 and 5: under Assumptions 1 and 2 (tubal rank $r$, tensor incoherence, i.i.d. Gaussian local sensing matrices), the truncated spectral initialization followed by Alt-ScalePGD-Min recovers an $\epsilon$-accurate estimate in $O(\log 1/\epsilon)$ iterations, with total per-slice sample complexity $O(\kappa^4 \mu^2 r n_3 \log n_3 (\kappa^4 r (n_1+n_2) + n_1 \log 1/\epsilon))$ with high probability. The companion result for Alt-PGD-Min gives iteration complexity $O(\kappa^2 \log 1/\epsilon)$ and establishes that the local-sensing model itself is tractable: exact $V$-updates decouple across slices and a projected gradient on $U$ converges linearly once $U_0$ is near $U^\star$. Experiments on synthetic tensors, video, and MRI report that Alt-ScalePGD-Min keeps its convergence rate when $\kappa$ grows, while the un-preconditioned version slows down as $\kappa^2$ predicts.

Load-bearing premise

The guarantees assume the spectral initialization lands inside a small contraction basin around the true factor space, and the initialization bound proved in the paper only guarantees that closeness when the condition number is small; for the large-$\kappa$ regime the advertised condition-number-free result relies on a warm start that is not proved.

Editorial extensions

If this is right

Video and dynamic-MRI volumes can be reconstructed from independent compressed measurements of individual frames, with no need to sense or hold the whole tensor at once.
Because Alt-ScalePGD-Min's iteration complexity is $O(\log 1/\epsilon)$ uniformly in the condition number, ill-conditioned low-tubal-rank tensors do not require a longer run just because they are harder to condition.
The per-slice sample-complexity formula contains a shared-factor term that decreases with the number of lateral slices $n_2$, so adding more frames or slices amortizes the cost of learning the common tensor factor $U$.
The exact minimization step for $V$ is decoupled across lateral slices, so each iteration can be parallelized across slices, matching distributed or streaming sensing architectures.
When the number of lateral slices is large, the analysis shows the total per-slice sample complexity drops to roughly $O(\kappa^4\mu^2 r n_1 n_3\log n_3 \log 1/\epsilon)$, which is close to the information-theoretic degrees of freedom up to small factors.

Reading between the lines

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

The same preconditioning idea should transfer to other tensor formats: replacing $(V*V^c)^{-1}$ by the inverse Gram tensor is cheap whenever the core size is small, so Tucker or tensor-train factorizations might inherit condition-number-free convergence.
The paper's fixed step size of 0.8 for Alt-ScalePGD-Min across all video and MRI experiments suggests the preconditioned update may be insensitive to $\kappa$ in practice, which would make it a plug-and-play solver for low-tubal-rank inverse problems beyond the measured setting.
Removing sample splitting, which the conclusion names as future work, would eliminate the $\log 1/\epsilon$ term from the sample complexity; a leave-one-out analysis is the natural route, as in matrix completion.
The proof of Theorem 5 relies on a small-basin contraction condition (inequality (67)) while the proved initialization bound is $\mathrm{Dis}(U_0,U_\star)\le 0.016\sqrt{r}\,\kappa^2$; for large $\kappa$ this gap needs a sharper initialization or a larger basin, so the advertised regime is most rigorous for bounded $\kappa$.

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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

Here's my take on Wu, Sun, Fan (arXiv:2412.17281). The interesting part is the model: recovering a low-tubal-rank tensor from independent Gaussian projections of each lateral slice. That's a sensible tensor analogue of the column-wise matrix sensing work by Vaswani and Nayer, and it is genuinely new as far as the cited literature goes. The two alternating algorithms are natural extensions, the preconditioned variant is a good idea, and the experiments (synthetic, video, MRI) show the method working, with the scaled version clearly faster on ill-conditioned data.

The proofs are serious and mostly self-contained, and the paper does a good job adapting the matrix proof skeleton to the t-product algebra. But there is a load-bearing gap in the theory as stated. Theorem 3 guarantees Dis(U0,U*) ≤ 0.016√r κ². Lemma 4's fact (5) — the lower bound on σ_min(V_t) — needs the perturbation term 0.36 Dis √r ||X*|| to be below about 0.06 σ*_min. Since ||X*|| = κ σ*_min, that requires Dis ≲ 1/(6√r κ). The initialization bound only satisfies this when r κ³ ≲ 10, and no such restriction appears in Theorems 4–5 or the corollaries. So the claimed κ-independent iteration complexity for Alt-ScalePGD-Min and the κ² log(1/ε) rate for Alt-PGD-Min are not established as stated for the ill-conditioned regime the paper advertises. This is a proof gap, not a contradiction; the experiments look consistent with the claims, but the theorem statements need a κ restriction or a sharper initialization.

There is also a second, smaller gap: the exact V-minimization step inverts an (r n3)×(r n3) matrix, which requires m_c ≥ r n3 per slice, but the corollaries' sample conditions don't include this. The supplementary complexity table has a mismatch in the κ power for Alt-ScalePGD-Min (κ^2 inside vs κ^4 in the main text), one lemma is attributed to a machine learning fine-tuning paper rather than a matrix analysis source, and no code is released. These are minor and fixable.

Overall: the local TCS model is a real contribution, the algorithm design is sensible, and the experiments give genuine support. The proof gap in the central theorem is the main issue. I'd send this to peer review, with the expectation of major revision. The authors should either prove a κ-robust initialization, add the missing κ restriction to all theorem statements, or weaken the claims. For a reader in tensor sensing or non-convex recovery, it's worth engaging with. I would not cite the κ-independent rate as proven until the gap is closed; I would cite the measurement model.

Referee Report

3 major / 6 minor

Summary. The paper introduces a tensor compressed sensing model in which each lateral slice of a low-tubal-rank tensor is sensed by an independent local measurement matrix, and formulates recovery as a nonconvex factored problem over two tensor factors. Two alternating-minimization algorithms are proposed: Alt-PGD-Min, which alternates a projected gradient step on the orthogonal factor with an exact least-squares update on the other factor, and Alt-ScalePGD-Min, which preconditions the gradient step by an approximate Hessian. The paper proves, under Gaussian local measurements and a tensor incoherence condition, that a truncated spectral initialization provides a warm start and that both algorithms converge linearly with sample complexity stated in terms of the tensor condition number kappa; the headline claim is that Alt-ScalePGD-Min has iteration complexity O(log(1/epsilon)) independent of kappa. Numerical experiments on synthetic tensors, video, and MRI data support the practical efficacy of the proposed methods.

Significance. If the theoretical claims were fully established, this would be a valuable contribution: the local slice-wise sensing model is new for tensors, the sample-splitting and truncation analysis is carefully designed, and the constants in the main bounds are fixed universal numbers rather than fitted parameters. The preconditioned method's claimed kappa-independent iteration complexity would be a substantial improvement over the kappa^2 dependence of unaccelerated factoring methods and over the sub-linear rates of convex tensor nuclear norm approaches. The experimental section is reasonably extensive and includes comparisons with the closest column-wise matrix sensing method. However, the central contraction proof contains a load-bearing smallness gap: the initialization bound proved in Theorem 3 is not strong enough to enter the contraction regime required by Lemma 4 and the proofs of Theorems 4 and 5 for large kappa, which is precisely the regime the preconditioned algorithm is advertised to address. The core idea is defensible and likely repairable, but the stated kappa-independent result is not currently proven.

major comments (3)

[Proof of Theorem 4; Lemma 4; Eq. (67)] The displayed lower bound for sigma_min(V_t) subtracts the perturbation 0.36*Dis(U_t,U*)*sqrt(r)*||X*|| from sigma*_min*sqrt(1-Dis^2(U_t,U*)). Since ||X*|| = kappa*sigma*_min, concluding sigma_min(V_t) >= 0.94*sigma*_min requires Dis(U_t,U*) to be at most of order 1/(6*sqrt(r)*kappa). The only initialization guarantee proved is Dis(U_0,U*) <= 0.016*sqrt(r)*kappa^2 (Theorem 3, Eq. (18)), which can satisfy that smallness requirement only when r*kappa^3 is bounded by an absolute constant. No such restriction is stated in Theorem 4, Theorem 5, or Corollaries 2-5; for the ill-conditioned regime emphasized by the paper the initialization bound is vacuous, since 0.016*sqrt(r)*kappa^2 can exceed 1. Consequently the contraction steps at Eq. (79) and Eq. (88) are not justified as stated, and the kappa-independent iteration complexity of Theorem 5 and Corollary 4 is not established.
[Alt-PGD-Min Algorithm; Eq. (23); Corollaries 2-5] The exact V-step computes v_{t,i} = (H_{t,i}H_{t,i}^c)^{-1} H_{t,i} y_i, which requires the matrix H_{t,i}H_{t,i}^c in R^{r*n3 x r*n3} to be nonsingular. This can hold only when the number of measurements per lateral slice satisfies m_c >= r*n3. The sample-size conditions in Corollary 2, Eq. (28), and in Corollaries 3-5 only impose m_c >= max{log n2, r log n3} together with a lower bound on m_c*n2; for large n2 this permits m_c to be much smaller than r*n3. The proof of Lemma 3 uses the same invertibility and the spectral lower bound sigma_min(Q_{t,i}Q_{t,i}^c) >= 0.86*m_c, so the per-slice identifiability condition m_c >= r*n3 is a missing necessary condition in the stated sample-complexity guarantees.
[Proof of Theorem 4; Eq. (79) and Eq. (88)] Step (d) of Eq. (79) estimates 1.41*eta*m*sqrt(r)*||X*||^2*Dis(U_t,U*) by 0.03, but substituting eta = 0.9/(m||X*||^2) and Dis <= 0.016*sqrt(r)*kappa^2 gives 1.41*0.9*0.016*r*kappa^2 = 0.0203*r*kappa^2, which is not an absolute constant. The subsequent inequality (1-x)(1+2x) >= 1 therefore requires r*kappa^2 to be small. In Theorem 5, Eq. (88), the analogous term is 1.6*0.9*sqrt(r)*Dis*kappa^2 <= 0.023*r*kappa^4, which is not bounded for large kappa. This is an independent manifestation of the same missing smallness condition and, by itself, invalidates the displayed contraction unless an explicit bound on r*kappa^2 or r*kappa^4 is imposed.

minor comments (6)

[Figures throughout] The main text refers to Figures 6-9 for the synthetic and video results, but the displayed figures in the manuscript are labeled Figures 1-4; the numbering should be made consistent.
[Eq. (67)] The perturbation term is printed as 0.36*Dis^2(U_t,U*)*sqrt(r)*||X*||, while Lemma 3 fact (3) gives 0.36*Dis(U_t,U*)*sqrt(r)*||X*||; the square on Dis appears to be a typo and should be corrected.
[Corollary 2] The step size is stated as eta = 0.8/(m||X*||), but Theorem 4 and the gradient normalization require eta = 0.8/(m||X*||^2); the missing square changes the dimensional consistency of the step size.
[Corollary 1] The success probability is stated as 1 - 1/(n1+n3)^10, while Theorem 3 contains the term c1(n1+n2) log n3; the indices n2 and n3 should be made consistent.
[Supplementary Table 1] The table reports the Alt-ScalePGD-Min sample complexity with an inner factor kappa^2*r*(n1+n2), whereas Corollary 5 and the abstract use kappa^4*r*(n1+n2); one of these is a typo.
[Lemma 13 and references] Lemma 13 cites (Kumar et al. 2022), a paper on fine-tuning pretrained features, for the standard inequality sigma_min(AB) >= sigma_min(A)*sigma_min(B); a standard linear algebra reference would be more appropriate.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: derivation is self-contained; the initialization-to-contraction gap is a proof-completeness concern, not a reduction to inputs.

full rationale

I walked the paper's claimed derivation chain. The truncated spectral initialization (Theorem 3) is proved from the t-SVD Wedin sin-theta theorem and sub-Gaussian/sub-exponential concentration arguments, with fixed universal constants (0.016, 0.84, 0.89, 1.4, etc.) chosen in the proof rather than fitted to any measured outcome. The contraction theorems (Theorems 4 and 5) assume a warm-start condition Dis(U_t, U_star) <= 0.016 sqrt(r) kappa^2, which is exactly the guarantee that Theorem 3 proves; this is standard local-convergence reasoning, not self-definition, because the contraction factor is strictly less than one and the proof shows the distance decreases iteration by iteration. The algorithm uses kappa, mu, and ||X_star|| to set the truncation threshold and step size; these are oracle quantities, which is a practical limitation, but the guarantees do not define the recovery target in terms of the algorithm's own output. The only self-citations (Wu & Sun 2024 and Wu & Fan 2024) appear in motivation, related work, and experiment-quality indices; none supplies a load-bearing theorem used in the convergence proofs. The proof gap identified by the reviewer concerns Eq. (67): the displayed bound sigma_min(V_t) >= sigma*_min sqrt(1 - Dis^2) - 0.36 Dis sqrt(r) ||X_star|| requires Dis to be of order 1/(sqrt(r) kappa) to reach 0.94 sigma*_min, while Theorem 3 only provides 0.016 sqrt(r) kappa^2, so the claimed kappa-independent rate is not fully established for large kappa. That is a correctness and proof-completeness issue, not circularity, because assuming a stronger warm-start condition would not make the conclusion identical to the assumption by construction. Therefore no circular step is present, and the central derivation is self-contained.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The theory rests on Gaussian slice-wise measurements, low tubal rank plus incoherence, sample splitting, and an implicit small-angle initialization condition that is not guaranteed by the initialization theorem for large kappa; the exact V-minimization also needs m_c >= r*n3, which is not stated.

free parameters (3)

truncation threshold alpha = C * kappa^2 * mu^2 * (sum y^2) / (m * n2)
Used in the spectral initialization (Eq. 15); the proof fixes alpha = kappa^2 * mu^2 * (1+epsilon1) * ||X*||_F^2 / n2, so the algorithm needs oracle kappa and mu. The constant C is universal, not fitted to the target result.
step size eta = c_eta / (m_c * ||X*||^2) for Alt-PGD-Min; c_eta / m_c for Alt-ScalePGD-Min, with c_eta <= 0.9
The theoretical step sizes require knowledge of ||X*|| for the non-scaled variant, and the scaled variant uses a constant step size; in experiments eta is tuned per dataset. The guarantees hold only for the chosen c_eta interval.
tubal rank r = known input
The algorithm takes r as input and all sample and iteration bounds scale with r; rank estimation is not addressed.

assumptions (6)

domain assumption Ground truth X* has tubal rank r << min(n1,n2,n3) and satisfies the incoherence condition max_i ||Z*(:,i,:)||_F <= mu * sqrt(r/n2) * ||X*|| (Assumption 1).
Defines the low-dimensional structure; needed for the spectral initialization bound and the V-subspace analysis.
domain assumption Each sensing tensor A_i has i.i.d. standard Gaussian entries (Assumption 2).
Gaussianity drives the sub-Gaussian and chi-square concentration used throughout Lemmas 3 and 5.
domain assumption Samples are split into 2T+1 independent groups, one fresh group per iteration (sample splitting).
Requires access to many independent measurements; experiments run without it, so the theory applies to a different data regime than the experiments.
ad hoc to paper The initialization satisfies Dis(U0,U*) <= c0 for a fixed small constant, or equivalently kappa is bounded by a constant (implicit in Lemma 4 and the proof of Theorem 4, Eq. 67).
The contraction constants (0.94, 1.1, etc.) are derived under a small-angle assumption; Theorem 3 only proves Dis <= 0.016*sqrt(r)*kappa^2, which meets the small-angle requirement only for kappa bounded by a constant.
ad hoc to paper m_c >= r * n3 so that H_{t,i}H_{t,i}^c is invertible in the exact V update (Eq. 23).
The least-squares solution (23) uses an inverse that exists with probability 1 only when the number of measurements per slice per iteration exceeds the r*n3 degrees of freedom of V(:,i,:); this condition is not stated in Corollaries 2 through 5.
standard math Standard tools: Wedin sin theta theorem, sub-exponential Bernstein inequalities, epsilon-net covering bounds, t-SVD and t-QR properties, and the matrix column-wise analysis of Vaswani (2024).
Used as black boxes in the proofs of Theorems 3, 4, and 5.

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Pith. "Pith review of Non-Convex Tensor Recovery from Local Measurements." pith.science (2026). https://pith.science/paper/WXXUQWAM

@misc{pith2026241217281,
  author       = {Pith},
  title        = {Pith review of: Non-Convex Tensor Recovery from Local Measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WXXUQWAM}},
  note         = {Machine review of arXiv:2412.17281}
}

abstract

Motivated by the settings where sensing the entire tensor is infeasible, this paper proposes a novel tensor compressed sensing model, where measurements are only obtained from sensing each lateral slice via mutually independent matrices. Leveraging the low tubal rank structure, we reparameterize the unknown tensor ${\boldsymbol {\mathcal X}}^\star$ using two compact tensor factors and formulate the recovery problem as a nonconvex minimization problem. To solve the problem, we first propose an alternating minimization algorithm, termed \textsf{Alt-PGD-Min}, that iteratively optimizes the two factors using a projected gradient descent and an exact minimization step, respectively. Despite nonconvexity, we prove that \textsf{Alt-PGD-Min} achieves $\epsilon$-accuracy recovery with $\mathcal O\left( \kappa^2 \log \frac{1}{\epsilon}\right)$ iteration complexity and $\mathcal O\left( \kappa^6rn_3\log n_3 \left( \kappa^2r\left(n_1 + n_2 \right) + n_1 \log \frac{1}{\epsilon}\right) \right)$ sample complexity, where $\kappa$ denotes tensor condition number of $\boldsymbol{\mathcal X}^\star$. To further accelerate the convergence, especially when the tensor is ill-conditioned with large $\kappa$, we prove \textsf{Alt-ScalePGD-Min} that preconditions the gradient update using an approximate Hessian that can be computed efficiently. We show that \textsf{Alt-ScalePGD-Min} achieves $\kappa$ independent iteration complexity $\mathcal O(\log \frac{1}{\epsilon})$ and improves the sample complexity to $\mathcal O\left( \kappa^4 rn_3 \log n_3 \left( \kappa^4r(n_1+n_2) + n_1 \log \frac{1}{\epsilon}\right) \right)$. Experiments validate the effectiveness of the proposed methods.

Figures

Figures reproduced from arXiv: 2412.17281 by the authors.

**Figure 1.** Spectral initialization with m0 = 200, mc = 100. random initialization, which has i.i.d. standard Gaussian entries. Without good initialization, we increase the mc slightly to mc = 120. The results depicted in [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 3.** Visualization of frame-7 in recovered videos. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 2.** Random initialization with mc = 120. Video Compressed Sensing We test the proposed TCS model (2) in the plane video sequence (only approximately low tubal rank) that has been used in previous work such as (Nayer, Narayanamurthy, and Vaswani 2019). It has 105 frames and each frame has been resized into 48 × 64. We use the same samples with size m = 1600 for initialization and iteration. We set r = 10 for three metho… view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: Comparison between LRcCS and Alt-PGD-Min under sufficiently large iterations 50000. [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]

**Figure 6.** Figure 6: Comparison of 35-th frame in recovered MRI with sample size m = 2000 for per frame. LRcCS: PSNR = 27.64, SSIM = 0.7888, FSIM = 0.8707. Alt-PGD-Min: PSNR = 24.58, SSIM = 0.6522, FSIM = 0.8018. Alt-ScalePGD-Min: PSNR = 28.46, SSIM = 0.8096, FSIM = 0.8800. (a) Original (b…

**Figure 7.** Figure 7: Comparison of 35-th frame in recovered MRI with sample size m = 2250 for per frame. LRcCS: PSNR= 27.66, SSIM = 0.7868, FSIM = 0.8708. Alt-PGD-Min: PSNR = 25.18, SSIM = 0.6783, FSIM = 0.8151. Alt-ScalePGD-Min: PSNR = 29.92, SSIM = 0.8583, FSIM = 0.9026. (a) Original (b)…

**Figure 8.** Figure 8: Comparison of 35-th frame in recovered MRI with sample size m = 2500 for per frame. LRcCS: PSNR = 28.08, SSIM = 0.8027, FSIM = 0.8766. Alt-PGD-Min: PSNR = 25.75, SSIM = 0.7029, FSIM = 0.8275. Alt-ScalePGD-Min: PSNR = 31.79, SSIM = 0.8977, FSIM = 0.9276. (a) m = 2000 (b…

**Figure 9.** Figure 9: PSNR evaluation for each frame under different sample size. [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]

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