Pith. sign in

REVIEW 3 major objections 4 minor 160 references

Jointly Sparse Blind Deconvolution via Riemannian Optimization

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read This paper proves that jointly sparse blind deconvolution can be solved by a Riemannian algorithm whose sample complexity grows like N^3 rather than N^9 when the sparsity level satisfies K^2 ≤ N.

desk verdict A serious landscape theorem for jointly sparse blind deconvolution, but the claimed sample-complexity win over Shi-Chi rests on an unproven hyperparameter restriction. read the letter →

arxiv 2608.04465 v1 pith:P625EF5X submitted 2026-08-05 math.OC cs.ITmath.IT

Wenlong Wang , Baiyang Guo , Zai Yang , Shixiang Chen , Junpeng Shi This is my paper · ORCID
classification math.OCcs.ITmath.IT MSC 90C2690C4894A12
keywords jointlysparseblinddeconvolutionRiemannianoptimizationsamplecomplexitynonconvexlandscapesecond-orderstationarypointsmoothℓ21objectivearraysignalprocessing
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

Jointly sparse blind deconvolution asks to recover an unknown filter and L input signals that share the same sparse support from their circulant convolutions, a model behind radar, passive localization, and fluorescence microscopy. The paper establishes that a smooth ℓ2,1-style objective on the unit sphere, rather than separate per-channel sparsity penalties, makes the optimization landscape benign: every sufficiently accurate second-order stationary point lies near the true inverse filter, provided enough channels are observed. The main quantitative claim is a sample-complexity bound of order $O(\kappa^8 K^6 N^3)$ for $K^2 \le N$, suppressing log factors, versus $O(\kappa^8 K^{-6} N^9)$ for the strongest prior non-joint analysis. If correct, joint sparsity buys a substantial reduction in the number of channels needed, and the paper proposes a Riemannian gradient-descent method with negative-curvature search that provably reaches the required stationary point.

What carries the argument

The load-bearing object is a smooth surrogate for the $\ell_{2,1}$ norm defined on the unit sphere: $\rho(h)=\sum_{n=1}^N (\frac{1}{L}\sum_{l=1}^L |y_l^\top \Gamma_n R h|^2+\epsilon)^{1/2}$, where $R$ is the whitening preconditioner $(\frac{1}{L}\sum_l C(y_l)^\top C(y_l))^{-1/2}$. The matrices $\Gamma_n$ encode circulant shifts, so rows of the recovered signal matrix are penalized jointly, and the preconditioner partially inverts the unknown filter. The geometric argument shows that for non-uniform support the asymptotic objective has only signed canonical basis vectors as global minimizers, and then controls two perturbations, the sample covariance error $E_\Omega$ and the preconditioning error $\Delta$, so that every approximate second-order stationary point is forced into a small neighborhood of such a vector. A negative-curvature direction $d=e_o - h_o h$ is used to rule out distant saddle points.

What would settle it

Generate inputs with a non-uniform common support but deterministic nonzero entries, such as all entries equal to one, run the proposed algorithm with a small number of channels below the theorem's bound, and check whether an approximate second-order stationary point exists far from every signed canonical basis vector.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 5: under a non-uniform common support, i.i.d. Gaussian nonzero entries, and an invertible filter with condition number $\kappa$, once $L \ge c_1 \kappa^8 \log^2 N (\sqrt{K} + \sqrt{\log L})^2 \max\{K^7 N^2, K^5 N^3, \epsilon^{-1}, \epsilon^{-1} K^{-2} N\}$, the estimate produced by RGD-NCS satisfies the error bound in Eq. (102) with probability at least $1 - 4L^{-8}$. Ignoring logarithmic factors and choosing $\epsilon \sim K^{-7}N^{-2}$, this is roughly $O(\kappa^8 K^6 N^3)$ samples when $K^2 \le N$, compared with $O(\kappa^8 K^{-6} N^9)$ for the per-channel analysis it compares against. The improvement comes from aggregating information across channels through the joint-sparsity objective, at the cost of a landscape with many saddle points that must be excluded by second-order arguments.

Load-bearing premise

The analysis assumes the nonzero entries of the input signals are independent standard Gaussian random variables across all channels; if those entries are deterministic, bounded, or heavy-tailed, the concentration bounds behind the landscape theorem have no guarantee.

Editorial extensions

If this is right

  • With $L$ satisfying (101), the recovered inverse filter obeys the explicit error bound (102), so both filter and input signals can be reconstructed to arbitrary accuracy by shrinking $\epsilon$ and the stationarity tolerances.
  • In the common regime $K^2 \le N$, joint sparsity lowers the sample requirement from order $N^9$ to order $N^3$ in the filter length, meaning far fewer antennas or snapshots are needed in array-processing applications.
  • The non-uniformity condition on the support is both necessary and sufficient in the asymptotic problem, so supports with cyclic symmetry are the only algebraic obstruction to unique recovery.
  • The same framework extends to complex-valued inputs, covering radar and array-signal models directly.

Reading between the lines

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

  • The i.i.d. Gaussian generative model is likely stronger than needed; a concentration analysis for sub-Gaussian or bounded nonzero entries would probably preserve the landscape conclusion with modified constants, but the paper does not prove this.
  • The numerical experiments show graceful behavior under noise and off-grid delays even though the theorems are noiseless and grid-aligned; a natural next step is a noise-aware objective with error bounds that depend explicitly on SNR and grid mismatch.
  • The same group-sparsity mechanism could be ported to other bilinear inverse problems, such as dictionary learning with common support across samples, where the circulant-structure arguments would need to be replaced by incoherence assumptions.
  • A direct testable prediction is that the success phase transition should shift left as the number of channels grows roughly like $N^3$ for fixed sparsity, which could be checked by Monte Carlo trials across $N$ and $L$.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies blind deconvolution of L channels that share a common filter and a common sparse support. It proposes a smooth approximation of the ℓ2,1-norm based objective on the unit sphere, after reparameterizing the problem through the inverse filter and applying a preconditioner built from the observations. Under a Gaussian generative model for the nonzeros and a non-uniform support assumption, it characterizes the landscape of the objective, proving that approximate second-order stationary points lie close to the target solution when the sample size satisfies (85). It then proposes RGD-NCS, proves that it reaches an approximate second-order stationary point in finitely many iterations, and combines the two results into a recovery guarantee in Theorem 5. A complex-valued extension and numerical experiments, including an array-processing application, are presented.

Significance. If the core comparison with Shi and Chi [15] is correct, the main theorem is a significant advance: it yields a finite-sample recovery guarantee whose dominant sample-complexity term is O(κ^8 K^6 N^3) in the regime K^2 ≤ N, versus the claimed O(κ^8 K^{-6} N^9) for [15]. The real-valued analysis is substantive: the landscape lemmas, the concentration bounds for EΩ and Δ, and the convergence proof for RGD-NCS are developed in detail, with the auxiliary technical estimates supplied in the supplement. The paper is also honest about the Gaussian and noiseless assumptions. However, the headline comparative claim rests on an unverified restriction on a parameter μ of [15], and the numerical experiments are run outside the theorem's stated regime; the complex-valued section also contains an apparent error in the complex gradient formula and stops short of proving the claimed complex guarantees. These issues do not obviously invalidate the real-valued theorem, but they do block acceptance in the current form.

major comments (3)
  1. [Section 3.3, paragraph after Eq. (101)] The claimed sample-complexity improvement over [15, Theorem 3] is load-bearing and is not established. The paper states that μ is an algorithmic hyperparameter 'whose order does not exceed that of θ' and then converts O(κ^8 θ^{-4} μ^{-2} N^3) into O(κ^8 K^{-6} N^9). No proof or citation is given for the restriction μ ≤ θ. If [15] allows μ = Ω(1), the same formula gives O(κ^8 K^{-4} N^7), which is better than the proposed O(κ^8 K^6 N^3) for part of the stated regime. Moreover, the paper's own implementation of [15] in Section 6.1 uses μ = min{10 N^{-5/4}, 0.05}, so for N = 64 and K = 1,2,3 one has μ > K/N, i.e., the numerical comparison with [15] is itself conducted outside the regime assumed in the theoretical comparison. The authors should either justify μ ≤ θ from the assumptions of [15], or recompute the comparison using the actually allowable range of μ and update the conclusions accordingly.
  2. [Section 6.1, Experiment 2] The numerical experiments do not operate in the regime of Theorem 5, so the claim that the experiments validate the theorem's quantitative sample-complexity prediction is too strong. With N = 64 and K = 3, Theorem 5 requires ε ≤ c2 K^{-7} N^{-2} ≈ 1.9 c2 and ϵg ≤ c3 K^{-7/2} N^{-1/2} ≈ 8 × 10^{-4} c3 (for O(1) constants), while the experiments set ε = min{10^5 N^{-5}, 10^{-4}} ≈ 9.3 × 10^{-5} and ϵg = 10^{-2}. Both values exceed the required ranges unless the unspecified constants are extraordinarily large. Thus Figure 2 demonstrates heuristic success outside the proven range; it does not quantitatively confirm Theorem 5. The authors should either rerun within the theorem's parameter regime or explicitly state that the experiments are robustness checks beyond the guarantee.
  3. [Section 5, Eq. (118)] The complex-valued gradient formula appears to be incorrect as written. For the term |y_l^T Γ_n R_C h|^2, the derivative with respect to the complex variable h involves the conjugate of y_l, i.e., the quadratic form should contain y_l^* y_l^T (equivalently y_l conjugate times y_l transpose), not y_l y_l^T. The printed expression R_C^H Γ_n^T (1/L Σ_l y_l y_l^T) Γ_n R_C h is not Hermitian and is not the Riemannian gradient of ρ_C under the metric in (117). Since Algorithm 1's complex variant and the Section 6.2 experiments rely on this formula, this needs to be corrected and verified. In addition, the section asserts that theoretical guarantees 'analogous to those established in Theorem 4 can be derived' but provides no proof; as written, the complex extension has no proven recovery guarantee and should be labeled accordingly.
minor comments (4)
  1. [Abstract and Section 1] The terms 'sample size' and 'number of channels' are both used for L; this is standard in the area, but a sentence explicitly fixing the notation would help avoid confusion with N, the filter length.
  2. [Section 2.2, Eq. (13)] The ambiguity model states that the filter is recovered up to a global scaling and a circulant shift. It would be helpful to state explicitly that the success criterion in Section 6.1 corresponds to this ambiguity model, since the criterion uses ∥g ⊛ ĝ_inv∥∞ / ∥g ⊛ ĝ_inv∥2.
  3. [Section 6.2] There are minor language issues, for example 'to a array signal processing application' in the opening sentence; these should be corrected in copyediting.
  4. [Section 7] The conclusion correctly notes that noisy and off-grid settings are outside the proven model, but this limitation should also be stated when the array-processing experiment is described, so that the experimental section does not give the impression that the theorem covers the application setting.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the theoretical bounds are derived from stated probabilistic assumptions and external concentration results, with no fitted parameter renamed as a prediction.

full rationale

The paper's central derivation chain is self-contained rather than circular. The optimization landscape theorems (Theorems 1-4) are proved from explicit assumptions on the i.i.d. Gaussian nonzero entries of P_Omega(x_l), with concentration of E_Omega taken from the external standard reference [58, Example 6.2] and the preconditioning deviation bound developed in the paper's own Lemma 7. Lemma 7 itself is proved using independent matrix concentration arguments, citing the moment-controlled Bernstein inequality from the external baseline [15], not from the present authors' prior work. The algorithm convergence guarantee in Proposition 8 is an independent smooth-manifold optimization argument, and Theorem 5 simply composes this convergence result with the landscape characterization; no parameter is fitted to data and then relabeled as a prediction. The comparison with [15] in Sections 3.2 and 3.3 uses an external theorem from [15], and the passage from theta approximately K/N and the algorithmic hyperparameter mu to the claimed rates is an interpretive comparison step. Even if the assumption mu = O(theta) is questionable or unsupported, that is a correctness or comparison-validity concern about an external baseline's parameter regime, not a circularity in the present derivation: the proposed sample-complexity bound does not reduce by construction to the comparison target, nor does it depend on [15] for its truth. Self-citations in the paper (references [35,36]) concern joint sparse frequency recovery and are contextual rather than load-bearing for Theorem 5. No instance of self-definition, fitted-input-called-prediction, or uniqueness-imported-from-authors was found. The honest finding is therefore a circularity score of 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central guarantee rests on a Gaussian generative model for sparse entries, an invertible circulant filter, a non-uniform support set, and standard concentration/circulant algebra. No new physical entities are introduced. The theoretical bounds are parameter-free in the sense that the constants are existential rather than fitted to data.

free parameters (3)
  • smoothing parameter ε = min(1e5 N^-5, 1e-4) in experiments; theory allows ε ≤ c2 K^-7 N^-2
    Hand-chosen smoothness parameter balancing approximation to the mixed ℓ2,1 norm and conditioning. The experimental value is far larger than the theoretical upper bound for representative N=64, K=27, so the reported experiments do not operate in the regime covered by Theorem 5.
  • first-order tolerance ϵg = 1e-2
    Algorithmic stopping tolerance. Theory requires ϵg ≤ c3 K^{-7/2} N^{-1/2}, which is much smaller than 1e-2 for the experimental sizes, so the experiments also fall outside the proven tolerance regime.
  • second-order tolerance ϵH = 1e-6 N^-3
    Algorithmic stopping tolerance for negative curvature. It scales with N only and does not match the theorem's K^{-7/2} scaling.
assumptions (5)
  • domain assumption The nonzero components PΩ(x_l), l=1,...,L, are i.i.d. Gaussian vectors with zero mean and covariance I.
    Stated at the start of Section 3 and used by Lemma 3 for concentration of EΩ and by Lemma 7 for moment bounds on C(x_l).
  • domain assumption C(g) is invertible with condition number κ.
    Used in Section 2.2 to define the inverse representation h, in the preconditioning construction, and in the sample complexity bounds of Theorems 4 and 5.
  • domain assumption The common support set Ω is non-uniform, i.e., it has no nontrivial cyclic symmetry.
    Proposition 1 shows this condition is necessary and sufficient for exact recovery in the asymptotic problem; all main theorems assume it.
  • domain assumption The observation model is noiseless circulant convolution, y_l = g ⊛ x_l.
    The theoretical analysis in Sections 3 and 4 has no noise term; noise is introduced only in the application experiment.
  • standard math Standard properties of circulant matrices, DFT diagonalization, and matrix Bernstein concentration inequalities.
    Invoked in Lemma 1, Lemma 7, Lemma 10 of the supplement, and throughout the Hessian calculations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Jointly Sparse Blind Deconvolution via Riemannian Optimization." pith.science (2026). https://pith.science/paper/P625EF5X

@misc{pith2026260804465,
  author       = {Pith},
  title        = {Pith review of: Jointly Sparse Blind Deconvolution via Riemannian Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P625EF5X}},
  note         = {Machine review of arXiv:2608.04465}
}
read the original abstract

Blind deconvolution has been widely applied in system identification and signal processing. While joint sparsity commonly arises in practical scenarios, effectively exploiting this structure to enhance recovery performance remains a challenging and largely open problem. In this paper, we propose a joint-sparsity-promoting optimization problem and develop a Riemannian optimization algorithm for its accurate and efficient solution. We further establish theoretical guarantees that characterize the non-asymptotic relationship between the estimation error and the sample complexity, showing that exploiting joint sparsity can significantly reduce the sample complexity required for successful recovery. Numerical experiments are provided that validate the theoretical results and demonstrate the effectiveness of the proposed approach.

Figures

Figures reproduced from arXiv: 2608.04465 by the authors.

Figure 1
Figure 1. Optimization landscapes corresponding to different sparsity-promoting objective functions. Darker colors indicate smaller objective values, whereas lighter colors indicate larger objective values. The red dots mark the target solutions. recover the true filter. In contrast, the global minimum of the exact ℓ2,1-based objective function remains aligned with the target solution. Moreover, the landscape induced by the p… view at source ↗
Figure 2
Figure 2. Phase-transition behavior of jointly sparse blind deconvolution under different parameter settings. White indicates a success rate of 100%, whereas black indicates a success rate of 0%. and direction-of-arrival (DOA) associated with the k-th path are denoted by τk, βk, and ωk, respectively. WLOG, we assume that τ1 ≤ τ2 ≤ · · · ≤ τK. The received signal at time t at the l-th antenna is modeled as yl (t) = X K k=1 βk … view at source ↗
Figure 3
Figure 3. Estimated delay spectra under different SNR levels for two parameter sets. TDOA estimation performance of the proposed algorithm under these two non-ideal factors through the estimated delay spectrum. The true power associated with the k-th path is given by |βk| 2 . The estimated delay spectrum is computed from the average row-wise energy of the recovered input signal matrix Xf. Specifically, let pe denote the estim… view at source ↗

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

160 extracted references · 76 canonical work pages

  1. [15]

    2009 IEEE Conference on Computer Vision and Pattern Recognition , pages=

    Understanding and evaluating blind deconvolution algorithms , author=. 2009 IEEE Conference on Computer Vision and Pattern Recognition , pages=. 2009 , organization=

    2009

  2. [1]

    N. J. Higham, Functions of matrices: theory and computation. 1em plus 0.5em minus 0.4em SIAM, 2008

    2008

  3. [2]

    Shi and Y

    L. Shi and Y. Chi, ``Manifold gradient descent solves multi-channel sparse blind deconvolution provably and efficiently,'' IEEE Transactions on Information Theory, vol. 67, no. 7, pp. 4784--4811, 2021

    2021

  4. [3]

    Vershynin, High-dimensional probability: An introduction with applications in data science

    R. Vershynin, High-dimensional probability: An introduction with applications in data science. 1em plus 0.5em minus 0.4em Cambridge university press, 2018, vol. 47

    2018

  5. [4]

    Sebah and X

    P. Sebah and X. Gourdon, ``Introduction to the gamma function,'' American Journal of Scientific Research, vol. 2, pp. 1--10, 2002

    2002

  6. [5]

    Boumal, An introduction to optimization on smooth manifolds

    N. Boumal, An introduction to optimization on smooth manifolds. 1em plus 0.5em minus 0.4em Cambridge University Press, 2023

    2023

  7. [6]

    2008 , publisher=

    Functions of matrices: theory and computation , author=. 2008 , publisher=

    2008

  8. [7]

    2019 , publisher=

    High-dimensional statistics: A non-asymptotic viewpoint , author=. 2019 , publisher=

    2019

Show all 160 references
  1. [8]

    IEEE Transactions on Information Theory , volume=

    Manifold gradient descent solves multi-channel sparse blind deconvolution provably and efficiently , author=. IEEE Transactions on Information Theory , volume=. 2021 , publisher=

    2021

  2. [9]

    Complete dictionary recovery over the sphere

    Sun, Ju and Qu, Qing and Wright, John , journal=. Complete dictionary recovery over the sphere. 2016 , publisher=

    2016

  3. [10]

    Advances in Neural Information Processing Systems , volume=

    Global geometry of multichannel sparse blind deconvolution on the sphere , author=. Advances in Neural Information Processing Systems , volume=

  4. [11]

    2018 , publisher=

    High-dimensional probability: An introduction with applications in data science , author=. 2018 , publisher=

    2018

  5. [12]

    Linear algebra and its applications , volume=

    Spectral decomposition of real circulant matrices , author=. Linear algebra and its applications , volume=. 2003 , publisher=

    2003

  6. [13]

    American Journal of Scientific Research , volume=

    Introduction to the gamma function , author=. American Journal of Scientific Research , volume=

  7. [14]

    Geophysics , volume=

    Multichannel blind deconvolution of seismic signals , author=. Geophysics , volume=. 1998 , publisher=

    1998

  8. [16]

    Blind Deconvolution Using Convex Programming , year=

    Ahmed, Ali and Recht, Benjamin and Romberg, Justin , journal=. Blind Deconvolution Using Convex Programming , year=

  9. [17]

    Advances in Neural Information Processing Systems , volume=

    A blind sparse deconvolution method for neural spike identification , author=. Advances in Neural Information Processing Systems , volume=

  10. [18]

    IEEE Journal on Selected Areas in Information Theory , volume=

    Dual-blind deconvolution for overlaid radar-communications systems , author=. IEEE Journal on Selected Areas in Information Theory , volume=. 2023 , publisher=

    2023

  11. [19]

    Leveraging Diversity and Sparsity in Blind Deconvolution , year=

    Ahmed, Ali and Demanet, Laurent , journal=. Leveraging Diversity and Sparsity in Blind Deconvolution , year=

  12. [20]

    SIAM Journal on Imaging Sciences , volume=

    Exact recovery of multichannel sparse blind deconvolution via gradient descent , author=. SIAM Journal on Imaging Sciences , volume=. 2020 , publisher=

    2020

  13. [21]

    IEEE Transactions on information Theory , volume=

    Blind identification and equalization based on second-order statistics: A time domain approach , author=. IEEE Transactions on information Theory , volume=. 1994 , publisher=

    1994

  14. [22]

    Multichannel Sparse Blind Deconvolution on the Sphere , year=

    Li, Yanjun and Bresler, Yoram , journal=. Multichannel Sparse Blind Deconvolution on the Sphere , year=

  15. [23]

    Identifiability in Bilinear Inverse Problems With Applications to Subspace or Sparsity-Constrained Blind Gain and Phase Calibration , year=

    Li, Yanjun and Lee, Kiryung and Bresler, Yoram , journal=. Identifiability in Bilinear Inverse Problems With Applications to Subspace or Sparsity-Constrained Blind Gain and Phase Calibration , year=

  16. [24]

    Self-Calibrated Super Resolution , year=

    Da Costa, Maxime Ferreira and Chi, Yuejie , booktitle=. Self-Calibrated Super Resolution , year=

  17. [25]

    Morrison Jr, Robert L and Do, Minh N and Munson, David C , year=

  18. [26]

    Stable Estimation of Pulses of Unknown Shape From Multiple Snapshots via

    Kalra, Meghna and Lee, Kiryung , journal=. Stable Estimation of Pulses of Unknown Shape From Multiple Snapshots via. 2024 , volume=

    2024

  19. [27]

    and Delaney, A.H

    Bresler, Y. and Delaney, A.H. , booktitle=. Resolution of overlapping echoes of unknown shape , year=

  20. [28]

    and Gunther, J.H

    Swindlehurst, A.L. and Gunther, J.H. , journal=. Methods for blind equalization and resolution of overlapping echoes of unknown shape , year=

  21. [29]

    Applied and Computational Harmonic Analysis , volume=

    Sensor calibration for off-the-grid spectral estimation , author=. Applied and Computational Harmonic Analysis , volume=. 2020 , publisher=

    2020

  22. [30]

    and Arguello, Henry , journal=

    Vargas, Edwin and Mishra, Kumar Vijay and Jacome, Roman and Sadler, Brian M. and Arguello, Henry , journal=. Dual-Blind Deconvolution for Overlaid Radar-Communications Systems , year=

  23. [31]

    Signal Processing , volume=

    Multi-antenna dual-blind deconvolution for joint radar-communications via soman minimization , author=. Signal Processing , volume=. 2024 , publisher=

    2024

  24. [32]

    ACM SIGGRAPH Asia 2009 papers , pages=

    Fast motion deblurring , author=. ACM SIGGRAPH Asia 2009 papers , pages=

    2009

  25. [33]

    Four short stories about

    Strohmer, Thomas , journal=. Four short stories about. 2002 , publisher=

    2002

  26. [34]

    arXiv preprint arXiv:0901.3403 , year=

    Distributed compressive sensing , author=. arXiv preprint arXiv:0901.3403 , year=

    arXiv

  27. [35]

    Spatial compressive sensing for

    Rossi, Marco and Haimovich, Alexander M and Eldar, Yonina C , journal=. Spatial compressive sensing for. 2013 , publisher=

    2013

  28. [36]

    Advances in Neural Information Processing Systems , volume=

    Finding a sparse vector in a subspace: Linear sparsity using alternating directions , author=. Advances in Neural Information Processing Systems , volume=

  29. [37]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=

    Model selection and estimation in regression with grouped variables , author=. Journal of the Royal Statistical Society Series B: Statistical Methodology , volume=. 2006 , publisher=

    2006

  30. [38]

    Proximal gradient method for nonsmooth optimization over the

    Chen, Shixiang and Ma, Shiqian and Man-Cho So, Anthony and Zhang, Tong , journal=. Proximal gradient method for nonsmooth optimization over the. 2020 , publisher=

    2020

  31. [39]

    Foundations of Computational Mathematics , volume=

    A geometric analysis of phase retrieval , author=. Foundations of Computational Mathematics , volume=. 2018 , publisher=

    2018

  32. [40]

    Fast and Guaranteed Blind Multichannel Deconvolution Under a Bilinear System Model , year=

    Lee, Kiryung and Tian, Ning and Romberg, Justin , journal=. Fast and Guaranteed Blind Multichannel Deconvolution Under a Bilinear System Model , year=

  33. [41]

    IEEE Transactions on Information Theory , volume=

    Average case analysis of multichannel sparse recovery using convex relaxation , author=. IEEE Transactions on Information Theory , volume=. 2009 , publisher=

    2009

  34. [42]

    Theoretical Results on Sparse Representations of Multiple-Measurement Vectors , year=

    Chen, Jie and Huo, Xiaoming , journal=. Theoretical Results on Sparse Representations of Multiple-Measurement Vectors , year=

  35. [43]

    Exact Joint Sparse Frequency Recovery via Optimization Methods , year=

    Yang, Zai and Xie, Lihua , journal=. Exact Joint Sparse Frequency Recovery via Optimization Methods , year=

  36. [44]

    and Xie, Lihua , journal=

    Yang, Zai and Tang, Jinhui and Eldar, Yonina C. and Xie, Lihua , journal=. On the Sample Complexity of Multichannel Frequency Estimation via Convex Optimization , year=

  37. [45]

    , journal=

    Yang, Dehui and Tang, Gongguo and Wakin, Michael B. , journal=. Super-Resolution of Complex Exponentials From Modulations With Unknown Waveforms , year=

  38. [46]

    Timely and Painless Breakups: Off-the-Grid Blind Message Recovery and Users’ Demixing , year=

    Daei, Sajad and Razavikia, Saeed and Skoglund, Mikael and Fodor, Gabor and Fischione, Carlo , journal=. Timely and Painless Breakups: Off-the-Grid Blind Message Recovery and Users’ Demixing , year=

  39. [47]

    Vectorized

    Chen, Jinchi and Gao, Weiguo and Mao, Sihan and Wei, Ke , journal=. Vectorized. 2022 , volume=

    2022

  40. [48]

    arXiv preprint arXiv:1501.06120 , year=

    A unified framework for identifiability analysis in bilinear inverse problems with applications to subspace and sparsity models , author=. arXiv preprint arXiv:1501.06120 , year=

    arXiv

  41. [49]

    IEEE Signal Processing Letters , volume=

    Blind deconvolution from multiple sparse inputs , author=. IEEE Signal Processing Letters , volume=. 2016 , publisher=

    2016

  42. [50]

    Wavelets and Sparsity XVII , volume=

    A note on the blind deconvolution of multiple sparse signals from unknown subspaces , author=. Wavelets and Sparsity XVII , volume=. 2017 , organization=

    2017

  43. [51]

    International conference on machine learning , pages=

    No spurious local minima in nonconvex low rank problems: A unified geometric analysis , author=. International conference on machine learning , pages=. 2017 , organization=

    2017

  44. [52]

    Inverse Problems , volume=

    Provable sample-efficient sparse phase retrieval initialized by truncated power method , author=. Inverse Problems , volume=. 2023 , publisher=

    2023

  45. [53]

    On the local linear rate of consensus on the

    Chen, Shixiang and Garcia, Alfredo and Hong, Mingyi and Shahrampour, Shahin , journal=. On the local linear rate of consensus on the. 2023 , publisher=

    2023

  46. [54]

    SIAM Journal on Optimization , volume=

    Accelerated methods for nonconvex optimization , author=. SIAM Journal on Optimization , volume=. 2018 , publisher=

    2018

  47. [55]

    2023 , publisher=

    An introduction to optimization on smooth manifolds , author=. 2023 , publisher=

    2023

  48. [56]

    Escaping from saddle points on

    Sun, Yue and Flammarion, Nicolas and Fazel, Maryam , journal=. Escaping from saddle points on

  49. [57]

    Advances in Neural Information Processing Systems , volume=

    Efficiently escaping saddle points on manifolds , author=. Advances in Neural Information Processing Systems , volume=

  50. [58]

    Trust-region methods on

    Absil, P-A and Baker, Christopher G and Gallivan, Kyle A , journal=. Trust-region methods on. 2007 , publisher=

    2007

  51. [59]

    Advances in Neural Information Processing Systems , volume=

    Adaptive negative curvature descent with applications in non-convex optimization , author=. Advances in Neural Information Processing Systems , volume=

  52. [60]

    Mathematical Programming , volume=

    Exploiting negative curvature in deterministic and stochastic optimization , author=. Mathematical Programming , volume=. 2019 , publisher=

    2019

  53. [61]

    Line search algorithms for locally

    Hosseini, Somayeh and Huang, Wen and Yousefpour, Rohollah , journal=. Line search algorithms for locally. 2018 , publisher=

    2018

  54. [62]

    Complete Dictionary Recovery Over the Sphere

    Sun, Ju and Qu, Qing and Wright, John , journal=. Complete Dictionary Recovery Over the Sphere. 2017 , volume=

    2017

  55. [63]

    Global identifiability of overcomplete dictionary learning via

    Sun, Yuchen and Huang, Kejun , booktitle=. Global identifiability of overcomplete dictionary learning via

  56. [64]

    and Sommer, Friedrich T

    Hillar, Christopher J. and Sommer, Friedrich T. , journal=. When Can Dictionary Learning Uniquely Recover Sparse Data From Subsamples? , year=

  57. [65]

    Dictionary Learning With Few Samples and Matrix Concentration , year=

    Luh, Kyle and Vu, Van , journal=. Dictionary Learning With Few Samples and Matrix Concentration , year=

  58. [66]

    SIAM Journal on Optimization , volume=

    Convergence Regions of Alternating Minimization Algorithms for Dictionary Learning , author=. SIAM Journal on Optimization , volume=. 2026 , publisher=

    2026

  59. [67]

    SIAM Journal on Mathematics of Data Science , volume=

    Simple alternating minimization provably solves complete dictionary learning , author=. SIAM Journal on Mathematics of Data Science , volume=. 2025 , publisher=

    2025

  60. [68]

    Inverse Problems , volume=

    A convergent blind deconvolution method for post-adaptive-optics astronomical imaging , author=. Inverse Problems , volume=. 2013 , publisher=

    2013

  61. [69]

    Neural blind deconvolution with

    Benfenati, A and Catozzi, A and Ruggiero, V , journal=. Neural blind deconvolution with. 2023 , publisher=

    2023

  62. [70]

    Inverse Problems , volume=

    Convex blind image deconvolution with inverse filtering , author=. Inverse Problems , volume=. 2018 , publisher=

    2018

  63. [71]

    Inverse Problems , volume=

    Multichannel blind deconvolution via maximum likelihood estimator: application in neural recordings , author=. Inverse Problems , volume=. 2019 , publisher=

    2019

  64. [72]

    Inverse Problems , volume=

    Generalized variational framework with minimax optimization for parametric blind deconvolution , author=. Inverse Problems , volume=. 2024 , publisher=

    2024

  65. [73]

    Toeplitz and circulant matrices:

    Gray, Robert M , year=. Toeplitz and circulant matrices:

  66. [74]

    1979 , publisher=

    Circulant matrices , author=. 1979 , publisher=

    1979

  67. [75]

    Algorithms for discrete

    Lu, RTMAC , year=. Algorithms for discrete

  68. [76]

    Discovering transforms:

    Bamieh, Bassam , journal=. Discovering transforms:

  69. [77]

    2010 , publisher=

    Principles of modern radar , author=. 2010 , publisher=

    2010

  70. [78]

    Laser & Photonics Reviews , volume=

    Computational adaptive optics for fluorescence microscopy via sparse blind deconvolution , author=. Laser & Photonics Reviews , volume=. 2025 , publisher=

    2025

  71. [79]

    and Nehorai, A

    Sarder, P. and Nehorai, A. , journal=. Deconvolution methods for. 2006 , volume=

    2006

  72. [80]

    Scientific reports , volume=

    Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery , author=. Scientific reports , volume=. 2013 , publisher=

    2013

  73. [81]

    Blind deconvolution of

    Soulez, Ferr. Blind deconvolution of. 2012 9th IEEE International Symposium on Biomedical Imaging (ISBI) , pages=. 2012 , organization=

    2012

  74. [82]

    Laser Applications to Chemical, Security and Environmental Analysis , pages=

    Superresolution with optically-motivated blind deconvolution , author=. Laser Applications to Chemical, Security and Environmental Analysis , pages=. 2016 , organization=

    2016

  75. [83]

    Plant Methods , volume=

    Multicolour three dimensional structured illumination microscopy of immunolabeled plant microtubules and associated proteins , author=. Plant Methods , volume=. 2019 , publisher=

    2019

  76. [84]

    On the determinants and inverses of circulant matrices with

    Shen, Shou-Qiang and Cen, Jian-Miao and Hao, Yong , journal=. On the determinants and inverses of circulant matrices with. 2011 , publisher=

    2011

  77. [85]

    Foundations of computational mathematics , volume=

    User-friendly tail bounds for sums of random matrices , author=. Foundations of computational mathematics , volume=. 2012 , publisher=

    2012

  78. [86]

    2022 , publisher=

    An introduction to passive radar , author=. 2022 , publisher=

    2022

  79. [87]

    SIAM Journal on Imaging Sciences , volume=

    Blind deconvolution by a steepest descent algorithm on a quotient manifold , author=. SIAM Journal on Imaging Sciences , volume=. 2018 , publisher=

    2018

  80. [88]

    Review of Scientific Instruments , volume=

    Contributed Review: Source-localization algorithms and applications using time of arrival and time difference of arrival measurements , author=. Review of Scientific Instruments , volume=. 2016 , publisher=

    2016

  81. [89]

    and Do, Minh N

    Morrison, Robert L. and Do, Minh N. and Munson, David C. , journal=. 2009 , volume=

    2009

  82. [90]

    Ho, K. C. and Sun, Ming , journal=. Passive Source Localization Using Time Differences of Arrival and Gain Ratios of Arrival , year=

  83. [91]

    and Chen, C.-K

    Gardner, W.A. and Chen, C.-K. , journal=. Signal-selective time-difference-of-arrival estimation for passive location of man-made signal sources in highly corruptive environments. 1992 , volume=

    1992

  84. [92]

    2026 , publisher=

    Cerwin, Stephen A and McMahan, Jesse T and Papadopoulos, Alexandros S and Piccini, Gerard N and Frissell, Nathaniel A and Collins, Kristina V and Montare, Aidan and Bilberry, Paul and Blackshear, Samuel and Themens, David R , journal=. 2026 , publisher=

    2026

  85. [93]

    A multipath-driven approach to

    Fabrizio, Giuseppe and Heitmann, Andrew , journal=. A multipath-driven approach to. 2013 , publisher=

    2013

  86. [94]

    Levin, Y

    A. Levin, Y. Weiss, F. Durand, and W. T. Freeman, ``Understanding and evaluating blind deconvolution algorithms,'' in 2009 IEEE Conference on Computer Vision and Pattern Recognition. 1em plus 0.5em minus 0.4em IEEE, 2009, pp. 1964--1971

    2009

  87. [95]

    Ahmed, B

    A. Ahmed, B. Recht, and J. Romberg, ``Blind deconvolution using convex programming,'' IEEE Transactions on Information Theory, vol. 60, no. 3, pp. 1711--1732, 2014

    2014

  88. [96]

    K. F. Kaaresen and T. Taxt, ``Multichannel blind deconvolution of seismic signals,'' Geophysics, vol. 63, no. 6, pp. 2093--2107, 1998

    1998

  89. [97]

    Prato, A

    M. Prato, A. L. Camera, S. Bonettini, and M. Bertero, ``A convergent blind deconvolution method for post-adaptive-optics astronomical imaging,'' Inverse Problems, vol. 29, no. 6, p. 065017, 2013

    2013

  90. [98]

    Benfenati, A

    A. Benfenati, A. Catozzi, and V. Ruggiero, ``Neural blind deconvolution with Poisson data,'' Inverse Problems, vol. 39, no. 5, p. 054003, 2023

    2023

  91. [99]

    Ekanadham, D

    C. Ekanadham, D. Tranchina, and E. Simoncelli, ``A blind sparse deconvolution method for neural spike identification,'' Advances in Neural Information Processing Systems, vol. 24, 2011

    2011

  92. [100]

    Zhang, H

    R. Zhang, H. Du, N. Zhou, Z. Zhou, H. Tang, J. Qian, Q. Chen, and C. Zuo, ``Computational adaptive optics for fluorescence microscopy via sparse blind deconvolution,'' Laser & Photonics Reviews, vol. 19, no. 17, p. 2500032, 2025

    2025

  93. [101]

    Sarder and A

    P. Sarder and A. Nehorai, ``Deconvolution methods for 3-D fluorescence microscopy images,'' IEEE Signal Processing Magazine, vol. 23, no. 3, pp. 32--45, 2006

    2006

  94. [102]

    Vargas, K

    E. Vargas, K. V. Mishra, R. Jacome, B. M. Sadler, and H. Arguello, ``Dual-blind deconvolution for overlaid radar-communications systems,'' IEEE Journal on Selected Areas in Information Theory, vol. 4, pp. 75--93, 2023

    2023

  95. [103]

    X.-G. Lv, F. Li, and T. Zeng, ``Convex blind image deconvolution with inverse filtering,'' Inverse Problems, vol. 34, no. 3, p. 035003, 2018

    2018

  96. [104]

    Ahmed and L

    A. Ahmed and L. Demanet, ``Leveraging diversity and sparsity in blind deconvolution,'' IEEE Transactions on Information Theory, vol. 64, no. 6, pp. 3975--4000, 2018

    2018

  97. [105]

    Li and Y

    Y. Li and Y. Bresler, ``Multichannel sparse blind deconvolution on the sphere,'' IEEE Transactions on Information Theory, vol. 65, no. 11, pp. 7415--7436, 2019

    2019

  98. [106]

    31, 2018

    ------, ``Global geometry of multichannel sparse blind deconvolution on the sphere,'' Advances in Neural Information Processing Systems, vol. 31, 2018

    2018

  99. [107]

    Q. Qu, X. Li, and Z. Zhu, ``Exact recovery of multichannel sparse blind deconvolution via gradient descent,'' SIAM Journal on Imaging Sciences, vol. 13, no. 3, pp. 1630--1652, 2020

    2020

  100. [108]

    K. Lee, N. Tian, and J. Romberg, ``Fast and guaranteed blind multichannel deconvolution under a bilinear system model,'' IEEE Transactions on Information Theory, vol. 64, no. 7, pp. 4792--4818, 2018

    2018

  101. [109]

    Akhavan, S

    S. Akhavan, S. Esmaeili, M. Kamarei, and H. Soltanian-Zadeh, ``Multichannel blind deconvolution via maximum likelihood estimator: application in neural recordings,'' Inverse Problems, vol. 35, no. 3, p. 035001, 2019

    2019

  102. [110]

    Q. Cao, D. Han, X. Wang, and W. Zhang, ``Generalized variational framework with minimax optimization for parametric blind deconvolution,'' Inverse Problems, vol. 40, no. 4, p. 045019, 2024

    2024

  103. [111]

    Bresler and A

    Y. Bresler and A. Delaney, ``Resolution of overlapping echoes of unknown shape,'' in International Conference on Acoustics, Speech, and Signal Processing,, 1989, pp. 2657--2660 vol.4

    1989

  104. [112]

    Swindlehurst and J

    A. Swindlehurst and J. Gunther, ``Methods for blind equalization and resolution of overlapping echoes of unknown shape,'' IEEE Transactions on Signal Processing, vol. 47, no. 5, pp. 1245--1254, 1999

    1999

  105. [113]

    Y. Li, K. Lee, and Y. Bresler, ``Identifiability in bilinear inverse problems with applications to subspace or sparsity-constrained blind gain and phase calibration,'' IEEE Transactions on Information Theory, vol. 63, no. 2, pp. 822--842, 2017

    2017

  106. [114]

    Kalra and K

    M. Kalra and K. Lee, ``Stable estimation of pulses of unknown shape from multiple snapshots via ESPRIT ,'' IEEE Transactions on Signal Processing, vol. 72, pp. 2637--2648, 2024

    2024

  107. [115]

    Y. C. Eldar, W. Liao, and S. Tang, ``Sensor calibration for off-the-grid spectral estimation,'' Applied and Computational Harmonic Analysis, vol. 48, no. 2, pp. 570--598, 2020

    2020

  108. [116]

    Rossi, A

    M. Rossi, A. M. Haimovich, and Y. C. Eldar, ``Spatial compressive sensing for MIMO radar,'' IEEE Transactions on Signal Processing, vol. 62, no. 2, pp. 419--430, 2013

    2013

  109. [117]

    Jacome, E

    R. Jacome, E. Vargas, K. V. Mishra, B. M. Sadler, and H. Arguello, ``Multi-antenna dual-blind deconvolution for joint radar-communications via soman minimization,'' Signal Processing, vol. 221, p. 109484, 2024

    2024

  110. [118]

    R. L. Morrison, M. N. Do, and D. C. Munson, `` MCA : A multichannel approach to SAR autofocus,'' IEEE Transactions on Image Processing, vol. 18, no. 4, pp. 840--853, 2009

    2009

  111. [119]

    J. Min, J. Jang, D. Keum, S.-W. Ryu, C. Choi, K.-H. Jeong, and J. C. Ye, ``Fluorescent microscopy beyond diffraction limits using speckle illumination and joint support recovery,'' Scientific reports, vol. 3, no. 1, p. 2075, 2013

    2013

  112. [120]

    Vavrdov \'a , O

    T. Vavrdov \'a , O. S amajov \'a , P. K r enek, M. Ove c ka, P. Flokov \'a , R. S naurov \'a , J. S amaj, and G. Komis, ``Multicolour three dimensional structured illumination microscopy of immunolabeled plant microtubules and associated proteins,'' Plant Methods, vol. 15, no....

    2019

  113. [121]

    Soulez, L

    F. Soulez, L. Denis, Y. Tourneur, and \'E . Thi \'e baut, ``Blind deconvolution of 3D data in wide field fluorescence microscopy,'' in 2012 9th IEEE International Symposium on Biomedical Imaging (ISBI). 1em plus 0.5em minus 0.4em IEEE, 2012, pp. 1735--1738

    2012

  114. [122]

    Soulez and M

    F. Soulez and M. Unser, ``Superresolution with optically-motivated blind deconvolution,'' in Laser Applications to Chemical, Security and Environmental Analysis. 1em plus 0.5em minus 0.4em Optica Publishing Group, 2016, pp. JT3A--38

    2016

  115. [123]

    Cho and S

    S. Cho and S. Lee, ``Fast motion deblurring,'' in ACM SIGGRAPH Asia 2009 papers, 2009, pp. 1--8

    2009

  116. [124]

    Strohmer, ``Four short stories about Toeplitz matrix calculations,'' Linear Algebra and its Applications, vol

    T. Strohmer, ``Four short stories about Toeplitz matrix calculations,'' Linear Algebra and its Applications, vol. 343, pp. 321--344, 2002

    2002

  117. [125]

    Chen and X

    J. Chen and X. Huo, ``Theoretical results on sparse representations of multiple-measurement vectors,'' IEEE Transactions on Signal Processing, vol. 54, no. 12, pp. 4634--4643, 2006

    2006

  118. [126]

    Y. C. Eldar and H. Rauhut, ``Average case analysis of multichannel sparse recovery using convex relaxation,'' IEEE Transactions on Information Theory, vol. 56, no. 1, pp. 505--519, 2009

    2009

  119. [127]

    Yang and L

    Z. Yang and L. Xie, ``Exact joint sparse frequency recovery via optimization methods,'' IEEE Transactions on Signal Processing, vol. 64, no. 19, pp. 5145--5157, 2016

    2016

  120. [128]

    Z. Yang, J. Tang, Y. C. Eldar, and L. Xie, ``On the sample complexity of multichannel frequency estimation via convex optimization,'' IEEE Transactions on Information Theory, vol. 65, no. 4, pp. 2302--2315, 2019

    2019

  121. [129]

    D. Yang, G. Tang, and M. B. Wakin, ``Super-resolution of complex exponentials from modulations with unknown waveforms,'' IEEE Transactions on Information Theory, vol. 62, no. 10, pp. 5809--5830, 2016

    2016

  122. [130]

    S. Daei, S. Razavikia, M. Skoglund, G. Fodor, and C. Fischione, ``Timely and painless breakups: Off-the-grid blind message recovery and users’ demixing,'' IEEE Transactions on Information Theory, vol. 71, no. 7, pp. 5226--5257, 2025

    2025

  123. [131]

    J. Chen, W. Gao, S. Mao, and K. Wei, ``Vectorized Hankel lift: A convex approach for blind super-resolution of point sources,'' IEEE Transactions on Information Theory, vol. 68, no. 12, pp. 8280--8309, 2022

    2022

  124. [132]

    Huang and P

    W. Huang and P. Hand, ``Blind deconvolution by a steepest descent algorithm on a quotient manifold,'' SIAM Journal on Imaging Sciences, vol. 11, no. 4, pp. 2757--2785, 2018

    2018

  125. [133]

    J. Sun, Q. Qu, and J. Wright, ``Complete dictionary recovery over the sphere I : Overview and the geometric picture,'' IEEE Transactions on Information Theory, vol. 63, no. 2, pp. 853--884, 2016

    2016

  126. [134]

    ------, ``Complete dictionary recovery over the sphere II : Recovery by Riemannian trust-region method,'' IEEE Transactions on Information Theory, vol. 63, no. 2, pp. 885--914, 2017

    2017

  127. [135]

    C. J. Hillar and F. T. Sommer, ``When can dictionary learning uniquely recover sparse data from subsamples?'' IEEE Transactions on Information Theory, vol. 61, no. 11, pp. 6290--6297, 2015

    2015

  128. [136]

    Luh and V

    K. Luh and V. Vu, ``Dictionary learning with few samples and matrix concentration,'' IEEE Transactions on Information Theory, vol. 62, no. 3, pp. 1516--1527, 2016

    2016

  129. [137]

    Sun and K

    Y. Sun and K. Huang, ``Global identifiability of overcomplete dictionary learning via l1 and volume minimization,'' in The Thirteenth International Conference on Learning Representations, 2025

    2025

  130. [138]

    Ruetz and K

    S. Ruetz and K. Schnass, ``Convergence regions of alternating minimization algorithms for dictionary learning,'' SIAM Journal on Optimization, vol. 36, no. 1, pp. 320--349, 2026

    2026

  131. [139]

    Liang, G

    G. Liang, G. Zhang, S. Fattahi, and R. Y. Zhang, ``Simple alternating minimization provably solves complete dictionary learning,'' SIAM Journal on Mathematics of Data Science, vol. 7, no. 3, pp. 855--883, 2025

    2025

  132. [140]

    Cosse, ``A note on the blind deconvolution of multiple sparse signals from unknown subspaces,'' in Wavelets and Sparsity XVII, vol

    A. Cosse, ``A note on the blind deconvolution of multiple sparse signals from unknown subspaces,'' in Wavelets and Sparsity XVII, vol. 10394. 1em plus 0.5em minus 0.4em SPIE, 2017, pp. 330--347

    2017

  133. [141]

    Wang and Y

    L. Wang and Y. Chi, ``Blind deconvolution from multiple sparse inputs,'' IEEE Signal Processing Letters, vol. 23, no. 10, pp. 1384--1388, 2016

    2016

  134. [142]

    R. Ge, C. Jin, and Y. Zheng, ``No spurious local minima in nonconvex low rank problems: A unified geometric analysis,'' in International conference on machine learning. 1em plus 0.5em minus 0.4em PMLR, 2017, pp. 1233--1242

    2017

  135. [143]

    J.-F. Cai, J. Li, and J. You, ``Provable sample-efficient sparse phase retrieval initialized by truncated power method,'' Inverse Problems, vol. 39, no. 7, p. 075008, 2023

    2023

  136. [144]

    Carmon, J

    Y. Carmon, J. C. Duchi, O. Hinder, and A. Sidford, ``Accelerated methods for nonconvex optimization,'' SIAM Journal on Optimization, vol. 28, no. 2, pp. 1751--1772, 2018

    2018

  137. [145]

    S. Chen, A. Garcia, M. Hong, and S. Shahrampour, ``On the local linear rate of consensus on the Stiefel manifold,'' IEEE Transactions on Automatic Control, vol. 69, no. 4, pp. 2324--2339, 2023

    2023

  138. [146]

    P. J. Davis, Circulant matrices. 1em plus 0.5em minus 0.4em Wiley New York, 1979, vol. 120

    1979

  139. [147]

    R. M. Gray, ``Toeplitz and circulant matrices: A review,'' 2006

    2006

  140. [148]

    Bamieh, ``Discovering transforms: A tutorial on circulant matrices, circular convolution, and the discrete fourier transform,'' arXiv preprint arXiv:1805.05533, 2018

    B. Bamieh, ``Discovering transforms: A tutorial on circulant matrices, circular convolution, and the discrete fourier transform,'' arXiv preprint arXiv:1805.05533, 2018

    2018 arXiv

  141. [149]

    Shen, J.-M

    S.-Q. Shen, J.-M. Cen, and Y. Hao, ``On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers,'' Applied Mathematics and Computation, vol. 217, no. 23, pp. 9790--9797, 2011

    2011

  142. [150]

    M. J. Wainwright, High-dimensional statistics: A non-asymptotic viewpoint. 1em plus 0.5em minus 0.4em Cambridge university press, 2019, vol. 48

    2019

  143. [151]

    M. A. Richards, J. Scheer, W. A. Holm, and W. L. Melvin, ``Principles of modern radar,'' 2010

    2010

  144. [152]

    Criscitiello and N

    C. Criscitiello and N. Boumal, ``Efficiently escaping saddle points on manifolds,'' Advances in Neural Information Processing Systems, vol. 32, 2019

    2019

  145. [153]

    Y. Sun, N. Flammarion, and M. Fazel, ``Escaping from saddle points on Riemannian manifolds,'' Advances in Neural Information Processing Systems, vol. 32, 2019

    2019

  146. [154]

    Absil, C

    P.-A. Absil, C. G. Baker, and K. A. Gallivan, ``Trust-region methods on Riemannian manifolds,'' Foundations of Computational Mathematics, vol. 7, no. 3, pp. 303--330, 2007

    2007

  147. [155]

    M. Liu, Z. Li, X. Wang, J. Yi, and T. Yang, ``Adaptive negative curvature descent with applications in non-convex optimization,'' Advances in Neural Information Processing Systems, vol. 31, 2018

    2018

  148. [156]

    F. E. Curtis and D. P. Robinson, ``Exploiting negative curvature in deterministic and stochastic optimization,'' Mathematical Programming, vol. 176, no. 1, pp. 69--94, 2019

    2019

  149. [157]

    J. A. Tropp, ``User-friendly tail bounds for sums of random matrices,'' Foundations of computational mathematics, vol. 12, no. 4, pp. 389--434, 2012

    2012

  150. [158]

    Hosseini, W

    S. Hosseini, W. Huang, and R. Yousefpour, ``Line search algorithms for locally Lipschitz functions on Riemannian manifolds,'' SIAM Journal on Optimization, vol. 28, no. 1, pp. 596--619, 2018

    2018

  151. [159]

    H. D. Griffiths and C. J. Baker, An introduction to passive radar. 1em plus 0.5em minus 0.4em Artech House, 2022

    2022

  152. [160]

    Fabrizio and A

    G. Fabrizio and A. Heitmann, ``A multipath-driven approach to HF geolocation,'' Signal Processing, vol. 93, no. 12, pp. 3487--3503, 2013

    2013

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.