{"id":648,"date":"2026-10-07T12:21:44","date_gmt":"2026-10-07T12:21:44","guid":{"rendered":"https:\/\/yanntraonmilin.perso.math.cnrs.fr\/?page_id=648"},"modified":"2026-10-07T14:40:45","modified_gmt":"2026-10-07T14:40:45","slug":"my-open-problems","status":"publish","type":"page","link":"https:\/\/yanntraonmilin.perso.math.cnrs.fr\/?page_id=648","title":{"rendered":"My open problems"},"content":{"rendered":"\n

<\/p>\n\n\n\n

Status:<\/strong> a question described as open in a paper is not necessarily still open today. I haven’t checked everything.<\/p>\n\n\n\n

Abbreviations:<\/strong> RIP = restricted isometry property; GPGD = generalized projected gradient descent; RED = regularization by denoising; SOR = stochastic orthogonal regularization; DPS = diffusion posterior sampling. PDF page numbers refer to the versions reviewed, including any cover sheets.<\/p>\n\n\n\n

Regularizer design<\/h2>\n\n\n\n
    \n
  1. Open question.<\/strong> Is the sufficient RIP threshold pointwise equal to the strongly sharp threshold for every model and regularizer?
    Source:<\/em> Stable recovery of low-dimensional cones in Hilbert spaces: One RIP to rule them all<\/a>. \u00a77, PDF pp.46\u201347.<\/li>\n\n\n\n
  2. Open question.<\/strong> For each fixed ambient dimension and sparsity, does the strongly sharp sparse-recovery threshold equal the weak family-wide threshold?
    Source:<\/em>     Stable recovery of low-dimensional cones in Hilbert spaces: One RIP to rule them all<\/a>. \u00a77, PDF p.47.<\/li>\n\n\n\n
  3. Open question.<\/strong> Is the proposed block-structured RIP threshold weakly sharp?
    Source:<\/em>     Stable recovery of low-dimensional cones in Hilbert spaces: One RIP to rule them all<\/a>. \u00a77, PDF p.47.<\/li>\n\n\n\n
  4. Open question.<\/strong> For a prescribed regularizer class, when is the best admissible RIP threshold positive?
    Source:<\/em>     Stable recovery of low-dimensional cones in Hilbert spaces: One RIP to rule them all<\/a>. \u00a77, PDF p.47.<\/li>\n\n\n\n
  5. Open question.<\/strong> Is the optimal regularizer attained, how can it be characterized, and is it unique? This is partially answered in A theory of optimal convex regularization for low-dimensional recovery<\/a>
    Source:<\/em>     Stable recovery of low-dimensional cones in Hilbert spaces: One RIP to rule them all<\/a>. \u00a77, PDF p.47.<\/li>\n\n\n\n
  6. Research direction.<\/strong> Can optimal-regularizer theory extend to Banach spaces and off-grid models?
    Source:<\/em>     A theory of optimal convex regularization for low-dimensional recovery<\/a>. \u00a72.3, PDF p.9.<\/li>\n\n\n\n
  7. Open question.<\/strong> Do descent-cone-volume and fixed-dimensional random-kernel criteria select the same optimal regularizers? This depends on the family of regularizers, still open for “almost all” convex regularizers
    Source:<\/em>     A theory of optimal convex regularization for low-dimensional recovery<\/a>. \u00a72.3, PDF p.9.<\/li>\n\n\n\n
  8. Open question.<\/strong> Can the sharp RIP threshold be computed explicitly for arbitrary regularizers, even for sparse or low-rank recovery?
    Source:<\/em>     A theory of optimal convex regularization for low-dimensional recovery<\/a>. \u00a73.1, PDF p.13.<\/li>\n\n\n\n
  9. Open question.<\/strong> Does the necessary threshold equal the sharp threshold, in special cases or generally?
    Source:<\/em>     A theory of optimal convex regularization for low-dimensional recovery<\/a>. \u00a73.1, PDF pp.15\u201316.<\/li>\n\n\n\n
  10. Research direction.<\/strong> Can nuclear-norm optimality be extended from symmetric to nonsymmetric matrices?
    Source:<\/em>     A theory of optimal convex regularization for low-dimensional recovery<\/a>. \u00a73.2, PDF p.17.<\/li>\n\n\n\n
  11. Open question.<\/strong> Is \u2113\u2081 uniquely optimal for each fixed sparsity greater than one?
    Source:<\/em>     A theory of optimal convex regularization for low-dimensional recovery<\/a>. \u00a73.2, PDF p.19.<\/li>\n\n\n\n
  12. Open question.<\/strong> For sparsity in levels and sparse-plus-low-rank models, what is optimal beyond weighted atomic-norm families?
    Source:<\/em>     A theory of optimal convex regularization for low-dimensional recovery<\/a>. \u00a74, PDF p.25.<\/li>\n\n\n\n
  13. Open question.<\/strong> Which properties of a compliance measure guarantee an optimal convex regularizer exists?
    Source:<\/em>     A theory of optimal convex regularization for low-dimensional recovery<\/a>. \u00a75, PDF p.25.<\/li>\n\n\n\n
  14. Research direction.<\/strong> How do compliance measures transform under model intersections, unions, and sums?
    Source:<\/em>     A theory of optimal convex regularization for low-dimensional recovery<\/a>. \u00a75, PDF p.25.<\/li>\n\n\n\n
  15. Research direction.<\/strong> How should regularizers trade exact identifiability against noise stability, possibly with measurement design?
    Source:<\/em>     A theory of optimal convex regularization for low-dimensional recovery<\/a>. \u00a75, PDF p.25.<\/li>\n\n\n\n
  16. Open question.<\/strong> Can optimal regularizers be constructed by optimizing the compliance quantities without guessing the answer?
    Source:<\/em>     A theory of optimal convex regularization for low-dimensional recovery<\/a>. \u00a75, PDF pp.25\u201326.<\/li>\n\n\n\n
  17. Research direction.<\/strong> What regularizers are optimal under statistical-dimension or Gaussian-width criteria for nonuniform recovery?
    Source:<\/em>     A theory of optimal convex regularization for low-dimensional recovery<\/a>. \u00a75, PDF p.26.<\/li>\n<\/ol>\n\n\n\n

    Hilbert-space recovery<\/h2>\n\n\n\n
      \n
    1. Research direction.<\/strong> Can the sufficient RIP threshold extend to non-Hilbert measurement norms and structured acquisitions such as rank-one projections?
      Source:<\/em>       Compressed sensing in Hilbert spaces<\/a>. \u00a75.3, PDF p.23.<\/li>\n\n\n\n
    2. Open question.<\/strong> What weaker necessary-and-sufficient geometric condition characterizes stable finite-dimensional embeddings?
      Source:<\/em>       Compressed sensing in Hilbert spaces<\/a>. \u00a75.4, PDF pp.23\u201324.<\/li>\n\n\n\n
    3. Open question.<\/strong> Can a sufficient regularizer-dependent RIP constant be established for kernel-Hilbert formulations of super-resolution?
      Source:<\/em>       Compressed sensing in Hilbert spaces<\/a>. \u00a75.5, PDF p.24.<\/li>\n\n\n\n
    4. Open question.<\/strong> Which usual Banach-space models have finite-dimensional normalized secant sets in suitable kernel metrics? This is partially answered in Compressive Statistical Learning with Random Feature Moments<\/a>
      Source:<\/em>       Compressed sensing in Hilbert spaces<\/a>. \u00a75.5, PDF p.24.<\/li>\n<\/ol>\n\n\n\n

      Optimal algorithms and RED<\/h2>\n\n\n\n
        \n
      1. Open question.<\/strong> Is orthogonal projection near-optimal for arbitrary unions of subspaces, and what projections are optimal for general models?
        Source:<\/em>         Towards optimal algorithms for the recovery of low-dimensional models with linear rates<\/a>. \u00a73.3, PDF pp.31\u201332.<\/li>\n\n\n\n
      2. Research direction.<\/strong> Do the sparse-projection near-optimality results extend to low-rank recovery and singular-value thresholding?
        Source:<\/em>         Towards optimal algorithms for the recovery of low-dimensional models with linear rates<\/a>. \u00a73.3, PDF p.31.<\/li>\n\n\n\n
      3. Open question.<\/strong> What algorithmic optimality criteria capture structured-noise stability as well as identifiability? some insights in From sparse recovery to plug-and-play priors, understanding trade-offs for stable recovery with generalized projected gradient descent<\/a>.
        Source:<\/em>         Towards optimal algorithms for the recovery of low-dimensional models with linear rates<\/a>. \u00a73.3, PDF p.32.<\/li>\n\n\n\n
      4. Research direction.<\/strong> Can the analysis cover other data-fit directions, including \u2113\u2081 regression, and implicit algorithms? some insights in From sparse recovery to plug-and-play priors, understanding trade-offs for stable recovery with generalized projected gradient descent<\/a>.
        Source:<\/em>         Towards optimal algorithms for the recovery of low-dimensional models with linear rates<\/a>. \u00a74.1, PDF p.33.<\/li>\n\n\n\n
      5. Open question.<\/strong> Within meaningful larger classes of averaged-direction methods, is GPGD optimal or can linear convergence be faster?
        Source:<\/em>         Towards optimal algorithms for the recovery of low-dimensional models with linear rates<\/a>. \u00a7\u00a74.2\u20134.3, PDF pp.34\u201335.<\/li>\n\n\n\n
      6. Research direction.<\/strong> Can approximate-projection analysis explain the observed difference between GM-RED and plug-and-play methods?
        Source:<\/em>         Towards optimal algorithms for the recovery of low-dimensional models with linear rates<\/a>. \u00a75.2.3, PDF p.40.<\/li>\n\n\n\n
      7. Research direction.<\/strong> How does optimality change if uniform linear convergence is required only after a finite burn-in?
        Source:<\/em>         Towards optimal algorithms for the recovery of low-dimensional models with linear rates<\/a>. Conclusion, PDF pp.46\u201347.<\/li>\n\n\n\n
      8. Open question.<\/strong> How can denoisers be trained with controlled restricted Lipschitz constants?
        Source:<\/em>         Towards optimal algorithms for the recovery of low-dimensional models with linear rates<\/a>. Conclusion, PDF p.47.<\/li>\n\n\n\n
      9. Open question.<\/strong> Can modified RED achieve stable recovery under generic restricted-Lipschitz projections without closeness to orthogonal projection?
        Source:<\/em>         A note on the convergence of RED algorithms under minimal hypotheses and open questions<\/a>. End of \u00a73, PDF p.4.<\/li>\n\n\n\n
      10. Open question.<\/strong> Can classical RED converge under restricted-isometry and restricted-Lipschitz assumptions with useful constants?
        Source:<\/em>         A note on the convergence of RED algorithms under minimal hypotheses and open questions<\/a>. \u00a74, PDF p.5.<\/li>\n\n\n\n
      11. Open question.<\/strong> Under what recovery assumptions is classical or modified RED preferable to GPGD?
        Source:<\/em>         A note on the convergence of RED algorithms under minimal hypotheses and open questions<\/a>. \u00a74, PDF p.5.<\/li>\n\n\n\n
      12. Research direction.<\/strong> Can the backprojection be estimated automatically for different structured noises, with corresponding guarantees?
        Source:<\/em>         From sparse recovery to plug-and-play priors, understanding trade-offs for stable recovery with generalized projected gradient descent<\/a>. Conclusion, PDF p.9.<\/li>\n\n\n\n
      13. Research direction.<\/strong> How should idempotence regularization be combined with SOR and other projection-training constraints?
        Source:<\/em>         From sparse recovery to plug-and-play priors, understanding trade-offs for stable recovery with generalized projected gradient descent<\/a>. Conclusion, PDF p.9.<\/li>\n<\/ol>\n\n\n\n

        Diffusion and learned priors<\/h2>\n\n\n\n
          \n
        1. Research direction.<\/strong> Can diffusion-prior recovery theory cover distributions supported on nonlinear manifolds?
          Source:<\/em>           A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm<\/a>. \u00a76, PDF p.10.<\/li>\n\n\n\n
        2. Research direction.<\/strong> Can the framework yield sample-complexity guarantees for DPS and diffusion-based posterior sampling or plug-and-play methods?
          Source:<\/em>           A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm<\/a>. \u00a76, PDF p.10.<\/li>\n\n\n\n
        3. Research direction.<\/strong> Can diffusion-prior guarantees extend to nonlinear forward operators, such as phase retrieval?
          Source:<\/em>           A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm<\/a>. \u00a76, PDF p.10.<\/li>\n\n\n\n
        4. Research direction.<\/strong> Can recovery be made provably robust to measurement noise and learned-prior mismatch?
          Source:<\/em>           A Recovery Theory for Diffusion Priors: Deterministic Analysis of the Implicit Prior Algorithm<\/a>. \u00a76, PDF p.10.<\/li>\n\n\n\n
        5. Open question.<\/strong> Can training control \u03a6_P, or can SOR itself be proved to control it?
          Source:<\/em>           Stochastic Orthogonal Regularization for deep projective priors<\/a>. \u00a72.2, PDF p.7.<\/li>\n\n\n\n
        6. Open question.<\/strong> What is the optimal sampling distribution for stochastic orthogonal regularization?
          Source:<\/em>           Stochastic Orthogonal Regularization for deep projective priors<\/a>. \u00a72.3, PDF p.7.<\/li>\n\n\n\n
        7. Research direction.<\/strong> Can SOR be adapted to other norms and reconstruction metrics?
          Source:<\/em>           Stochastic Orthogonal Regularization for deep projective priors<\/a>. Discussion, PDF p.15.<\/li>\n\n\n\n
        8. Research direction.<\/strong> How can SOR be extended to variational autoencoders, diffusion models, and nondeterministic priors?
          Source:<\/em>           Stochastic Orthogonal Regularization for deep projective priors<\/a>. Conclusion, PDF p.16.<\/li>\n\n\n\n
        9. Research direction.<\/strong> How should approximate, rather than exact, membership in an atomic model be quantified?
          Source:<\/em>           Disentangled latent representations of images with atomic autoencoders<\/a>. \u00a7II, PDF p.3.<\/li>\n\n\n\n
        10. Open question.<\/strong> Can atomic disentanglement be proved, using a rigorous notion of irreducible atom-generating functions?
          Source:<\/em>           Disentangled latent representations of images with atomic autoencoders<\/a>; Max-sparsity atomic autoencoders with application to inverse problems<\/a>. Conclusions, PDF pp.5 and 12 respectively.<\/li>\n\n\n\n
        11. Research direction.<\/strong> Can atomic autoencoders yield higher-level semantic disentanglement?
          Source:<\/em>           Disentangled latent representations of images with atomic autoencoders<\/a>. Conclusion, PDF p.5.<\/li>\n\n\n\n
        12. Research direction.<\/strong> Can normalizing latent blocks improve navigation through the learned representation?
          Source:<\/em>           Disentangled latent representations of images with atomic autoencoders<\/a>. Conclusion, PDF p.5.<\/li>\n\n\n\n
        13. Research direction.<\/strong> Can learned atomic representations be given bounded-distortion guarantees?
          Source:<\/em>           Disentangled latent representations of images with atomic autoencoders<\/a>. Conclusion, PDF p.5.<\/li>\n\n\n\n
        14. Research direction.<\/strong> Can max-sparsity atomic autoencoders be extended to photorealistic, textured images?
          Source:<\/em>           Max-sparsity atomic autoencoders with application to inverse problems<\/a>. Conclusion, PDF p.12.<\/li>\n\n\n\n
        15. Research direction.<\/strong> Can post-training quantization bounds extend to Vision Transformers, including normalization and multi-head attention?
          Source:<\/em>           On the impact of the parametrization of deep convolutional neural networks on post-training quantization<\/a>. \u00a76, PDF p.10.<\/li>\n<\/ol>\n\n\n\n

          Off-grid spikes and nonconvex recovery<\/h2>\n\n\n\n
            \n
          1. Research direction.<\/strong> How should basin theory handle overestimated spike counts and non-isolated sets of minimizers?
            Source:<\/em>             The basins of attraction of the global minimizers of the non-convex sparse spike estimation problem<\/a>. Remark 2.2, PDF p.12.<\/li>\n\n\n\n
          2. Open question.<\/strong> Can full convergence and measurement guarantees avoid the exponential-in-dimension grid cost of initialization?
            Source:<\/em>             Projected gradient descent for non-convex sparse spike estimation<\/a>; An algorithm for non-convex off-the-grid sparse spike estimation with a minimum separation constraint<\/a>. Conclusions, PDF pp.4 and 3 respectively.<\/li>\n\n\n\n
          3. Research direction.<\/strong> Can alternative pruning, dimension reduction, or quasi-Newton acceleration improve sparse-spike reconstruction?
            Source:<\/em>             Projected gradient descent for non-convex sparse spike estimation<\/a>. Conclusion, PDF p.4.<\/li>\n\n\n\n
          4. Research direction.<\/strong> What stability guarantees hold for practical spike-recovery algorithms under noise and model mismatch?
            Source:<\/em>             Projected gradient descent for non-convex sparse spike estimation<\/a>; An algorithm for non-convex off-the-grid sparse spike estimation with a minimum separation constraint<\/a>; Estimation of off-the-grid sparse spikes with over-parametrized projected gradient descent: theory and application<\/a>. Conclusions, PDF pp.4, 3, and 30 respectively.<\/li>\n\n\n\n
          5. Open question.<\/strong> Can restrictive RIP and measurement-complexity requirements be reduced toward finite-dimensional sparse-recovery scales?
            Source:<\/em>             Estimation of off-the-grid sparse spikes with over-parametrized projected gradient descent: theory and application<\/a>. \u00a73, PDF p.14; conclusion, p.30.<\/li>\n\n\n\n
          6. Research direction.<\/strong> Can over-parameterized recovery theory extend beyond Gaussian kernels?
            Source:<\/em>             Estimation of off-the-grid sparse spikes with over-parametrized projected gradient descent: theory and application<\/a>. Conclusion, PDF p.30.<\/li>\n\n\n\n
          7. Research direction.<\/strong> Can the required projection distance be estimated in practice instead of supplied as an oracle quantity?
            Source:<\/em>             Estimation of off-the-grid sparse spikes with over-parametrized projected gradient descent: theory and application<\/a>. Conclusion, PDF p.30.<\/li>\n\n\n\n
          8. Conjecture.<\/strong> Can the ambient-dimension dependence of the basin bound be removed for Gaussian measurements with high probability?
            Source:<\/em>             On strong basins of attractions for non-convex sparse spike estimation: upper and lower bounds<\/a>. PDF p.6.<\/li>\n\n\n\n
          9. Research direction.<\/strong> Can nonasymptotic upper and lower basin bounds match the basins observed experimentally?
            Source:<\/em>             On strong basins of attractions for non-convex sparse spike estimation: upper and lower bounds<\/a>. \u00a76, PDF pp.11\u201312.<\/li>\n\n\n\n
          10. Research direction.<\/strong> Can direct Hessian analysis handle several spikes and structured measurement operators where RIP is poorly suited?
            Source:<\/em>             On strong basins of attractions for non-convex sparse spike estimation: upper and lower bounds<\/a>. \u00a76, PDF pp.11\u201312.<\/li>\n\n\n\n
          11. Research direction.<\/strong> Can efficient initializations give quantitative end-to-end guarantees for super-resolution and Gaussian-mixture recovery?
            Source:<\/em>             The basins of attraction of the global minimizers of non-convex inverse problems with low-dimensional models in infinite dimension<\/a>. \u00a74, PDF p.18; \u00a75, p.19.<\/li>\n\n\n\n
          12. Research direction.<\/strong> Can basin theory be applied to low-rank tensor recovery?
            Source:<\/em>             The basins of attraction of the global minimizers of non-convex inverse problems with low-dimensional models in infinite dimension<\/a>. \u00a75, PDF p.19.<\/li>\n\n\n\n
          13. Research direction.<\/strong> Can basin theory and suitable RIP guarantees be established for learned neural generative models?
            Source:<\/em>             The basins of attraction of the global minimizers of non-convex inverse problems with low-dimensional models in infinite dimension<\/a>. \u00a75, PDF p.19.<\/li>\n\n\n\n
          14. Research direction.<\/strong> Can nonuniform recovery analysis substantially improve the quantitative basin bounds?
            Source:<\/em>             The basins of attraction of the global minimizers of non-convex inverse problems with low-dimensional models in infinite dimension<\/a>. \u00a75, PDF p.19.<\/li>\n\n\n\n
          15. Research direction.<\/strong> Which alternative greedy block-selection rules improve projected block-coordinate descent?
            Source:<\/em>             Projected Block Coordinate Descent for sparse spike estimation.<\/a>. \u00a7IV, PDF p.5.<\/li>\n\n\n\n
          16. Research direction.<\/strong> Can projected block-coordinate updates be parallelized effectively?
            Source:<\/em>             Projected Block Coordinate Descent for sparse spike estimation.<\/a>. \u00a7IV, PDF p.5.<\/li>\n\n\n\n
          17. Research direction.<\/strong> Can sketched reconstruction operate on full 3D data rather than separate 2D slices?
            Source:<\/em>             Sketched over-parametrized projected gradient descent for sparse spike estimation<\/a>. Conclusion, PDF p.5.<\/li>\n\n\n\n
          18. Open question.<\/strong> Why does sketching sometimes slightly improve reconstruction accuracy?
            Source:<\/em>             Sketched over-parametrized projected gradient descent for sparse spike estimation<\/a>. Conclusion, PDF p.5.<\/li>\n<\/ol>\n\n\n\n

            Compressive statistical learning<\/h2>\n\n\n\n
              \n
            1. Open question.<\/strong> Can practical compressive-learning optimization algorithms be proved efficient and accurate?
              Source:<\/em>               Compressive Statistical Learning with Random Feature Moments<\/a>; Statistical Learning Guarantees for Compressive Clustering and Compressive Mixture Modeling<\/a>. PDF p.12; \u00a77, PDF p.29 respectively.<\/li>\n\n\n\n
            2. Open question.<\/strong> Can model-bias bounds and oracle inequalities be sharpened and extended to other losses?
              Source:<\/em>               Compressive Statistical Learning with Random Feature Moments<\/a>. Conclusion, PDF p.24.<\/li>\n\n\n\n
            3. Open question.<\/strong> Can compressive statistical learning achieve fast excess-risk rates under suitable assumptions?
              Source:<\/em>               Compressive Statistical Learning with Random Feature Moments<\/a>. Conclusion, PDF p.24.<\/li>\n\n\n\n
            4. Research direction.<\/strong> Can sketching theory explain statistical-information preservation through neural-network layers and pooling?
              Source:<\/em>               Compressive Statistical Learning with Random Feature Moments<\/a>. Conclusion, PDF p.24.<\/li>\n\n\n\n
            5. Research direction.<\/strong> What trade-offs connect differential privacy and learning accuracy from perturbed or masked sketches?
              Source:<\/em>               Compressive Statistical Learning with Random Feature Moments<\/a>. Conclusion, PDF p.24.<\/li>\n\n\n\n
            6. Research direction.<\/strong> How should task-compatible kernels and sketches be designed for kernel PCA, clustering, classification, and regression?
              Source:<\/em>               Compressive Statistical Learning with Random Feature Moments<\/a>. Conclusion, PDF pp.24\u201325.<\/li>\n\n\n\n
            7. Open question.<\/strong> How universal is a fixed kernel: which learning tasks can share one database sketch?
              Source:<\/em>               Compressive Statistical Learning with Random Feature Moments<\/a>. Conclusion, PDF p.25.<\/li>\n\n\n\n
            8. Research direction.<\/strong> Can sketches based on U-statistics support ranking and tasks involving pairs or tuples of samples?
              Source:<\/em>               Compressive Statistical Learning with Random Feature Moments<\/a>. Conclusion, PDF p.25.<\/li>\n\n\n\n
            9. Open question.<\/strong> Can sufficient sketch sizes approach O(kd), rather than O(k\u00b2d), and match empirical phase transitions?
              Source:<\/em>               Statistical Learning Guarantees for Compressive Clustering and Compressive Mixture Modeling<\/a>; Compressive learning for patch-based image denoising<\/a>. \u00a73.4, PDF p.11 and \u00a77, p.29; \u00a76, PDF pp.18\u201319 respectively.<\/li>\n\n\n\n
            10. Open question.<\/strong> What lower bounds show whether sketch-size and mixture-separation assumptions are necessary?
              Source:<\/em>               Statistical Learning Guarantees for Compressive Clustering and Compressive Mixture Modeling<\/a>. \u00a77, PDF p.29.<\/li>\n\n\n\n
            11. Research direction.<\/strong> Can structural centroid constraints yield sharper bias control and faster compressive clustering?
              Source:<\/em>               Statistical Learning Guarantees for Compressive Clustering and Compressive Mixture Modeling<\/a>. \u00a73, PDF p.8.<\/li>\n\n\n\n
            12. Open question.<\/strong> Can overmodeling with an increasing number of Dirac components yield statistical consistency?
              Source:<\/em>               Statistical Learning Guarantees for Compressive Clustering and Compressive Mixture Modeling<\/a>. \u00a73, PDF p.9.<\/li>\n\n\n\n
            13. Research direction.<\/strong> Can higher-dimensional convex or lifted approaches give provable Fourier-sketch decoders without prohibitive computation?
              Source:<\/em>               Statistical Learning Guarantees for Compressive Clustering and Compressive Mixture Modeling<\/a>. \u00a73.3, PDF p.11; \u00a77, p.29.<\/li>\n\n\n\n
            14. Open question.<\/strong> Can Gaussian-mixture bias terms, including the generalized KL contribution, be controlled as sharply as in clustering?
              Source:<\/em>               Statistical Learning Guarantees for Compressive Clustering and Compressive Mixture Modeling<\/a>. \u00a74, PDF p.13.<\/li>\n\n\n\n
            15. Research direction.<\/strong> Can guarantees be proved for fast structured approximations to Gaussian sketching matrices by analyzing their kernels?
              Source:<\/em>               Statistical Learning Guarantees for Compressive Clustering and Compressive Mixture Modeling<\/a>. \u00a77, PDF p.29.<\/li>\n<\/ol>\n\n\n\n

              Compressive image models<\/h2>\n\n\n\n
                \n
              1. Research direction.<\/strong> Can low-rank covariance parameters be recovered with Frobenius-error guarantees from sketches?
                Source:<\/em>                 Compressive learning for patch-based image denoising<\/a>. \u00a75.2, PDF p.13.<\/li>\n\n\n\n
              2. Open question.<\/strong> What separation notion and RIP theory fit zero-mean mixtures with differently oriented covariance subspaces?
                Source:<\/em>                 Sketched learning for image denoising<\/a>; Compressive learning for patch-based image denoising<\/a>. \u00a73.1, PDF p.6; \u00a77, PDF p.22 respectively.<\/li>\n\n\n\n
              3. Research direction.<\/strong> Can Fourier-frequency distributions be adapted specifically to zero-mean Gaussian mixtures?
                Source:<\/em>                 Sketched learning for image denoising<\/a>. Conclusion, PDF p.10.<\/li>\n\n\n\n
              4. Research direction.<\/strong> Can LR-COMP be accelerated using sparse-spike recovery algorithms with guarantees?
                Source:<\/em>                 Sketched learning for image denoising<\/a>. Conclusion, PDF p.10.<\/li>\n\n\n\n
              5. Research direction.<\/strong> Can sketch scale and covariance ranks be selected automatically, allowing different ranks per component?
                Source:<\/em>                 Compressive learning for patch-based image denoising<\/a>. \u00a75.2, PDF p.13; \u00a77, p.22.<\/li>\n\n\n\n
              6. Research direction.<\/strong> Can compressive patch models extend to generalized Gaussian mixtures, other inverse problems, and video denoising?
                Source:<\/em>                 Sketched learning for image denoising<\/a>; Compressive learning for patch-based image denoising<\/a>. \u00a72.1, PDF p.4; \u00a77, PDF p.22 respectively.<\/li>\n\n\n\n
              7. Open question.<\/strong> Which sketching operators permit recovery of general neural-network-parametrized distributions?
                Source:<\/em>                 Compressive learning of deep regularization for denoising<\/a>. Conclusion, PDF p.12.<\/li>\n\n\n\n
              8. Research direction.<\/strong> Can batch-less deep compressive learning scale to larger patches and other linear inverse problems?
                Source:<\/em>                 Batch-less stochastic gradient descent for compressive learning of deep regularization for image denoising<\/a>. PDF pp.11\u201313.<\/li>\n<\/ol>\n\n\n\n

                Image decomposition<\/h2>\n\n\n\n
                  \n
                1. Open question.<\/strong> What identifiability and incoherence theory supports gradient-sparse plus low-patch-rank decomposition?
                  Source:<\/em>                   Adaptive parameter selection for gradient-sparse + low patch-rank recovery: application to image decomposition<\/a>. \u00a7IV, PDF p.6.<\/li>\n\n\n\n
                2. Open question.<\/strong> Can convergence of the adaptive parameter-selection scheme be proved?
                  Source:<\/em>                   Adaptive parameter selection for gradient-sparse + low patch-rank recovery: application to image decomposition<\/a>. \u00a7IV, PDF p.6.<\/li>\n\n\n\n
                3. Research direction.<\/strong> Can the tuning strategy be extended to other decomposition models, such as the block nuclear-norm model?
                  Source:<\/em>                   Adaptive parameter selection for gradient-sparse + low patch-rank recovery: application to image decomposition<\/a>. \u00a7IV, PDF p.6.<\/li>\n\n\n\n
                4. Research direction.<\/strong> Can learned joint low-dimensional regularizers receive recovery guarantees for broader inverse problems?
                  Source:<\/em>                   Joint structure-texture low dimensional modeling for image decomposition with a plug and play framework<\/a>. \u00a77, PDF pp.23\u201324.<\/li>\n\n\n\n
                5. Research direction.<\/strong> Can the joint model use piecewise-continuous structures, dictionary-sparse or mixed textures, and texture datasets?
                  Source:<\/em>                   Joint structure-texture low dimensional modeling for image decomposition with a plug and play framework<\/a>. \u00a77, PDF p.24.<\/li>\n\n\n\n
                6. Research direction.<\/strong> Can joint learning handle three or more components, such as jump\u2013oscillation\u2013trend decompositions?
                  Source:<\/em>                   Joint structure-texture low dimensional modeling for image decomposition with a plug and play framework<\/a>. \u00a77, PDF p.24.<\/li>\n\n\n\n
                7. Research direction.<\/strong> Can learned convex or variational generative priors reduce the computation and memory cost of joint decomposition?
                  Source:<\/em>                   Joint structure-texture low dimensional modeling for image decomposition with a plug and play framework<\/a>. \u00a77, PDF p.24.<\/li>\n<\/ol>\n\n\n\n

                  Other inverse problems and applications<\/h2>\n\n\n\n
                    \n
                  1. Open question.<\/strong> For a fixed kernel, which annihilating matrix has the best RIP constant or order?
                    Source:<\/em>                     Robust Multi-image Processing With Optimal Sparse Regularization<\/a>. \u00a73.4.2, PDF p.7.<\/li>\n\n\n\n
                  2. Research direction.<\/strong> Can sparse-outlier guarantees be extended to additional dense measurement noise?
                    Source:<\/em>                     Robust Multi-image Processing With Optimal Sparse Regularization<\/a>. \u00a75.2.2, PDF p.14.<\/li>\n\n\n\n
                  3. Research direction.<\/strong> Can dictionary or nonlocal sparse priors with favorable non-concentration properties improve outlier removal?
                    Source:<\/em>                     Robust Multi-image Processing With Optimal Sparse Regularization<\/a>. \u00a75.2.3, PDF pp.14\u201315.<\/li>\n\n\n\n
                  4. Open question.<\/strong> Why does the semidefinite lifting approach to phase unmixing perform well in underdetermined cases?
                    Source:<\/em>                     Phase Unmixing : Multichannel Source Separation with Magnitude Constraints<\/a>. \u00a74, PDF p.4.<\/li>\n\n\n\n
                  5. Research direction.<\/strong> Can faster SDP solvers reduce the very large iteration counts in high-SNR underdetermined phase unmixing?
                    Source:<\/em>                     Phase Unmixing : Multichannel Source Separation with Magnitude Constraints<\/a>. \u00a73, PDF p.4.<\/li>\n\n\n\n
                  6. Research direction.<\/strong> Can phase unmixing handle unknown mixing matrices, flexible magnitude constraints, and additional phase structure?
                    Source:<\/em>                     Phase Unmixing : Multichannel Source Separation with Magnitude Constraints<\/a>. \u00a74, PDF p.4.<\/li>\n\n\n\n
                  7. Research direction.<\/strong> Can annotated data train better keep\/discard classifiers and player detectors robust to motion and scale changes?
                    Source:<\/em>                     D\u00e9coupage automatique de vid\u00e9os de sport amateur par d\u00e9tection de personnes et analyse de contenu colorim\u00e9trique<\/a>. \u00a75, PDF p.7.<\/li>\n\n\n\n
                  8. Research direction.<\/strong> Can player detection be combined with optical flow for better action tracking?
                    Source:<\/em>                     Piecewise linear prediction model for action tracking in sports<\/a>. Conclusion, PDF pp.5\u20136.<\/li>\n\n\n\n
                  9. Research direction.<\/strong> Can wide-angle player-motion tracking prevent losing the game action and improve reacquisition?
                    Source:<\/em>                     Real-time multi-sport action tracking with convolutional neural networks<\/a>. \u00a7IV, PDF p.6.<\/li>\n<\/ol>\n\n\n\n

                    Historical questions and later follow-ups<\/h2>\n\n\n\n

                    These questions were raised in earlier papers and received subsequent answers or partial answers under specified assumptions.<\/p>\n\n\n\n

                      \n
                    1. Historical question.<\/strong> Is \u2113\u2081 optimal beyond weighted \u2113\u2081 norms, and is the nuclear norm analogously optimal?
                      Source:<\/em>                       Optimality of 1-norm regularization among weighted 1-norms for sparse recovery: a case study on how to find optimal regularizations<\/a>; Is the 1-norm the best convex sparse regularization?<\/a>. Conclusions.
                      Follow-up:<\/em> Later                       A theory of optimal convex regularization for low-dimensional recovery<\/a> proves optimality among coercive continuous convex regularizers for specified RIP criteria, with stated model restrictions.<\/li>\n\n\n\n
                    2. Historical question.<\/strong> Can over-parameterized off-grid projected gradient descent receive theoretical recovery guarantees?
                      Source:<\/em>                       Fast off-the-grid sparse recovery with over-parametrized projected gradient descent<\/a>. \u00a7IV, PDF p.5.
                      Follow-up:<\/em> Addressed in a specified setting by                       Estimation of off-the-grid sparse spikes with over-parametrized projected gradient descent: theory and application<\/a>; practical initialization and projection assumptions still matter.<\/li>\n\n\n\n
                    3. Historical question.<\/strong> Can strong basins and geometric convergence be characterized for general parametrized low-dimensional models?
                      Source:<\/em>                       The basins of attraction of the global minimizers of non-convex inverse problems with low-dimensional models in infinite dimension<\/a>. Introduction, PDF p.4.
                      Follow-up:<\/em> Later                       On strong basins of attractions for non-convex sparse spike estimation: upper and lower bounds<\/a> advances the sparse-spike case; broader models remain separate.<\/li>\n\n\n\n
                    4. Historical question.<\/strong> Can information preservation by compressive k-means sketches be proved?
                      Source:<\/em>                       Compressive K-means<\/a>. \u00a75, PDF p.4.
                      Follow-up:<\/em> Later                       Compressive Statistical Learning with Random Feature Moments<\/a> and Statistical Learning Guarantees for Compressive Clustering and Compressive Mixture Modeling<\/a> establish guarantees under specified assumptions.<\/li>\n\n\n\n
                    5. Historical question.<\/strong> Can deep compressive learning avoid high-dimensional grid discretization and use fast sketching?
                      Source:<\/em>                       Compressive learning of deep regularization for denoising<\/a>. Conclusion, PDF p.12.
                      Follow-up:<\/em> Batch-less SGD in                       Batch-less stochastic gradient descent for compressive learning of deep regularization for image denoising<\/a> addresses the grid obstacle; scaling remains relevant.<\/li>\n\n\n\n
                    6. Historical question.<\/strong> Can neural motion prediction trained on the collected data improve sports action tracking?
                      Source:<\/em>                       Piecewise linear prediction model for action tracking in sports<\/a>. Conclusion, PDF pp.5\u20136.
                      Follow-up:<\/em> Followed up directly by the CNN tracking method in                       Real-time multi-sport action tracking with convolutional neural networks<\/a>.<\/li>\n<\/ol>\n\n\n\n

                      <\/h2>\n\n\n\n

                      <\/p>\n","protected":false},"excerpt":{"rendered":"

                      Status: a question described as open in a paper is not necessarily still open today. I haven’t checked everything. Abbreviations: RIP = restricted isometry property; GPGD = generalized projected gradient descent; RED = regularization by denoising; SOR = stochastic orthogonal regularization; DPS = diffusion posterior sampling. PDF page numbers refer to the versions reviewed, including […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":10,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-648","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/yanntraonmilin.perso.math.cnrs.fr\/index.php?rest_route=\/wp\/v2\/pages\/648","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/yanntraonmilin.perso.math.cnrs.fr\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/yanntraonmilin.perso.math.cnrs.fr\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/yanntraonmilin.perso.math.cnrs.fr\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/yanntraonmilin.perso.math.cnrs.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=648"}],"version-history":[{"count":8,"href":"https:\/\/yanntraonmilin.perso.math.cnrs.fr\/index.php?rest_route=\/wp\/v2\/pages\/648\/revisions"}],"predecessor-version":[{"id":657,"href":"https:\/\/yanntraonmilin.perso.math.cnrs.fr\/index.php?rest_route=\/wp\/v2\/pages\/648\/revisions\/657"}],"wp:attachment":[{"href":"https:\/\/yanntraonmilin.perso.math.cnrs.fr\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=648"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}