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Publications

Submitted Preprints

F. Krahmer and D. Stöger. On the convex geometry of blind deconvolution and matrix completion, 2019.
BibTeX:
@unpublished{KS19,
  author = {Krahmer, Felix and Stöger, Dominik},
  title = {On the convex geometry of blind deconvolution and matrix completion},
  year = {2019},
  note = {arXiv preprint arXiv:1902.11156}
}
B. Choi, M. Iwen and F. Krahmer. Sparse Harmonic Transforms: A New Class of Sublinear-time Algorithms for Learning Functions of Many Variables, 2018.
BibTeX:
@unpublished{CIK18,
  author = {Choi, Bosu and Iwen, Mark and Krahmer, Felix},
  title = {Sparse Harmonic Transforms: A New Class of Sublinear-time Algorithms for Learning Functions of Many Variables},
  year = {2018},
  note = {arXiv preprint arXiv:1808.04932}
}
M. Iwen, F. Krahmer, Krause-Solberg Sara and J. Maly. On Recovery Guarantees for One-Bit Compressed Sensing on Manifolds, 2018.
BibTeX:
@unpublished{IKKM18,
  author = {Iwen, Mark and Krahmer, Felix and Krause-Solberg, Sara, and Maly, Johhannes},
  title = {On Recovery Guarantees for One-Bit Compressed Sensing on Manifolds},
  year = {2018},
  note = {arXiv preprint arXiv:1807.06490}
}

Refereed Journal Articles

J.-M. Feng, F. Krahmer and R. Saab. Quantized Compressed Sensing for Partial Random Circulant Matrices, Appl. Comput. Harmon. Anal., to appear.
BibTeX:
@article{FKS17,
  author = {Feng, J.-M. and Krahmer, F. and Saab, R.},
  title = {Quantized Compressed Sensing for Partial Random Circulant Matrices},
  journal = {Appl. Comput. Harmon. Anal.},
  year = {to appear},
  url = {https://arxiv.org/abs/1702.04711}
}
J. Geppert, F. Krahmer and D. Stöger. Sparse Power Factorization: Balancing peakiness and sample complexity, Adv. Comput. Math., to appear.
BibTeX:
@article{GKS19,
  author = {Geppert, Jakob and Krahmer, Felix and Stöger, Dominik},
  title = {Sparse Power Factorization: Balancing peakiness and sample complexity},
  journal = {Adv. Comput. Math.},
  year = {to appear},
  note = {arXiv preprint arXiv:1804.09097}
}
B. Bringmann, D. Cremers, F. Krahmer and M. Moeller. The Homotopy Method Revisited: Computing Solution Paths of ℓ1-Regularized Problems, Math. Comp., 82(313):2343-2364, 2018.
BibTeX:
@article{BCKM18,
  author = {Bringmann, B. and Cremers, D. and Krahmer, F. and Moeller, M.},
  title = {The Homotopy Method Revisited: Computing Solution Paths of ℓ1-Regularized Problems},
  journal = {Math. Comp.},
  year = {2018},
  volume = {82},
  number = {313},
  pages = {2343-2364},
  url = {http://www.ams.org/journals/mcom/2018-87-313/S0025-5718-2017-03287-7/}
}
P. Jung, F. Krahmer and D. Stöger. Blind Demixing and Deconvolution at Near-Optimal Rate, IEEE Trans. Inform. Theory, 64(2):704-727, 2018.
Abstract
We consider simultaneous blind deconvolution of r source signals from its noisy superposition, a problem also referred to blind demixing and deconvolution. This signal processing problem occurs in the context of the Internet of Things where a massive number of sensors sporadically communicate only short messages over unknown channels. We show that robust recovery of message and channel vectors can be achieved via convex optimization when random linear encoding using i.i.d. complex Gaussian matrices is used at the devices and the number of required measurements at the receiver scales with the degrees of freedom of the overall estimation problem. Since the scaling is linear in r our result significantly improves over recent works.
BibTeX:
@article{JKS17,
  author = {Jung, P. and Krahmer, F. and Stöger, D.},
  title = {Blind Demixing and Deconvolution at Near-Optimal Rate},
  journal = {IEEE Trans. Inform. Theory},
  year = {2018},
  volume = {64},
  number = {2},
  pages = {704-727},
  url = {https://arxiv.org/abs/1704.04178}
}
F. Krahmer and Y. Liu. Phase Retrieval Without Small-Ball Probability Assumptions, IEEE Trans. Inform. Theory, 64(1):485-500, 2018.
BibTeX:
@article{KL18,
  author = {Felix Krahmer and Yi-Kai Liu},
  title = {Phase Retrieval Without Small-Ball Probability Assumptions},
  journal = {IEEE Trans. Inform. Theory},
  year = {2018},
  volume = {64},
  number = {1},
  pages = {485-500},
  url = {http://arxiv.org/abs/1604.07281}
}
K. Lee, F. Krahmer and J. Romberg. Spectral methods for passive imaging: Nonasymptotic performance and robustness, SIAM J. Imag. Sci., 11(3):2110-2164, SIAM, 2018.
BibTeX:
@article{LKR18,
  author = {Lee, Kiryung and Krahmer, Felix and Romberg, Justin},
  title = {Spectral methods for passive imaging: Nonasymptotic performance and robustness},
  journal = {SIAM J. Imag. Sci.},
  publisher = {SIAM},
  year = {2018},
  volume = {11},
  number = {3},
  pages = {2110--2164}
}
D. Gross, F. Krahmer and R. Kueng. Improved Recovery Guarantees for Phase Retrieval from Coded Diffraction Patterns, Appl. Comput. Harmon. Anal., 42(1):37 - 64, 2017.
Abstract
In this work we analyze the problem of phase retrieval from Fourier measurements with random diffraction patterns. To this end, we consider the recently introduced PhaseLift algorithm, which expresses the problem in the language of convex optimization. We provide recovery guarantees which require O ( log^2 d ) different diffraction patterns, thus improving on recent results by Candès et al. [1], which demand O ( log ^4 d ) different patterns.
BibTeX:
@article{GKK15,
  author = {D. Gross and F. Krahmer and R. Kueng},
  title = {Improved Recovery Guarantees for Phase Retrieval from Coded Diffraction Patterns},
  journal = {Appl. Comput. Harmon. Anal.},
  year = {2017},
  volume = {42},
  number = {1},
  pages = {37 - 64},
  url = {http://arxiv.org/abs/1402.6286},
  doi = {10.1016/j.acha.2015.05.004}
}
M. Kech and F. Krahmer. Optimal Injectivity Conditions for Bilinear Inverse Problems with Applications to Identifiability of Deconvolution Problems, SIAM Journal on Applied Algebra and Geometry, 1(1):20-37, 2017.
Abstract
We study identifiability for bilinear inverse problems under sparsity and subspace constraints. We show that, up to a global scaling ambiguity, almost all such maps are injective on the set of pairs of sparse vectors if the number of measurements m exceeds 2(s_1+s_2)-2, where s_1 and s_2 denote the sparsity of the two input vectors, and injective on the set of pairs of vectors lying in known subspaces of dimensions n_1 and n_2 if m≥ 2(n_1+n_2)-4. We also prove that both these bounds are tight in the sense that one cannot have injectivity for a smaller number of measurements. Our proof technique draws from algebraic geometry. As an application we derive optimal identifiability conditions for the deconvolution problem, thus improving on recent work of Li, Lee, and Bresler [Y. Li, K. Lee, and Y. Bresler, Identifiability and Stability in Blind Deconvolution under Minimal Assumptions, preprint, https://arxiv.org/abs/1507.01308, 2015].
BibTeX:
@article{KK17,
  author = {Michael Kech and Felix Krahmer},
  title = {Optimal Injectivity Conditions for Bilinear Inverse Problems with Applications to Identifiability of Deconvolution Problems},
  journal = {SIAM Journal on Applied Algebra and Geometry},
  year = {2017},
  volume = {1},
  number = {1},
  pages = {20-37},
  url = {http://arxiv.org/abs/1603.07316},
  doi = {10.1137/16M1067469}
}
A. Israel, F. Krahmer and R. Ward. An arithmetic-geometric mean inequality for products of three matrices, Linear Algebra and its Applications, 488:1-12, 2016.
BibTeX:
@article{IKW14,
  author = {Israel, Arie and Krahmer, Felix and Ward, Rachel},
  title = {An arithmetic-geometric mean inequality for products of three matrices},
  journal = {Linear Algebra and its Applications},
  year = {2016},
  volume = {488},
  pages = {1--12},
  url = {http://arxiv.org/abs/1411.0333},
  doi = {10.1016/j.laa.2015.09.013}
}
F. Krahmer and R. Ward. A unified framework for linear dimensionality reduction in L1, Results in Mathematics, 70(1):209-231, 2016.
BibTeX:
@article{KW14,
  author = {Krahmer, Felix and Ward, Rachel},
  title = {A unified framework for linear dimensionality reduction in L1},
  journal = {Results in Mathematics},
  year = {2016},
  volume = {70},
  number = {1},
  pages = {209--231},
  url = {http://arxiv.org/abs/1405.1332v5},
  doi = {10.1007/s00025-015-0475-x}
}
M. Iwen and F. Krahmer. Fast Subspace Approximation via Greedy Least-Squares, Constr. Approx., 42(2):281-301, 2015.
BibTeX:
@article{IK13,
  author = {Iwen, M. and Krahmer, F.},
  title = {Fast Subspace Approximation via Greedy Least-Squares},
  journal = {Constr. Approx.},
  year = {2015},
  volume = {42},
  number = {2},
  pages = {281--301},
  doi = {10.1007/s00365-014-9273-z}
}
F. Krahmer, D. Needell and R. Ward. Compressive Sensing with Redundant Dictionaries and Structured Measurements, SIAM J. Math. Anal., 47(6):4606-4629, 2015.
BibTeX:
@article{KNW15,
  author = {Krahmer, Felix and Needell, Deanna and Ward, Rachel},
  title = {Compressive Sensing with Redundant Dictionaries and Structured Measurements},
  journal = {SIAM J. Math. Anal.},
  year = {2015},
  volume = {47},
  number = {6},
  pages = {4606--4629},
  url = {http://arxiv.org/abs/1501.03208},
  doi = {10.1137/151005245}
}
M. Sandbichler, F. Krahmer, T. Berer, P. Burgholzer and M. Haltmeier. A Novel Compressed Sensing Scheme for Photoacoustic Tomography, SIAM J. Appl. Math., 75(6):2475-2494, 2015.
BibTeX:
@article{SKBBH15,
  author = {Sandbichler, M. and Krahmer, F. and Berer, T. and Burgholzer, P. and Haltmeier, M.},
  title = {A Novel Compressed Sensing Scheme for Photoacoustic Tomography},
  journal = {SIAM J. Appl. Math.},
  year = {2015},
  volume = {75},
  number = {6},
  pages = {2475--2494},
  url = {http://arxiv.org/abs/1501.04305},
  doi = {10.1137/141001408}
}
D. Gross, F. Krahmer and R. Kueng. A Partial Derandomization of PhaseLift using Spherical Designs, J Fourier Anal. Appl., 21(2):229-266, 2015.
BibTeX:
@article{GKK13,
  author = {Gross, D. and Krahmer, F. and Kueng, R.},
  title = {A Partial Derandomization of !PhaseLift using Spherical Designs},
  journal = {J Fourier Anal. Appl.},
  year = {2014},
  volume = {21},
  number = {2},
  pages = {229--266},
  url = {http://arxiv.org/abs/1310.2267},
  doi = {10.1007/s00041-014-9361-2}
}
J. Feng and F. Krahmer. An RIP approach to Sigma-Delta quantization for compressed sensing, IEEE Signal Process. Lett., 21(11):1351-1355, 2014.
BibTeX:
@article{FK14,
  author = {Feng, J. and Krahmer, F},
  title = {An RIP approach to Sigma-Delta quantization for compressed sensing},
  journal = {IEEE Signal Process. Lett.},
  year = {2014},
  volume = {21},
  number = {11},
  pages = {1351--1355},
  doi = {10.1109/LSP.2014.2336700}
}
F. Krahmer, G. Kutyniok and J. Lemvig. Sparse Matrices in Frame Theory, Computational Statistics, 29:547-568, 2014.
BibTeX:
@article{KKL12b,
  author = {Krahmer, F. and Kutyniok, G. and Lemvig, J.},
  title = {Sparse Matrices in Frame Theory},
  journal = {Computational Statistics},
  year = {2014},
  volume = {29},
  pages = {547--568},
  doi = {10.1007/s00180-013-0446-1}
}
F. Krahmer, S. Mendelson and H. Rauhut. Suprema of Chaos Processes and the Restricted Isometry Property, Comm. Pure Appl. Math., 67(11):1877-1904, 2014.
BibTeX:
@article{KMR12,
  author = {Krahmer, F. and Mendelson, S. and Rauhut, H.},
  title = {Suprema of Chaos Processes and the Restricted Isometry Property},
  journal = {Comm. Pure Appl. Math.},
  year = {2014},
  volume = {67},
  number = {11},
  pages = {1877-1904},
  url = {http://arxiv.org/abs/1207.0235},
  doi = {10.1002/cpa.21504}
}
F.. Krahmer and G.. Pfander. Local sampling and approximation of operators with bandlimited Kohn-Nirenberg symbols, Constr. Approx., 39(3):541-572, 2014.
BibTeX:
@article{KP13,
  author = {Krahmer, F. and Pfander, G.},
  title = {Local sampling and approximation of operators with bandlimited Kohn-Nirenberg symbols},
  journal = {Constr. Approx.},
  year = {2014},
  volume = {39},
  number = {3},
  pages = {541-572},
  doi = {10.1007/s00365-014-9228-4}
}
F. Krahmer and H. Rauhut. Structured random measurements in signal processing, GAMM-Mitteilungen, 37(2):217-238, 2014.
BibTeX:
@inproceedings{KR14,
  author = {Krahmer, F. and Rauhut, H.},
  title = {Structured random measurements in signal processing},
  journal = {GAMM-Mitteilungen},
  year = {2014},
  volume = {37},
  number = {2},
  pages = {217--238},
  doi = {10.1002/gamm.201410010}
}
F. Krahmer, R. Saab and Ö. Yilmaz. Sigma-Delta quantization of sub-Gaussian frame expansions and its application to compressed sensing, Inform. Inference, 3(1):40-58, 2014.
BibTeX:
@article{KSY13,
  author = {Krahmer, F. and Saab, R. and Yilmaz, Ö.},
  title = {Sigma-Delta quantization of sub-Gaussian frame expansions and its application to compressed sensing},
  journal = {Inform. Inference},
  year = {2014},
  volume = {3},
  number = {1},
  pages = {40-58},
  doi = {10.1093/imaiai/iat007}
}
F. Krahmer and R. Ward. Stable and robust sampling strategies for compressive imaging, IEEE Trans. Image Proc., 23(2):612-622, 2014.
BibTeX:
@article{KW13,
  author = {F. Krahmer and R. Ward},
  title = {Stable and robust sampling strategies for compressive imaging},
  journal = {IEEE Trans. Image Proc.},
  year = {2014},
  volume = {23},
  number = {2},
  pages = {612--622},
  doi = {10.1109/TIP.2013.2288004}
}
F. Krahmer, G. Kutyniok and J. Lemvig. Sparsity and spectral properties of dual frames, Linear Algebra and its Applications, 439(4):982 - 998, 2013.
BibTeX:
@article{KKL13,
  author = {Krahmer, F. and Kutyniok, G. and Lemvig, J.},
  title = {Sparsity and spectral properties of dual frames},
  journal = {Linear Algebra and its Applications},
  year = {2013},
  volume = {439},
  number = {4},
  pages = {982 - 998},
  doi = {10.1016/j.laa.2012.10.016}
}
M. Burr and F. Krahmer. SqFreeEVAL: An (almost) optimal real-root isolation algorithm, Journal of Symbolic Computation, 47(2):153-166, 2012.
BibTeX:
@article{BK11,
  author = {Burr, M. and Krahmer, F.},
  title = {SqFreeEVAL: An (almost) optimal real-root isolation algorithm},
  journal = {Journal of Symbolic Computation},
  year = {2012},
  volume = {47},
  number = {2},
  pages = {153--166},
  doi = {10.1016/j.jsc.2011.08.022}
}
F. Krahmer, R. Saab and R. Ward. Root-exponential accuracy for coarse quantization of finite frame expansions, IEEE J. Inf. Theo., 58(2):1069-1079, 2012.
BibTeX:
@article{KSW12,
  author = {Krahmer, F. and Saab, R. and Ward, R.},
  title = {Root-exponential accuracy for coarse quantization of finite frame expansions},
  journal = {IEEE J. Inf. Theo.},
  year = {2012},
  volume = {58},
  number = {2},
  pages = {1069--1079},
  doi = {10.1109/TIT.2011.2168942}
}
F. Krahmer and R. Ward. Lower bounds for the error decay incurred by coarse quantization schemes, Appl. Comput. Harmonic Anal., 32(1):131-138, 2012.
BibTeX:
@article{KW12,
  author = {Krahmer, F. and Ward, R.},
  title = {Lower bounds for the error decay incurred by coarse quantization schemes},
  journal = {Appl. Comput. Harmonic Anal.},
  year = {2012},
  volume = {32},
  number = {1},
  pages = {131--138},
  doi = {10.1016/j.acha.2011.06.003}
}
P. Casazza, A. Heinecke, F. Krahmer and G. Kutyniok. Optimally sparse frames, IEEE J. Inf. Theo., 57(11):7279-7287, 2011.
BibTeX:
@article{CHKK11,
  author = {Casazza, P. and Heinecke, A. and Krahmer, F. and Kutyniok, G.},
  title = {Optimally sparse frames},
  journal = {IEEE J. Inf. Theo.},
  year = {2011},
  volume = {57},
  number = {11},
  pages = {7279--7287},
  doi = {10.1109/TIT.2011.2160521}
}
P. Deift, C.S. Güntürk and F. Krahmer. An Optimal Family of Exponentially Accurate One-Bit Sigma-Delta Quantization Schemes, Comm. Pure Appl. Math., 64(7):883-919, 2011.
BibTeX:
@article{DGK11,
  author = {Percy Deift and C. Sinan Güntürk and Felix Krahmer},
  title = {An Optimal Family of Exponentially Accurate One-Bit Sigma-Delta Quantization Schemes},
  journal = {Comm. Pure Appl. Math.},
  year = {2011},
  volume = {64},
  number = {7},
  pages = {883--919},
  doi = {10.1002/cpa.20367}
}
F. Krahmer and R. Ward. New and improved Johnson-Lindenstrauss embeddings via the Restricted Isometry Property, SIAM J. Math. Anal., 43(3):1269-1281, SIAM, 2011.
BibTeX:
@article{KW11,
  author = {Krahmer, F. and Ward, R.},
  title = {New and improved Johnson-Lindenstrauss embeddings via the Restricted Isometry Property},
  journal = {SIAM J. Math. Anal.},
  publisher = {SIAM},
  year = {2011},
  volume = {43},
  number = {3},
  pages = {1269--1281},
  doi = {10.1137/100810447}
}
F. Krahmer, G.E. Pfander and P. Rashkov. Uncertainty in time-frequency representations on finite abelian groups and applications, Appl. Comput. Harmon. Anal., 25(2):209-225, 2008.
BibTeX:
@article{KraPfaRa08,
  author = {Krahmer, Felix and Pfander, Götz E. and Rashkov, Peter},
  title = {Uncertainty in time-frequency representations on finite abelian groups and applications},
  journal = {Appl. Comput. Harmon. Anal.},
  year = {2008},
  volume = {25},
  number = {2},
  pages = {209--225},
  doi = {10.1016/j.acha.2007.09.008}
}

PhD Thesis

F. Krahmer. Novel Schemes for Sigma-Delta Modulation: From Improved Exponential Accuracy to Low-Complexity Design. Thesis at: New York University, 2009.
BibTeX:
@phdthesis{KrThesis,
  author = {Felix Krahmer},
  title = {Novel Schemes for Sigma-Delta Modulation: From Improved Exponential Accuracy to Low-Complexity Design},
  school = {New York University},
  year = {2009},
  url = {http://num.math.uni-goettingen.de/ f.krahmer/Thesis_Felix_Krahmer.pdf}
}

Technical Reports

F. Krahmer, C. Kümmerle and H. Rauhut. A Quotient Property for Matrices with Heavy-Tailed Entries and its Application to Noise-Blind Compressed Sensing, 2018.
Abstract
For a large class of random matrices A with i.i.d. entries we show that the 1-quotient property holds with probability exponentially close to 1. In contrast to previous results, our analysis does not require concentration of the entrywise distributions. We provide a unified proof that recovers corresponding previous re- sults for (sub-)Gaussian and Weibull distributions. Our findings generalize known results on the geometry of random polytopes, providing lower bounds on the size of the largest Euclidean ball contained in the centrally symmetric polytope spanned by the columns of A.
At the same time, our results establish robustness of noise-blind l1-decoders for recovering sparse vectors x from underdetermined, noisy linear measurements y=Ax+w under the weakest possible assumptions on the entrywise distributions that allow for recovery with optimal sample complexity even in the noiseless case. Our analysis predicts superior robustness behavior for measurement matrices with super-Gaussian entries, which we confirm by numerical experiments.
BibTeX:
@unpublished{KrahmerKuemmerleRauhut18,
  author = {Felix Krahmer, Christian Kümmerle, Holger Rauhut},
  title = {A Quotient Property for Matrices with Heavy-Tailed Entries and its Application to Noise-Blind Compressed Sensing},
  year = {2018},
  url = {https://arxiv.org/abs/1806.04261}
}
M. Burr, F. Krahmer and C. Yap. Continuous Amortization: A non-probabilistic adaptive analysis technique. Thesis at: Electronic Colloquium on Computational Complexity, 2009.
BibTeX:
@techreport{BuKrYa09,
  author = {Burr, Michael and Krahmer, Felix and Yap, Chee},
  title = {Continuous Amortization: A non-probabilistic adaptive analysis technique},
  school = {Electronic Colloquium on Computational Complexity},
  year = {2009},
  note = {Technical report}
}
F. Krahmer, G. Pfander and P. Rashkov. Support size conditions for time-frequency representations on finite Abelian groups. Thesis at: Jacobs University, Bremen, 2007.
BibTeX:
@techreport{KraPfaRa07,
  author = {Krahmer, F. and Pfander, G. and Rashkov, P.},
  title = {Support size conditions for time-frequency representations on finite Abelian groups},
  school = {Jacobs University, Bremen},
  year = {2007},
  note = {Technical report}
}
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