Anders Christian Hansen

Professor II - Flere komplekse variable, logikk og operatoralgebraer

$Bilde av Anders Christian Hansen$

English version of this page

E-post anderch@math.uio.no

Mobiltelefon +44 1223 760403

Rom 702

Brukernavn

Besøksadresse Moltke Moes vei 35 Niels Henrik Abels hus 0851 Oslo

Postadresse Postboks 1053 Blindern 0316 Oslo

Last ned visittkort

Research interests:

Functional Analysis (applied), Artificial Intelligence, Foundations of Computational Mathematics, Solvability Complexity Index hierarchy, Generalised Hardness of Approximation, Optimisation, Inverse Problems, Medical Imaging, Operator/Spectral Theory, Numerical Analysis, Computational Harmonic Analysis, PDEs, Compressed Sensing, Mathematical Signal Processing, Sampling Theory, Geometric Integration, Operator Algebras

Emneord: Operatoralgebraer, Matematikk, AI

Colbrook, Matthew J.; Antun, Vegard & Hansen, Anders Christian (2022). The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and Smale's 18th problem. Proceedings of the National Academy of Sciences of the United States of America. ISSN 0027-8424. 119(12). doi: 10.1073/pnas.2107151119. Fulltekst i vitenarkiv
Adcock, Ben; Antun, Vegard & Hansen, Anders Christian (2021). Uniform recovery in infinite-dimensional compressed sensing and applications to structured binary sampling. Applied and Computational Harmonic Analysis. ISSN 1063-5203. 55, s. 1–40. doi: 10.1016/j.acha.2021.04.001. Fulltekst i vitenarkiv
Schoormans, Jasper; Strijkers, Gustav J.; Hansen, Anders Christian; Nederveen, Aart & Coolen, Bram (2020). Compressed sensing MRI with variable density averaging (CS-VDA) outperforms full sampling at low SNR. Physics in Medicine and Biology. ISSN 0031-9155. doi: 10.1088/1361-6560/ab63b7.
Antun, Vegard; Renna, Francesco; Poon, Clarice; Adcock, Ben & Hansen, Anders Christian (2020). On instabilities of deep learning in image reconstruction and the potential costs of AI. Proceedings of the National Academy of Sciences of the United States of America. ISSN 0027-8424. 117(48), s. 30088–30095. doi: 10.1073/pnas.1907377117. Fulltekst i vitenarkiv Vis sammendrag
Deep learning, due to its unprecedented success in tasks such as image classification, has emerged as a new tool in image reconstruction with potential to change the field. In this paper we demonstrate a crucial phenomenon: deep learning typically yields unstable methods for image reconstruction. The instabilities usually occur in several forms: (1) certain tiny, almost undetectable perturbations, both in the image and sampling domain, may result in severe artefacts in the reconstruction, (2) a small structural change, for example a tumour, may not be captured in the reconstructed image and (3) (a counterintuitive type of instability) more samples may yield poorer performance. Our new stability test with algorithms and easy to use software detects the instability phenomena. The test is aimed at researchers to test their networks for instabilities and for government agencies, such as the Food and Drug Administration (FDA), to secure safe use of deep learning methods.
Colbrook, Matthew & Hansen, Anders Christian (2019). On the infinite-dimensional QR algorithm. Numerische Mathematik. ISSN 0029-599X. 143, s. 17–83. doi: 10.1007/s00211-019-01047-5.
Colbrook, Matthew; Bogdan, Roman & Hansen, Anders Christian (2019). How to compute spectra with error control. Physical Review Letters. ISSN 0031-9007. 122(25). doi: 10.1103/PhysRevLett.122.250201.
Thesing, Laura & Hansen, Anders Christian (2018). Linear reconstructions and the analysis of the stable sampling rate. Sampling Theory in Signal and Image Processing. ISSN 1530-6429. 17, s. 103–126. doi: 10.1007/BF03549616.
Hansen, Anders Christian & Thesing, L (2018). On the stable sampling rate for binary measurements and wavelet reconstruction. Applied and Computational Harmonic Analysis. ISSN 1063-5203. s. 1–25. doi: 10.1016/j.acha.2018.08.004.
Hansen, Anders Christian & Nevanlinna, Olavi (2017). Complexity issues in computing spectra, pseudospectra and resolvents. Banach Center Publications. ISSN 0137-6934.
Bastounis, Alexander; Adcock, Ben & Hansen, Anders Christian (2017). From Global to Local: Getting More from Compressed Sensing. SIAM News. ISSN 0036-1437.
Bastounis, Alexander & Hansen, Anders Christian (2017). On the absence of uniform recovery in many real-world applications of compressed sensing and the restricted isometry property and nullspace property in levels. SIAM Journal of Imaging Sciences. ISSN 1936-4954. 10(1), s. 335–371. doi: 10.1137/15M1043972.
Adcock, Ben; Hansen, Anders Christian; Poon, Clarice & Roman, Bogdan (2017). Breaking the coherence barrier: A new theory for compressed sensing. Forum of Mathematics, Sigma. ISSN 2050-5094. 5, s. 1–84. doi: 10.1017/fms.2016.32.
Jones, Alexander Daniel; Tamtogl, Anton; Calvo-Almaz, Irene & Hansen, Anders Christian (2016). Continuous compressed sensing of inelastic and quasielastic Helium Atom Scattering spectra. Scientific Reports. ISSN 2045-2322. s. 1–11. doi: 10.1038/srep27776.
Adcock, Ben; Gataric, Milana & Hansen, Anders Christian (2016). Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples. Applied and Computational Harmonic Analysis. ISSN 1063-5203. 42(3), s. 508–535. doi: 10.1016/j.acha.2015.09.006.
Adcock, Ben; Hansen, Anders Christian & Roman, Bogdan (2016). A note on compressed sensing of structured sparse wavelet coefficients from subsampled Fourier measurements. IEEE Signal Processing Letters. ISSN 1070-9908. 23(5), s. 732–736. doi: 10.1109/LSP.2016.2550101.
Adcock, Ben; Gataric, Milana & Hansen, Anders Christian (2016). Density Theorems for Nonuniform Sampling of Bandlimited Functions Using Derivatives or Bunched Measurements. Journal of Fourier Analysis and Applications. ISSN 1069-5869. s. 1–37. doi: 10.1007/s00041-016-9504-8.
Jones, Alexander Daniel; Adcock, Ben & Hansen, Anders Christian (2016). On asymptotic incoherence and its implications for compressed sensing of inverse problems. IEEE Transactions on Information Theory. ISSN 0018-9448. 62(2), s. 1020–1037. doi: 10.1109/TIT.2015.2508562.
Adcock, Ben & Hansen, Anders Christian (2015). Generalized sampling and the stable and accurate reconstruction of piecewise analytic functions from their Fourier coefficients. Mathematics of Computation. ISSN 0025-5718. 84(291), s. 237–270. doi: 10.1090/s0025-5718-2014-02860-3.
Adcock, Ben; Gataric, Milana & Hansen, Anders Christian (2015). Recovering piecewise smooth functions from nonuniform Fourier measurements. Lecture Notes in Computational Science and Engineering. ISSN 1439-7358. 106, s. 117–125. doi: 10.1007/978-3-319-19800-2_8.
Adcock, Ben; Hansen, Anders Christian; Kutyniok, Gitta & Ma, Jackie (2015). Linear stable sampling rate: Optimality of 2D wavelet reconstructions from Fourier measurements. SIAM Journal on Mathematical Analysis. ISSN 0036-1410. 47(2), s. 1196–1233. doi: 10.1137/140959365.
Adcock, Ben & Hansen, Anders Christian (2015). Generalized Sampling and Infinite Dimensional Compressed Sensing. Foundations of Computational Mathematics. ISSN 1615-3375. doi: 10.1007/s10208-015-9276-6.
Ben-Artzi, Jonathan; Hansen, Anders Christian; Nevanlinna, Olavi & Seidel, Markus (2015). New barriers in complexity theory: On the solvability complexity index and the towers of algorithms. Comptes rendus. Mathematique. ISSN 1631-073X. 353(10), s. 931–936. doi: 10.1016/j.crma.2015.08.002.

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Adcock, Ben & Hansen, Anders Christian (2021). Compressive Imaging: Structure, Sampling, Learning. Cambridge University Press. ISBN 978-1-108-42161-4. 613 s.
Hansen, Anders Christian & Roman, Bogdan (2020). Structure and Optimisation in Computational Harmonic Analysis: On Key Aspects in Sparse Regularisation. Springer Nature. ISBN 978-3-030-61886-5. 50 s.
Hansen, Anders Christian; Calderbank, Robert; Roman, Bogdan & Thesing, Laura (2019). On Reconstructing Functions from Binary Measurements. Birkhäuser Verlag. ISBN 978-3-319-73073-8. 33 s.
Bastounis, Alexander & Hansen, Anders Christian (2015). On random and deterministic compressed sensing and the Restricted Isometry Property in levels. IEEE conference proceedings. ISBN 978-1-4673-7353-1. 587 s.
Adcock, Ben; Hansen, Anders Christian & Gataric, Milana (2015). Stable nonuniform sampling with weighted Fourier frames and recovery in arbitrary spaces. IEEE conference proceedings. ISBN 978-1-4673-7353-1. 587 s.
Adcock, Ben; Hansen, Anders Christian & Roman, Bogdan (2015). The Quest for Optimal Sampling: Computationally Efficient, Structure-Exploiting Measurements for Compressed Sensing. Birkhäuser Verlag. ISBN 978-3-319-16042-9. 468 s.

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Antun, Vegard; Colbrook, Matthew & Hansen, Anders Christian (2022). Mathematical paradoxes unearth the boundaries of AI. TheScienceBreaker. ISSN 2571-9262. doi: 10.25250/thescbr.brk653.
Antun, Vegard; Colbrook, Matthew J. & Hansen, Anders Christian (2022). Proving existence is not enough: Mathematical paradoxes unravel the limits of neural networks in AI. SIAM News. ISSN 0036-1437. 55(4), s. 1–4.
Antun, Vegard; Gottschling, Nina; Hansen, Anders Christian & Adcock, Ben (2021). Deep learning in scientific computing: Understanding the instability mystery. SIAM News. ISSN 0036-1437. 54(2), s. 3–5.
Hansen, Anders Christian & Thesing, Laura (2017). Sampling from binary measurements - on reconstructions from Walsh coefficients.

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Publisert 26. mars 2015 13:38 - Sist endret 29. sep. 2023 12:05

Anders Christian Hansen

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