Anders Christian Hansen

• Ferreira, Nádia Gonçalves; Thomas, Owen Matthew Truscott; Gjesdal, Kjell Inge; Müller, Stig; Oldenburg, Jan & Syversen, Ingrid Framås [Vis alle 8 forfattere av denne artikkelen] (2026). Artificial intelligence in prostate MRI: Comparative diagnostic performance in a high-prevalence cohort. Acta Radiologica. ISSN 0284-1851. doi: 10.1177/02841851261470455. Fulltekst i vitenarkiv
• Gazdag, Luca Eva; Bastounis, Alexander & Hansen, Anders Christian (2026). Generalised Hardness of Approximation and the SCI Hierarchy –On Determining the Boundaries of Training Algorithms in AI. Foundations of Computational Mathematics. ISSN 1615-3375. doi: 10.1007/s10208-026-09764-8. Fulltekst i vitenarkiv
• Gazdag, Luca Eva; Antun, Vegard & Hansen, Anders Christian (2025). Instability in deep learning – when algorithms cannot compute uncertainty quantifications for neural networks. European journal of applied mathematics. ISSN 0956-7925. 37(1), s. 290–312. doi: 10.1017/S095679252510017X. Fulltekst i vitenarkiv
• 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 (PNAS). ISSN 0027-8424. 119(12). doi: 10.1073/pnas.2107151119. Fulltekst i vitenarkiv Vis sammendrag

  Significance Instability is the Achilles’ heel of modern artificial intelligence (AI) and a paradox, with training algorithms finding unstable neural networks (NNs) despite the existence of stable ones. This foundational issue relates to Smale’s 18th mathematical problem for the 21st century on the limits of AI. By expanding methodologies initiated by Gödel and Turing, we demonstrate limitations on the existence of (even randomized) algorithms for computing NNs. Despite numerous existence results of NNs with great approximation properties, only in specific cases do there also exist algorithms that can compute them. We initiate a classification theory on which NNs can be trained and introduce NNs that—under suitable conditions—are robust to perturbations and exponentially accurate in the number of hidden layers.

• 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 Vis sammendrag

  Infinite-dimensional compressed sensing deals with the recovery of analog signals (functions) from linear measurements, often in the form of integral transforms such as the Fourier transform. This framework is well-suited to many real-world inverse problems, which are typically modeled in infinite-dimensional spaces, and where the application of finite-dimensional approaches can lead to noticeable artefacts. Another typical feature of such problems is that the signals are not only sparse in some dictionary, but possess a so-called local sparsity in levels structure. Consequently, the sampling scheme should be designed so as to exploit this additional structure. In this paper, we introduce a series of uniform recovery guarantees for infinite-dimensional compressed sensing based on sparsity in levels and so-called multilevel random subsampling. By using a weighted -regularizer we derive measurement conditions that are sharp up to log factors, in the sense that they agree with the best known measurement conditions for oracle estimators in which the support is known a priori. These guarantees also apply in finite dimensions, and improve existing results for unweighted -regularization. To illustrate our results, we consider the problem of binary sampling with the Walsh transform using orthogonal wavelets. Binary sampling is an important mechanism for certain imaging modalities. Through carefully estimating the local coherence between the Walsh and wavelet bases, we derive the first known recovery guarantees for this problem.

• 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 (PNAS). 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.

• 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. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• Bastounis, Alexander; Adcock, Ben & Hansen, Anders Christian (2017). From Global to Local: Getting More from Compressed Sensing. SIAM News. ISSN 0036-1437. Fulltekst i vitenarkiv
• Adcock, Ben; Hansen, Anders Christian; Poon, Clarice & Roman, Bogdan (2017). Breaking the coherence barrier: A new theory for compressed sensing. Forum of Mathematics, Sigma. 5, s. 1–84. doi: 10.1017/fms.2016.32. Fulltekst i vitenarkiv
• 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. 10(1), s. 335–371. doi: 10.1137/15M1043972. Fulltekst i vitenarkiv
• Hansen, Anders Christian & Nevanlinna, Olavi (2017). Complexity issues in computing spectra, pseudospectra and resolvents. Banach Center Publications. ISSN 0137-6934. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• 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. s. 1–11. doi: 10.1038/srep27776. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv

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• Adcock, Ben & Hansen, Anders Christian (2021). Compressive Imaging: Structure, Sampling, Learning. Cambridge University Press. ISBN 9781108421614. 613 s. Fulltekst i vitenarkiv
• Hansen, Anders Christian & Roman, Bogdan (2020). Structure and Optimisation in Computational Harmonic Analysis: On Key Aspects in Sparse Regularisation. Springer Nature. ISBN 9783030618865. 50 s. Fulltekst i vitenarkiv
• Hansen, Anders Christian; Calderbank, Robert; Roman, Bogdan & Thesing, Laura (2019). On Reconstructing Functions from Binary Measurements. Birkhäuser Verlag. ISBN 9783319730738. 33 s. Fulltekst i vitenarkiv
• Bastounis, Alexander & Hansen, Anders Christian (2015). On random and deterministic compressed sensing and the Restricted Isometry Property in levels. IEEE (Institute of Electrical and Electronics Engineers). ISBN 9781467373531. 587 s. Fulltekst i vitenarkiv
• Adcock, Ben; Hansen, Anders Christian & Roman, Bogdan (2015). The Quest for Optimal Sampling: Computationally Efficient, Structure-Exploiting Measurements for Compressed Sensing. Birkhäuser Verlag. ISBN 9783319160429. 468 s. Fulltekst i vitenarkiv
• Adcock, Ben; Hansen, Anders Christian & Gataric, Milana (2015). Stable nonuniform sampling with weighted Fourier frames and recovery in arbitrary spaces. IEEE (Institute of Electrical and Electronics Engineers). ISBN 9781467373531. 587 s. Fulltekst i vitenarkiv

Se alle arbeider i NVA

• Antun, Vegard; Colbrook, Matthew & Hansen, Anders Christian (2022). Mathematical paradoxes unearth the boundaries of AI. TheScienceBreaker. ISSN 2571-9262. doi: 10.25250/thescbr.brk653. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• 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. Fulltekst i vitenarkiv
• Hansen, Anders Christian & Thesing, Laura (2017). Sampling from binary measurements - on reconstructions from Walsh coefficients. Fulltekst i vitenarkiv

Se alle arbeider i NVA