# Professor Anders C. Hansen

Anders leads the Applied Functional and Harmonic Analysis group within the Faculty of Mathematics at the University of Cambridge and Department of Applied Mathematics and Theoretical Physics (DAMTP). He is a Professor of mathematics at the University of Cambridge, Professor of Mathematics at the University of Oslo, a Royal Society University Research Fellow and also a Fellow of Peterhouse.

*Email:* ach70@cam.ac.uk

*Tel:* +44 1223 760403

*Office:* F2.01

## Upcoming and Recent Events

- Organizing the workshop
*Computational mathematics in computer assisted proofs (Sept 12-16, 2022)*together with*Charles Fefferman*and*Svetlana Jitomirskaya.* - Speaking (online) at
*IST (Jan. 20, 2022).* - Organizing the workshop
*Interpretability, safety, and security in AI (Dec 13-15, 2021)*together with*Rich Baraniuk*,*Miguel Rodrigues*and*Adrian Weller.* - Speaking (online) at the
*University of Leicester (Oct. 14, 2021).* - Speaking at
*EPFL (Sept. 21, 2021).* - Speaking (online) at the
*University of Chicago Mathematics Colloquium (April 7, 2021).* - Speaking (online) at the
*Cambridge Science Festival (March 29, 2021).* - Speaking (online) at the
*One World IMAGing and INvErse problems (IMAGINE) seminar (Feb 17, 2021).* - Speaking (online) at the
*Gran Sasso Science Institute (Jan 28, 2021).* - Speaking (online) at
*XAI: Explaining what goes on inside DNN/AI (Oct 20, 2020).* - Speaking (online) at the
*Max Planck Institute of Molecular Cell Biology and Genetics (Sept 24, 2020).* - Speaking (online) at the
*Mathematics of Machine Learning, LMS-Bath Symposium (Aug 6, 2020).**Watch the talk.* - Speaking (online) at the
*One World Seminar Series on the Mathematics of Machine Learning (July 5, 2020).**Watch the talk.* - Speaking at the
*University of Minnesota, Applied and Computational Math Colloquium (Feb. 3 2020)* - Invited speaker at
*Computational Harmonic Analysis and Data Science, Banff International Research Station (Nov 2019)**.* - Speaking at
*EPFL, Imaging in the Age of Machine Learning*(Oct 25, 2019) - Speaking at the
*University of Pittsburgh, Algebra-combinatorics-geometry seminar*(Sept 26, 2019) - Invited speaker at
*Workshop on Harmonic analysis and Machine Learning (Sept 2019)**.* - Invited speaker at
*Algorithms and Complexity for Continuous Problems, Dagstuhl (Aug 2019)**.* - Plenary speaker at
*National Academy of Sciences, Arthur M. Sackler Colloquim: The Science of Deep Learning, Washington D.C. (March 2019)**.* - Speaking at
*Imperial College/University College London, Numerical Analysis Seminar (Feb. 20 2019)* - Invited speaker at
*Variational methods and optimization in imaging, Institut Henri*(Feb. 2019)*Poincaré**.* - Speaking at Imperial College, Pure Analysis Seminar (Jan. 10 2019).

## Prizes and Awards

2. 2018 IMA Prize in Mathematics and Applications.

3. Leverhulme Prize in Mathematics and Statistics 2017.

4. Royal Society University Research Fellow 2012.

## News

1. SIAM News reports (front page) on our work from the paper "The mathematics of adversarial attacks in AI -- Why deep learning is unstable despite the existence of stable neural networks" in the recent October edition: *Deep Learning: What Could Go Wrong?.*

2. Our book *Compressive Imaging: Structure, Sampling, Learning* (with B. Adcock) is out on Cambridge University Press.

3. SIAM News reports on our work on deep learning in scientific computing in the recent March edition: *Deep Learning in Scientific Computing: Understanding the Instability Mystery.*

4. Our paper *On instabilities of deep learning in image reconstruction and the potential costs of AI** *is now published in Proc. Natl. Acad. Sci. Here is some of the press coverage:
*Cambridge University News, *
* Physics World, *
* EurekAlert, *
* The Register, *
* Health Care Business, *
* Radiology Business, *
* Science Daily, *
* Psychology Today, *
* Government Computing, *
* Diagnostic Imaging, *
* News Medical, *
* Press Release Point, *
* Tech Xplore, *
* Aunt Minnie, *
* My Science, *
* Digit, *
* The Talking Machines, *
* MC.AI, *
* Rama on Healthcare, *
* News8PLus, *
* Genethique, *
* Healthcare in Europe, *
* AuntminnieEurope, *
* Newsbreak, *
* AI Development Hub, *
* FirstWord MedTech, *
* AI Daily. *

5. Our paper *How to compute spectra with error control** *is on the cover of the last June edition of Physical Review Letters.

6. The Sackler Colloquium at the US National Academy of Sciences: "The Science of deep learning". Watch the presentation "On instabilities in deep learning - Does AI come at a cost?"

7. SIAM News has our work on the Restricted Isometry Property in Levels in compressed sensing on the front page of the October edition: *From Global to Local: Getting More from Compressed Sensing.*

8. Our new Nature, Sci. Rep. paper *Continuous compressed sensing of inelastic and quasielastic Helium Atom Scattering spectra** *on continuous/infinite-dimensional compressed sensing in surface scattering is now published. This is the first time compressed sensing is used in Helium Atom Scattering and the first time infinite-dimensional compressed sensing is implemented with real data (see the papers *Generalized Sampling and Infinite Dimensional Compressed Sensing* and *Breaking the coherence barrier: A new theory for compressed sensing* where the theoretical ideas were launched).

9. Siemens validated in practice, using a modified MRI machine, the asymptotic sparsity, asymptotic incoherence and high resolution concepts introduced by our work (see Breaking the coherence barrier: A new theory for compressed sensing and also On asymptotic structure in compressed sensing). From their conclusion:

“[...] The image resolution has been greatly improved [...]. Current results practically demonstrated that it is possible to break the coherence barrier by increasing the spatial resolution in MR acquisitions. This likewise implies that the full potential of the compressed sensing is unleashed only if asymptotic sparsity and asymptotic incoherence is achieved.”

Their work Novel Sampling Strategies for Sparse MR Image Reconstruction was published in May 2014 in the Proceedings of the International Society for Magnetic Resonance in Medicine.

## Students and Post-Docs

**Phd Students:** 1. Clarice Poon (graduated 2015), 2. Milana Gataric (graduated 2016), 3. Alexander Jones (graduated 2016), 4. Alexander Bastounis (graduated 2018), 5. Vegard Antun (graduated 2020), 6. Matt Colbrook (graduated 2020), 7. Laura Thesing (graduating 2021), 8. Simon Becker (graduating 2021), 9. Nina Gottschling (graduating 2022), 10. Paolo Campodonico (graduating 2023), 11. David Liu (graduating 2023), 12. Luca Gazdag (graduating 2024), 13. Parley Yang (graduating 2024).

**Post-docs:** 1. Jonathan Ben-Artzi ( 2011-2014, PhD: Brown University), 2. Bogdan Roman ( 2013-2016, 2016-2019, PhD: University of Cambridge), 3. Priscilla Canizares (2015-2016, PhD: Autonomous University of Barcelona), 4. Milana Gataric (2015-2016, PhD: University of Cambridge), 5. Francesco Renna (2016-2018, PhD: University of Padova), 6. Alexander Bastounis (2019-2021, PhD: University of Cambridge), 7. Vegard Antun (2020-, PhD: University of Oslo).

## Teaching

Part III course on Compressed Sensing.

## Research Interests

Functional Analysis (applied), Foundations of Computations, Artificial Intelligence, Compressed Sensing, Optimisation, Operator/Spectral Theory, Numerical Analysis, Computational Harmonic Analysis, Mathematical Signal Processing, Sampling Theory, Inverse Problems, Medical Imaging, Geometric Integration, Operator Algebras

## Editor

Proceedings of the Royal Society Series A (2014-2020)

Networks & Heterogeneous Media (2021- )

SIAM Journal on Imaging Sciences (2022- )

## Books

- B. Adcock, A. C. Hansen,
*Compressive Imaging: Structure, Sampling, Learning*,

Cambridge University Press (2021)

## Selected Papers

- A. C. Hansen,
*On the Solvability Complexity Index, the n-Pseudospectrum and Approximations of Spectra of Operators*,

J. Amer. Math. Soc. - J. Ben-Artzi, M. Colbrook, A. C. Hansen, O. Nevanlinna, M. Seidel,
*Computing spectra - On the Solvability Complexity Index hierarchy and towers of algorithms**.* - V. Antun, F. Renna, C. Poon, B. Adcock, A. C. Hansen,
*On instabilities of deep learning in image reconstruction and the potential costs of AI*,2020, no. 5, 201907377

Proc. Natl. Acad. Sci. *M. Colbrook,*B. Roman, A. C. Hansen,*How to compute spectra with error control*,122, 250201 (front cover)

Phys. Rev. Lett.- B. Adcock, A. C. Hansen, C. Poon, B. Roman,
*Breaking the coherence barrier: A new theory for compressed sensing,*

*Forum of Mathematics, Sigma*5(4):1-84 - B. Adcock, A. C. Hansen,
*Generalized Sampling and Infinite Dimensional Compressed Sensing*,

*Found. Comp. Math.*16, no. 5, 1263-1323 - A. Bastounis, B. Adcock, A. C. Hansen,
*From Global to Local: Getting More from Compressed Sensing*,

SIAM News, 50, no. 8 October 2017 (front cover) - V. Antun, N. Gottschling, A. C. Hansen, B. Adcock,
*Deep Learning in Scientific Computing: Understanding the Instability Mystery*.

SIAM News, 54, no. 2 March 2021

## Papers in Chronological Order

- A. Bastounis, A. C. Hansen, V. Vlacic,
*The mathematics of adversarial attacks in AI -- Why deep learning is unstable despite the existence of stable neural networks.* - A. Bastounis, A. C. Hansen, V. Vlacic,
*The extended Smale's 9th problem -- On computational barriers and paradoxes in estimation, regularisation, computer-assisted proofs and learning.* - S. Becker, A. C. Hansen,
*Computing solutions of Schrodinger equations on unbounded domains - On the brink of numerical algorithms.* - N. Gottschling, V. Antun, B. Adcock, A. C. Hansen,
*The troublesome kernel: why deep learning for inverse problems is typically unstable.* - J. Ben-Artzi, M. Colbrook, A. C. Hansen, O. Nevanlinna, M. Seidel,
*Computing spectra - On the Solvability Complexity Index hierarchy and towers of algorithms**.* - A. Bastounis, A. C. Hansen, D. Higham, I. Tyukin, V. Vlacic,
*Deep Learning: What Could Go Wrong?*
SIAM News, 54, no. 8 October 2021 - V. Antun, M, Colbrook, A. C. Hansen,
*Can stable and accurate neural networks be computed? - On the barriers of deep learning and Smale's 18th problem.*
Proc. Natl. Acad. Sci. (to appear)
- T. Loss, M, Colbrook, A. C. Hansen,
*Stratified Sampling Based Compressed Sensing for Structured Signals.*
IEEE Trans. Signal Process.
- M. Colbrook, A. C. Hansen,
*The foundations of spectral computations via the Solvability Complexity Index hierarchy: Part I.*
J. Eur. Math. Soc.
- L. Thesing, V. Antun, A. C. Hansen,
*What do AI algorithms actually learn - On false structures in deep learning.* - V. Antun, N. Gottschling, A. C. Hansen, B. Adcock,
*Deep Learning in Scientific Computing: Understanding the Instability Mystery*.

SIAM News, 54, no. 2 March 2021 - L. Thesing, A. C. Hansen,
*Non-uniform recovery guarantees for binary measurements and infinite-dimensional compressed sensing.*
J. Fourier Anal. Appl. (to appear)
- B. Adcock, V. Antun, A. C. Hansen, Uniform recovery in infinite-dimensional compressed sensing and applications to structured binary sampling Appl. Comput. Harmon. Anal. (to appear)
- V. Antun, F. Renna, C. Poon, B. Adcock, A. C. Hansen,
*On instabilities of deep learning in image reconstruction and the potential costs of AI*,2020, no. 5, 201907377

Proc. Natl. Acad. Sci. - J. Schoormans, G. J. Strijkers, A. C. Hansen, A. J. Nederveen, B. F. Coolen,
*Compressed Sensing MRI with Variable Density Averaging (CS-VDA) Outperforms Full Sampling at Low SNR.*
Phys. Med. Biol. *M. Colbrook,*B. Roman, A. C. Hansen,*How to compute spectra with error control*,122, 250201

Phys. Rev. Lett.- A. C. Hansen, B. Roman,
*On structure and optimisation in computational harmonic analysis - The key aspects in sparse regularisation*,

Springer*(to appear)* *M. Colbrook, A. C. Hansen, On the Infinite-dimensional QR Algorithm,*

(to appear)*Numerische Mathematik*- R. Calderbank, A. C. Hansen, L. Thesing, B. Roman
*On reconstructions from measurements with binary functions,*

*Springer*(to appear) - L. Thesing, A. C. Hansen,
*Linear reconstructions and the analysis of the stable sampling rate,*

*Sampl. Theory Signal Image Process.*(to appear) - A. C. Hansen, L. Thesing,
*On the Stable Sampling rate for binary measurements and wavelet reconstruction,*

Appl. Comput. Harmon. Anal.*(to appear)* - A. Bastounis, B. Adcock, A. C. Hansen,
*From Global to Local: Getting More from Compressed Sensing*,

SIAM News, 50, no. 8 October 2017 - A. C. Hansen, L. Thesing,
*Sampling from binary measurements - On Reconstructions from Walsh coefficients,*

*IEEE 2017 Int. Conf. on Samp. Theory and Appl.*256-260 (2017) - A. Bastounis, A. C. Hansen,
*On the absence of uniform recovery in many real-world applications of compressed sensing and the RIP & nullspace property in levels.*10(1):335-371

SIAM Jour. Imag. Scienc. - B. Adcock, A. C. Hansen, C. Poon, B. Roman,
*Breaking the coherence barrier: A new theory for compressed sensing,*

*Forum of Mathematics, Sigma*5(4):1-84 - A. C. Hansen, O. Nevanlinna,
*Complexity Issues in Computing Spectra, Pseudospectra and Resolvents*,

*Banach Centre Pub.*112:171-194 - B. Adcock, M. Gataric, A. C. Hansen,
*Density theorems for nonuniform sampling of bandlimited functions using derivatives or bunched measurements,*

*J. Fourier Anal. Appl. 23(6):1311-1347* *B. Adcock, A. C. Hansen, B. Roman,**A note on compressed sensing of structured sparse wavelet coefficients from subsampled Fourier measurements**,*

*IEEE Signal Process. Lett.**23(5):732 - 736*

*A. Jones , A. Tamtogl, I. Calvo-Almazan, A. C. Hansen,**Continuous compressed sensing of inelastic and quasielastic Helium Atom Scattering spectra**,*

*Nature, Sci. Rep. 6, Art. num.: 27776**A. Jones , A. Tamtogl, I. Calvo-Almazan, A. C. Hansen,**Continuous compressed sensing of inelastic and quasielastic Helium Atom Scattering spectra**(supplementary material),*

6, Art. num.: 27776*Nature, Sci. Rep.**J. Ben-Artzi, A. C. Hansen, O. Nevanlinna, M. Seidel,**New barriers in complexity theory: On the Solvability Complexity Index and towers of algorithms**,*

353, no. 10, 931-936*C. R. Acad. Sci. Paris Sér. I Math.**J. Ben-Artzi, A. C. Hansen, O. Nevanlinna, M. Seidel,**The Solvability Complexity Index - Computer science and logic meet scientific computing**.**B. Adcock, M. Gataric, A. C. Hansen, Recovering piecewise smooth functions from nonuniform Fourier measuremets,*

Springer Lect. Notes in Comp. Sci. and Eng. 2015- A. Bastounis, A. C. Hansen,
*On random and deterministic compressed sensing and the Restricted Isometry Property in Levels,*

*IEEE 2015 Int. Conf. on Samp. Theory and Appl.* - B. Adcock, A. C. Hansen, M. Gataric,
*Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples,*42(3):508-535

Appl. Comput. Harmon. Anal. - B. Adcock, M. Gataric, A. C. Hansen,
*Stable nonuniform sampling with weighted Fourier frames and recovery in arbitrary spaces,*

*IEEE 2015 Int. Conf. on Samp. Theory and Appl.* - B. Adcock, A. C. Hansen, A. Jones,
*On asymptotic incoherence and its implications for compressed sensing for inverse problems,*62, no. 2, 1020-1032

IEEE Trans. Inf. Theory, - B. Roman, B. Adcock, A. C. Hansen,
*On asymptotic structure in compressed sensing.* - B. Adcock, G. Kutyniok, A. C. Hansen, J. Ma,
*Linear Stable Sampling Rate: Optimality of 2D Wavelet Reconstructions from Fourier Measurements*,47(2), 1196–1233

SIAM J. Math. Anal. - B. Adcock, A. C. Hansen,
*Generalized Sampling and Infinite Dimensional Compressed Sensing*,

*Found. Comp. Math.*16, no. 5, 1263-1323 - B. Adcock, A. C. Hansen, B. Roman
*The quest for optimal sampling: computationally efficient, structure-exploiting measurements for compressed sensing*,

Springer2015*,* - B. Roman, A. Bastounis, B. Adcock, A. C. Hansen,
*On fundamentals of models and sampling in compressed sensing.* - A. Jones, B. Adcock, A. C. Hansen
*Analyzing the structure of multidimensional compressed sensing problems through coherence*. - B. Adcock, M. Gataric, A. C. Hansen,
*On stable reconstructions from univariate nonuniform Fourier measurements*,7(3):1690-1723

SIAM Jour. Imag. Scienc. - B. Adcock, A. C. Hansen, B. Roman, G. Teschke,
*Generalized sampling: stable reconstructions, inverse problems and compressed sensing over the continuum*,

Adv. in Imag. and Electr. Phys. - B. Adcock, A. C. Hansen, A. Shadrin,
*A stability barrier for reconstructions from Fourier samples**,*52, no. 1, 125-139

SIAM Jour. on Num. Anal. - B. Adcock, A. C. Hansen, C. Poon, B. Roman,
*Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing*,

Proc. of the 10th Int. Conf. on Samp. Theory and Appl., 2013 - B. Adcock, A. C. Hansen, C. Poon,
*Optimal wavelet reconstructions from Fourier samples via generalized sampling*,

Proc. of the 10th Int. Conf. on Samp. Theory and Appl., 2013 - B. Adcock, A. C. Hansen, C. Poon,
*Beyond Consistent Reconstructions: Optimality and Sharp Bounds for Generalized Sampling, and Application to the Uniform Resampling Problem,*

SIAM J. Math. Anal. - B. Adcock, A. C. Hansen, C. Poon,
*On optimal wavelet reconstructions from Fourier samples: linearity and universality of the stable sampling rate**,*36, no. 3, 387-415

Appl. Comput. Harmon. Anal. - B. Adcock, A. C. Hansen,
*Generalized sampling and the stable and accurate reconstruction of piecewise analytic functions from their Fourier coefficients*,

Math. Comp. 84, 237-270 - B. Adcock, A. C. Hansen, E. Herrholz, G. Teschke,
*Generalized Sampling: Extensions to Frames and Inverse and Ill-Posed Problems*,

Inverse Prob. - B. Adcock, A. C. Hansen,
*Reduced Consistency Sampling in Hilbert Spaces*,

*Proc. of the 9th Int. Conf. on Samp. Theory and Appl., 2011* - B. Adcock, A. C. Hansen,
*Stable reconstructions in Hilbert spaces and the resolution of the Gibbs phenomenon*,

Appl. Comput. Harmon. Anal. - B. Adcock, A. C. Hansen,
*A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases*,

J. Fourier Anal. Appl. - A. C. Hansen,
*A theoretical framework for backward error analysis on manifolds*,

*J. Geom. Mech.*3, no. 1, 81 - 111 - A. C. Hansen,
*On the Solvability Complexity Index, the n-Pseudospectrum and Approximations of Spectra of Operators*,

J. Amer. Math. Soc. - A. C. Hansen, J. Strain,
*On the order of deferred correction*,

Appl. Numer. Math. - A. C. Hansen,
*Infinite dimensional numerical linear algebra; theory and applications*,

*Proc. R. Soc. Lond. Ser. A.*466, no. 2124, 3539-3559 - A. C. Hansen,
*On the approximation of spectra of linear operators on Hilbert spaces*,

J. Funct. Anal. - A. C. Hansen, J. Strain,
*Convergence theory for spectral deferred correction*,

Preprint, UC Berkeley

*(to appear)*## Previous Events

- Invited speaker at
*Analysis and Computation in High Dimensions, Hausdorff Institute (Oct. 2018)**.* - Invited speaker at
*Measuring the Complexity of Computational Content: From Combinatorial Problems to Analysis, Dagstuhl (Sept. 2018)**.* - Invited speaker at the
*Algebraic and geometric aspects of numerical methods for differential equations, Mittag-Leffler Institute*(July 5 2018) - Invited speaker at Isaac Newton Institute (May 24 2018)
- Speaking at the University of Oslo (May 14-16 2018, slides).
- Speaking at the
*University of Manchester*(May 4 2018). - Invited speaker at
*Banff Research Station*(April 25 2018). - Invited speaker at the
*Institut Henri Poincaré*(Feb 12 2018). - Organizing the program
*Approximation, sampling and compression in data science, Isaac Newton Institute (Jan-June 2019).* - Organizing the workshop
*Mathematics of data: Structured representations for sensing, approximation and learning, Alan Turing Institute (May 27-May 31, 2019)*. - Speaking at LMU Munich (Jan 31, 2018).
- Organizing the workshop
*Inverse Problems Network Meeting 2, Isaac Newton Institute (Nov 23-Nov 24, 2017)*. - Speaking at the University of Warwick (Nov 15, 2017).
- Invited speaker at
*Generative models, parameter learning and sparsity, Isaac Newton Institute (2017)*. - Plenary speaker at the
*Fourteenth International Conference on Computability and Complexity in Analysis (2017).* - Plenary speaker at
*SPARS (2017).* - Plenary speaker at
*Structured Regularization for High-Dimensional Data Analysis,**Institut Henri Poincaré (2017).* - Keynote speaker at
*FoCM:*Approximation Theory Workshop*(2017)**.* - Invited speaker at
*FoCM:*Information-Based Complexity Workshop*(2017)**.* - Invited speaker at
*Multiscale and High-Dimensional Problems, Oberwolfach**(2017)**.* - Plenary speaker at
*The 14th International workshop on Quantum Chromodynamics (QCD) in extreme conditions (2016)**.* - Plenary speaker at
*Strobl16: Time-Frequency Analysis and Related Topics (2016)**.* - Plenary speaker at
*Computational and Analytic Problems in Spectral Theory (2016)**.* - Invited speaker at
*Low Complexity Models in Signal Processing, Hausdorff Institute (2016)**.* - Plenary speaker at
*The Bath/RAL Numerical Analysis Day (2015)**.* - Plenary speaker at
*UCL-Duke Workshop on Sensing and Analysis of High-Dimensional Data (2014)**.* - Plenary speaker at
*Pseudospectra of operators: spectral singularities, semiclassics, pencils and random matrices (2014)**.* - Invited speaker at
*FoCM:*Real Number Complexity Workshop*(2014)**.* - Plenary speaker at
*iTWIST'14 (2014)**.* - Plenary speaker at
*French-German Conference on Mathematical Image Analysis,**Institut Henri Poincaré (2014)**.* - Invited speaker at
*The 5th International Conference on Computational Harmonic Analysis**(2014)**.* - Invited speaker at
*Compressed sensing and its Applications**(2013)**.* - Plenary speaker at
*Sparse Representation of Functions: Analytic and Computational Aspects**(2012)**.* - Plenary speaker at
*Sparsity, Localization and Dictionary Learning**(2012)**.*

## Thesis

A. C. Hansen, *On the approximation of spectra of linear Hilbert space operators*, PhD Thesis.

## Student Awards

- Smith-Knight/Rayleigh-Knight Prize 2007,
*On the approximation of spectra and pseudospectra of linear operators on Hilbert spaces* - John Butcher Award 2007 (joint with T. Schmelzer (Oxford)),
*A theoretical framework for backward error analysis on manifolds*.