Linear inverse problems with nonnegativity constraints: singularity of optimisers. (English) Zbl 1532.65027
In the paper under review, the authors study linear inverse problems of the type \(Ax = y\), where \(A\) is a linear operator, and \(y\) lies in a suitable linear space. In many applications, for example medical imaging and Positron Emission Tomography, \(x\) is a nonnegative number.
In this paper, the authors focus on the case where the noise model leads to maximum likelihood estimation through general divergences, which cover a wide range of common noise statistics such as Gaussian and Poisson.
Thinking of the unknown \(x\) as a function (say \(f\)) in some functional space \(X\) of functions over a compact set \(K\in \mathbb R^p\) and where the operator \(A\) is a linear mapping from \(X\) to \(\mathbb R^m\), leads to the optimisation problem \(\min_{f\geq 0}D(y,A\mu)\). This non-negativity constraint has been shown to cause sparsity in various contexts in optimal control and optimization.
The authors prove several key results and provide several numerical examples. The paper is well written with a good set of references.
In this paper, the authors focus on the case where the noise model leads to maximum likelihood estimation through general divergences, which cover a wide range of common noise statistics such as Gaussian and Poisson.
Thinking of the unknown \(x\) as a function (say \(f\)) in some functional space \(X\) of functions over a compact set \(K\in \mathbb R^p\) and where the operator \(A\) is a linear mapping from \(X\) to \(\mathbb R^m\), leads to the optimisation problem \(\min_{f\geq 0}D(y,A\mu)\). This non-negativity constraint has been shown to cause sparsity in various contexts in optimal control and optimization.
The authors prove several key results and provide several numerical examples. The paper is well written with a good set of references.
Reviewer: Steven B. Damelin (Ann Arbor)
MSC:
| 65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
| 49N45 | Inverse problems in optimal control |
| 94A08 | Image processing (compression, reconstruction, etc.) in information and communication theory |
References:
| [1] | J. H. O. Adler Kohr Öktem, ODL-a Python framework for rapid prototyping in inverse problems, Royal Institute of Technology (2017) |
| [2] | C. A. Y. D. V. F. P. Boyer Chambolle Castro Duval De Gournay Weiss, On representer theorems and convex regularization, SIAM Journal on Optimization, 29, 1260-1281 (2019) · Zbl 1423.49036 · doi:10.1137/18M1200750 |
| [3] | K. Bredies and M. Carioni, Sparsity of solutions for variational inverse problems with finite-dimensional data, Calculus of Variations and Partial Differential Equations, 59 (2020), Paper No. 14, 26 pp. · Zbl 1430.49036 |
| [4] | H. Brezis, Functional Aanalysis, Sobolev Spaces and Partial Differential Equations, Springer Science & Business Media, 2010. · Zbl 1218.46002 |
| [5] | Y. C. T. N. C. S. M.-J. C. Cavalcanti Oberlin Dobigeon Févotte Stute Ribeiro Tauber, Factor analysis of dynamic PET images: Beyond Gaussian noise, IEEE Transactions on Medical Imaging (2019) |
| [6] | A. T. Chambolle Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, Journal of Mathematical Imaging and Vision, 40, 120-145 (2010) · Zbl 1255.68217 · doi:10.1007/s10851-010-0251-1 |
| [7] | A. S.-i. Cichocki Amari, Families of alpha-beta-and gamma-divergences: Flexible and robust measures of similarities, Entropy, 12, 1532-1568 (2010) · Zbl 1229.94030 · doi:10.3390/e12061532 |
| [8] | C. B. E. Clason Kaltenbacher Resmerita, Regularization of ill-posed problems with non-negative solutions, Splitting Algorithms, Modern Operator Theory, and Applications, 113-135 (2019) |
| [9] | C. A. Clason Schiela, Optimal control of elliptic equations with positive measures, ESAIM: Control, Optimisation and Calculus of Variations, 23, 217-240 (2017) · Zbl 1365.49005 · doi:10.1051/cocv/2015046 |
| [10] | T. Debarre, Q. Denoyelle and J. Fageot, On the uniqueness of solutions for the basis pursuit in the continuum, Inverse Problems, 38 (2022), 125005, 19 pp, arXiv: 2009.11855. · Zbl 1504.94031 |
| [11] | Q. V. G. Denoyelle Duval Peyré, Support recovery for sparse super-resolution of positive measures, Journal of Fourier Analysis and Applications, 23, 1153-1194 (2017) · Zbl 1417.65223 · doi:10.1007/s00041-016-9502-x |
| [12] | A. R. De Pierro, On the relation between the ISRA and the EM algorithm for positron emission tomography, IEEE Transactions on Medical Imaging, 12, 328-333 (1993) · doi:10.1109/42.232263 |
| [13] | C. J. Févotte Idier, Algorithms for nonnegative matrix factorization with the \(\beta \)-divergence, Neural Computation, 23, 2421-2456 (2011) · Zbl 1231.65072 · doi:10.1162/NECO_a_00168 |
| [14] | T. T. Georgiou, Solution of the general moment problem via a one-parameter imbedding, IEEE Transactions on Automatic Control, 50, 811-826 (2005) · Zbl 1365.93096 · doi:10.1109/TAC.2005.849212 |
| [15] | S. C. D. Henrot Soussen Brie, Fast positive deconvolution of hyperspectral images, IEEE Transactions on Image Processing, 22, 828-833 (2012) · Zbl 1373.94162 · doi:10.1109/TIP.2012.2216280 |
| [16] | S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions, Springer Science & Business Media, 2002. · Zbl 1015.26030 |
| [17] | D. D. H. S. Lee Seung, Algorithms for non-negative matrix factorization, Advances in Neural Information Processing Systems, 556-562 (2001) |
| [18] | J. E. E. Lohéac Trélat Zuazua, Minimal controllability time for the heat equation under unilateral state or control constraints, Mathematical Models and Methods in Applied Sciences, 27, 1587-1644 (2017) · Zbl 1370.35156 · doi:10.1142/S0218202517500270 |
| [19] | B. M. J. Mair Rao Anderson, Positron emission tomography, Borel measures and weak convergence, Inverse Problems, 12, 965-976 (1996) · Zbl 0862.60098 · doi:10.1088/0266-5611/12/6/011 |
| [20] | O. C. O. Öktem Pouchol Verdier, Spatiotemporal PET reconstruction using ML-EM with learned diffeomorphic deformation, International Workshop on Machine Learning for Medical Image Reconstruction, 151-162 (2019) |
| [21] | W. J. Palenstijn, Astra toolbox, Jan. 2012. |
| [22] | C. Pouchol and O. Verdier, The ML-EM algorithm in continuum: Sparse measure solutions, Inverse Problems, 36 (2020), 035013, 26 pp. · Zbl 1436.62392 |
| [23] | C. O. Pouchol Verdier, Statistical model and ML-EM algorithm for emission tomography with known movement, Journal of Mathematical Imaging and Vision, 63, 650-663 (2021) · Zbl 1524.62107 · doi:10.1007/s10851-021-01021-7 |
| [24] | E. H. W. A. N. Resmerita Engl Iusem, The expectation-maximization algorithm for ill-posed integral equations: A convergence analysis, Inverse Problems, 23, 2575-2588 (2007) · Zbl 1282.65173 · doi:10.1088/0266-5611/23/6/019 |
| [25] | R. T. Rockafellar, Extension of Fenchel’s duality theorem for convex functions, Duke Mathematical Journal, 33, 81-89 (1966) · Zbl 0138.09301 |
| [26] | L. A. Y. Shepp Vardi, Maximum likelihood reconstruction for emission tomography, IEEE Transactions on Medical Imaging, 1, 113-122 (1982) |
| [27] | U. A. T. Y. K. Simsekli Cemgil Yilmaz, Learning the beta-divergence in tweedie compound poisson matrix factorization models, International Conference on Machine Learning, 1409-1417 (2013) |
| [28] | Y. L. L. Vardi Shepp Kaufman, A statistical model for positron emission tomography, Journal of the American statistical Association, 80, 8-37 (1985) · Zbl 0561.62094 · doi:10.1080/01621459.1985.10477119 |
| [29] | Z. E. Yang Oja, Unified development of multiplicative algorithms for linear and quadratic nonnegative matrix factorization, IEEE Transactions on Neural Networks, 22, 1878-1891 (2011) |
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