ICTEAM > ISPGroup > Research@ISPGroup > ## Bilinear and Biconvex Inverse Problems for Computational Sensing SystemsRecent years have been characterised by a surge of efforts into new In particular, for many inverse problems arising in the acquisition of natural and man-made signals, the principle of However, the physical implementation of computational sensing strategies is far from being simple or stable, and is a subject of ongoing research. Since the inverse problem that maps the measurements collected by the sensing device to an estimate of the original signal typically requires a very accurate numerical model of the physical sensing system, it is often the case that non-trivial calibration procedures are required to complete or refine the available information describing the sensing system. In general terms, if the sensing system is represented by a linear operator {$\mathcal A$} that produces some measurements {$\boldsymbol y = \mathcal A(\boldsymbol x)$}, it is often the case that there is a mismatch between the model {$\mathcal A$} and its implementation {$\mathcal A_{\boldsymbol t}$}, that is {$\boldsymbol y = \mathcal A_{\boldsymbol t}(\boldsymbol x) \neq \mathcal A (\boldsymbol x)$} where the missing parameters {$\boldsymbol t$} are part of the unknowns and are actually mixed with the signal. Rather than using known training signals {$\boldsymbol x$} to fix part of the unknowns in {$\boldsymbol y = \mathcal A_{\boldsymbol t}(\boldsymbol x)$} and be able to retrieve {$\boldsymbol t$}, we focus on what are commonly referred to as a The common aspect of such a broad class of sensing models is in that, when we attempt to solve the inverse problem
{$$(\hat{\boldsymbol x}, \hat{\boldsymbol t}) := {\rm argmin}_{(\boldsymbol \xi, \boldsymbol \theta) \in \mathbb D_{\boldsymbol \xi} \times \mathbb D_{\boldsymbol \theta}} f(\boldsymbol \xi, \boldsymbol \theta)$$}
that minimises the discrepancy {$$f(\boldsymbol \xi, \boldsymbol \theta)
:= \|\boldsymbol y - \mathcal A_{\boldsymbol \theta}(\boldsymbol \xi)\|^2_2$$} between the observations and the solution
on some domain {$\mathbb D_{\boldsymbol \xi} \times \mathbb D_{\boldsymbol \theta}$} we are faced with a In this branch of our research on sensing and imaging we aim at addressing such non-convex objectives {$f(\boldsymbol \xi, \boldsymbol \theta)$} under a
Intuitively and roughly speaking, if even in such an infinite-sample regime the problem is non-convex, there is no hope of solving it with simple methods; conversely, if there exists a non-asymptotic regime for which which we can bound {$$\vert f(\boldsymbol \xi, \boldsymbol \theta) - \mathbb E_{\boldsymbol a} f(\boldsymbol \xi, \boldsymbol \theta) \vert \leq \delta$$} for some arbitrarily small {$\delta$}, it is sensible that the optimisation problem will be (at least locally) convex, and enjoy for finite-sample regimes of the properties of the problem in expectation. With this rationale in mind we mention, among others, many recent breakthroughs in the context of phase retrieval (Candès, Strohmer, Voroninski 2014; Candès, Li, Soltanolkotabi, 2015; White, Sanghavi, Ward, 2015; Sun, Qu, Wright, 2016), covariance estimation and sketching (Chen, Chi, Goldsmith 2015), blind deconvolution and demixing (Ahmed, Recht, Romberg, 2014; Ahmed, Cosse, Demanet, 2015; Bahmani and Romberg, 2015; Ling and Strohmer 2016; Li, Ling, Strohmer, Wei 2016) as well as blind calibration (Friedlander and Strohmer 2014; Ling and Strohmer 2015; Cambareri and Jacques 2016). The most important aspect of such approaches with respect to the previous literature ( Note that some of these works adopt a
In particular, we have focused on a special instance of blind calibration, that is solving the bilinear system
{$$\boldsymbol y_l = {\rm diag}({\boldsymbol g}) {\boldsymbol A}_l {\boldsymbol x}, \ l = 1, \ldots, p$$}
in {$(\boldsymbol x, \boldsymbol g)$} where {${\boldsymbol A}_l \in \mathbb R^{m \times n}$} are independent and identically distributed (i.i.d.) random matrices with i.i.d. sub-Gaussian entries. In this case, we have solved
{$$ \mathop{{\rm argmin}}_{({\boldsymbol \xi},{\boldsymbol \gamma}) \in \Sigma \times {\mathcal B}} \textstyle\frac{1}{2mp} {\sum_{l=1}^{p} \left\Vert {\rm diag}({\boldsymbol \gamma}) {\boldsymbol A}_l {\boldsymbol \xi} - \boldsymbol y_l\right\Vert^2_2} $$}
where {$\Sigma \subseteq \mathbb R^n$} is a subspace of a signal domain and {$\mathcal B \subset \mathbb R^m$} is a closed convex set of When {$\boldsymbol g$} is not too far from unity ( In conclusion, we believe that this research, and its potential extension to blind deconvolution is critical for compressive imaging strategies, which are essentially hindered by the realisation of the sensing operator in the physical domain, and whose mismatch with the mathematical model often takes the form of a convolution kernel ( ## References:## a) Own Contributions:**2016.**"Through the Haze: A Non-Convex Approach to Blind Calibration for Linear Random Sensing Models" V. Cambareri and L. Jacques Submitted to Information and Inference: A Journal of the IMA, arXiv:1610.09028**2016.**"A Greedy Blind Calibration Method for Compressed Sensing with Unknown Sensor Gains" V. Cambareri and L. Jacques Submitted to ICASSP'17, arXiv:1610.02851**2016.**"Non-Convex Blind Calibration for Compressed Sensing via Iterative Hard Thresholding" V. Cambareri, L. Jacques Accepted in the 11th "IMA International Conference on Mathematics in Signal Processing", Birmingham, UK (12-14/12/2016) dial:2078.1/178320**2016.**"A Non-Convex Blind Calibration Method for Randomised Sensing Strategies" V. Cambareri, L. Jacques 2016 4th International Workshop on Compressed Sensing Theory and its Applications to Radar, Sonar and Remote Sensing (CoSeRa) (invited) arXiv:1605.02615, doi:10.1109/CoSeRa.2016.7745690, dial:2078.1/178303**2016.**"A Non-Convex Approach to Blind Calibration for Linear Random Sensing Models" V. Cambareri, L. Jacques Proceedings of the "international Traveling Workshop on Interactions between Sparse models and Technology (iTWIST'16)", Aalborg, Denmark, 2016 arXiv:1609.04167, dial:2078.1/178307**2016.**"A Non-Convex Approach to Blind Calibration from Linear Sub-Gaussian Random Measurements" V. Cambareri and L. Jacques 37th WIC Symposium on Information Theory in the Benelux, 6th joint WIC/IEEE SP Symposium on Information Theory and Signal Processing in the Benelux dial:2078.1/178326
## b) Related Work:- Ahmed, Ali, Augustin Cosse, and Laurent Demanet. "A convex approach to blind deconvolution with diverse inputs." Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2015 IEEE 6th International Workshop on. IEEE, 2015.
- Ahmed, Ali, Benjamin Recht, and Justin Romberg. "Blind deconvolution using convex programming." IEEE Transactions on Information Theory 60.3 (2014): 1711-1732.
- Ahmed, Ali, Felix Krahmer, and Justin Romberg. "Empirical Chaos Processes and Blind Deconvolution." arXiv preprint (2016).
- Bahmani, Sohail, and Justin Romberg. "Lifting for blind deconvolution in random mask imaging: Identifiability and convex relaxation." SIAM Journal on Imaging Sciences 8.4 (2015): 2203-2238.
- Balzano, Laura, and Robert Nowak. "Blind calibration of sensor networks." Proceedings of the 6th international conference on Information processing in sensor networks. ACM, 2007.
- Bilen, Çağdaş, et al. "Convex optimization approaches for blind sensor calibration using sparsity." IEEE Transactions on Signal Processing 62.18 (2014): 4847-4856.
- Candes, Emmanuel J., Thomas Strohmer, and Vladislav Voroninski. "Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming." Communications on Pure and Applied Mathematics 66.8 (2013): 1241-1274.
- Candes, Emmanuel J., Xiaodong Li, and Mahdi Soltanolkotabi. "Phase retrieval via Wirtinger flow: Theory and algorithms." IEEE Transactions on Information Theory 61.4 (2015): 1985-2007.
- Chen, Yuxin, Yuejie Chi, and Andrea J. Goldsmith. "Exact and stable covariance estimation from quadratic sampling via convex programming." IEEE Transactions on Information Theory 61.7 (2015): 4034-4059.
- Friedlander, Benjamin, and Thomas Strohmer. "Bilinear compressed sensing for array self-calibration." 2014 48th Asilomar Conference on Signals, Systems and Computers. IEEE, 2014.
- Li, Xiaodong, et al. "Rapid, robust, and reliable blind deconvolution via nonconvex optimization." arXiv preprint arXiv:1606.04933 (2016).
- Ling, Shuyang, and Thomas Strohmer. "Blind deconvolution meets blind demixing: Algorithms and performance bounds." arXiv preprint arXiv:1512.07730 (2015).
- Ling, Shuyang, and Thomas Strohmer. "Self-calibration and biconvex compressive sensing." Inverse Problems 31.11 (2015): 115002.
- Ling, Shuyang, and Thomas Strohmer. "Self-Calibration via Linear Least Squares." arXiv preprint arXiv:1611.04196 (2016).
- Sanghavi, Sujay, Rachel Ward, and Chris D. White. "The local convexity of solving systems of quadratic equations." Results in Mathematics (2016): 1-40.
- Sun, Ju, Qing Qu, and John Wright. "A geometric analysis of phase retrieval." arXiv preprint arXiv:1602.06664 (2016).
**2016**"Towards Non-Convex Blind Calibration for Compressed Sensing", FNRS Wavelet Contact Group, 16 November 2016
n/a ## Projects## CategoriesSensingImaging CompressedSensing ResearchCategoryAll ## ISPGroup Participant(s)Last updated September 19, 2017, at 08:43 AM |