Uniqueness in bilinear inverse problems with applications to subspace and joint sparsity models

Yanjun Li, Kiryung Lee, Yoram Bresler

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Bilinear inverse problems (BIPs), the resolution of two vectors given their image under a bilinear mapping, arise in many applications, such as blind deconvolution and dictionary learning. However, there are few results on the uniqueness of solutions to BIPs. For example, blind gain and phase calibration (BGPC) is a structured bilinear inverse problem, which arises in inverse rendering in computational relighting, in blind gain and phase calibration in sensor array processing, in multichannel blind deconvolution (MBD), etc. It is interesting to study the uniqueness of solutions to such problems. In this paper, we define identifiability of a bilinear inverse problem up to a group of transformations. We derive conditions under which the solutions can be uniquely determined up to the transformation group. Then we apply these results to BGPC and give sufficient conditions for unique recovery under subspace or joint sparsity constraints. For BGPC with joint sparsity constraints, we develop a procedure to determine the relevant transformation groups. We also give necessary conditions in the form of tight lower bounds on sample complexities, and demonstrate the tightness by numerical experiments.

Original languageEnglish (US)
Title of host publication2015 International Conference on Sampling Theory and Applications, SampTA 2015
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages568-572
Number of pages5
ISBN (Electronic)9781467373531
DOIs
StatePublished - Jul 2 2015
Event11th International Conference on Sampling Theory and Applications, SampTA 2015 - Washington, United States
Duration: May 25 2015May 29 2015

Publication series

Name2015 International Conference on Sampling Theory and Applications, SampTA 2015

Other

Other11th International Conference on Sampling Theory and Applications, SampTA 2015
CountryUnited States
CityWashington
Period5/25/155/29/15

Fingerprint

Inverse problems
Sparsity
Inverse Problem
Calibration
Uniqueness
Subspace
Blind Deconvolution
Transformation group
Uniqueness of Solutions
Deconvolution
Array processing
Sensor Array
Tightness
Identifiability
Sensor arrays
Glossaries
Model
Rendering
Recovery
Numerical Experiment

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Signal Processing
  • Statistics and Probability

Cite this

Li, Y., Lee, K., & Bresler, Y. (2015). Uniqueness in bilinear inverse problems with applications to subspace and joint sparsity models. In 2015 International Conference on Sampling Theory and Applications, SampTA 2015 (pp. 568-572). [7148955] (2015 International Conference on Sampling Theory and Applications, SampTA 2015). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/SAMPTA.2015.7148955

Uniqueness in bilinear inverse problems with applications to subspace and joint sparsity models. / Li, Yanjun; Lee, Kiryung; Bresler, Yoram.

2015 International Conference on Sampling Theory and Applications, SampTA 2015. Institute of Electrical and Electronics Engineers Inc., 2015. p. 568-572 7148955 (2015 International Conference on Sampling Theory and Applications, SampTA 2015).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Li, Y, Lee, K & Bresler, Y 2015, Uniqueness in bilinear inverse problems with applications to subspace and joint sparsity models. in 2015 International Conference on Sampling Theory and Applications, SampTA 2015., 7148955, 2015 International Conference on Sampling Theory and Applications, SampTA 2015, Institute of Electrical and Electronics Engineers Inc., pp. 568-572, 11th International Conference on Sampling Theory and Applications, SampTA 2015, Washington, United States, 5/25/15. https://doi.org/10.1109/SAMPTA.2015.7148955
Li Y, Lee K, Bresler Y. Uniqueness in bilinear inverse problems with applications to subspace and joint sparsity models. In 2015 International Conference on Sampling Theory and Applications, SampTA 2015. Institute of Electrical and Electronics Engineers Inc. 2015. p. 568-572. 7148955. (2015 International Conference on Sampling Theory and Applications, SampTA 2015). https://doi.org/10.1109/SAMPTA.2015.7148955
Li, Yanjun ; Lee, Kiryung ; Bresler, Yoram. / Uniqueness in bilinear inverse problems with applications to subspace and joint sparsity models. 2015 International Conference on Sampling Theory and Applications, SampTA 2015. Institute of Electrical and Electronics Engineers Inc., 2015. pp. 568-572 (2015 International Conference on Sampling Theory and Applications, SampTA 2015).
@inproceedings{3abea060eba04acfaa8844218cf62c0b,
title = "Uniqueness in bilinear inverse problems with applications to subspace and joint sparsity models",
abstract = "Bilinear inverse problems (BIPs), the resolution of two vectors given their image under a bilinear mapping, arise in many applications, such as blind deconvolution and dictionary learning. However, there are few results on the uniqueness of solutions to BIPs. For example, blind gain and phase calibration (BGPC) is a structured bilinear inverse problem, which arises in inverse rendering in computational relighting, in blind gain and phase calibration in sensor array processing, in multichannel blind deconvolution (MBD), etc. It is interesting to study the uniqueness of solutions to such problems. In this paper, we define identifiability of a bilinear inverse problem up to a group of transformations. We derive conditions under which the solutions can be uniquely determined up to the transformation group. Then we apply these results to BGPC and give sufficient conditions for unique recovery under subspace or joint sparsity constraints. For BGPC with joint sparsity constraints, we develop a procedure to determine the relevant transformation groups. We also give necessary conditions in the form of tight lower bounds on sample complexities, and demonstrate the tightness by numerical experiments.",
author = "Yanjun Li and Kiryung Lee and Yoram Bresler",
year = "2015",
month = "7",
day = "2",
doi = "10.1109/SAMPTA.2015.7148955",
language = "English (US)",
series = "2015 International Conference on Sampling Theory and Applications, SampTA 2015",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
pages = "568--572",
booktitle = "2015 International Conference on Sampling Theory and Applications, SampTA 2015",
address = "United States",

}

TY - GEN

T1 - Uniqueness in bilinear inverse problems with applications to subspace and joint sparsity models

AU - Li, Yanjun

AU - Lee, Kiryung

AU - Bresler, Yoram

PY - 2015/7/2

Y1 - 2015/7/2

N2 - Bilinear inverse problems (BIPs), the resolution of two vectors given their image under a bilinear mapping, arise in many applications, such as blind deconvolution and dictionary learning. However, there are few results on the uniqueness of solutions to BIPs. For example, blind gain and phase calibration (BGPC) is a structured bilinear inverse problem, which arises in inverse rendering in computational relighting, in blind gain and phase calibration in sensor array processing, in multichannel blind deconvolution (MBD), etc. It is interesting to study the uniqueness of solutions to such problems. In this paper, we define identifiability of a bilinear inverse problem up to a group of transformations. We derive conditions under which the solutions can be uniquely determined up to the transformation group. Then we apply these results to BGPC and give sufficient conditions for unique recovery under subspace or joint sparsity constraints. For BGPC with joint sparsity constraints, we develop a procedure to determine the relevant transformation groups. We also give necessary conditions in the form of tight lower bounds on sample complexities, and demonstrate the tightness by numerical experiments.

AB - Bilinear inverse problems (BIPs), the resolution of two vectors given their image under a bilinear mapping, arise in many applications, such as blind deconvolution and dictionary learning. However, there are few results on the uniqueness of solutions to BIPs. For example, blind gain and phase calibration (BGPC) is a structured bilinear inverse problem, which arises in inverse rendering in computational relighting, in blind gain and phase calibration in sensor array processing, in multichannel blind deconvolution (MBD), etc. It is interesting to study the uniqueness of solutions to such problems. In this paper, we define identifiability of a bilinear inverse problem up to a group of transformations. We derive conditions under which the solutions can be uniquely determined up to the transformation group. Then we apply these results to BGPC and give sufficient conditions for unique recovery under subspace or joint sparsity constraints. For BGPC with joint sparsity constraints, we develop a procedure to determine the relevant transformation groups. We also give necessary conditions in the form of tight lower bounds on sample complexities, and demonstrate the tightness by numerical experiments.

UR - http://www.scopus.com/inward/record.url?scp=84941140133&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84941140133&partnerID=8YFLogxK

U2 - 10.1109/SAMPTA.2015.7148955

DO - 10.1109/SAMPTA.2015.7148955

M3 - Conference contribution

T3 - 2015 International Conference on Sampling Theory and Applications, SampTA 2015

SP - 568

EP - 572

BT - 2015 International Conference on Sampling Theory and Applications, SampTA 2015

PB - Institute of Electrical and Electronics Engineers Inc.

ER -