Abstract
Quantifying the parameters and corresponding uncertainties of hundreds of strongly lensed quasar systems holds the key to resolving one of the most important scientific questions: the Hubble constant (\(H_{0}\)) tension. The commonly used Markov chain Monte Carlo (MCMC) method has been too time-consuming to achieve this goal, yet recent work has shown that convolution neural networks (CNNs) can be an alternative with seven orders of magnitude improvement in speed. With 31,200 simulated strongly lensed quasar images, we explore the usage of Vision Transformer (ViT) for simulated strong gravitational lensing for the first time. We show that ViT could reach competitive results compared with CNNs, and is specifically good at some lensing parameters, including the most important mass-related parameters such as the center of lens \(\theta _{1}\) and \(\theta _{2}\), the ellipticities \(e_1\) and \(e_2\), and the radial power-law slope \(\gamma '\). With this promising preliminary result, we believe the ViT (or attention-based) network architecture can be an important tool for strong lensing science for the next generation of surveys. The open source of our code and data is in https://github.com/kuanweih/strong_lensing_vit_resnet.
K.-W. Huang, G.C.-F. Chen and Y.-Y. Lin—Equal contribution
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References
Perlmutter, S., et al.: Measurements of omega and lambda from 42 high-redshift supernovae. Astrophys. J. 517, 565–586 (1999)
Riess, A.G., et al.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116, 1009–1038 (1998)
Hinshaw, G., et al.: Nine-year Wilkinson microwave anisotropy probe (WMAP) observations: cosmological parameter results. Astrophys. J. Suppl. Ser. 208, 19 (2013)
Riess, A.G., et al.: Cosmic distances calibrated to 1% precision with Gaia EDR3 parallaxes and Hubble space telescope photometry of 75 Milky Way Cepheids confirm tension with \(\Lambda \)CDM. Astrophys. J. Lett. 908(1), L6 (2021)
Planck Collaboration, N., et al.: Planck 2018 results. VI. Cosmological parameters. Astron. Astrophys. 641, A6 (2020)
Freedman, W.L., et al.: Calibration of the tip of the red giant branch. Astrophys. J. 891(1), 57 (2020)
Wong, K.C., et al.: H0LiCOW – XIII. A 2.4 per cent measurement of H\(_{0}\) from lensed quasars: 5.3\(\sigma \) tension between early- and late-Universe probes. Monthly Notices R. Astron. Soc. 498(1), 1420–1439 (2020)
Treu, T., Marshall, P.J.: Time delay cosmography. Astron. Astrophys. Rev. 24, 11 (2016)
Suyu, S.H., Chang, T.-C., Courbin, F., Okumura, T.: Cosmological distance indicators. Space Sci. Rev. 214(5), 91 (2018)
Refsdal, S.: On the possibility of determining Hubble’s parameter and the masses of galaxies from the gravitational lens effect. Mon. Not. R. Astron. Soc. 128, 307 (1964)
Wong, K.C., et al.: H0LiCOW - IV. Lens mass model of HE 0435–1223 and blind measurement of its time-delay distance for cosmology. Mon. Not. R. Astron. Soc. 465, 4895–4913 (2017)
S. Birrer, et al. H0LiCOW - IX. Cosmographic analysis of the doubly imaged quasar SDSS 1206+4332 and a new measurement of the Hubble constant. Mon. Not. R. Astron. Soc. 484, 4726–4753 (2019)
Rusu, C.E., et al.: H0LiCOW XII. Lens mass model of WFI2033-4723 and blind measurement of its time-delay distance and H\(_{0}\). Mon. Not. R. Astron. Soc. 498(1), 1440–1468 (2020)
Chen, G.C.-F., et al.: Constraining the microlensing effect on time delays with a new time-delay prediction model in H\(_{0}\) measurements. Mon. Not. R. Astron. Soc. 481(1), 1115–1125 (2018)
Chen, G.C.F., et al.: SHARP - VIII. J 0924+0219 lens mass distribution and time-delay prediction through adaptive-optics imaging. Mon. Not. R. Astron. Soc. 513, 2349–2359 (2022)
Shajib, A.J., et al.: STRIDES: a 3.9 per cent measurement of the Hubble constant from the strong lens system DES J0408–5354. Mon. Not. R. Astron. Soc. 494(4), 6072–6102 (2020)
LSST Science Collaboration, et al.: LSST Science Book, Version 2.0. arXiv e-prints, page arXiv:0912.0201, December 2009
Oguri, M., Marshall, P.J.: Gravitationally lensed quasars and supernovae in future wide-field optical imaging surveys. Mon. Not. R. Astron. Soc. 405, 2579–2593 (2010)
Hezaveh, Y.D., Levasseur, L.P., Marshall, P.J.: Fast automated analysis of strong gravitational lenses with convolutional neural networks. Nature. 548(7669), 555–557 (2017)
Levasseur, L.P., Hezaveh, Y.D., Wechsler, R.H.: Uncertainties in parameters estimated with neural networks: application to strong gravitational lensing. Astrophys. J. Lett. 850(1), L7 (2017)
Brehmer, J., Mishra-Sharma, S., Hermans, J., Louppe, G., Cranmer, K.: Mining for dark matter substructure: inferring sub halo population properties from strong lenses with machine learning. Astrophys. J. 886(1), 49 (2019)
Wagner-Carena, S., et al.: Hierarchical inference with Bayesian neural networks: an application to strong gravitational lensing. Astrophys. J. 909(2), 187 (2021)
Lin, J.Y.-Y., Yu, H., Morningstar, W., Peng, J., Holder, G.: Hunting for dark matter Subhalos in strong gravitational lensing with neural networks. In: 34th Conference on Neural Information Processing Systems, October 2020
Park, J.W., et al.: Large-scale gravitational lens modeling with Bayesian neural networks for accurate and precise inference of the Hubble constant. Astrophys. J. 910(1), 39 (2021)
Morgan, R., Nord, B., Birrer, S., Lin, J.Y.-Y., Poh, J.: Deeplenstronomy: a dataset simulation package for strong gravitational lensing. J. Open Source Softw. 6(58), 2854 (2021)
Morningstar, W.R., et al.: Analyzing interferometric observations of strong gravitational lenses with recurrent and convolutional neural networks. arXiv preprint arXiv:1808.00011 (2018)
Coogan, A., Karchev, K., Weniger, C.: Targeted likelihood-free inference of dark matter substructure in strongly-lensed galaxies. In 34th Conference on Neural Information Processing Systems, October 2020
Ostdiek, B., Rivero, A.D., Dvorkin, C.: Extracting the subhalo mass function from strong lens images with image segmentation. Astrophys. J. 927(1), 3 (2022)
Ostdiek, B., Rivero, A.D., Dvorkin, C.: Image segmentation for analyzing galaxy-galaxy strong lensing systems. Astron. Astrophys. 657, L14 (2022)
Thuruthipilly, H., Zadrozny, A., Pollo, A.: Finding strong gravitational lenses through self-attention. arXiv preprint arXiv:2110.09202 (2021)
Vaswani, A., et al.: Attention is all you need. In: Advances in Neural Information Processing Systems, vol. 30 (2017)
Dosovitskiy, A., et al.: An image is worth 16x16 words: Transformers for image recognition at scale. In: 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, 3–7 May 2021. OpenReview.net, 2021
Paul, S., Chen, P.-Y.: Vision transformers are robust learners. In: AAAI (2022)
Raghu, M., Unterthiner, T., Kornblith, S., Zhang, C., Dosovitskiy, A.: Do vision transformers see like convolutional neural networks? In: Beygelzimer, A., Dauphin, Y., Liang, P., Wortman Vaughan, J. (eds.) Advances in Neural Information Processing Systems (2021)
Birrer, S., Amara, A.: lenstronomy: multi-purpose gravitational lens modelling software package. Phys. Dark Univ. 22, 189–201 (2018)
Birrer, S., et al.: Lenstronomy ii: a gravitational lensing software ecosystem. J. Open Sourc. Softw. 6(62), 3283 (2021)
Suyu, S.H., et al.: Two accurate time-delay distances from strong lensing: implications for cosmology. Astrophys. J. 766, 70 (2013)
Barkana, R.: Fast calculation of a family of elliptical mass gravitational lens models. Astrophys. J. 502, 531 (1998)
Sérsic, J.L.: Atlas de galaxias Australes. Observatorio Astronomico, Cordoba, Argentina (1968)
Krist, J.E., Hook, R.N.: NICMOS PSF variations and tiny Tim simulations. In: Casertano, S., Jedrzejewski, R., Keyes, T., Stevens, M. (eds.) The 1997 HST Calibration Workshop with a New Generation of Instruments, p. 192, January 1997
Chen, G.C.-F., et al.: SHARP - III. First use of adaptive-optics imaging to constrain cosmology with gravitational lens time delays. Mon. Not. R. Astron. Soc. 462, 3457–3475 (2016)
Chen, G.C.-F., et al.: A SHARP view of H0LiCOW: H\(_{0}\) from three time-delay gravitational lens systems with adaptive optics imaging. Mon. Not. R. Astron. Soc. 490(2), 1743–1773 (2019)
Geoff C.-F. Chen, Treu, T., Fassnacht, C.D., Ragland, S., Schmidt, T., Suyu, S.H.: Point spread function reconstruction of adaptive-optics imaging: meeting the astrometric requirements for time-delay cosmography. Mon. Not. R. Astron. Soc. 508(1), 755–761 (2021)
Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., Li, F.-F.: Imagenet: a large-scale hierarchical image database. In: 2009 IEEE Conference on Computer Vision and Pattern Recognition, pp. 248–255 (2009)
Wolf, T., et al.: Transformers: state-of-the-art natural language processing. In: Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations, pp. 38–45. Association for Computational Linguistics, October 2020
He, K., Zhang, X., Ren, S., Sun, J.: Deep Residual Learning for Image Recognition. arXiv e-prints, arXiv:1512.03385, December 2015
Paszke, A., et al.: Pytorch: an imperative style, high-performance deep learning library. In: Wallach, H., Larochelle, H., Beygelzimer, A., d’Alché-Buc, F., Fox, E., Garnett, R., (eds.), Advances in Neural Information Processing Systems, vol. 32, pp. 8024–8035. Curran Associates Inc (2019)
Gal, Y., Ghahramani, Z.: Dropout as a Bayesian approximation: representing model uncertainty in deep learning. In: Balcan, M.F., Weinberger, K.Q. (eds.), Proceedings of The 33rd International Conference on Machine Learning, volume 48 of Proceedings of Machine Learning Research, pp. 1050–1059. PMLR, New York, New York, USA, 20–22 June 2016
Kendall, A., Gal, Y.: What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision? arXiv e-prints. arXiv:1703.04977, March 2017
Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. arXiv e-prints. arXiv:1412.6980, December 2014
Abdalla, E., et al.: Cosmology intertwined: a review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies. J. High Energy Astrophys. 34, 49–211 (2022)
Suyu, S.H., et al.: The Hubble constant and new discoveries in cosmology. ArXiv e-prints. arxiv:1202.4459, February 2012
Falco, E.E., Gorenstein, M.V., Shapiro, I.I.: On model-dependent bounds on H(0) from gravitational images Application of Q0957 + 561A.B. Astrophys. J. Lett. 289, L1–L4 (1985)
Gorenstein, M.V., Falco, E.E., Shapiro, I.I.: Degeneracies in parameter estimates for models of gravitational lens systems. Astrophys. J. 327, 693 (1988)
Schneider, P., Sluse, D.: Mass-sheet degeneracy, power-law models and external convergence: impact on the determination of the Hubble constant from gravitational lensing. Astron. Astrophys. 559, A37 (2013)
Xu, D., et al.: Lens galaxies in the Illustris simulation: power-law models and the bias of the Hubble constant from time delays. Mon. Not. R. Astron. Soc. 456, 739–755 (2016)
Gomer, M., Williams, L.L.R.: Galaxy-lens determination of H\(_{0}\): constraining density slope in the context of the mass sheet degeneracy. J. Cosmol. Astropart. Phys. 2020(11), 045 (2020)
Kochanek, C.S.: Over constrained gravitational lens models and the Hubble constant. Mon. Not. R. Astron. Soc. 493(2), 1725–1735 (2020)
Blum, K., Castorina, E., Simonović, M.: Could quasar lensing time delays hint to a core component in Halos, instead of H\(_{0}\) tension? Astrophys. J. Lett. 892(2), L27 (2020)
Millon, M., et al.: TDCOSMO. I. An exploration of systematic uncertainties in the inference of H\(_{0}\) from time-delay cosmography. Astron. Astrophys. 639, A101 (2020)
Ding, X., et al.: Time delay lens modelling challenge. Mon. Not. R. Astron. Soc. 503(1), 1096–1123 (2021)
Birrer, S., et al.: TDCOSMO. IV. Hierarchical time-delay cosmography – joint inference of the Hubble constant and galaxy density profiles. Astron. Astrophys. 643, A165 (2020)
Chen, G.C.-F., Fassnacht, C.D., Suyu, S.H., Yıldırım, A., Komatsu, E., Bernal, J.L.: TDCOSMO. VI. Distance measurements in time-delay cosmography under the mass-sheet transformation. Astron. Astrophys. 652, A7 (2021)
Tagore, A.S., et al.: Reducing biases on H\(_{0}\) measurements using strong lensing and galaxy dynamics: results from the EAGLE simulation. Mon. Not. R. Astron. Soc. 474(3), 3403–3422 (2018)
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Huang, KW. et al. (2023). Strong Gravitational Lensing Parameter Estimation with Vision Transformer. In: Karlinsky, L., Michaeli, T., Nishino, K. (eds) Computer Vision – ECCV 2022 Workshops. ECCV 2022. Lecture Notes in Computer Science, vol 13801. Springer, Cham. https://doi.org/10.1007/978-3-031-25056-9_10
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