* [32] Q. Xie, Q. Zhao, D. Meng, and Z. Xu, “Kronecker-basis-representation based tensor sparsity and its applications to tensor recovery,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 40, no. 8, pp. 1888–1902, 2018.
  * [33] Y. Chen, W. He, N. Yokoya, and T. Huang, “Hyperspectral image restoration using weighted group sparsity-regularized low-rank tensor decomposition,” IEEE Trans. Cybern., vol. 50, no. 8, pp. 3556–3570, 2020\.
  * [34] Y. Chen, W. He, N. Yokoya, T. Huang, and X. Zhao, “Nonlocal tensor-ring decomposition for hyperspectral image denoising,” IEEE Trans. Geosci. Remote Sens., vol. 58, no. 2, pp. 1348–1362, 2020.
  * [35] Y. Chang, L. Yan, H. Fang, S. Zhong, and W. Liao, “Hsi-denet: Hyperspectral image restoration via convolutional neural network,” IEEE Trans. Geosci. Remote Sens., pp. 1–16, 2018.
  * [36] Q. Yuan, Q. Zhang, J. Li, H. Shen, and L. Zhang, “Hyperspectral image denoising employing a spatial-spectral deep residual convolutional neural network,” IEEE Trans. Geosci. Remote Sens., pp. 1–14, 2018.
  * [37] X. Liu, S. Bourennane, and C. Fossati, “Denoising of hyperspectral images using the parafac model and statistical performance analysis,” IEEE Trans. Geosci. Remote Sens., vol. 50, no. 10, pp. 3717–3724, 2012.
  * [38] Y. Chang, L. Yan, and S. Zhong, “Hyper-laplacian regularized unidirectional low-rank tensor recovery for multispectral image denoising,” in Proc. CVPR, pp. 4260–4268, 2017.
  * [39] N. Renard, S. Bourennane, and J. Blanc-Talon, “Denoising and dimensionality reduction using multilinear tools for hyperspectral images,” IEEE Geosci. Remote Sens. Lett., vol. 5, pp. 138–142, April 2008.
  * [40] H. Zhang, L. Liu, W. He, and L. Zhang, “Hyperspectral image denoising with total variation regularization and nonlocal low-rank tensor decomposition,” IEEE Trans. Geosci. Remote Sens., vol. 58, pp. 3071–3084, May 2020.
  * [41] L. Zhuang and J. M. Bioucas-Dias, “Fast hyperspectral image denoising and inpainting based on low-rank and sparse representations,” IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., vol. 11, no. 3, pp. 730–742, 2018.
  * [42] G. Chen and S.-E. Qian, “Denoising of hyperspectral imagery using principal component analysis and wavelet shrinkage,” IEEE Trans. Geosci. Remote Sens., vol. 49, pp. 973–980, Mar. 2011.
  * [43] T. Zhou and D. Tao, “Godec: Randomized low-rank and sparse matrix decomposition in noisy case,” in Proc. ICML, 2011.
  * [44] Y. Chen, Y. Guo, Y. Wang, D. Wang, C. Peng, and G. He, “Denoising of hyperspectral images using nonconvex low rank matrix approximation,” IEEE Trans. Geosci. Remote Sens., vol. 55, no. 9, pp. 5366–5380, 2017.
  * [45] H. Ye, H. Li, B. Yang, F. Cao, and Y. Tang, “A novel rank approximation method for mixture noise removal of hyperspectral images,” IEEE Trans. Geosci. Remote Sens., vol. 57, no. 7, pp. 4457–4469, 2019.
  * [46] T. Xie, S. Li, and B. Sun, “Hyperspectral images denoising via nonconvex regularized low-rank and sparse matrix decomposition,” IEEE Trans. Image Process., vol. 29, pp. 44–56, 2020.
  * [47] H. Fan, C. Li, Y. Guo, G. Kuang, and J. Ma, “Spatial-spectral total variation regularized low-rank tensor decomposition for hyperspectral image denoising,” IEEE Trans. Geosci. Remote Sens., vol. 56, no. 10, pp. 6196–6213, 2018.
  * [48] Y. Wang, J. Peng, Q. Zhao, Y. Leung, X. Zhao, and D. Meng, “Hyperspectral image restoration via total variation regularized low-rank tensor decomposition,” IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., vol. 11, no. 4, pp. 1227–1243, 2018.
  * [49] W. He, H. Zhang, H. Shen, and L. Zhang, “Hyperspectral image denoising using local low-rank matrix recovery and global spatial-spectral total variation,” IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., vol. 11, no. 3, pp. 713–729, 2018.
  * [50] H. Zeng, X. Xie, H. Cui, H. Yin, and J. Ning, “Hyperspectral image restoration via global l_{1-2} spatial-spectral total variation regularized local low-rank tensor recovery,” IEEE Trans. Geosci. Remote Sens., pp. 1–17, 2020.
  * [51] L. Sun, B. Jeon, Y. Zheng, and Z. Wu, “Hyperspectral image restoration using low-rank representation on spectral difference image,” IEEE Geosci. Remote Sens. Lett., vol. 14, no. 7, pp. 1151–1155, 2017.
  * [52] J. Peng, Q. Xie, Q. Zhao, Y. Wang, L. Yee, and D. Meng, “Enhanced 3dtv regularization and its applications on hsi denoising and compressed sensing,” IEEE Tran. Image Process., vol. 29, pp. 7889–7903, 2020.
  * [53] Y. Zheng, T. Huang, X. Zhao, Y. Chen, and W. He, “Double-factor-regularized low-rank tensor factorization for mixed noise removal in hyperspectral image,” IEEE Trans. Geosci. Remote Sens., pp. 1–15, 2020.
  * [54] K. Alonso, M. Bachmann, K. Burch, E. Carmona, D. Cerra, R. De los Reyes, D. Dietrich, U. Heiden, A. Hölderlin, J. Ickes, et al., “Data products, quality and validation of the dlr earth sensing imaging spectrometer (desis),” Sensors, vol. 19, no. 20, p. 4471, 2019.
  * [55] W. Li, F. Feng, H. Li, and Q. Du, “Discriminant analysis-based dimension reduction for hyperspectral image classification: A survey of the most recent advances and an experimental comparison of different techniques,” IEEE Geosci. Remote Sens. Mag., vol. 6, no. 1, pp. 15–34, 2018.
  * [56] B. Rasti, D. Hong, R. Hang, P. Ghamisi, X. Kang, J. Chanussot, and J. Benediktsson, “Feature extraction for hyperspectral imagery: The evolution from shallow to deep (overview and toolbox),” IEEE Geosci. Remote Sens. Mag., vol. 8, no. 4, pp. 60–88, 2020.
  * [57] X. Wu, D. Hong, J. Tian, J. Chanussot, W. Li, and R. Tao, “Orsim detector: A novel object detection framework in optical remote sensing imagery using spatial-frequency channel features,” IEEE Trans. Geosci. Remote Sens., vol. 57, no. 7, pp. 5146–5158, 2019.
  * [58] X. Wu, D. Hong, J. Chanussot, Y. Xu, R. Tao, and Y. Wang, “Fourier-based rotation-invariant feature boosting: An efficient framework for geospatial object detection,” IEEE Geosci. Remote Sens. Lett., vol. 17, no. 2, pp. 302–306, 2020.
  * [59] D. D. Lee and H. S. Seung, “Algorithms for non-negative matrix factorization,” in Proc. NIPS, pp. 556–562, 2001.
  * [60] N. Gillis and R. J. Plemmons, “Sparse nonnegative matrix underapproximation and its application to hyperspectral image analysis,” Linear Algebra its Appl., vol. 438, no. 10, pp. 3991–4007, 2013.
  * [61] W. Yan, B. Zhang, and Z. Yang, “A novel regularized nonnegative matrix factorization for spectral-spatial dimension reduction of hyperspectral imagery,” IEEE Access, vol. 6, pp. 77953–77964, 2018.
  * [62] J. Wen, J. E. Fowler, M. He, Y.-Q. Zhao, C. Deng, and V. Menon, “Orthogonal nonnegative matrix factorization combining multiple features for spectral–spatial dimensionality reduction of hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens., vol. 54, no. 7, pp. 4272–4286, 2016.
  * [63] B. Rasti, M. O. Ulfarsson, and J. R. Sveinsson, “Hyperspectral feature extraction using total variation component analysis,” IEEE Trans. Geosci. Remote Sens., vol. 54, no. 12, pp. 6976–6985, 2016.
  * [64] J. An, X. Zhang, H. Zhou, and L. Jiao, “Tensor-based low-rank graph with multimanifold regularization for dimensionality reduction of hyperspectral images,” IEEE Trans. Geosci. Remote Sens., vol. 56, no. 8, pp. 4731–4746, 2018.
  * [65] M. Belkin and P. Niyogi, “Laplacian eigenmaps for dimensionality reduction and data representation,” Neural Comput., vol. 15, no. 6, pp. 1373–1396, 2003\.
  * [66] M. Aharon, M. Elad, and A. Bruckstein, “K-svd: An algorithm for designing overcomplete dictionaries for sparse representation,” IEEE Trans. Signal Process., vol. 54, no. 11, pp. 4311–4322, 2006.
  * [67] F. Nie, H. Huang, X. Cai, and C. H. Ding, “Efficient and robust feature selection via joint $\ell$2,1-norms minimization,” in Proc. NIPS, pp. 1813–1821, 2010.
  * [68] B. Recht, M. Fazel, and P. A. Parrilo, “Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization,” SIAM Review, vol. 52, no. 3, pp. 471–501, 2010.
  * [69] J. Wu, Z. Lin, and H. Zha, “Joint latent subspace learning and regression for cross-modal retrieval,” in Proc. ACM SIGIR, pp. 917–920, 2017.
  * [70] L. Ma, M. M. Crawford, and J. Tian, “Local manifold learning-based $k$-nearest-neighbor for hyperspectral image classification,” IEEE Trans. Geosci. Remote Sens., vol. 48, no. 11, pp. 4099–4109, 2010.
  * [71] S. T. Roweis and L. K. Saul, “Nonlinear dimensionality reduction by locally linear embedding,” Science, vol. 290, no. 5500, pp. 2323–2326, 2000.
  * [72] Z. Zhang and H. Zha, “Principal manifolds and nonlinear dimensionality reduction via tangent space alignment,” SIAM J. Sci. Comput., vol. 26, no. 1, pp. 313–338, 2004.
  * [73] H. Huang, F. Luo, J. Liu, and Y. Yang, “Dimensionality reduction of hyperspectral images based on sparse discriminant manifold embedding,” ISPRS J. Photogramm. Remote Sens., vol. 106, pp. 42–54, 2015.
  * [74] W. He, H. Zhang, L. Zhang, W. Philips, and W. Liao, “Weighted sparse graph based dimensionality reduction for hyperspectral images,” IEEE Geosci. Remote Sens. Lett., vol. 13, no. 5, pp. 686–690, 2016.
  * [75] D. Hong, N. Yokoya, and X. X. Zhu, “Learning a robust local manifold representation for hyperspectral dimensionality reduction,” IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., vol. 10, no. 6, pp. 2960–2975, 2017\.
  * [76] J. An, X. Zhang, H. Zhou, J. Feng, and L. Jiao, “Patch tensor-based sparse and low-rank graph for hyperspectral images dimensionality reduction,” IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., vol. 11, no. 7, pp. 2513–2527, 2018\.
  * [77] N. H. Ly, Q. Du, and J. E. Fowler, “Sparse graph-based discriminant analysis for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens., vol. 52, no. 7, pp. 3872–3884, 2013.
  * [78] N. H. Ly, Q. Du, and J. E. Fowler, “Collaborative graph-based discriminant analysis for hyperspectral imagery,” IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., vol. 7, no. 6, pp. 2688–2696, 2014.
  * [79] M. Imani and H. Ghassemian, “Feature space discriminant analysis for hyperspectral data feature reduction,” ISPRS J. Photogramm. Remote Sens., vol. 102, pp. 1–13, 2015.
  * [80] H. Huang, Y. Duan, H. He, G. Shi, and F. Luo, “Spatial-spectral local discriminant projection for dimensionality reduction of hyperspectral image,” ISPRS J. Photogramm. Remote Sens., vol. 156, pp. 77–93, 2019.
  * [81] D. Hong, L. Gao, J. Yao, B. Zhang, P. Antonio, and J. Chanussot, “Graph convolutional networks for hyperspectral image classification,” IEEE Trans. Geosci. Remote Sens., 2020.  DOI: 10.1109/TGRS.2020.3015157.
  * [82] D. Hong, Regression-Induced Representation Learning and Its Optimizer: A Novel Paradigm to Revisit Hyperspectral Imagery Analysis.  PhD thesis, Technische Universität München, 2019.
  * [83] S. Ji and J. Ye, “Linear dimensionality reduction for multi-label classification,” in Proc. IJCAI, 2009.
  * [84] D. Hong, N. Yokoya, J. Xu, and X. Zhu, “Joint & progressive learning from high-dimensional data for multi-label classification,” in Proc. ECCV, pp. 469–484, 2018.
  * [85] X. He and P. Niyogi, “Locality preserving projections,” in Proc. NIPS, pp. 153–160, 2004.
  * [86] W. Liao, A. Pizurica, P. Scheunders, W. Philips, and Y. Pi, “Semisupervised local discriminant analysis for feature extraction in hyperspectral images,” IEEE Trans. Geosci. Remote Sens., vol. 51, no. 1, pp. 184–198, 2012.
  * [87] M. Zhao, Z. Zhang, T. W. Chow, and B. Li, “A general soft label based linear discriminant analysis for semi-supervised dimensionality reduction,” Neural Netw., vol. 55, pp. 83–97, 2014.
  * [88] H. Wu and S. Prasad, “Semi-supervised dimensionality reduction of hyperspectral imagery using pseudo-labels,” Pattern Recognit., vol. 74, pp. 212–224, 2018.
  * [89] D. Hong, N. Yokoya, J. Chanussot, J. Xu, and X. X. Zhu, “Learning to propagate labels on graphs: An iterative multitask regression framework for semi-supervised hyperspectral dimensionality reduction,” ISPRS J. Photogramm. Remote Sens., vol. 158, pp. 35–49, 2019.
  * [90] D. Hong, N. Yokoya, J. Chanussot, J. Xu, and X. X. Zhu, “Joint and progressive subspace analysis (jpsa) with spatial-spectral manifold alignment for semi-supervised hyperspectral dimensionality reduction,” IEEE Trans. Cybern., 2020.  DOI: 10.1109/TCYB.2020.3028931.
  * [91] R. Hang, Q. Liu, D. Hong, and P. Ghamisi, “Cascaded recurrent neural networks for hyperspectral image classification,” IEEE Trans. Geosci. Remote Sens., vol. 57, no. 8, pp. 5384–5394, 2019.
  * [92] E. Levina and P. J. Bickel, “Maximum likelihood estimation of intrinsic dimension,” in Proc. NIPS, pp. 777–784, 2005.
  * [93] J. M. Bioucas-Dias and J. M. Nascimento, “Hyperspectral subspace identification,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 8, pp. 2435–2445, 2008.
  * [94] J. M. Bioucas-Dias, A. Plaza, N. Dobigeon, M. Parente, Q. Du, P. Gader, and J. Chanussot, “Hyperspectral unmixing overview: Geometrical, statistical, and sparse regression-based approaches,” IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., vol. 5, no. 2, pp. 354–379, 2012.
  * [95] N. Keshava and J. F. Mustard, “Spectral unmixing,” IEEE Signal Process. Mag., vol. 19, no. 1, pp. 44–57, 2002.
  * [96] Z. Yang, G. Zhou, S. Xie, S. Ding, J.-M. Yang, and J. Zhang, “Blind spectral unmixing based on sparse nonnegative matrix factorization,” IEEE Tran. Image Process., vol. 20, no. 4, pp. 1112–1125, 2010.
  * [97] Y. Qian, S. Jia, J. Zhou, and A. Robles-Kelly, “Hyperspectral unmixing via $l_{1/2}$ sparsity-constrained nonnegative matrix factorization,” IEEE Trans. Geosci. Remote Sens., vol. 49, no. 11, pp. 4282–4297, 2011.
  * [98] J. Sigurdsson, M. O. Ulfarsson, and J. R. Sveinsson, “Hyperspectral unmixing with $l_{q}$ regularization,” IEEE Trans. Geosci. Remote Sens., vol. 52, no. 11, pp. 6793–6806, 2014.
  * [99] P. Thouvenin, N. Dobigeon, and J. Tourneret, “Hyperspectral unmixing with spectral variability using a perturbed linear mixing model,” IEEE Trans. Signal Process., vol. 64, no. 2, pp. 525–538, 2015.
  * [100] L. Drumetz, M. Veganzones, S. Henrot, R. Phlypo, J. Chanussot, and C. Jutten, “Blind hyperspectral unmixing using an extended linear mixing model to address spectral variability,” IEEE Trans. Image Process., vol. 25, no. 8, pp. 3890–3905, 2016.
  * [101] W. He, H. Zhang, and L. Zhang, “Total variation regularized reweighted sparse nonnegative matrix factorization for hyperspectral unmixing,” IEEE Trans. Geosci. Remote Sens., vol. 55, no. 7, pp. 3909–3921, 2017.
  * [102] J. Yao, D. Meng, Q. Zhao, W. Cao, and Z. Xu, “Nonconvex-sparsity and nonlocal-smoothness-based blind hyperspectral unmixing,” IEEE Trans. Image Process., vol. 28, no. 6, pp. 2991–3006, 2019.
  * [103] J. Liu, J. Zhang, Y. Gao, C. Zhang, and Z. Li, “Enhancing spectral unmixing by local neighborhood weights,” IEEE J. Sel. Top. Appl. Earth Obs. and Remote Sens., vol. 5, no. 5, pp. 1545–1552, 2012.
  * [104] X. Lu, H. Wu, Y. Yuan, P. Yan, and X. Li, “Manifold regularized sparse nmf for hyperspectral unmixing,” IEEE Trans. Geosci. Remote Sens., vol. 51, no. 5, pp. 2815–2826, 2012.
  * [105] W. Wang, Y. Qian, and Y. Y. Tang, “Hypergraph-regularized sparse nmf for hyperspectral unmixing,” IEEE J. Sel. Top. Appl. Earth Obs. and Remote Sens., vol. 9, no. 2, pp. 681–694, 2016.
  * [106] J. Qin, H. Lee, J. T. Chi, L. Drumetz, J. Chanussot, Y. Lou, and A. L. Bertozzi, “Blind hyperspectral unmixing based on graph total variation regularization,” IEEE Trans. Geosci. Remote Sens., 2020.  DOI: 10.1109/TGRS.2020.3020810.
  * [107] Y. Qian, F. Xiong, S. Zeng, J. Zhou, and Y. Y. Tang, “Matrix-vector nonnegative tensor factorization for blind unmixing of hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens., vol. 55, no. 3, pp. 1776–1792, 2016.
  * [108] T. Imbiriba, R. A. Borsoi, and J. C. M. Bermudez, “Low-rank tensor modeling for hyperspectral unmixing accounting for spectral variability,” IEEE Trans. Geosci. Remote Sens., vol. 58, no. 3, pp. 1833–1842, 2019.
  * [109] B. Palsson, J. Sigurdsson, J. R. Sveinsson, and M. O. Ulfarsson, “Hyperspectral unmixing using a neural network autoencoder,” IEEE Access, vol. 6, pp. 25646–25656, 2018.
  * [110] Y. Su, A. Marinoni, J. Li, J. Plaza, and P. Gamba, “Stacked nonnegative sparse autoencoders for robust hyperspectral unmixing,” IEEE Geosci. Remote Sens.Lett., vol. 15, no. 9, pp. 1427–1431, 2018.
  * [111] B. Palsson, J. R. Sveinsson, and M. O. Ulfarsson, “Spectral-spatial hyperspectral unmixing using multitask learning,” IEEE Access, vol. 7, pp. 148861–148872, 2019.
  * [112] Z. Han, D. Hong, L. Gao, B. Zhang, and J. Chanussot, “Deep half-siamese networks for hyperspectral unmixing,” IEEE Geosci. Remote Sens.Lett., 2020\.  DOI:10.1109/LGRS.2020.3011941.
  * [113] J. M. Nascimento and J. M. Dias, “Vertex component analysis: A fast algorithm to unmix hyperspectral data,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 4, pp. 898–910, 2005.
  * [114] C.-I. Chang and A. Plaza, “A fast iterative algorithm for implementation of pixel purity index,” IEEE Geosci. Remote Sens. Lett., vol. 3, no. 1, pp. 63–67, 2006.
  * [115] M. E. Winter, “N-findr: An algorithm for fast autonomous spectral end-member determination in hyperspectral data,” in Imaging Spectrometry V, vol. 3753, pp. 266–275, International Society for Optics and Photonics, 1999\.
  * [116] D. C. Heinz and C.-I. Chang, “Fully constrained least squares linear spectral mixture analysis method for material quantification in hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens., vol. 39, no. 3, pp. 529–545, 2001.
  * [117] R. Heylen, D. Burazerovic, and P. Scheunders, “Fully constrained least squares spectral unmixing by simplex projection,” IEEE Trans. Geosci. Remote Sens., vol. 49, no. 11, pp. 4112–4122, 2011.
  * [118] J. M. Bioucas-Dias and M. A. Figueiredo, “Alternating direction algorithms for constrained sparse regression: Application to hyperspectral unmixing,” in Proc. WHISPERS, pp. 1–4, IEEE, 2010.
  * [119] M.-D. Iordache, J. M. Bioucas-Dias, and A. Plaza, “Total variation spatial regularization for sparse hyperspectral unmixing,” IEEE Trans. Geosci. Remote Sens., vol. 50, no. 11, pp. 4484–4502, 2012.
  * [120] M.-D. Iordache, J. M. Bioucas-Dias, and A. Plaza, “Collaborative sparse regression for hyperspectral unmixing,” IEEE Trans. Geosci. Remote Sens., vol. 52, no. 1, pp. 341–354, 2013.
  * [121] X. Fu, W. Ma, J. Bioucas-Dias, and T. Chan, “Semiblind hyperspectral unmixing in the presence of spectral library mismatches,” IEEE Trans. Geosci. Remote Sens., vol. 54, no. 9, pp. 5171–5184, 2016.
  * [122] J. Huang, T.-Z. Huang, L.-J. Deng, and X.-L. Zhao, “Joint-sparse-blocks and low-rank representation for hyperspectral unmixing,” IEEE Trans. Geosci. Remote Sens., vol. 57, no. 4, pp. 2419–2438, 2018.
  * [123] D. Hong and X. Zhu, “SULoRA: Subspace unmixing with low-rank attribute embedding for hyperspectral data analysis,” IEEE J. Sel. Topics Signal Process., vol. 12, no. 6, pp. 1351–1363, 2018.
  * [124] D. Hong, N. Yokoya, J. Chanussot, and X. Zhu, “An augmented linear mixing model to address spectral variability for hyperspectral unmixing,” IEEE Trans. Image Process., vol. 28, no. 4, pp. 1923–1938, 2019.
  * [125] R. Heylen, M. Parente, and P. Gader, “A review of nonlinear hyperspectral unmixing methods,” IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., vol. 7, no. 6, pp. 1844–1868, 2014.
  * [126] N. Yokoya, C. Grohnfeldt, and J. Chanussot, “Hyperspectral and multispectral data fusion: A comparative review of the recent literature,” IEEE Geosci. Remote Sens. Mag., vol. 5, no. 2, pp. 29–56, 2017.
  * [127] M. T. Eismann and R. C. Hardie, “Application of the stochastic mixing model to hyperspectral resolution enhancement,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 9, pp. 1924–1933, 2004.
  * [128] N. Yokoya, T. Yairi, and A. Iwasaki, “Coupled nonnegative matrix factorization unmixing for hyperspectral and multispectral data fusion,” IEEE Trans. Geosci. Remote Sens., vol. 50, no. 2, pp. 528–537, 2012.
  * [129] Q. Wei, J. Bioucas-Dias, N. Dobigeon, and J. Tourneret, “Hyperspectral and multispectral image fusion based on a sparse representation,” IEEE Trans. Geosci. Remote Sens., vol. 53, no. 7, pp. 3658–3668, 2015.
  * [130] M. Simoes, J. Bioucas‐Dias, L. B. Almeida, and J. Chanussot, “A convex formulation for hyperspectral image superresolution via subspace-based regularization,” IEEE Trans. Geosci. Remote Sens., vol. 53, no. 6, pp. 3373–3388, 2015.
  * [131] W. He, Y. Chen, N. Yokoya, C. Li, and Q. Zhao, “Hyperspectral super-resolution via coupled tensor ring factorization,” arXiv preprint arXiv:2001.01547, 2020.
  * [132] L. Loncan, L. B. De Almeida, J. M. Bioucas-Dias, X. Briottet, J. Chanussot, N. Dobigeon, S. Fabre, W. Liao, G. A. Licciardi, M. Simoes, et al., “Hyperspectral pansharpening: A review,” IEEE Geosci. Remote Sens. Mag., vol. 3, no. 3, pp. 27–46, 2015.
  * [133] N. Akhtar, F. Shafait, and A. Mian, “Sparse spatio-spectral representation for hyperspectral image super-resolution,” in Proc. ECCV, pp. 63–78, Springer, 2014.
  * [134] N. Akhtar, F. Shafait, and A. Mian, “Bayesian sparse representation for hyperspectral image super resolution,” in Proc. CVPR, pp. 3631–3640, 2015\.
  * [135] W. Dong, F. Fu, G. Shi, X. Cao, J. Wu, G. Li, and X. Li, “Hyperspectral image super-resolution via non-negative structured sparse representation,” IEEE Trans. Image Process., vol. 25, no. 5, pp. 2337–2352, 2016.
  * [136] K. Wang, Y. Wang, X. L. Zhao, J. C. W. Chan, Z. Xu, and D. Meng, “Hyperspectral and multispectral image fusion via nonlocal low-rank tensor decomposition and spectral unmixing,” IEEE Trans. Geosci. Remote Sens., vol. 58, no. 11, pp. 7654–7671, 2020.
  * [137] Q. Wei, N. Dobigeon, and J.-Y. Tourneret, “Fast fusion of multi-band images based on solving a sylvester equation,” IEEE Trans. Image Process., vol. 24, no. 11, pp. 4109–4121, 2015.
  * [138] R. Kawakami, Y. Matsushita, J. Wright, M. Ben-Ezra, Y.-W. Tai, and K. Ikeuchi, “High-resolution hyperspectral imaging via matrix factorization,” in Proc. CVPR, pp. 2329–2336, IEEE, 2011.
  * [139] R. Wu, H.-T. Wai, and W.-K. Ma, “Hybrid inexact bcd for coupled structured matrix factorization in hyperspectral super-resolution,” IEEE Trans. Signal Process., vol. 68, pp. 1728–1743, 2020.
  * [140] S. Li, R. Dian, L. Fang, and J. M. Bioucas-Dias, “Fusing hyperspectral and multispectral images via coupled sparse tensor factorization,” IEEE Trans. Image Process., vol. 27, no. 8, pp. 4118–4130, 2018.
  * [141] C. I. Kanatsoulis, X. Fu, N. D. Sidiropoulos, and W.-K. Ma, “Hyperspectral super-resolution: A coupled tensor factorization approach,” IEEE Trans. Signal Process., vol. 66, no. 24, pp. 6503–6517, 2018.
  * [142] R. Dian, S. Li, and L. Fang, “Learning a low tensor-train rank representation for hyperspectral image super-resolution,” IEEE Trans. Neural Netw. Learn. Syst., vol. 30, no. 9, pp. 2672–2683, 2019.
  * [143] Y. Xu, Z. Wu, J. Chanussot, and Z. Wei, “Hyperspectral images super-resolution via learning high-order coupled tensor ring representation,” IEEE Trans. Neural Netw. Learn. Syst., 2020.
  * [144] R. Dian, L. Fang, and S. Li, “Hyperspectral image super-resolution via non-local sparse tensor factorization,” in Proc. CVPR, pp. 5344–5353, 2017\.
  * [145] Y. Wang, X. Chen, Z. Han, S. He, et al., “Hyperspectral image super-resolution via nonlocal low-rank tensor approximation and total variation regularization,” Remote Sens., vol. 9, no. 12, p. 1286, 2017\.
  * [146] Y. Xu, Z. Wu, J. Chanussot, and Z. Wei, “Nonlocal patch tensor sparse representation for hyperspectral image super-resolution,” IEEE Trans. Image Process., vol. 28, no. 6, pp. 3034–3047, 2019.
  * [147] S. Mei, X. Yuan, J. Ji, Y. Zhang, S. Wan, and Q. Du, “Hyperspectral image spatial super-resolution via 3d full convolutional neural network,” Remote Sens., vol. 9, no. 11, p. 1139, 2017.
  * [148] J. M. Haut, R. Fernandez-Beltran, M. E. Paoletti, J. Plaza, A. Plaza, and F. Pla, “A new deep generative network for unsupervised remote sensing single-image super-resolution,” IEEE Trans. Geosci.Remote Sens., vol. 56, no. 11, pp. 6792–6810, 2018.
  * [149] X. Liu, C. Deng, J. Chanussot, D. Hong, and B. Zhao, “Stfnet: A two-stream convolutional neural network for spatiotemporal image fusion,” IEEE Trans. Geosci. Remote Sens., vol. 57, no. 9, pp. 6552–6564, 2019.
  * [150] W. Liu and J. Lee, “An efficient residual learning neural network for hyperspectral image superresolution,” IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens., vol. 12, no. 4, pp. 1240–1253, 2019.
  * [151] K. Zheng, L. Gao, W. Liao, D. Hong, B. Zhang, X. Cui, and J. Chanussot, “Coupled convolutional neural network with adaptive response function learning for unsupervised hyperspectral super resolution,” IEEE Trans. Geosci. Remote Sens., 2020.
  * [152] T. Uezato, D. Hong, N. Yokoya, and W. He, “Guided deep decoder: Unsupervised image pair fusion,” in Proc. ECCV, pp. 87–102, Springer, 2020.
  * [153] J. Yao, D. Hong, J. Chanussot, D. Meng, X. Zhu, and Z. Xu, “Cross-attention in coupled unmixing nets for unsupervised hyperspectral super-resolution,” in Proc. ECCV, pp. 208–224, Springer, 2020.
  * [154] C. Lanaras, E. Baltsavias, and K. Schindler, “Hyperspectral super-resolution by coupled spectral unmixing,” in Proc. ICCV, pp. 3586–3594, 2015.
  * [155] L. Wald, “Quality of high resolution synthesised images: Is there a simple criterion?,” in Proc. ICFED, pp. 99–103, 2000.
  * [156] Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process., vol. 13, no. 4, pp. 600–612, 2004.
  * [157] M. Dalla Mura, S. Prasad, F. Pacifici, P. Gamba, J. Chanussot, and J. A. Benediktsson, “Challenges and opportunities of multimodality and data fusion in remote sensing,” Proc. IEEE, vol. 103, no. 9, pp. 1585–1601, 2015.
  * [158] D. Hong, L. Gao, N. Yokoya, J. Yao, J. Chanussot, D. Qian, and B. Zhang, “More diverse means better: Multimodal deep learning meets remote-sensing imagery classification,” IEEE Trans. Geosci. Remote Sens., 2020.  DOI: 10.1109/TGRS.2020.3016820.
  * [159] D. Hong, N. Yokoya, G.-S. Xia, J. Chanussot, and X. X. Zhu, “X-modalnet: A semi-supervised deep cross-modal network for classification of remote sensing data,” ISPRS J. Photogramm. Remote Sens., vol. 167, pp. 12–23, 2020.
  * [160] J. Ngiam, A. Khosla, M. Kim, J. Nam, H. Lee, and A. Ng, “Multimodal deep learning,” in Proc. ICML, pp. 689–696, 2011.
  * [161] C. Wang and S. Mahadevan, “A general framework for manifold alignment.,” in Proc. AAAI, pp. 79–86, 2009.
  * [162] G. Matasci, M. Volpi, D. Tuia, and M. Kanevski, “Transfer component analysis for domain adaptation in image classification,” in Image and Signal Processing for Remote Sensing XVII, vol. 8180, p. 81800F, International Society for Optics and Photonics, 2011.
  * [163] D. Tuia, M. Volpi, M. Trolliet, and G. Camps-Valls, “Semisupervised manifold alignment of multimodal remote sensing images,” IEEE Trans. Geosci. Remote Sens., vol. 52, no. 12, pp. 7708–7720, 2014.
  * [164] C. Wang and S. Mahadevan, “Heterogeneous domain adaptation using manifold alignment,” in Proc. IJCAI, vol. 22, p. 1541, 2011.
  * [165] G. Matasci, M. Volpi, M. Kanevski, L. Bruzzone, and D. Tuia, “Semisupervised transfer component analysis for domain adaptation in remote sensing image classification,” IEEE Trans. Geosci. Remote Sens., vol. 53, no. 7, pp. 3550–3564, 2015.
  * [166] S. J. Pan, I. W. Tsang, J. T. Kwok, and Q. Yang, “Domain adaptation via transfer component analysis,” IEEE Trans. Neural Netw., vol. 22, no. 2, pp. 199–210, 2011.
  * [167] D. Tuia and G. Camps-Valls, “Kernel manifold alignment for domain adaptation,” PloS One, vol. 11, no. 2, p. e0148655, 2016.
  * [168] J. Hu, D. Hong, Y. Wang, and X. X. Zhu, “A comparative review of manifold learning techniques for hyperspectral and polarimetric sar image fusion,” Remote Sens., vol. 11, no. 6, p. 681, 2019.
  * [169] J. Hu, D. Hong, and X. X. Zhu, “Mima: Mapper-induced manifold alignment for semi-supervised fusion of optical image and polarimetric sar data,” IEEE Trans. Geosci. Remote Sens., vol. 57, no. 11, pp. 9025–9040, 2019.
  * [170] D. Hong, N. Yokoya, J. Chanussot, and X. X. Zhu, “CoSpace: Common subspace learning from hyperspectral-multispectral correspondences,” IEEE Trans. Geos. Remote Sens., vol. 57, no. 7, pp. 4349–4359, 2019.
  * [171] D. Hong, N. Yokoya, N. Ge, J. Chanussot, and X. Zhu, “Learnable manifold alignment (LeMA): A semi-supervised cross-modality learning framework for land cover and land use classification,” ISPRS J. Photogramm. Remote Sens., vol. 147, pp. 193–205, 2019.
  * [172] D. Hong, J. Chanussot, N. Yokoya, J. Kang, and X. X. Zhu, “Learning-shared cross-modality representation using multispectral-lidar and hyperspectral data,” IEEE Geosci. Remote Sens. Lett., vol. 17, no. 8, pp. 1470–1474, 2020.
  * [173] D. Hong, J. Kang, N. Yokoya, and J. Chanussot, “Graph-induced aligned learning on subspaces for hyperspectral and multispectral data,” IEEE Trans. Geosci. Remote Sens., 2020.  DOI: 10.1109/TGRS.2020.3021140.
  * [174] J. R. Hershey, J. L. Roux, and F. Weninger, “Deep unfolding: Model-based inspiration of novel deep architectures,” arXiv preprint arXiv:1409.2574, 2014.
  * [175] S. Diamond, V. Sitzmann, F. Heide, and G. Wetzstein, “Unrolled optimization with deep priors,” arXiv preprint arXiv:1705.08041, 2017.

|  Danfeng Hong (S’16–M’19) received the M.Sc. degree (summa cum laude) in
computer vision, College of Information Engineering, Qingdao University,
Qingdao, China, in 2015, the Dr. -Ing degree (summa cum laude) in Signal
Processing in Earth Observation (SiPEO), Technical University of Munich (TUM),
Munich, Germany, in 2019. Since 2015, he worked as a Research Associate at the
Remote Sensing Technology Institute (IMF), German Aerospace Center (DLR),
Oberpfaffenhofen, Germany. Currently, he is a research scientist and leads a
Spectral Vision working group at IMF, DLR, and also an adjunct scientist in
GIPSA-lab, Grenoble INP, CNRS, Univ. Grenoble Alpes, Grenoble, France. His
research interests include signal / image processing and analysis,
hyperspectral remote sensing, machine / deep learning, artificial intelligence
and their applications in Earth Vision.
---|---
|  Wei He (S’14-M’17) received the B.S. degree in School of Mathematics and
statistics and the Ph.D degree in Surveying, Mapping and Remote Sensing
(LIESMARS) from Wuhan University, Wuhan, China, in 2012 and 2017,
respectively. He was a researcher at RIKEN Center for Advanced Intelligence
Project, from 2018 to 2020. He is currently a research scientist with the
Geoinformatics unit, RIKEN Center for Advanced Intelligence Project, Japan.
His research interests include image quality improvement, remote sensing image
processing, matrix/tensor analysis and deep learning.
---|---
|  Naoto Yokoya (S’10–M’13) received the M.Eng. and Ph.D. degrees from the
Department of Aeronautics and Astronautics, the University of Tokyo, Tokyo,
Japan, in 2010 and 2013, respectively. He is currently a Lecturer at the
University of Tokyo and a Unit Leader at the RIKEN Center for Advanced
Intelligence Project, Tokyo, Japan, where he leads the Geoinformatics Unit. He
was an Assistant Professor at the University of Tokyo from 2013 to 2017. In
2015-2017, he was an Alexander von Humboldt Fellow, working at the German
Aerospace Center (DLR), Oberpfaffenhofen, and Technical University of Munich
(TUM), Munich, Germany. His research is focused on the development of image
processing, data fusion, and machine learning algorithms for understanding
remote sensing images, with applications to disaster management. Dr. Yokoya
won the first place in the 2017 IEEE Geoscience and Remote Sensing Society
(GRSS) Data Fusion Contest organized by the Image Analysis and Data Fusion
Technical Committee (IADF TC). He is the Chair (2019-2021) and was a Co-Chair
(2017-2019) of IEEE GRSS IADF TC and also the secretary of the IEEE GRSS All
Japan Joint Chapter since 2018. He is an Associate Editor for the IEEE Journal
of Selected Topics in Applied Earth Observations and Remote Sensing (JSTARS)
since 2018. He is/was a Guest Editor for the IEEE JSTARS in 2015-2016, for
Remote Sensing in 2016-2020, and for the IEEE Geoscience and Remote Sensing
Letters (GRSL) in 2018-2019.
---|---
|  Jing Yao received the B.Sc. degree from Northwest University, Xi’an, China,
in 2014, and the Ph.D. degree in the School of Mathematics and Statistics,
Xi’an Jiaotong University, Xi’an, China, in 2021. He is currently an Assistant
Professor with the Key Laboratory of Digital Earth Science, Aerospace
Information Research Institute, Chinese Academy of Sciences, Beijing, China.
From 2019 to 2020, he was a visiting student at Signal Processing in Earth
Observation (SiPEO), Technical University of Munich (TUM), Munich, Germany,
and at the Remote Sensing Technology Institute (IMF), German Aerospace Center
(DLR), Oberpfaffenhofen, Germany. His research interests include low-rank
modeling, hyperspectral image analysis and deep learning-based image
processing methods.
---|---
|  Lianru Gao (M’12–SM’18) received the B.S. degree in civil engineering from
Tsinghua University, Beijing, China, in 2002, the Ph.D. degree in cartography
and geographic information system from Institute of Remote Sensing
Applications, Chinese Academy of Sciences (CAS), Beijing, China, in 2007. He
is currently a Professor with the Key Laboratory of Digital Earth Science,
Aerospace Information Research Institute, CAS. He also has been a visiting
scholar at the University of Extremadura, Cáceres, Spain, in 2014, and at the
Mississippi State University (MSU), Starkville, USA, in 2016. His research
focuses on hyperspectral image processing and information extraction. In last
ten years, he was the PI of 10 scientific research projects at national and
ministerial levels, including projects by the National Natural Science
Foundation of China (2010-2012, 2016-2019, 2018-2020), and by the Key Research
Program of the CAS (2013-2015). He has published more than 160 peer-reviewed
papers, and there are more than 80 journal papers included by SCI. He was
coauthor of an academic book entitled “Hyperspectral Image Classification And
Target Detection”. He obtained 28 National Invention Patents in China. He was
awarded the Outstanding Science and Technology Achievement Prize of the CAS in
2016, and was supported by the China National Science Fund for Excellent Young
Scholars in 2017, and won the Second Prize of The State Scientific and
Technological Progress Award in 2018. He received the recognition of the Best
Reviewers of the IEEE Journal of Selected Topics in Applied Earth Observations
and Remote Sensing in 2015, and the Best Reviewers of the IEEE Transactions on
Geoscience and Remote Sensing in 2017.
---|---
|  Jocelyn Chanussot (M’04–SM’04–F’12) received the M.Sc. degree in electrical
engineering from the Grenoble Institute of Technology (Grenoble INP),
Grenoble, France, in 1995, and the Ph.D. degree from the Université de Savoie,
Annecy, France, in 1998. Since 1999, he has been with Grenoble INP, where he
is currently a Professor of signal and image processing. His research
interests include image analysis, hyperspectral remote sensing, data fusion,
machine learning and artificial intelligence. He has been a visiting scholar
at Stanford University (USA), KTH (Sweden) and NUS (Singapore). Since 2013, he
is an Adjunct Professor of the University of Iceland. In 2015-2017, he was a
visiting professor at the University of California, Los Angeles (UCLA). He
holds the AXA chair in remote sensing and is an Adjunct professor at the
Chinese Academy of Sciences, Aerospace Information research Institute,
Beijing. Dr. Chanussot is the founding President of IEEE Geoscience and Remote
Sensing French chapter (2007-2010) which received the 2010 IEEE GRS-S Chapter
Excellence Award. He has received multiple outstanding paper awards. He was
the Vice-President of the IEEE Geoscience and Remote Sensing Society, in
charge of meetings and symposia (2017-2019). He was the General Chair of the
first IEEE GRSS Workshop on Hyperspectral Image and Signal Processing,
Evolution in Remote sensing (WHISPERS). He was the Chair (2009-2011) and
Cochair of the GRS Data Fusion Technical Committee (2005-2008). He was a
member of the Machine Learning for Signal Processing Technical Committee of
the IEEE Signal Processing Society (2006-2008) and the Program Chair of the
IEEE International Workshop on Machine Learning for Signal Processing (2009).
He is an Associate Editor for the IEEE Transactions on Geoscience and Remote
Sensing, the IEEE Transactions on Image Processing and the Proceedings of the
IEEE. He was the Editor-in-Chief of the IEEE Journal of Selected Topics in
Applied Earth Observations and Remote Sensing (2011-2015). In 2014 he served
as a Guest Editor for the IEEE Signal Processing Magazine. He is a Fellow of
the IEEE, a member of the Institut Universitaire de France (2012-2017) and a
Highly Cited Researcher (Clarivate Analytics/Thomson Reuters).
---|---
|  Liangpei Zhang (M’06-SM’08-F’19) received the B.S. degree in physics from
Hunan Normal University, Changsha, China, in 1982, the M.S. degree in optics
from the Xi’an Institute of Optics and Precision Mechanics, Chinese Academy of
Sciences, Xi’an, China, in 1988, and the Ph.D. degree in photogrammetry and
remote sensing from Wuhan University, Wuhan, China, in 1998. He is a “Chang-
Jiang Schola” chair professor appointed by the ministry of education of China
in state key laboratory of information engineering in surveying, mapping, and
remote sensing (LIESMARS), Wuhan University. He was a principal scientist for
the China state key basic research project (2011-2016) appointed by the
ministry of national science and technology of China to lead the remote
sensing program in China. He has published more than 700 research papers and
five books. He is the Institute for Scientific Information (ISI) highly cited
author. He is the holder of 30 patents. His research interests include
hyperspectral remote sensing, high-resolution remote sensing, image
processing, and artificial intelligence. Dr. Zhang is a Fellow of Institute of
Electrical and Electronic Engineers (IEEE) and the Institution of Engineering
and Technology (IET). He was a recipient of the 2010 best paper Boeing award,
the 2013 best paper ERDAS award from the American society of photogrammetry
and remote sensing (ASPRS) and 2016 best paper theoretical innovation award
from the international society for optics and photonics (SPIE). His research
teams won the top three prizes of the IEEE GRSS 2014 Data Fusion Contest, and
his students have been selected as the winners or finalists of the IEEE
International Geoscience and Remote Sensing Symposium (IGARSS) student paper
contest in recent years. He also serves as an associate editor or editor of
more than ten international journals. Dr. Zhang is currently serving as an
associate editor of the IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING.
Dr. Zhang is the founding chair of IEEE Geoscience and Remote Sensing Society
(GRSS) Wuhan Chapter.
---|---
|  Xiao Xiang Zhu (M’12–SM’14–F’20) received the Master (M.Sc.) degree, her
doctor of engineering (Dr.-Ing.) degree and her “Habilitation” in the field of
signal processing from Technical University of Munich (TUM), Munich, Germany,
in 2008, 2011 and 2013, respectively. She is currently the Professor for Data
Science in Earth Observation (former: Signal Processing in Earth Observation)
at Technical University of Munich (TUM) and the Head of the Department “EO
Data Science” at the Remote Sensing Technology Institute, German Aerospace
Center (DLR). Since 2019, Zhu is a co-coordinator of the Munich Data Science
Research School (www.mu-ds.de). Since 2019 She also heads the Helmholtz
Artificial Intelligence – Research Field “Aeronautics, Space and Transport”.
Since May 2020, she is the director of the international future AI lab ”AI4EO
– Artificial Intelligence for Earth Observation: Reasoning, Uncertainties,
Ethics and Beyond”, Munich, Germany. Since October 2020, she also serves as a
co-director of the Munich Data Science Institute (MDSI), TUM. Prof. Zhu was a
guest scientist or visiting professor at the Italian National Research Council
(CNR-IREA), Naples, Italy, Fudan University, Shanghai, China, the University
of Tokyo, Tokyo, Japan and University of California, Los Angeles, United
States in 2009, 2014, 2015 and 2016, respectively. Her main research interests
are remote sensing and Earth observation, signal processing, machine learning
and data science, with a special application focus on global urban mapping.
Dr. Zhu is a member of young academy (Junge Akademie/Junges Kolleg) at the
Berlin-Brandenburg Academy of Sciences and Humanities and the German National
Academy of Sciences Leopoldina and the Bavarian Academy of Sciences and
Humanities. She is a Fellow of IEEE, serves as an associate Editor of IEEE
Transactions on Geoscience and Remote Sensing and as the area editor
responsible for special issues of IEEE Signal Processing Magazine.
---|---


Using Machine Learning
to disentangle LHC signatures of Dark Matter candidates

C. K. Khosa1, V. Sanz1,2,3, M. Soughton1

1 Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH,
UK

2 Departament de Física Teòrica and IFIC, Universitat de València-CSIC,
E-46100, Burjassot, Spain

3 Alan Turing Institute, British Library, 96 Euston Road, London NW1 2DB, UK

*<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
## Abstract

We study the prospects of characterising Dark Matter at colliders using
Machine Learning (ML) techniques. We focus on the monojet and missing
transverse energy (MET) channel and propose a set of benchmark models for the
study: a typical WIMP Dark Matter candidate in the form of a SUSY neutralino,
a pseudo-Goldstone impostor in the shape of an Axion-Like Particle, and a
light Dark Matter impostor whose interactions are mediated by a heavy
particle. All these benchmarks are tensioned against each other, and against
the main SM background ($Z$+jets). Our analysis uses both the leading-order
kinematic features as well as the information of an additional hard jet. We
explore different representations of the data, from a simple event data sample
with values of kinematic variables fed into a Logistic Regression algorithm or
a Fully Connected Neural Network, to a transformation of the data into images
related to probability distributions, fed to Deep and Convolutional Neural
Networks. We also study the robustness of our method against including
detector effects, dropping kinematic variables, or changing the number of
events per image. In the case of signals with more combinatorial possibilities
(events with more than one hard jet), the most crucial data features are
selected by performing a Principal Component Analysis. We compare the
performance of all these methods, and find that using the 2D images of the
combined information of multiple events significantly improves the
discrimination performance.

###### Contents

  1. 1 Introduction
  2. 2 Description of Benchmark Models
  3. 3 Kinematic Distributions
  4. 4 DM Characterization using Machine Learning
    1. 4.1 Logistic Regression
    2. 4.2 Neural Networks-kinematic features
    3. 4.3 DNN with 2D histograms
      1. 4.3.1 Monojet
      2. 4.3.2 Dijet
    4. 4.4 CNN with 2D histograms
  5. 5 Conclusions and Outlook
  6. 6 Acknowledgments
  7. A Analysis set-up
    1. A.1 Leading-order(LO) parton-level simulation
    2. A.2 Leading-order and Next-to-Leading order (NLO) detector-level simulation
  8. B PCA correlations
  9. C Signal to background analysis
  10. D Classifier performance at different levels of signal

##  1 Introduction

After the Higgs boson discovery [1, 2] at the Large Hadron Collider (LHC),
some of the discovery potential focus has shifted towards Dark Matter (DM)
searches. The discovery of DM and its characterisation would have profound
consequences in Particle Physics, Cosmology and Astrophysics and the LHC could
be the key to it. In spite of having experimental evidence of the presence of
DM, we do not know what is its true nature, its mass scale, spin and
interactions; or even if DM is a particle or a whole sector of new particles
and interactions as in the SM.

The unknown properties of DM open the possibility of many different types of
DM candidates being consistent with the DM relic density determination from
the Cosmic Microwave Background (CMB), as well as other astrophysical
constraints. To further explore DM, searches are being conducted in three main
directions: underground experiments aiming to directly detect the interaction
of DM with nuclei (direct detection) [3], astrophysical observatories
searching for an excess of light or charged particles in the sky (indirect
detection) [4], and collider searches for imprints of DM in collisions of
protons and leptons (collider searches) [5].

Among the hypothesized DM candidates, the category of Weakly-Interacting
Massive Particles (WIMPs) enjoy a privileged position, as a WIMP DM could in
turn link to other issues plaguing the Standard Model (SM) of Particle
Physics. Indeed, the WIMP paradigm is realised in many extensions of the SM,
such as Supersymmetry (SUSY), where the WIMP is typically a new stable
Majorana fermion with electroweak couplings and mass linked to the breaking of
SUSY.

Typical DM collider searches are based on the idea that a stable and neutral
particle, if produced at colliders, would leave the detector without
resistance. Hence, the collider strategy is to search for traces of the DM
presence via the associated production of other particles, namely the
identification of singular objects within the detector (mono searches), where
a single object could be a jet, W or Z boson, top quark, photon or $t\bar{t}$
pair. The motivation for using these channels is that DM candidates (which
cannot be directly detected) could be exposed through a momentum-mismatch in
the final state, where the detected objects appear to recoil against nothing
[6]. Note that collider stability is the most common assumption for DM
searches but that a variety of DM scenarios could also be indirectly probed at
the LHC via displaced track and vertices signatures using the long lived
particle searches, see e.g. ref. [7] and references therein.

In this paper we focus on collider-stable particles, i.e. particles which do
not interact with the detector and do not decay before leaving it, hence
manifesting as missing momentum and mimicking DM particles. Note that a
collider-stable particle could be completely stable (as is thought to be the
case for DM particles) or be unstable but with a lifetime long enough such
that it is not likely to decay before it leaves the detector. In our search
for New Physics at the LHC, we must then attempt to remove this degeneracy.
This task is even more challenging than analysing the signal and background
for DM searches because here we are concerned with two or more (unknown)
physics models.

The challenge of disentangling different DM candidates would likely benefit
from the implementation of new and more sophisticated techniques, beyond the
conventional search strategies. This difficult task calls for the use of
Machine Learning (ML), which is emerging as a powerful tool for New Physics
searches at LHC (see e.g. the recent review by Radovic et al. [8] and
references therein).

More conventional ML methods like boosted decision trees (BDTs) have already
been incorporated into many data analysis packages which made a significant
impact in the analysis. Several recent studies have shown exciting
applications of the ML methods for various tasks, like constraining Wilson
coefficients of higher-dimensional operators in the EFT framework [9, 10, 11],
top-tagging [12, 13], cosmological phase transitions [14], parameter exclusion
in SUSY models [15], quark-gluon tagging [16] etc. ML techniques have also
been used recently for non-collider DM searches using substructure probes [17,
18], for cosmological DM [19] and in direct detection experiments [20].

In this work, we will apply ML techniques to explore the possibility of
disentangling different scenarios for DM and impostors. Initially, we will
focus on supervised methods and leave for the future a more model-independent
unsupervised or semi-supervised explorations. We will consider the canonical
search for DM: events with one jet recoiling against missing transverse energy
(monojet searches [21, 22]), and use Machine Learning techniques in DM signal
characterization.

The main focus of this paper is to explore the potential of ML techniques for
the characterization of the discovered DM candidate 111However in the appendix
we also perform the classification task for the SM background versus signal
benchmarks.. Therefore our task is to explore the features of the signal and
to be able to distinguish one signal from another. We will compare the
features of DM signals from different Beyond the Standard Model (BSM) models.
Specifically, we will use as benchmarks of comparison three types of models: a
heavy WIMP dark matter from SUSY [23], Axion-Like Particles (ALPs) [24, 25]
and a simplified DM model with a spin-1 mediator. These will provide enough
variety of characteristics to analyse differences and degeneracies among
models.

Note we will base our analysis solely on differential information, not on
overall cross-sections. The reason to restrict the analysis on kinematics
alone is due to the freedom one has on the values of couplings in each model
which impact the production cross section and the branching ratios to specific
final states. For example, the production cross section of SUSY WIMPs depends
strongly on its nature, e.g. Bino-like or Higgsino-like would lead to
different values on the total number of events, yet differential kinematics
(divided by the total number of events) would not sizeably change. By
restricting the analysis to differential information, we can draw more robust
conclusions about the ability to distinguish different scenarios. Note though
that discovery potential does depend on both discrimination power and
production cross section, and this work focuses on the aspects of the analysis
which can then be translated into different benchmarks.

The paper is organised as follows. In the next section 2 we describe the
models which we use for benchmarking DM scenarios. In Sec. 3, we discuss
kinematic features in the monojet signal, both at leading-order in the QCD
expansion (LO) and next-to-leading-order (NLO) and show differences between
the SUSY benchmark and the SM background. In section 4 we describe the ML
methods and address the WIMP DM characterization using ML methods considered.
In the last section, we discuss our findings and conclude.

Figure 1: Feynman diagrams for monojet process in the linear ALP (top), EFT
framework with a massive spin-1 mediator (middle) and SUSY WIMP DM cases
(bottom).

##  2 Description of Benchmark Models

In this section we describe our choices of benchmark models. These models span
a large enough range of kinematic features to compare with SM processes, as
well as to see the strength and limitations of the task of disentangling
different DM scenarios.

Model | Mass | Type of coupling
---|---|---
SUSY1 |  $m_{\tilde{\chi}^{0}}$= 100 GeV | Bino-like
SUSY2 |  $m_{\tilde{\chi}^{0}}$= 200 GeV | Bino-like
SUSY3 |  $m_{\tilde{\chi}^{0}}$= 300 GeV | Bino-like
ALP | negligible | gluon-ALP
EFT | negligible | 4-fermion
Table 1: Summary of benchmark models.

We first consider a set of three benchmarks in the WIMP DM scenario based on
Bino-like SUSY neutralinos and with masses in the range 100 to 300 GeV, see
Table 1. We did not consider heavier WIMPs as they would likely be very hard
to find at the LHC in monojet events. The cross section of production in
monojet final states decreases quickly with the WIMP mass [26].

To contrast against the WIMP we consider two alternative cases. The first is
an Axion-Like Particle (or ALP) which is a paradigm for signatures from
pseudo-Goldstone bosons. They can be light and collider-stable due to the
derivative (suppressed) nature of their couplings. ALPs themselves could be a
DM particle or can be a DM mediator (see e.g. [27]). These exotic particles
are constrained by Astrophysics as well as colliders in a complementary
fashion [24, 28]. ALPs are also searched in axion experiments, which are
designed to target their couplings with the photons.

In this work, we do not restrict the ALP to be a DM candidate, as it could
decay after being produced, just not inside the detector 222If the ALP decays
inside the detector, its detection would still be difficult due to its
lightness. Nevertheless, one can still search for its effects on tails of
distributions [29].. It escapes detection as it has no charge under
$SU(3)_{c}\times U(1)_{Y}$, hence its signatures are the same as those of DM
in mono-searches. For the monojet channel, the ALP relevant couplings are to
gluons, as couplings to quarks are mass-suppressed:

$\displaystyle\mathcal{L}_{a}\supset-\frac{g_{agg}}{2}a\,\text{Tr}\left[G_{\mu\nu}\tilde{G}^{\mu\nu}\right].$
(1)

Note that the ALP-gluon coupling has the following bound [24, 25]

$g_{agg}\lesssim 1.1\cdot
10^{-5}\,\mathrm{GeV^{-1}}\quad(90\%\,\,\text{C.L.})\quad\text{for}\quad
m_{a}\lesssim 60\,\mathrm{MeV}$ (2)

Note that coupling of ALPs with photons or massive particles would lead to
mono-photon, -W, -Z, -top and -Higgs signatures.

We consider a second alternative scenario to WIMP DM based on light DM
produced from the decay of the heavy mediator. We label this EFT DM, as the
effective interaction between the SM and DM is via a four-fermion higher-order
operator. The simplified model Lagrangian which describes the interaction of
DM ($\chi$) with the vector mediator ($Y$) is given by

$\mathcal{L}_{Y}=\bar{\chi}\gamma_{\mu}g^{V}_{\chi}\chi Y^{\mu}.$ (3)

and similarly for the interaction between the mediator and quarks. Note that
the kinematic distribution of events is not very sensitive to the dark matter
mass (see Fig. A1 in the appendix) in the limit of $m_{Y}\gg m_{\chi}$.

Within these models, monojet signatures would result from the processes shown
in Fig. 1: a pair of DM particles (or a single ALP) produced in association
with one initial state radiation (ISR) gluon or quark.

Finally, we consider the dominant SM background given by $Z$+jets, where the
$Z$ boson decays to neutrinos.

Figure 2: Monojet detector-level 1D histograms for WIMP, EFT and ALP
scenarios, as well as the SM background.

##  3 Kinematic Distributions

After presenting our benchmark scenarios for New Physics, in this section we
describe the kinematic features which the Machine Learning algorithms will be
able to optimize over. The aim of this section is to explain some of the
features that the ML results will show.

The input for the ML studies is a list of events with their kinematic features
simulated using a particular benchmark model. We perform both parton- and
detector-level simulations to generate the data samples. More details related
to event generation can be found in Appendix A.

The kinematic features of those events are generically multi-dimensional, but
one can project into 1D and 2D kinematic space. For example, in Fig. 2 we show
a set of 1D distributions for the monojet process 333The overall behaviour is
indeed maintained from parton-level to detector-level.. Note that the jet
transverse momentum in SUSY1 (lightest SUSY WIMP) and ALP is very similar, and
also close to the SM background distribution, whereas the other SUSY scenarios
and the EFT exhibit a harder spectrum and easier discrimination with respect
to the SM. Additionally, the pseudorapidity $\eta$ distribution of the jet is
very similar for the ALP and SM background cases. As the EFT case leads to
harder $p_{T}$ spectrum and easier to differentiate from ALPs than the light
SUSY case. We will see how the ML techniques will exhibit a similar trend.
ALPs will be hard to separate from SUSY1, or light DM, whereas EFT and SUSY3
will have the most overlap, as both exhibit similar hard spectra.

At the level of 1D distributions, one cannot distinguish any preferred
direction of the azimuthal angle $\phi$ distribution for both the New Physics
signals and SM background processes.

Figure 3: Detector-level 1D histograms for the dijet process for WIMP, EFT and
ALP scenarios, as well as the SM background.

Additional information can be obtained when moving from 1D to 2D
distributions, as we will show in section 4.3 (where we show the event
distribution in the 2D plane of $p_{T}^{j}$ vs $\eta$) and is discussed (in
the Bayesian context) in Ref. [30]. Different models are compared to each
other in this plane. The characteristics of these rough features, broadening
in pseudorapidity and $p_{T}$ reach, and what conclusions one can draw from
them, depend on level of accuracy of our simulation. To explore robustness
against showering and detector effects, we promoted the simulation to the
Pythia and Delphes level.

Finally, we discuss how one would use another source of information from these
monojet topologies. Monojet events do often contain an additional hard jet.
Splitting or additional ISR emission is not extremely rare, and with an
additional object in the final state more information can be extracted of the
DM nature. To account for the additional jet, we simulate detector-level
events for the monojet (LO) and dijet (NLO) production. For all the cases, we
consider data samples of 50K events with one additional jet of $p_{T}>$ 25
GeV. 1D histograms for the constructed kinematic variable are shown in Fig. 3.
An additional jet provides a new source of kinematic information, enhancing
the discriminating power of our study. For example, notice that the
distributions of $\Delta\phi_{j_{1}j_{2}}$ ($j_{1}$: leading jet, $j_{2}$:
sub-leading jet) and $\Delta\phi_{MET}^{j_{2}}$ are quite characteristic for
the signals and background.

##  4 DM Characterization using Machine Learning

In this section we start our discussion on the use of supervised Machine
Learning techniques for Dark Matter signal characterization, i.e. we focus on
disentangling the different DM signals from each other rather than the
discovery of any DM signal from the background. In other words, we are
assuming that an excess of events has been identified at the LHC, and one is
trying to unveil its true origin. We then ask the question: Could there be
contamination of the SUSY imposters in the vanilla SUSY WIMP events? Assuming
a similar size of signal events, the main question we shall endeavour to
answer is then whether it is possible to disentangle an electroweak-scale WIMP
DM signal from an ALP or an EFT signal with very light DM.

The input data we used for the analysis consists of three kinematic features
of the monojet i.e. $p_{T}$, $\eta_{j}$ and $\phi_{j}$. We use this data both
as an array input and also in the form of 2D histograms to build a ML
algorithm.

In the following sections, we will compare the performance of different
methods for parton-level and detector-level simulations. For this analysis, we
will consider the kinematics of the leading jet ($p_{T}>130$ GeV) for the
monojet analysis and in addition to this also of the sub-leading jet for the
dijet data sample. For the reasons mentioned in Sec. 2, we do not use the
cross-section information, so a balanced dataset is considered for all the
classes. Data samples are divided in 70$\%$ $:$ 30$\%$ proportions for the
training and test samples444A portion of the data sample which will not be
shown to the network while training. Rather, it will be used to assess the
performance of the network as on the unseen data set..

###  4.1 Logistic Regression

As a first step, we perform logistic regression for the monojet data without
any data processing. We used SGDclassifier of sklearn python library with a
log-loss function. For the parton level events, Receiver operating
characteristic (ROC) curves for SUSY signals vs ALP and EFT signals are shown
in Fig 4. We consider the three benchmark values for the neutralino mass in
the WIMP scenario described in Sec. 2. We can see that the AUC555AUC (area
under the ROC curve) is a measure of the algorithm’s performance. value varies
from 0.50 to 0.64 for the SUSY3 versus EFT and SUSY3 versus ALP, and the other
four cases lie in-between. In other words, one can easily separate heavy WIMPs
from the ALP monojet, however EFT monojet would not be efficiently classified.
The regularization parameter $\lambda=10^{-5}$ is used for all the cases.

Figure 4: ROC curves using logistic regression for different WIMP scenarios
versus ALP/EFT signals (monojet process).

###  4.2 Neural Networks-kinematic features

We investigate the classification accuracy using Deep Neural Networks (DNN)
with the same input features i.e. $p_{T}$, $\eta$ and $\phi$ of the jet. We
used five fully-connected hidden layers for the network as including more
layers does not improve the performance.

For all the layers, except the final one, the number of neurons is equal to
the number of data features that are considered. For the intermediate layers,
a ReLU activation function is used, whereas we use a ‘sigmoid’ function for
the output layer. We considered the binary cross-entropy loss function. The
‘dropouts’666It regularizes the network by deactivating the fraction of
neurons. option is also activated, with 0.2 as the optimized choice. Finally,
the ‘adadelta’ optimizer777It is used to update NN weights for loss function
minimization., and batch-size and epochs are set to 500 and 300, respectively.

Before discussing the performance of DNN for the signal classification, we
discuss some aspects of the analysis when using DNNs with the three kinematic
inputs: the first aspect relates to the robustness of the analysis with
changes in the level of simulation detail. We see from the similarity between
figures A3 and A4 in Appendix C for the signal versus background scenario that
the DNN only performs slightly worse when using the more noisy detector-level
data than the pure parton-level data. We also compared the performance of DNNs
with just two variables (by excluding the $\phi$ variable) and found that
there is no degradation in the classification accuracy. This is expected, as
the three input variables are redundant once energy-momentum conservation
within the event is taken into account888We also explore the benefit of
considering dijet processes in addition to monojet for the signal versus
background scenario. As mentioned previously, dijet searches have the
potential to provide more information on the nature of DM particles. In this
case we consider eight features:
$p_{T}^{j_{1}},p_{T}^{j_{2}},\eta^{j_{1}},\eta^{j_{2}}$, MET,
$\Delta\phi_{j_{1}j_{2}}$, $\Delta\phi_{MET}^{j_{1}}$, and
$\Delta\phi_{MET}^{j_{2}}$. Using the DNN with the same architecture for this
data sample we get indeed an enhancement in the classification accuracy as
shown in Fig. A5 in Appendix C.

Figure 5: ROC curves using Neural Network for different WIMP scenarios versus
ALP/EFT signals using the raw kinematic features (monojet process).

We use DNNs with kinematic features as input for the classification of
different signals. As mentioned earlier, classification accuracy is not very
different for the detector level simulated data, hence we consider parton-
level data in this section. In Fig. 5, we show the ROC curves using the Neural
Network with kinematic features for the WIMP signal, where the other new
physics signals are taken as competitors. The AUC values for WIMP SUSY3 versus
EFT, and SUSY1 versus axion signal are 0.50 and 0.58 respectively (same as
logistic regression case). Therefore the likelihood to misidentifying light
WIMPs and axions is very high, and the same is true for heavy WIMP and EFT DM.

Figure 6: Probability of misidentifying ALPs or EFT DM scenarios with a true
SUSY WIMP as a function of the neutralino mass.

Another way to express this statement is shown in Fig. 6, where we plot the
likelihood that a true WIMP signal is misidentified for an ALP or EFT DM
scenario. The likelihood is computed as the false positive rate for the
optimal point on the ROC curve. Since we are not considering the weight of the
cross-section for different signals, we find the optimal point by minimizing
the Euclidean distance from the (1=TPR999TPR is fraction of signal events
correctly identified by the algorithm, and FPR is the fraction of background
events identified as signal events by the algorithm.,0=FPR) value.

###  4.3 DNN with 2D histograms

As mentioned earlier, the information in the monojet event is saturated by
choosing two variables, $p_{T}^{j}$ and $\eta^{j}$. Therefore, inspired by the
use of Convolutional Neural Networks (CNNs) in the classification of images,
we construct ‘images’ from 2D histograms using $p_{T}$ and $\eta$ of the jet.

Figure 7: Illustrative 2D histograms for SUSY1 (left) and ALP case (right)
averaging over 200 events.

One has choices on how to group the events into images. The simulated dataset
contains $N_{\text{Tot}}$ total number of events, and is divided into
$N_{\text{Images}}$ number of images, such that each image contains
$r=N_{\text{Tot}}/N_{\text{Images}}$ number of events.

Creating a number of ‘images’ to train a network is a powerful tool since each
image itself is giving an approximation of the joint Probability Density
Function (PDF) distribution for both $p_{T}$ and $\eta$. The degree to which
each image approximates the true theoretical PDF depends on the number of
events $r$ chosen to be in an image. For a fixed total number of events
(determined by the LHC integrated luminosity) there will then be a trade-off
between $r$ and $N_{\text{Images}}$ that will affect the accuracy of the
model. We shall see that by viewing the input data in a different form, the
algorithm is able to learn more information of the distributions.

We will examine this method in both the monojet case and the dijet case.

####  4.3.1 Monojet

Before analysing the data with a CNN, we first attempt solving the problem
with a DNN. This is achieved by decomposing the 2D histogram into a 1D array
with values corresponding to the normalised number of events in each bin. Note
that whilst this may seem like we are just reconstructing the original
distributions, this is not the case since this data now contains correlations
from individual distributions. A few illustrative pictures of these plots are
shown in Fig. 7.

We consider $p_{T}$ and $\eta$ in range [130, 2000] GeV and [-4 to 4],
respectively with 29 $\times$ 29 bins. We use the information of event density
in this grid as an input for a DNN. The network is well optimised with two
fully connected hidden layers, both consisting of twenty neurons with a ReLU
activation function and a softmax activation function for the output layer.
After investigation, we do not find over-training to be an issue, so we only
include a small number of dropout neurons. The network is trained for 300
epochs with a batch size of 500. We evaluate the network’s performance through
accuracy (found from finding whether the predicted result (ranging from 0 to
1) round to 0 or to 1). This is done within the training dataset whilst
training, then with the validation dataset whilst evaluating how the network
is training, and finally with the test dataset.

We find that averaging over more number of events per image improves the
classification accuracy. As an illustration of this effect, in Fig. 8, we show
the accuracy plot for WIMP SUSY3 versus ALPs data and how it depends on the
number of events injected in one picture for the fixed data sample. The
different colored curves correspond to varying the total data sample. For a
fixed data sample, the number of images a DNN is trained and tested is reduced
when we increase the number of events per image. As we know that with more
events per image the input data approaches closer to the theoretical PDF for
the event distributions and this is clearly reflected in the increase in
accuracy with number of events per image. Nevertheless, at about 50 events per
image, the accuracy plateaus and no much more gain is obtained by bunching
more events per image.

Figure 8: Accuracy versus number of events per image, with varying total
number of events for SUSY3 versus ALP. We use fully-connected NNs with two
hidden layers for different data sample size. The monojet data set is
considered for this plot.

We also studied the dependence of the accuracy with the size of the data
sample. Once we have enough data to train the DNN, adding more events does not
improve accuracy, whereas the accuracy decreases for smaller values of the
ratio $r$. Clearly, for larger values of $r$, the increase of sample size from
10K to 800K is more important. The only limitation of the sample size is Monte
Carlo generation, and one can observe no substantial differences once the
sample size is above 100K.

We find that for $r=1$ we obtain the same result as for the DNN using directly
the kinematic features. However for $r>1$ we see a clear improvement in
performance, due to feeding the network a different format of the data which
allows the algorithm to learn more from the type of events present in each BSM
model. This bunching of events $r\neq 1$ could be seen as a way to connect
event-by-event analyses with PDF distributions of the data in the 2D plane.

In Fig. 8 central values of accuracy are plotted for different total number of
events. With total number of 100K data sample, the DNN performance is very
stable, but it starts becoming unstable for the 50K data sample. We also note
that using less than 20k total events leads to large errors in accuracy. We do
not show the accuracy for values of $r>200$ since we are keeping the total
number of events fixed, there are not enough images to produce reliable
results. ROC curves for all signal-to-signal combinations using DNN with 2D
histograms are shown in Fig. 9 for the monojet data sample.

As mentioned earlier, the analysis has been conducted with the assumption that
the number of different type of signal events are the same and that we can
fully decompose events into purely signal and background. This has allowed us
to investigate disentangling the various DM model presented, however it is not
a realistic scenario. We therefore also explore how effectively the
classifiers can disentangle DM models with varying levels of signal in
Appendix D.

Figure 9: ROC curves (DNN with 2D histograms, $r=20$) for SUSY WIMP benchmarks
versus other signals, for the monojet data set (parton-level) with 50K events
are used for each signal.

####  4.3.2 Dijet

We also consider the Next-to-Leading-Order (NLO) processes in which two
parton-level jets in addition to the 1-jet process are produced. We also
simulate detector-level for the dijet processes, and use both the leading and
sub-leading jet. More details about how two jet events are selected can be
found in the Appendix A.

Since for the dijet process we have eight different features, there are 28
possible variations of 2D histograms that we could produce. If this were to be
fully taken into account, then it might prove fruitful to combine the results
of those 28 ML algorithms together in some manner. Alternatively, one could
produce three (or higher)-dimensional histograms and deconvolve them for the
DNN, or use a three (or higher)-dimensional CNN. However, we shall not take
this approach here to avoid over-complicating the analysis and obscuring the
physical insight which is our primary aim. Instead, we perform a Principal
Component Analysis (PCA) to the dijet data in order to determine which
features are more relevant.

Figure 10: The explained variance ratio of the new principal component axes
(with the number of principal component axes being 8, the same number of
original features), highlighting the relative importance of the principal
components in terms of capturing variance in the datasets.

Figure 10 (and Fig. A2 in the Appendix) shows that whilst all but one feature
columns contribute to the variance of the dataset, a significant amount of the
variance within the datasets is captured within the first three principal
component axes. By finding the correlation between the original features and
the new principal component axes, we can determine which features are most
important in terms of holding information. We expect an increase in
classification performance if we were to use the principal component axes as
new feature columns, however we choose instead to use the original feature
columns that correspond most strongly to the most important principal
component axes, as we are primarily interested in establishing which features
are most relevant for our 2D analysis as a proof-of-concept.

The correlations can be seen in Table 2 for ALP and the SUSY3 benchmark, and
for other cases in Tables B1, B2, B3, and B4 of the Appendix B. One can see
from the table that the three most important features are the same for all
datasets, namely $p_{T}^{j_{1}}$, $\Delta\phi_{j_{1}j_{2}}$, and
$\Delta\phi_{\text{MET}}^{j_{1}}$.

ALP PCA correlations
---
| $p_{T}^{j_{1}}$ | $p_{T}^{j_{2}}$ | $\eta_{j_{1}}$ | $\eta_{j_{2}}$ | $\Delta\phi_{j_{1}j_{2}}$ | MET | $\Delta\phi_{\text{MET}}^{j_{1}}$ | $\Delta\phi_{\text{MET}}^{j_{2}}$
PC-1 | 0.66 | 0.36 | 0.01 | 0.00 | -0.01 | 0.67 | 0.01 | 0.01
PC-2 | -0.01 | 0.00 | -0.02 | -0.02 | -0.38 | -0.01 | 0.76 | 0.52
PC-3 | 0.01 | -0.01 | 0.05 | 0.04 | -0.76 | 0.00 | 0.06 | -0.64
PC-4 | 0.00 | 0.00 | -0.70 | -0.71 | -0.04 | 0.00 | -0.02 | -0.06
PC-5 | -0.29 | 0.93 | 0.05 | -0.04 | -0.01 | -0.22 | -0.01 | -0.01
PC-6 | 0.02 | -0.06 | 0.71 | -0.71 | 0.00 | 0.01 | -0.00 | 0.00
PC-7 | 0.70 | 0.05 | 0.00 | -0.00 | 0.00 | -0.71 | -0.00 | 0.00
PC-8 | 0.00 | -0.00 | 0.00 | 0.00 | 0.52 | -0.00 | 0.64 | -0.56

SUSY3, $M_{\tilde{\chi}_{1}^{0}}=300$ GeV PCA correlations
---
| $p_{T}^{j_{1}}$ | $p_{T}^{j_{2}}$ | $\eta_{j_{1}}$ | $\eta_{j_{2}}$ | $\Delta\phi_{j_{1}j_{2}}$ | MET | $\Delta\phi_{\text{MET}}^{j_{1}}$ | $\Delta\phi_{\text{MET}}^{j_{2}}$
PC-1 | 0.67 | 0.32 | 0.00 | 0.01 | 0.00 | 0.67 | -0.00 | -0.00
PC-2 | -0.00 | -0.01 | 0.01 | -0.01 | 0.32 | -0.00 | -0.76 | -0.57
PC-3 | 0.00 | -0.00 | -0.07 | -0.06 | -0.80 | 0.00 | 0.11 | -0.58
PC-4 | -0.01 | 0.02 | 0.70 | 0.70 | -0.07 | -0.01 | 0.01 | -0.05
PC-5 | -0.22 | 0.95 | -0.00 | -0.02 | 0.00 | -0.23 | -0.01 | -0.00
PC-6 | -0.00 | 0.01 | -0.71 | 0.71 | 0.01 | -0.00 | -0.01 | -0.00
PC-7 | 0.71 | -0.01 | -0.00 | 0.00 | -0.00 | -0.71 | 0.00 | 0.00
PC-8 | 0.00 | 0.00 | 0.00 | -0.00 | 0.50 | 0.00 | 0.64 | -0.58

Table 2: PCA original feature to principal component correlations for the ALP
and SUSY3 benchmark (rounded to 2 d.p.).

We do find an increase in classifier performance when using the
$p_{T}^{j_{1}}$ and $\Delta\phi_{\text{MET}}^{j_{1}}$ distributions over
$p_{T}^{j_{1}}$ and $\eta_{j_{1}}$. The ROC curves for a DNN (with the same
architecture as before) are shown in Fig. 11.

Figure 11: ROC curves (DNN with 2D histograms, $r=20$) for SUSY WIMP versus
signal for the dijet detector-level simulation. For these plots, 50K events
are used for each process.

We note that using dijet data with the 2D histogram approach performs slightly
worse than when using the monojet data showing that more information is
contained within the $p_{T}^{j}$ and $\eta_{j}$ for a monojet process than the
$p_{T}^{j_{1}}$ and $\Delta\phi_{\text{MET}}^{j_{1}}$ of one jet in a dijet
process and therefore it is still useful to include the other features of a
dijet event, even if they do offer diminishing returns on performance after
the second. Note that our results are not intended as a comprehensive
evaluation but rather as evidence that it is possible and prudent to include
dijet events for the 2D histogram method.

###  4.4 CNN with 2D histograms

After transforming our events into histograms/images (which are the
approximations of PDFs) and processing them using a DNN, we move onto applying
CNNs to them.101010CNNs have been used in many HEP analyses, see e.g. [31,
32]. We use a CNN with two convolutional layers, two max-pooling layers and
one dense flatten hidden layer, with ReLu activation function for all the
cases. ROC curves are shown in Figs. 12 and 13, for monojet and dijet
processes respectively. As mentioned earlier, CNNs are able to retain the
information of spatial correlations and usually perform better than DNNs for
the image data. But in our case, because images are constructed from highly
processed information (PDFs are already an abstract concept) we find no
sizeable improvement respect to the DNN results.

Figure 12: ROC curves (CNN with 2D histograms, $r=20$) for SUSY WIMP
benchmarks versus other signals, for the monojet case with 50K events are used
for each signal. Figure 13: ROC curves (CNN with 2D histograms, $r=20$) for
signal versus signal for the dijet simulation. For these plots, 50K events are
used for each process. Figure 14: Classifier output P(SUSY) from the DNN run
for monojet SUSY3 and ALP with $r=20$. Figure 15: ALP cross-section bound for
$\sigma_{SUSY}=0.1$ fb and $S_{SUSY}/S_{ALP}$=2.

##  5 Conclusions and Outlook

In this paper, we explored supervised Machine Learning algorithms to
disentangle different DM candidates, assuming an excess is observed in the
channels involving a jet recoiling against missing transverse energy. We also
considered the topology including an additional hard jet. For this analysis,
we compared simulations with and without detector-level effect, and found that
there is no significant degradation in the discrimination performance when we
account for realistic detector-level effects.

We used different ML approaches to the classification problem: logistic
regression, DNNs and CNNs. We found that for the kinematic variables NN
performs slightly better than logistic regression.

We then created histograms with variable number of events, which can be seen
as approximations to the probability density of events in a kinematic 2D
plane. We feed these histograms to a DNN and a CNN. The processing of these
histograms offers a better classification accuracy than lists of events with
values of kinematic observables. We show that this method can produce good
results for the characterisation of different DM models. We found a trade-off
between the accuracy and number of events averaged for one histogram ($r$). In
a more realistic situation, one would expect a small sample of DM events, and
therefore the question of how learning changes at low values of $r$ is key.

We also investigated processes with an additional hard jet to estimate the
accuracy gain as compared to the monojet topology, as these topologies could
contain more information despite a lower cross section. In this case, we
performed a PCA analysis to decide the most important combination of features,
and then constructed 2D histograms of those although we find that when using
the 2D histogram method we obtain slightly worse performance when using dijet
events with only two variables however we do demonstrate that it is possible
to incorporate the additional information from dijet events into our methods.

One can translate the techniques proposed here into limits of impostor cross
section by incorporating the information of the cross-section of specific
model benchmarks. An example on how one could translate this analysis into a
limit-setting procedure can be understood as follows. In Fig. 14, we shows the
classifier output of one of the algorithms we have trained, the DNN run for a
choice of $r$. This algorithm output can be re-interpreted as an impostor
cross-section limit, see Fig. 15. To produce this figure we assumed a fixed
value of the WIMP cross section and plotted the impostor cross section which
would be excluded using a similar analysis of Ref. [33].

Moreover, this analysis could be extended for other channels in collider dark
matter searches, e.g. for the Vector Boson Fusion (VBF) topology where we have
two additional forward jets. It would be interesting to compare the
performance of proposed methods for the dijet and VBF case. A combined
analysis of different channels may offer enhanced sensitivity which will be
investigated in the near future.

Finally, the promising results of this analysis adds a further motivation to
explore the possibility of using unsupervised techniques.

##  6 Acknowledgments

C.K.K. wishes to acknowledge support from the Royal Society-SERB Newton
International Fellowship (NF171488). The work of V.S. is supported by the
Science Technology and Facilities Council (STFC) under grant number
ST/P000819/1. M.S. wishes to acknowledge the support by the Data Intensive
Science Center in the South East Physics Network (DISCnet), an extension of
the STFC, under grant number ST/P006760/1. We would also like to thank
Alexander Belyaev for his comments on the levels of signal and background.

##  Appendix A Analysis set-up

###  A.1 Leading-order(LO) parton-level simulation

We generate parton-level events for monojet and missing energy signals and
centre-of-mass energy $\sqrt{s}=14$ TeV using MadGraph$\\_$aMC$@$NLO v2.6.3.2
[34]. For SUSY WIMP, we used MSSM-SLHA2 model (all the SUSY spectrum is set to
be very heavy except the lightest neutralino). We use the Feynrules model file
[35, 36] for the linear ALPs [37] and spin-one mediator case
(DMsimp$\\_$s$\\_$spin1 [38] model) in the EFT framework.

Using these models, we generate the following processes

$pp\rightarrow aj\qquad\mbox{ALPs}$ (4)
$pp\rightarrow\chi\bar{\chi}j\qquad\mbox{EFT, spin-1 mediator}$ (5)
$pp\rightarrow\tilde{\chi}^{0}_{1}\tilde{\chi}^{0}_{1}j\qquad\mbox{SUSY-WIMP}$
(6)

For SM monojet background, we consider the following dominant background:

$pp\rightarrow Zj(Z\rightarrow\nu\bar{\nu}).$ (7)

For all the processes 400K events are generated using a cut of $p_{T}>130$ GeV
for the jet $p_{T}$. The following kinematic variables are constructed;

$p_{T}^{j}({\text{MET}}),\eta^{j},\phi^{j}.$

###  A.2 Leading-order and Next-to-Leading order (NLO) detector-level
simulation

We generate both monojet and dijet $+$ MET processes. As earlier, hard
processes are generated using Madgraph and pythia [39] is used for
hadronization and showering. Detector effects are incorporated using Delphes
[40] with the default ATLAS card111111As a check we compared our background
simulation ($\sqrt{s}=13$ TeV) with the latest ATLAS monojet paper[41] and
found out that we get 15$\%$ more events than reported in the ATLAS paper.. A
generation level cut of $p_{T}^{j}>130$ GeV is used for leading jet $p_{T}$.
For the dijet case, while analysing the root file, in addition to the leading
jet $p_{T}$ cut we also demand $p_{T}^{j}>25$ GeV for the sub-leading jet.

We have used 200K events for all the cases after root file analysis. To avoid
the double-counting from showering, a jet merging scheme (MLM) with xqcut 20
GeV is applied. For the monojet, we consider the same three kinematic features
as in the parton-level monojet events. For the dijet, we construct the
following kinematic variables :

  * •

$p_{T}^{j_{1}}$: transverse momentum of the leading jet.

  * •

$p_{T}^{j_{2}}$ transverse momentum of the sub-leading jet.

  * •

$\eta^{j_{1}}$: pseudo-rapidity of the leading jet.

  * •

$\eta^{j_{2}}$: pseudo-rapidity of the sub-leading jet.

  * •

MET: missing transverse momentum.

  * •

$\Delta\phi_{j_{1}j_{2}}$: angular separation between the leading and sub-
leading jet.

  * •

$\Delta\phi_{\text{MET}}^{j_{1}}$: angular separation between the MET and
leading jet.

  * •

$\Delta\phi_{\text{MET}}^{j_{2}}$: angular separation between the MET and sub-
leading jet.

For the dijet sample, we consider 50K events for all the processes.

Figure A1: The $p_{T}$, $\eta$, and $\phi$ distributions for the monojet and
MET process in case of EFT spin-1 mediator (at parton-level) for fixed
mediator mass and various dark matter masses.

##  Appendix B PCA correlations

Figure A2: The variance ratio of the new principal component axes (with the
number of principal component axes being 8, the same number of original
features), highlighting the relative importance of the principal components in
terms of capturing variance in the datasets.

Background PCA correlations
---
| $p_{T}^{j_{1}}$ | $p_{T}^{j_{2}}$ | $\eta_{j_{1}}$ | $\eta_{j_{2}}$ | $\Delta\phi_{j_{1}j_{2}}$ | MET | $\Delta\phi_{\text{MET}}^{j_{1}}$ | $\Delta\phi_{\text{MET}}^{j_{2}}$
PC-1 | 0.67 | 0.41 | -0.01 | -0.01 | -0.01 | 0.62 | 0.00 | -0.00
PC-2 | 0.00 | 0.00 | 0.01 | 0.00 | 0.45 | -0.00 | -0.77 | -0.45
PC-3 | -0.00 | -0.00 | 0.09 | 0.10 | -0.70 | -0.00 | 0.00 | -0.70
PC-4 | -0.01 | 0.00 | -0.70 | -0.70 | -0.09 | -0.01 | -0.01 | -0.10
PC-5 | -0.12 | 0.89 | -0.01 | 0.02 | -0.00 | -0.45 | 0.00 | 0.00
PC-6 | -0.01 | 0.01 | 0.71 | -0.71 | -0.01 | -0.01 | 0.01 | -0.00
PC-7 | 0.73 | -0.23 | 0.00 | 0.00 | 0.00 | -0.65 | 0.00 | 0.00
PC-8 | 0.00 | 0.00 | -0.00 | -0.00 | 0.54 | -0.00 | 0.64 | -0.54

Table B1: PCA original feature to principal component correlations for the SM
background (rounded to 2 d.p.).

EFT PCA correlations
---
| $p_{T}^{j_{1}}$ | $p_{T}^{j_{2}}$ | $\eta_{j_{1}}$ | $\eta_{j_{2}}$ | $\Delta\phi_{j_{1}j_{2}}$ | MET | $\Delta\phi_{\text{MET}}^{j_{1}}$ | $\Delta\phi_{\text{MET}}^{j_{2}}$
PC-1 | 0.67 | 0.34 | -0.00 | 0.00 | 0.00 | 0.66 | 0.01 | 0.02
PC-2 | -0.01 | -0.01 | -0.00 | 0.01 | -0.33 | -0.01 | 0.76 | 0.56
PC-3 | 0.01 | -0.00 | 0.00 | 0.01 | -0.80 | 0.01 | 0.10 | -0.59
PC-4 | -0.00 | 0.01 | 0.71 | 0.71 | 0.01 | -0.00 | -0.00 | 0.00
PC-5 | -0.22 | 0.94 | -0.03 | 0.02 | -0.01 | -0.27 | 0.00 | -0.00
PC-6 | 0.01 | -0.04 | -0.71 | 0.71 | 0.01 | 0.01 | -0.01 | 0.00
PC-7 | -0.71 | 0.04 | 0.00 | -0.00 | -0.00 | 0.70 | -0.00 | -0.00
PC-8 | -0.00 | 0.00 | -0.00 | 0.00 | 0.51 | 0.00 | 0.64 | -0.57

Table B2: PCA original features to principal component correlations for the
EFT simplified framework (rounded to 2 d.p.).

SUSY1, $M_{\tilde{\chi}_{1}^{0}}=100$ GeV PCA correlations
---
| $p_{T}^{j_{1}}$ | $p_{T}^{j_{2}}$ | $\eta_{j_{1}}$ | $\eta_{j_{2}}$ | $\Delta\phi_{j_{1}j_{2}}$ | MET | $\Delta\phi_{\text{MET}}^{j_{1}}$ | $\Delta\phi_{\text{MET}}^{j_{2}}$
PC-1 | 0.67 | 0.35 | 0.01 | 0.00 | -0.00 | 0.66 | 0.01 | 0.00
PC-2 | 0.00 | 0.01 | -0.01 | -0.01 | 0.37 | 0.00 | -0.76 | -0.53
PC-3 | 0.00 | -0.02 | -0.01 | -0.01 | 0.77 | 0.01 | -0.07 | 0.63
PC-4 | -0.00 | -0.01 | 0.71 | 0.71 | 0.01 | -0.00 | -0.01 | -0.00
PC-5 | -0.21 | 0.93 | -0.03 | 0.04 | 0.01 | -0.28 | 0.00 | 0.01
PC-6 | -0.01 | 0.05 | 0.71 | -0.71 | 0.00 | -0.02 | -0.00 | -0.00
PC-7 | 0.71 | -0.05 | -0.00 | -0.00 | 0.00 | -0.70 | -0.00 | 0.00
PC-8 | 0.00 | 0.00 | 0.00 | 0.00 | -0.52 | -0.00 | -0.65 | 0.56

Table B3: PCA original features to principal component correlations for the
SUSY1 case (rounded to 2 d.p.).

SUSY2, $M_{\tilde{\chi}_{1}^{0}}=200$ GeV PCA correlations
---
| $p_{T}^{j_{1}}$ | $p_{T}^{j_{2}}$ | $\eta_{j_{1}}$ | $\eta_{j_{2}}$ | $\Delta\phi_{j_{1}j_{2}}$ | MET | $\Delta\phi_{\text{MET}}^{j_{1}}$ | $\Delta\phi_{\text{MET}}^{j_{2}}$
PC-1 | 0.67 | 0.32 | 0.00 | -0.00 | -0.01 | 0.67 | 0.01 | 0.01
PC-2 | 0.01 | 0.01 | 0.00 | -0.01 | 0.34 | 0.01 | -0.76 | -0.56
PC-3 | -0.00 | 0.01 | 0.03 | 0.06 | 0.79 | -0.00 | -0.09 | 0.60
PC-4 | 0.00 | -0.01 | 0.71 | 0.70 | -0.05 | 0.00 | 0.00 | -0.04
PC-5 | -0.22 | 0.95 | 0.01 | 0.01 | -0.01 | -0.24 | 0.00 | -0.01
PC-6 | 0.00 | 0.00 | -0.71 | 0.71 | -0.01 | 0.00 | -0.00 | -0.01
PC-7 | 0.71 | -0.01 | 0.00 | -0.00 | 0.00 | -0.71 | -0.00 | -0.00
PC-8 | 0.00 | 0.00 | -0.00 | 0.00 | 0.51 | -0.00 | 0.65 | -0.57

Table B4: PCA original features to principal component correlations for the
SUSY2 case (rounded to 2 d.p.).

##  Appendix C Signal to background analysis

We show here results for signal to background classification. Note that these
results are not intended to be used for discovery since we are assuming the
same number of signal and background events which would not be the case for
real data, but rather to demonstrate the performance of our methods and
provide a reference for how well it is possible to distinguish the different
signals from background depending on the signal kinematics.

Figure A3: ROC curves using Neural Networks for different signals versus
background for the monojet parton-level analysis.

In Fig. A3, we show the ROC curves for various signals at parton-level,
whereas the classification accuracy for the detector-level monojet and dijet
event generation is shown in Fig A4 and A5 respectively. ALPs are more
difficult to pick up from the SM background, a task that becomes simpler and
simpler as we increase the mass of the DM particle. SUSY3 and EFT, as
expected, exhibit very similar performances, and in Fig. A3 the two lines
overlap each other. Nevertheless, the SUSY3 benchmark corresponds to a DM
particle of 300 GeV, whereas the EFT corresponds to a DM particle with
negligible mass.

Figure A4: ROC curves using Neural Networks for different signals versus
background for the monojet detector-level simulated data. Figure A5: Neural
Network dijet detector-level ROC curves for different signals versus
background.

##  Appendix D Classifier performance at different levels of signal

Figure A6: DNN for 2D histogram accuracy for differing number of signal events
for trained for 100 epochs.

Here we present how the DNN and CNN for 2D histograms perform for different
levels of signal. We run the classifiers as in sections 4.3 and 4.4 but this
time we construct the histograms from a mixture of both signal and background
events. We set the total number of events per histogram to be fixed and change
the ratio of number of signal events $N_{\text{Sig}}$ to background events
$N_{\text{BG}}$ and show the model accuracy in Figure A6 for when
distinguishing between SUSY WIMP events and ALP and EFT signals for various
event ratios. Note that in the limit $N_{\text{Sig}}/N_{\text{BG}}\to\infty$
we reduce to the previous situation where we are comparing only signal with no
background. We see that it is harder to distinguish SUSY signals from the EFT
than the ALP, especially around the $m_{\tilde{\chi}^{0}}$= 300 GeV range.

The plot shows us that we require comparable levels of signal to background
for good performance, illustrating that this method is not suitable for
discovery since discovery would necessitate good performance for
$N_{\text{Sig}}\ll N_{\text{BG}}$ (it would be expected that performance
increases with total number of events - perhaps with a very large number of
events the algorithm would be much more sensitive for lower levels of signal
but that is beyond the scope of this paper.

The point of having the mixing between signal and background is to demonstrate
how effectively we can disentangle DM candidates beyond the perfect idealised
scenario where $N_{\text{Sig}}=N_{\text{BG}}$ \- assuming a discovery of
signal is made it is good to know benchmarks for performance since we would
not be able to fully separate signal from background in a realistic scenario.
Knowing these benchmarks is also useful for when constructing unsupervised
algorithms where it is useful to know the number of events required for good
performance.

## References

  * [1] G. Aad et al. [ATLAS], Phys. Lett. B 716 (2012), 1-29, [arXiv:1207.7214 [hep-ex]].
  * [2] S. Chatrchyan et al. [CMS], Phys. Lett. B 716 (2012), 30-61, [arXiv:1207.7235 [hep-ex]].
  * [3] See the following review and references therein: M. Schumann, J. Phys. G 46 (2019) no.10, 103003 doi:10.1088/1361-6471/ab2ea5 [arXiv:1903.03026 [astro-ph.CO]].
  * [4] See the following pedagogical lectures and references therein: T. R. Slatyer, doi:10.1142/9789813233348_0005 [arXiv:1710.05137 [hep-ph]].
  * [5] See the following review and references therein: A. Boveia and C. Doglioni, Ann. Rev. Nucl. Part. Sci. 68 (2018), 429-459 doi:10.1146/annurev-nucl-101917-021008 [arXiv:1810.12238 [hep-ex]].
  * [6] M. Beltran, D. Hooper, E. W. Kolb, Z. A. C. Krusberg and T. M. P. Tait, JHEP 09 (2010), 037, [arXiv:1002.4137 [hep-ph]].
  * [7] J. M. No, P. Tunney and B. Zaldivar, arXiv:1908.11387 [hep-ph].
  * [8] A. Radovic et al., Nature 560 (2018) no.7716, 41.
  * [9] J. Brehmer, K. Cranmer, G. Louppe and J. Pavez, Phys. Rev. D 98 (2018) no.5, 052004, [arXiv:1805.00020 [hep-ph]].
  * [10] J. Brehmer, K. Cranmer, G. Louppe and J. Pavez, Phys. Rev. Lett. 121 (2018) no.11, 111801, [arXiv:1805.00013 [hep-ph]].
  * [11] F. F. Freitas, C. K. Khosa and V. Sanz, Phys. Rev. D 100 (2019) 035040, [arXiv:1902.05803 [hep-ph]].
  * [12] G. Kasieczka, T. Plehn, M. Russell and T. Schell, JHEP 1705 (2017) 006, [arXiv:1701.08784 [hep-ph]].
  * [13] G. Kasieczka et al., SciPost Phys. 7 (2019) 014, [arXiv:1902.09914 [hep-ph]].
  * [14] M. L. Piscopo, M. Spannowsky and P. Waite, Phys. Rev. D 100 (2019) no.1, 016002, [arXiv:1902.05563 [hep-ph]].
  * [15] S. Caron, J. S. Kim, K. Rolbiecki, R. Ruiz de Austri and B. Stienen, Eur. Phys. J. C 77 (2017) no.4, 257, [arXiv:1605.02797 [hep-ph]].
  * [16] G. Kasieczka, N. Kiefer, T. Plehn and J. M. Thompson, SciPost Phys. 6 (2019) 069, [arXiv:1812.09223 [hep-ph]].
  * [17] J. Brehmer, S. Mishra-Sharma, J. Hermans, G. Louppe and K. Cranmer, arXiv:1909.02005 [astro-ph.CO].
  * [18] S. Alexander, S. Gleyzer, E. McDonough, M. W. Toomey and E. Usai, arXiv:1909.07346 [astro-ph.CO].
  * [19] C. Escamilla-Rivera, M. A. C. Quintero and S. Capozziello, arXiv:1910.02788 [astro-ph.CO].
  * [20] U. Simola, B. Pelssers, D. Barge, J. Conrad and J. Corander, JINST 14 (2019) no.03, P03004, [arXiv:1810.09930 [astro-ph.IM]].
  * [21] ATLAS-CONF-2020-048.
  * [22] A. M. Sirunyan et al. Phys. Rev. D 97 (2018) no.9, 092005, [arXiv:1712.02345 [hep-ex]].
  * [23] J. R. Ellis, J. S. Hagelin, D. V. Nanopoulos, K. A. Olive and M. Srednicki, Nucl. Phys. B 238 (1984), 453-476,
  * [24] K. Mimasu and V. Sanz, JHEP 1506 (2015) 173, [arXiv:1409.4792 [hep-ph]].
  * [25] I. Brivio, M. B. Gavela, L. Merlo, K. Mimasu, J. M. No, R. del Rey and V. Sanz, Eur. Phys. J. C 77 (2017) no.8, 572, [arXiv:1701.05379 [hep-ph]].
  * [26] D. Barducci, A. Belyaev, A. K. M. Bharucha, W. Porod and V. Sanz, JHEP 1507 (2015) 066, [arXiv:1504.02472 [hep-ph]].
  * [27] H. M. Lee, M. Park and V. Sanz, JHEP 1303 (2013) 052, [arXiv:1212.5647 [hep-ph]].
  * [28] M. Bauer, M. Neubert and A. Thamm, JHEP 12 (2017), 044, [arXiv:1708.00443 [hep-ph]].
  * [29] M. B. Gavela, J. M. No, V. Sanz and J. F. de Trocóniz, arXiv:1905.12953 [hep-ph].
  * [30] F. Ferreira, S. Fichet and V. Sanz, Phys. Lett. B 778, 35 (2018) doi:10.1016/j.physletb.2018.01.008 [arXiv:1702.05106 [hep-ph]].
  * [31] G. Kasieczka, T. Plehn, M. Russell and T. Schell, JHEP 05 (2017), 006, [arXiv:1701.08784 [hep-ph]].
  * [32] S. Macaluso and D. Shih, JHEP 10 (2018), 121, [arXiv:1803.00107 [hep-ph]].
  * [33] D. Barducci, A. Belyaev, A. K. M. Bharucha, W. Porod and V. Sanz, JHEP 07 (2015), 066, [arXiv:1504.02472 [hep-ph]].
  * [34] J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer and T. Stelzer, JHEP 1106 (2011) 128, [arXiv:1106.0522 [hep-ph]].
  * [35] N. D. Christensen and C. Duhr, Comput. Phys. Commun. 180 (2009) 1614, [arXiv:0806.4194 [hep-ph]].
  * [36] C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer and T. Reiter, Comput. Phys. Commun. 183 (2012) 1201, [arXiv:1108.2040 [hep-ph]].
  * [37] http://feynrules.irmp.ucl.ac.be/wiki/ALPsEFT
  * [38] http://feynrules.irmp.ucl.ac.be/wiki/DMsimp
  * [39] T. Sjostrand, S. Mrenna and P. Z. Skands, Comput. Phys. Commun. 178 (2008) 852, [arXiv:0710.3820 [hep-ph]].
  * [40] J. de Favereau et al. [DELPHES 3 Collaboration], JHEP 1402 (2014) 057, [arXiv:1307.6346 [hep-ex]].
  * [41] [ATLAS], ATLAS-CONF-2020-048.