{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:10:01Z","timestamp":1760235001732,"version":"build-2065373602"},"reference-count":47,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2021,7,8]],"date-time":"2021-07-08T00:00:00Z","timestamp":1625702400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Future Internet"],"abstract":"<jats:p>Flow-based generative models have recently become one of the most efficient approaches to model data generation. Indeed, they are constructed with a sequence of invertible and tractable transformations. Glow first introduced a simple type of generative flow using an invertible 1\u00d71 convolution. However, the 1\u00d71 convolution suffers from limited flexibility compared to the standard convolutions. In this paper, we propose a novel invertible n\u00d7n convolution approach that overcomes the limitations of the invertible 1\u00d71 convolution. In addition, our proposed network is not only tractable and invertible but also uses fewer parameters than standard convolutions. The experiments on CIFAR-10, ImageNet and Celeb-HQ datasets, have shown that our invertible n\u00d7n convolution helps to improve the performance of generative models significantly.<\/jats:p>","DOI":"10.3390\/fi13070179","type":"journal-article","created":{"date-parts":[[2021,7,8]],"date-time":"2021-07-08T10:42:17Z","timestamp":1625740937000},"page":"179","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Fast Flow Reconstruction via Robust Invertible n \u00d7 n Convolution"],"prefix":"10.3390","volume":"13","author":[{"given":"Thanh-Dat","family":"Truong","sequence":"first","affiliation":[{"name":"Computer Vision and Image Understanding Lab, Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR 72501, USA"}]},{"given":"Chi Nhan","family":"Duong","sequence":"additional","affiliation":[{"name":"Department of Computer Science and Software Engineering, Concordia University, Montreal, QC H3G 2V4, Canada"}]},{"given":"Minh-Triet","family":"Tran","sequence":"additional","affiliation":[{"name":"Faculty of Information Technology, University of Science, VNU-HCM, Ho Chi Minh 721337, Vietnam"}]},{"given":"Ngan","family":"Le","sequence":"additional","affiliation":[{"name":"Computer Vision and Image Understanding Lab, Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR 72501, USA"}]},{"given":"Khoa","family":"Luu","sequence":"additional","affiliation":[{"name":"Computer Vision and Image Understanding Lab, Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR 72501, USA"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"He, K., Zhang, X., Ren, S., and Sun, J. (2016, January 27\u201330). Deep residual learning for image recognition. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Las Vegas, NV, USA.","DOI":"10.1109\/CVPR.2016.90"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Huang, G., Liu, Z., Van Der Maaten, L., and Weinberger, K.Q. (2017, January 21\u201326). Densely connected convolutional networks. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Honolulu, HI, USA.","DOI":"10.1109\/CVPR.2017.243"},{"key":"ref_3","unstructured":"Sun, S., Pang, J., Shi, J., Yi, S., and Ouyang, W. (2021, July 08). FishNet: A Versatile Backbone for Image, Region, and Pixel Level Prediction. Available online: https:\/\/arxiv.org\/abs\/1901.03495."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Liu, W., Anguelov, D., Erhan, D., Szegedy, C., Reed, S., Fu, C.Y., and Berg, A.C. (2016). Ssd: Single shot multibox detector. Lecture Notes in Computer Science Proceedings of the European Conference on Computer Vision, Amsterdam, The Netherlands, 8\u201316 October 2016, Springer.","DOI":"10.1007\/978-3-319-46448-0_2"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Redmon, J., Divvala, S., Girshick, R., and Farhadi, A. (2016, January 27\u201330). You only look once: Unified, real-time object detection. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Las Vegas, NV, USA.","DOI":"10.1109\/CVPR.2016.91"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Lin, T.Y., Goyal, P., Girshick, R., He, K., and Doll\u00e1r, P. (2017, January 22\u201329). Focal loss for dense object detection. Proceedings of the IEEE International Conference on Computer Vision, Venice, Italy.","DOI":"10.1109\/ICCV.2017.324"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Luu, K., Seshadri, K., Savvides, M., Bui, T., and Suen, C. (2011, January 11\u201313). Contourlet Appearance Model for Facial Age Estimation. Proceedings of the 2011 International Joint Conference on Biometrics (IJCB), Washington, DC, USA.","DOI":"10.1109\/IJCB.2011.6117601"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3843","DOI":"10.1016\/j.patcog.2015.05.021","article-title":"Facial Aging and Asymmetry Decomposition Based Approaches to Identification of Twins","volume":"48","author":"Le","year":"2015","journal-title":"Pattern Recognit."},{"key":"ref_9","first-page":"4780","article-title":"Spartans: Single-sample Periocular-based Alignment-robust Recognition Technique Applied to Non-frontal Scenarios","volume":"12","author":"Xu","year":"2015","journal-title":"IEEE Trans. Image Process."},{"key":"ref_10","unstructured":"Xu, J., Luu, K., Savvides, M., Bui, T., and Suen, C. (2011, January 11\u201313). Investigating Age Invariant Face Recognition Based on Periocular Biometrics. Proceedings of the 2011 International Joint Conference on Biometrics (IJCB), Washington, DC, USA."},{"key":"ref_11","unstructured":"Duong, C., Quach, K., Luu, K., and Le, H.K. (2011, January 22\u201327). Fine Tuning Age Estimation with Global and Local Facial Features. Proceedings of the 36th International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Prague, Czech Republic."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Luu, K., Bui, T.K., and Suen, C. (2009, January 28\u201330). Age Estimation using Active Appearance Models and Support Vector Machine Regression. Proceedings of the 2009 IEEE 3rd International Conference on Biometrics: Theory, Applications, and Systems, Washington, DC, USA.","DOI":"10.1109\/BTAS.2009.5339053"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Luu, K., Bui, T., and Suen, C. (2011, January 21\u201325). Kernel Spectral Regression of Perceived Age from Hybrid Facial Features. Proceedings of the 2011 IEEE International Conference on Automatic Face and Gesture Recognition (FG), Santa Barbara, CA, USA.","DOI":"10.1109\/FG.2011.5771334"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Chen, C., Yang, W., Wang, Y., Ricanek, K., and Luu, K. (2011, January 21\u201325). Facial Feature Fusion and Model Selection for Age Estimation. Proceedings of the 2011 IEEE International Conference on Automatic Face and Gesture Recognition (FG), Santa Barbara, CA, USA.","DOI":"10.1109\/FG.2011.5771398"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"834","DOI":"10.1109\/TPAMI.2017.2699184","article-title":"Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs","volume":"40","author":"Chen","year":"2017","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Long, J., Shelhamer, E., and Darrell, T. (2015, January 7\u201312). Fully convolutional networks for semantic segmentation. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Boston, MA, USA.","DOI":"10.1109\/CVPR.2015.7298965"},{"key":"ref_17","unstructured":"Luu, K.K., Bui, T., and Suen, C. (2008, January 2\u20135). The Familial Face Database: A Longitudinal Study of Family-based Growth and Development on Face Recognition. Proceedings of the Robust Biometrics: Understanding Science and Technology, Marriott Waikiki, HI, USA."},{"key":"ref_18","unstructured":"Luu, K. (June, January 31). Computer Approaches for Face Aging Problems. Proceedings of the 23th Canadian Conference On Artificial Intelligence (CAI), Ottawa, ON, Canada."},{"key":"ref_19","unstructured":"Mirza, M., and Osindero, S. (2014). Conditional generative adversarial nets. arXiv."},{"key":"ref_20","unstructured":"Karras, T., Aila, T., Laine, S., and Lehtinen, J. (2017). Progressive growing of gans for improved quality, stability, and variation. arXiv."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Duong, C., Luu, K., Quach, K., and Bui, T. (2016, January 27\u201330). Longitudinal Face Modeling via Temporal Deep Restricted Boltzmann Machines. Proceedings of the 2016 IEEE Conference On Computer Vision And Pattern Recognition (CVPR), Las Vegas, NV, USA.","DOI":"10.1109\/CVPR.2016.622"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Duong, C., Quach, K., Luu, K., Le, T., and Savvides, M. (2017, January 22\u201329). Temporal Non-volume Preserving Approach to Facial Age-Progression and Age-Invariant Face Recognition. Proceedings of the 2017 IEEE International Conference On Computer Vision (ICCV), Venice, Italy.","DOI":"10.1109\/ICCV.2017.403"},{"key":"ref_23","unstructured":"Mattia, F.D., Galeone, P., Simoni, M.D., and Ghelfi, E. (2019). A Survey on GANs for Anomaly Detection. arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Yu, J., Lin, Z., Yang, J., Shen, X., Lu, X., and Huang, T.S. (2018, January 18\u201323). Generative image inpainting with contextual attention. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Salt Lake City, UT, USA.","DOI":"10.1109\/CVPR.2018.00577"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Karras, T., Laine, S., and Aila, T. (2019, January 15\u201320). A style-based generator architecture for generative adversarial networks. Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition, Long Beach, CA, USA.","DOI":"10.1109\/CVPR.2019.00453"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Ledig, C., Theis, L., Husz\u00e1r, F., Caballero, J., Cunningham, A., Acosta, A., Aitken, A., Tejani, A., Totz, J., and Wang, Z. (2017, January 21\u201326). Photo-realistic single image super-resolution using a generative adversarial network. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Honolulu, HI, USA.","DOI":"10.1109\/CVPR.2017.19"},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Duong, C., Luu, K., Quach, K., Nguyen, N., Patterson, E., Bui, T., and Le, N. (2019, January 16\u201320). Automatic Face Aging in Videos via Deep Reinforcement Learning. Proceedings of the 2019 IEEE\/CVF Conference On Computer Vision And Pattern Recognition (CVPR), Long Beach, CA, USA.","DOI":"10.1109\/CVPR.2019.01025"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"437","DOI":"10.1007\/s11263-018-1113-3","article-title":"Deep Appearance Models: A Deep Boltzmann Machine Approach for Face Modeling","volume":"127","author":"Duong","year":"2019","journal-title":"Int. J. Comput. Vis."},{"key":"ref_29","unstructured":"Bengio, S., Wallach, H., Larochelle, H., Grauman, K., Cesa-Bianchi, N., and Garnett, R. (2018). Glow: Generative Flow with Invertible 1x1 Convolutions. Advances in Neural Information Processing Systems 31, Curran Associates, Inc."},{"key":"ref_30","unstructured":"Lee, D.D., Sugiyama, M., Luxburg, U.V., Guyon, I., and Garnett, R. (2016). Improved Variational Inference with Inverse Autoregressive Flow. Advances in Neural Information Processing Systems 29, Curran Associates, Inc."},{"key":"ref_31","unstructured":"Dinh, L., Krueger, D., and Bengio, Y. (2015). NICE: Non-linear Independent Components Estimation. arXiv."},{"key":"ref_32","unstructured":"Dinh, L., Sohl-Dickstein, J., and Bengio, S. (2017, January 24\u201326). Density estimation using Real NVP. Proceedings of the 3rd International Conference on Learning Representations, ICLR, Toulon, France."},{"key":"ref_33","unstructured":"Ghahramani, Z., Welling, M., Cortes, C., Lawrence, N.D., and Weinberger, K.Q. (2014). Generative Adversarial Nets. Advances in Neural Information Processing Systems 27, Curran Associates, Inc."},{"key":"ref_34","unstructured":"Kingma, D.P., and Welling, M. (2014, January 14\u201316). Auto-Encoding Variational Bayes. Proceedings of the 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada."},{"key":"ref_35","unstructured":"Hoogeboom, E., van den Berg, R., and Welling, M. (2019). Emerging Convolutions for Generative Normalizing Flows. arxiv."},{"key":"ref_36","unstructured":"Guyon, I., Luxburg, U.V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., and Garnett, R. (2017). Masked Autoregressive Flow for Density Estimation. Advances in Neural Information Processing Systems 30, Curran Associates, Inc."},{"key":"ref_37","first-page":"573","article-title":"Invertible Residual Networks","volume":"Volume 97","author":"Chaudhuri","year":"2019","journal-title":"Proceedings of the 36th International Conference on Machine Learning"},{"key":"ref_38","unstructured":"Kim, H., Papamakarios, G., and Mnih, A. (2021). The Lipschitz Constant of Self-Attention. arXiv."},{"key":"ref_39","unstructured":"Chen, R.T., Behrmann, J., Duvenaud, D., and Jacobsen, J.H. (2019). Residual flows for invertible generative modeling. arXiv."},{"key":"ref_40","unstructured":"Krizhevsky, A. (2021, July 08). Learning Multiple Layers of Features from Tiny Images. Available online: http:\/\/www.cs.toronto.edu\/~kriz\/learning-features-2009-TR.pdf."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Deng, J., Dong, W., Socher, R., Li, L.J., Li, K., and Fei-Fei, L. (2009, January 20\u201325). ImageNet: A Large-Scale Hierarchical Image Database. Proceedings of Computer Vision and Pattern Recognition (CVPR), Miami, FL, USA.","DOI":"10.1109\/CVPR.2009.5206848"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Liu, Z., Luo, P., Wang, X., and Tang, X. (2015, January 7\u201313). Deep Learning Face Attributes in the Wild. Proceedings of International Conference on Computer Vision (ICCV), Santiago, Chile.","DOI":"10.1109\/ICCV.2015.425"},{"key":"ref_43","unstructured":"Ho, J., Chen, X., Srinivas, A., Duan, Y., and Abbeel, P. (2019). Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design. arXiv."},{"key":"ref_44","unstructured":"Guyon, I., Luxburg, U.V., Bengio, S., Wallach, H., Fergus, R., Vishwanathan, S., and Garnett, R. (2017). Attention is All you Need. Advances in Neural Information Processing Systems, Curran Associates, Inc."},{"key":"ref_45","first-page":"881","article-title":"MADE: Masked Autoencoder for Distribution Estimation","volume":"Volume 37","author":"Bach","year":"2015","journal-title":"Proceedings of the 32nd International Conference on Machine Learning"},{"key":"ref_46","doi-asserted-by":"crossref","unstructured":"Truong, D., Duong, C.N., Luu, K., Tran, M., and Le, N. (2020, January 13\u201315). Domain Generalization via Universal Non-volume Preserving Approach. Proceedings of the 2020 17th Conference On Computer And Robot Vision (CRV), Ottawa, ON, Canada.","DOI":"10.1109\/CRV50864.2020.00021"},{"key":"ref_47","unstructured":"Kingma, D.P., and Ba, J. (2015, January 7\u20139). Adam: A Method for Stochastic Optimization. Proceedings of the 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA."}],"container-title":["Future Internet"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-5903\/13\/7\/179\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:27:51Z","timestamp":1760164071000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-5903\/13\/7\/179"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,8]]},"references-count":47,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2021,7]]}},"alternative-id":["fi13070179"],"URL":"https:\/\/doi.org\/10.3390\/fi13070179","relation":{},"ISSN":["1999-5903"],"issn-type":[{"type":"electronic","value":"1999-5903"}],"subject":[],"published":{"date-parts":[[2021,7,8]]}}}