{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:19:38Z","timestamp":1740107978363,"version":"3.37.3"},"reference-count":58,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T00:00:00Z","timestamp":1682035200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T00:00:00Z","timestamp":1682035200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100016378","name":"Technische Universit\u00e4t Dortmund","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100016378","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Int J Softw Tools Technol Transfer"],"published-print":{"date-parts":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>TADS are a novel, concise white-box representation of neural networks. In this paper, we apply TADS to the problem of neural network verification, using them to generate either proofs or concise error characterizations for desirable neural network properties. In a case study, we consider the robustness of neural networks to adversarial attacks, i.e., small changes to an input that drastically change a neural networks perception, and show that TADS can be used to provide precise diagnostics on how and where robustness errors a occur. We achieve these results by introducing <jats:italic>Precondition Projection<\/jats:italic>, a technique that yields a TADS describing network behavior precisely on a given subset of its input space, and combining it with PCA, a traditional, well-understood dimensionality reduction technique. We show that PCA is easily compatible with TADS. All analyses can be implemented in a straightforward fashion using the rich algebraic properties of TADS, demonstrating the utility of the TADS framework for neural network explainability and verification. While TADS do not yet scale as efficiently as state-of-the-art neural network verifiers, we show that, using PCA-based simplifications, they can still scale to medium-sized problems and yield concise explanations for potential errors that can be used for other purposes such as debugging a network or generating new training samples.<\/jats:p>","DOI":"10.1007\/s10009-023-00701-6","type":"journal-article","created":{"date-parts":[[2023,4,21]],"date-time":"2023-04-21T15:11:34Z","timestamp":1682089894000},"page":"355-374","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["The power of typed affine decision structures: a case study"],"prefix":"10.1007","volume":"25","author":[{"given":"Gerrit","family":"Nolte","sequence":"first","affiliation":[]},{"given":"Maximilian","family":"Schl\u00fcter","sequence":"additional","affiliation":[]},{"given":"Alnis","family":"Murtovi","sequence":"additional","affiliation":[]},{"given":"Bernhard","family":"Steffen","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,4,21]]},"reference":[{"issue":"4","key":"701_CR1","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1002\/wics.101","volume":"2","author":"H. Abdi","year":"2010","unstructured":"Abdi, H., Williams, L.J.: Principal component analysis. Wiley Interdiscip. Rev.: Comput. Stat. 2(4), 433\u2013459 (2010)","journal-title":"Wiley Interdiscip. Rev.: Comput. Stat."},{"key":"701_CR2","doi-asserted-by":"publisher","first-page":"52138","DOI":"10.1109\/ACCESS.2018.2870052","volume":"6","author":"A. Adadi","year":"2018","unstructured":"Adadi, A., Berrada, M.: Peeking inside the black-box: a survey on explainable artificial intelligence (xai). IEEE Access 6, 52138\u201352160 (2018)","journal-title":"IEEE Access"},{"issue":"1\u20132","key":"701_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1561\/2500000051","volume":"7","author":"A. Albarghouthi","year":"2021","unstructured":"Albarghouthi, A., et al.: Introduction to neural network verification. Found. Trends Program. Lang. 7(1\u20132), 1\u2013157 (2021)","journal-title":"Found. Trends Program. Lang."},{"key":"701_CR4","unstructured":"Arora, R., Basu, A., Mianjy, P., Mukherjee, A.: Understanding deep neural networks with rectified linear units. Arxiv preprint (2016). arXiv:1611.01491"},{"key":"701_CR5","doi-asserted-by":"publisher","DOI":"10.1007\/b97662","volume-title":"Linear Algebra Done Right","author":"S. Axler","year":"1997","unstructured":"Axler, S.: Linear Algebra Done Right. Springer, Berlin (1997)"},{"key":"701_CR6","unstructured":"Bak, S., Liu, C., Johnson, T.: The second international verification of neural networks competition (vnn-comp 2021): summary and results. Arxiv preprint (2021). arXiv:2109.00498"},{"issue":"8","key":"701_CR7","doi-asserted-by":"publisher","first-page":"1553","DOI":"10.1109\/TNNLS.2013.2293637","volume":"25","author":"M. Bianchini","year":"2014","unstructured":"Bianchini, M., Scarselli, F.: On the complexity of neural network classifiers: a comparison between shallow and deep architectures. IEEE Trans. Neural Netw. Learn. Syst. 25(8), 1553\u20131565 (2014)","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"issue":"9","key":"701_CR8","doi-asserted-by":"publisher","first-page":"2812","DOI":"10.1039\/C3AY41907J","volume":"6","author":"R. Bro","year":"2014","unstructured":"Bro, R., Smilde, A.K.: Principal component analysis. Anal. Methods 6(9), 2812\u20132831 (2014)","journal-title":"Anal. Methods"},{"key":"701_CR9","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1148-8","volume-title":"An Introduction to Convex Polytopes","author":"A. Brondsted","year":"1983","unstructured":"Brondsted, A.: An Introduction to Convex Polytopes, first edn. Springer, New York, NY (1983). https:\/\/doi.org\/10.1007\/978-1-4612-1148-8","edition":"1"},{"key":"701_CR10","first-page":"1877","volume":"33","author":"T. Brown","year":"2020","unstructured":"Brown, T., Mann, B., Ryder, N., Subbiah, M., Kaplan, J.D., Dhariwal, P., Neelakantan, A., Shyam, P., Sastry, G., Askell, A., et al.: Language models are few-shot learners. Adv. Neural Inf. Process. Syst. 33, 1877\u20131901 (2020)","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"701_CR11","volume-title":"Advances in Neural Information Processing Systems","author":"R.R. Bunel","year":"2018","unstructured":"Bunel, R.R., Turkaslan, I., Torr, P., Kohli, P., Mudigonda, P.K.: A unified view of piecewise linear neural network verification. In: Advances in Neural Information Processing Systems, vol.\u00a031 (2018)"},{"key":"701_CR12","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1109\/SP.2017.49","volume-title":"2017 IEEE Symposium on Security and Privacy (SP)","author":"N. Carlini","year":"2017","unstructured":"Carlini, N., Wagner, D.: Towards evaluating the robustness of neural networks. In: 2017 IEEE Symposium on Security and Privacy (SP) pp.\u00a039\u201357. IEEE Comput. Soc., Los Alamitos (2017)"},{"key":"701_CR13","doi-asserted-by":"publisher","first-page":"1244","DOI":"10.1145\/3219819.3220063","volume-title":"Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining","author":"L. Chu","year":"2018","unstructured":"Chu, L., Hu, X., Hu, J., Wang, L., Pei, J.: Exact and consistent interpretation for piecewise linear neural networks: a closed form solution. In: Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, pp.\u00a01244\u20131253 (2018)"},{"issue":"4","key":"701_CR14","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1093\/logcom\/2.4.511","volume":"2","author":"P. Cousot","year":"1992","unstructured":"Cousot, P., Cousot, R.: Abstract interpretation frameworks. J. Log. Comput. 2(4), 511\u2013547 (1992)","journal-title":"J. Log. Comput."},{"issue":"3","key":"701_CR15","doi-asserted-by":"publisher","first-page":"250","DOI":"10.1093\/comjnl\/8.3.250","volume":"8","author":"R.J. Dakin","year":"1965","unstructured":"Dakin, R.J.: A tree-search algorithm for mixed integer programming problems. Comput. J. 8(3), 250\u2013255 (1965)","journal-title":"Comput. J."},{"issue":"6","key":"701_CR16","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1109\/MSP.2012.2211477","volume":"29","author":"L. Deng","year":"2012","unstructured":"Deng, L.: The mnist database of handwritten digit images for machine learning research. IEEE Signal Process. Mag. 29(6), 141\u2013142 (2012)","journal-title":"IEEE Signal Process. Mag."},{"key":"701_CR17","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/978-3-030-53288-8_3","volume-title":"International Conference on Computer Aided Verification","author":"Y.Y. Elboher","year":"2020","unstructured":"Elboher, Y.Y., Gottschlich, J., Katz, G.: An abstraction-based framework for neural network verification. In: International Conference on Computer Aided Verification, pp.\u00a043\u201365. Springer, Berlin (2020)"},{"key":"701_CR18","volume-title":"Advances in Neural Information Processing Systems","author":"M. Fazlyab","year":"2019","unstructured":"Fazlyab, M., Robey, A., Hassani, H., Morari, M., Pappas, G.: Efficient and accurate estimation of lipschitz constants for deep neural networks. In: Advances in Neural Information Processing Systems, vol.\u00a032 (2019)"},{"key":"701_CR19","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1109\/SP.2018.00058","volume-title":"2018 IEEE Symposium on Security and Privacy (SP)","author":"T. Gehr","year":"2018","unstructured":"Gehr, T., Mirman, M., Drachsler-Cohen, D., Tsankov, P., Chaudhuri, S., Vechev, M.: Ai2: safety and robustness certification of neural networks with abstract interpretation. In: 2018 IEEE Symposium on Security and Privacy (SP), pp.\u00a03\u201318. IEEE Comput. Soc., Los Alamitos (2018)"},{"key":"701_CR20","volume-title":"Deep Learning","author":"I. Goodfellow","year":"2016","unstructured":"Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press, Cambridge (2016). http:\/\/www.deeplearningbook.org"},{"key":"701_CR21","unstructured":"Goodfellow, I.J., Shlens, J., Szegedy, C.: Explaining and harnessing adversarial examples. Arxiv preprint (2014). arXiv:1412.6572"},{"issue":"3","key":"701_CR22","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1080\/02331939408844018","volume":"31","author":"V.V. Gorokhovik","year":"1994","unstructured":"Gorokhovik, V.V., Zorko, O.I., Birkhoff, G.: Piecewise affine functions and polyhedral sets. Optimization 31(3), 209\u2013221 (1994)","journal-title":"Optimization"},{"key":"701_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/s10009-021-00635-x","author":"F. Gossen","year":"2021","unstructured":"Gossen, F., Steffen, B.: Algebraic aggregation of random forests: towards explainability and rapid evaluation. Int. J. Softw. Tools Technol. Transf. (2021). https:\/\/doi.org\/10.1007\/s10009-021-00635-x.","journal-title":"Int. J. Softw. Tools Technol. Transf."},{"key":"701_CR24","doi-asserted-by":"publisher","first-page":"354","DOI":"10.1016\/j.patcog.2017.10.013","volume":"77","author":"J. Gu","year":"2018","unstructured":"Gu, J., Wang, Z., Kuen, J., Ma, L., Shahroudy, A., Shuai, B., Liu, T., Wang, X., Wang, G., Cai, J., et al.: Recent advances in convolutional neural networks. Pattern Recognit. 77, 354\u2013377 (2018)","journal-title":"Pattern Recognit."},{"key":"701_CR25","first-page":"12","volume":"116","author":"R. Guidotti","year":"2019","unstructured":"Guidotti, R., Monreale, A., Pedreschi, D.: The ai black box explanation problem. ERCIM News 116, 12\u201313 (2019)","journal-title":"ERCIM News"},{"issue":"5","key":"701_CR26","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1145\/3236009","volume":"51","author":"R. Guidotti","year":"2018","unstructured":"Guidotti, R., Monreale, A., Ruggieri, S., Turini, F., Giannotti, F., Pedreschi, D.: A survey of methods for explaining black box models. ACM Comput. Surv. 51(5), 93 (2018). https:\/\/doi.org\/10.1145\/3236009.","journal-title":"ACM Comput. Surv."},{"key":"701_CR27","volume-title":"Advances in Neural Information Processing Systems","author":"B. Han","year":"2018","unstructured":"Han, B., Yao, Q., Yu, X., Niu, G., Xu, M., Hu, W., Tsang, I., Sugiyama, M.: Co-teaching: Robust training of deep neural networks with extremely noisy labels. In: Advances in Neural Information Processing Systems, vol.\u00a031. (2018)"},{"key":"701_CR28","series-title":"PMLR Proceedings of Machine Learning Research","first-page":"2596","volume-title":"Proceedings of the 36th International Conference on Machine Learning","author":"B. Hanin","year":"2019","unstructured":"Hanin, B., Rolnick, D.: Complexity of linear regions in deep networks. In: Chaudhuri, K., Salakhutdinov, R. (eds.) Proceedings of the 36th International Conference on Machine Learning. PMLR Proceedings of Machine Learning Research, vol.\u00a097, pp.\u00a02596\u20132604. (2019). https:\/\/proceedings.mlr.press\/v97\/hanin19a.html"},{"key":"701_CR29","volume-title":"Advances in Neural Information Processing Systems","author":"B. Hanin","year":"2019","unstructured":"Hanin, B., Rolnick, D.: Deep relu networks have surprisingly few activation patterns. In: Advances in Neural Information Processing Systems, vol.\u00a032. (2019)"},{"key":"701_CR30","unstructured":"Hinz, P.: Using activation histograms to bound the number of affine regions in ReLU feed-forward neural networks. Arxiv (2021). arXiv:2103.17174"},{"key":"701_CR31","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/978-3-319-63387-9_5","volume-title":"International Conference on Computer Aided Verification","author":"G. Katz","year":"2017","unstructured":"Katz, G., Barrett, C., Dill, D.L., Julian, K., Kochenderfer, M.J.: Reluplex: an efficient smt solver for verifying deep neural networks. In: International Conference on Computer Aided Verification, pp.\u00a097\u2013117. Springer, Berlin (2017)"},{"key":"701_CR32","unstructured":"Kingma, D.P., Ba, J.: Adam: a method for stochastic optimization. Arxiv preprint (2014). arXiv:1412.6980"},{"key":"701_CR33","volume-title":"Adversarial Examples in the Physical World","author":"A. Kurakin","year":"2016","unstructured":"Kurakin, A., Goodfellow, I., Bengio, S., et al.: Adversarial Examples in the Physical World (2016)"},{"key":"701_CR34","unstructured":"Leofante, F., Narodytska, N., Pulina, L., Tacchella, A.: Automated verification of neural networks: advances, challenges and perspectives (2018). ArXiv preprint. arXiv:1805.09938"},{"key":"701_CR35","volume-title":"Proceedings of the AAAI Conference on Artificial Intelligence","author":"B. Luo","year":"2018","unstructured":"Luo, B., Liu, Y., Wei, L., Xu, Q.: Towards imperceptible and robust adversarial example attacks against neural networks. In: Proceedings of the AAAI Conference on Artificial Intelligence, vol.\u00a032 (2018)"},{"key":"701_CR36","volume-title":"Introduction to Tropical Geometry","author":"D. Maclagan","year":"2021","unstructured":"Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry, vol.\u00a0161. Am. Math. Soc., Providence (2021)"},{"key":"701_CR37","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/978-3-030-94946-4","volume-title":"Metric Spaces","author":"R. Magnus","year":"2022","unstructured":"Magnus, R.: Metric spaces. In: Metric Spaces, pp.\u00a01\u201327. Springer, Berlin (2022)"},{"issue":"5","key":"701_CR38","doi-asserted-by":"publisher","first-page":"728","DOI":"10.1109\/JPROC.2021.3065238","volume":"109","author":"P. Maragos","year":"2021","unstructured":"Maragos, P., Charisopoulos, V., Theodosis, E.: Tropical geometry and machine learning. Proc. IEEE 109(5), 728\u2013755 (2021)","journal-title":"Proc. IEEE"},{"key":"701_CR39","volume-title":"Advances in Neural Information Processing Systems","author":"G.F. Montufar","year":"2014","unstructured":"Montufar, G.F., Pascanu, R., Cho, K., Bengio, Y.: On the number of linear regions of deep neural networks. In: Advances in Neural Information Processing Systems vol.\u00a027 (2014)"},{"key":"701_CR40","doi-asserted-by":"publisher","first-page":"607","DOI":"10.1145\/3351095.3372850","volume-title":"Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency","author":"R.K. Mothilal","year":"2020","unstructured":"Mothilal, R.K., Sharma, A., Tan, C.: Explaining machine learning classifiers through diverse counterfactual explanations. In: Proceedings of the 2020 Conference on Fairness, Accountability, and Transparency, pp.\u00a0607\u2013617 (2020)"},{"issue":"5","key":"701_CR41","doi-asserted-by":"publisher","first-page":"1283","DOI":"10.1016\/j.ejc.2009.11.005","volume":"31","author":"S. Ovchinnikov","year":"2010","unstructured":"Ovchinnikov, S.: Discrete piecewise linear functions. Eur. J. Comb. 31(5), 1283\u20131294 (2010)","journal-title":"Eur. J. Comb."},{"key":"701_CR42","doi-asserted-by":"publisher","first-page":"582","DOI":"10.1109\/SP.2016.41","volume-title":"2016 IEEE Symposium on Security and Privacy (SP)","author":"N. Papernot","year":"2016","unstructured":"Papernot, N., McDaniel, P., Wu, X., Jha, S., Swami, A.: Distillation as a defense to adversarial perturbations against deep neural networks. In: 2016 IEEE Symposium on Security and Privacy (SP), pp.\u00a0582\u2013597. IEEE Comput. Soc., Los Alamitos (2016)"},{"key":"701_CR43","unstructured":"Pascanu, R., Montufar, G., Bengio, Y.: On the number of response regions of deep feed forward networks with piece-wise linear activations. Arxiv preprint (2013). arXiv:1312.6098"},{"key":"701_CR44","series-title":"PMLR","first-page":"2847","volume-title":"International Conference on Machine Learning","author":"M. Raghu","year":"2017","unstructured":"Raghu, M., Poole, B., Kleinberg, J., Ganguli, S., Sohl-Dickstein, J.: On the expressive power of deep neural networks. In: International Conference on Machine Learning. PMLR, pp.\u00a02847\u20132854. (2017)"},{"key":"701_CR45","doi-asserted-by":"publisher","unstructured":"Schl\u00fcter, M., Nolte, G., Murtovi, A., Bernhard, S.: Towards rigorous understanding of Neural Networks via semantics-preserving transformations. Int. J. Softw. Tools Technol. Transf. (2023, in press). https:\/\/doi.org\/10.1007\/s10009-023-00700-7","DOI":"10.1007\/s10009-023-00700-7"},{"key":"701_CR46","series-title":"PMLR","first-page":"4558","volume-title":"International Conference on Machine Learning","author":"T. Serra","year":"2018","unstructured":"Serra, T., Tjandraatmadja, C., Ramalingam, S.: Bounding and counting linear regions of deep neural networks. In: International Conference on Machine Learning. PMLR, pp.\u00a04558\u20134566. (2018)"},{"key":"701_CR47","series-title":"POPL","first-page":"1","volume-title":"Proceedings of the ACM on Programming Languages","author":"G. Singh","year":"2019","unstructured":"Singh, G., Gehr, T., P\u00fcschel, M., Vechev, M.: An abstract domain for certifying neural networks. In: Proceedings of the ACM on Programming Languages. POPL vol.\u00a03, pp.\u00a01\u201330 (2019)"},{"key":"701_CR48","unstructured":"Sudjianto, A., Knauth, W., Singh, R., Yang, Z., Zhang, A.: Unwrapping the black box of deep ReLU networks: Interpretability, diagnostics, and simplification. Arxiv (2020). arXiv:2011.04041"},{"key":"701_CR49","unstructured":"Szegedy, C., Zaremba, W., Sutskever, I., Bruna, J., Erhan, D., Goodfellow, I., Fergus, R.: Intriguing properties of neural networks. Arxiv preprint (2013). arXiv:1312.6199"},{"key":"701_CR50","volume-title":"Pattern Recognition","author":"S. Theodoridis","year":"2006","unstructured":"Theodoridis, S., Koutroumbas, K.: Pattern Recognition. Elsevier, Amsterdam (2006)"},{"issue":"7782","key":"701_CR51","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1038\/s41586-019-1724-z","volume":"575","author":"O. Vinyals","year":"2019","unstructured":"Vinyals, O., Babuschkin, I., Czarnecki, W.M., Mathieu, M., Dudzik, A., Chung, J., Choi, D.H., Powell, R., Ewalds, T., Georgiev, P., et al.: Grandmaster level in starcraft ii using multi-agent reinforcement learning. Nature 575(7782), 350\u2013354 (2019)","journal-title":"Nature"},{"key":"701_CR52","unstructured":"Wang, S., Chen, Y., Abdou, A., Jana, S.: Mixtrain: Scalable training of verifiably robust neural networks. Arxiv preprint (2018). arXiv:1811.02625"},{"key":"701_CR53","first-page":"29909","volume":"34","author":"S. Wang","year":"2021","unstructured":"Wang, S., Zhang, H., Xu, K., Lin, X., Jana, S., Hsieh, C.J., Kolter, J.Z.: Beta-crown: efficient bound propagation with per-neuron split constraints for neural network robustness verification. Adv. Neural Inf. Process. Syst. 34, 29909\u201329921 (2021)","journal-title":"Adv. Neural Inf. Process. Syst."},{"issue":"1\u20133","key":"701_CR54","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/0169-7439(87)80084-9","volume":"2","author":"S. Wold","year":"1987","unstructured":"Wold, S., Esbensen, K., Geladi, P.: Principal component analysis. Chemom. Intell. Lab. Syst. 2(1\u20133), 37\u201352 (1987)","journal-title":"Chemom. Intell. Lab. Syst."},{"key":"701_CR55","doi-asserted-by":"crossref","unstructured":"Woo, S., Lee, C.L.: Decision boundary formation of deep convolution networks with ReLU. In: 2018 IEEE 16th Intl. Conf. on Dependable, Autonomic and Secure Computing, 16th Intl. Conf. on Pervasive Intelligence and Computing, 4th Intl. Conf. on Big Data Intelligence and Computing and Cyber Science and Technology Congress (DASC\/PiCom\/DataCom\/CyberSciTech), pp.\u00a0885\u2013888. IEEE (2018)","DOI":"10.1109\/DASC\/PiCom\/DataCom\/CyberSciTec.2018.00-13"},{"key":"701_CR56","series-title":"PMLR","first-page":"5824","volume-title":"International Conference on Machine Learning","author":"L. Zhang","year":"2018","unstructured":"Zhang, L., Naitzat, G., Lim, L.H.: Tropical geometry of deep neural networks. In: International Conference on Machine Learning. PMLR, pp.\u00a05824\u20135832 (2018)"},{"key":"701_CR57","unstructured":"Zhang, X., Wu, D.: Empirical studies on the properties of linear regions in deep neural networks. Arxiv preprint (2020). arXiv:2001.01072"},{"key":"701_CR58","first-page":"4480","volume-title":"Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition","author":"S. Zheng","year":"2016","unstructured":"Zheng, S., Song, Y., Leung, T., Goodfellow, I.: Improving the robustness of deep neural networks via stability training. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp.\u00a04480\u20134488 (2016)"}],"container-title":["International Journal on Software Tools for Technology Transfer"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10009-023-00701-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10009-023-00701-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10009-023-00701-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,19]],"date-time":"2023-09-19T09:09:13Z","timestamp":1695114553000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10009-023-00701-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,21]]},"references-count":58,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,6]]}},"alternative-id":["701"],"URL":"https:\/\/doi.org\/10.1007\/s10009-023-00701-6","relation":{},"ISSN":["1433-2779","1433-2787"],"issn-type":[{"type":"print","value":"1433-2779"},{"type":"electronic","value":"1433-2787"}],"subject":[],"published":{"date-parts":[[2023,4,21]]},"assertion":[{"value":"19 February 2023","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 April 2023","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}