Abstract
Convolutional neural networks used in real-world recognition must be able to detect inputs that are Out-of-Distribution (OoD) with respect to the known or training data. A popular, simple method is to detect OoD inputs using confidence scores based on the Mahalanobis distance from known data. However, this procedure involves estimating the multivariate normal (MVN) density of high dimensional data using the insufficient number of observations (e.g., the dimensionality of features at the last two layers in the ResNet-101 model are 2048 and 1024, with ca. 1000–5000 examples per class for density estimation). In this work, we analyze the instability of parametric estimates of MVN density in high dimensionality and analyze the impact of this on the performance of Mahalanobis distance-based OoD detection. We show that this effect makes Mahalanobis distance-based methods ineffective for near OoD data. We show that the minimum distance from known data beyond which outliers are detectable depends on the dimensionality and number of training samples and decreases with the growing size of the training dataset. We also analyzed the performance of modifications of the Mahalanobis distance method used to minimize density fitting errors, such as using a common covariance matrix for all classes or diagonal covariance matrices. On OoD benchmarks (on CIFAR-10, CIFAR-100, SVHN, and Noise datasets), using representations from the DenseNet or ResNet models, we show that none of these methods should be considered universally superior.
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References
Bendale, A., Boult, T.: Towards Open World Recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1893–1902 (2015)
Breunig, M.M., Kriegel, H.P., Ng, R.T., Sander, J.: Lof: identifying density-based local outliers. SIGMOD Rec. 29(2), 93–104 (2000). https://doi.org/10.1145/335191.335388
Chakraborty, A., Alam, M., Dey, V., Chattopadhyay, A., Mukhopadhyay, D.: A survey on adversarial attacks and defences. CAAI Trans. Intell. Technol. 6(1), 25–45 (2021). https://doi.org/10.1049/cit2.12028
Eykholt, K., et al.: Robust physical-world attacks on deep learning visual classification. In: 2018 IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2018, Salt Lake City, UT, USA, 18–22 June 2018, pp. 1625–1634. IEEE Computer Society (2018). https://doi.org/10.1109/CVPR.2018.00175
Feng, D., Rosenbaum, L., Dietmayer, K.: Towards safe autonomous driving: capture uncertainty in the deep neural network for lidar 3d vehicle detection. In: 2018 21st International Conference on Intelligent Transportation Systems (ITSC), pp. 3266–3273. IEEE (2018)
Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, New York (2009). https://doi.org/10.1007/978-0-387-21606-5
He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 770–778 (2016)
Hendrycks, D., Mazeika, M., Dietterich, T.: Deep anomaly detection with outlier exposure. Proceedings of the International Conference on Learning Representations (2019)
Hendrycks, D., Zhao, K., Basart, S., Steinhardt, J., Song, D.: Natural adversarial examples. arXiv preprint arXiv:1907.07174 (2019)
Huang, G., Liu, Z., Van Der Maaten, L., Weinberger, K.Q.: Densely connected convolutional networks. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 4700–4708 (2017)
Lee, K., Lee, K., Lee, H., Shin, J.: A simple unified framework for detecting out-of-distribution samples and adversarial attacks. In: Proceedings of the 32nd International Conference on Neural Information Processing Systems, pp. 7167–7177. NIPS 2018, Curran Associates Inc., Red Hook, NY (2018)
Lin, M., Chen, Q., Yan, S.: Network in network. arXiv preprint arXiv:1312.4400 (2013)
McAllister, R., et al.: Concrete problems for autonomous vehicle safety: Advantages of bayesian deep learning. In: Proceedings of the 26th International Joint Conference on Artificial Intelligence, pp. 4745–4753. IJCAI 2017, AAAI Press (2017)
Nguyen, A., Yosinski, J., Clune, J.: Deep neural networks are easily fooled: high confidence predictions for unrecognizable images. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 427–436 (2015)
Pham, H., Dai, Z., Xie, Q., Luong, M.T., Le, Q.V.: Meta pseudo labels. In: IEEE Conference on Computer Vision and Pattern Recognition (2021). https://arxiv.org/abs/2003.10580
Ren, J., Fort, S., Liu, J., Roy, A.G., Padhy, S., Lakshminarayanan, B.: A simple fix to mahalanobis distance for improving near-ood detection. arXiv preprint arXiv:2106.09022 (2021)
Rousseeuw, P.J.: Least median of squares regression. J. Am. Stat. Assoc. 79(388), 871–880 (1984)
Sehwag, V., Chiang, M., Mittal, P.: Ssd: a unified framework for self-supervised outlier detection. In: International Conference on Learning Representations (2021). https://openreview.net/forum?id=v5gjXpmR8J
Sharif, M., Bhagavatula, S., Bauer, L., Reiter, M.K.: Accessorize to a crime: real and stealthy attacks on state-of-the-art face recognition. In: Proceedings of the 2016 ACM SIGSAC Conference on Computer and Communications Security, pp. 1528–1540 (2016)
Zhou, Z., Firestone, C.: Humans can decipher adversarial images. Nature Commun. 10(1), 1–9 (2019)
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Maciejewski, H., Walkowiak, T., Szyc, K. (2022). Out-of-Distribution Detection in High-Dimensional Data Using Mahalanobis Distance - Critical Analysis. In: Groen, D., de Mulatier, C., Paszynski, M., Krzhizhanovskaya, V.V., Dongarra, J.J., Sloot, P.M.A. (eds) Computational Science – ICCS 2022. ICCS 2022. Lecture Notes in Computer Science, vol 13350. Springer, Cham. https://doi.org/10.1007/978-3-031-08751-6_19
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