Quadric Hypersurface Intersection for Manifold Learning in Feature Space

Fedor Pavutnitskiy, Sergei O. Ivanov, Evgeniy Abramov, Viacheslav Borovitskiy, Artem Klochkov, Viktor Vyalov, Anatolii Zaikovskii, Aleksandr Petiushko
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:10999-11013, 2022.

Abstract

The knowledge that data lies close to a particular submanifold of the ambient Euclidean space may be useful in a number of ways. For instance, one may want to automatically mark any point far away from the submanifold as an outlier or to use the geometry to come up with a better distance metric. Manifold learning problems are often posed in a very high dimension, e.g. for spaces of images or spaces of words. Today, with deep representation learning on the rise in areas such as computer vision and natural language processing, many problems of this kind may be transformed into problems of moderately high dimension, typically of the order of hundreds. Motivated by this, we propose a manifold learning technique suitable for moderately high dimension and large datasets. The manifold is learned from the training data in the form of an intersection of quadric hypersurfaces—simple but expressive objects. At test time, this manifold can be used to introduce a computationally efficient outlier score for arbitrary new data points and to improve a given similarity metric by incorporating the learned geometric structure into it.

Cite this Paper

BibTeX
@InProceedings{pmlr-v151-pavutnitskiy22a, title = { Quadric Hypersurface Intersection for Manifold Learning in Feature Space }, author = {Pavutnitskiy, Fedor and Ivanov, Sergei O. and Abramov, Evgeniy and Borovitskiy, Viacheslav and Klochkov, Artem and Vyalov, Viktor and Zaikovskii, Anatolii and Petiushko, Aleksandr}, booktitle = {Proceedings of The 25th International Conference on Artificial Intelligence and Statistics}, pages = {10999--11013}, year = {2022}, editor = {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel}, volume = {151}, series = {Proceedings of Machine Learning Research}, month = {28--30 Mar}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v151/pavutnitskiy22a/pavutnitskiy22a.pdf}, url = {https://proceedings.mlr.press/v151/pavutnitskiy22a.html}, abstract = { The knowledge that data lies close to a particular submanifold of the ambient Euclidean space may be useful in a number of ways. For instance, one may want to automatically mark any point far away from the submanifold as an outlier or to use the geometry to come up with a better distance metric. Manifold learning problems are often posed in a very high dimension, e.g. for spaces of images or spaces of words. Today, with deep representation learning on the rise in areas such as computer vision and natural language processing, many problems of this kind may be transformed into problems of moderately high dimension, typically of the order of hundreds. Motivated by this, we propose a manifold learning technique suitable for moderately high dimension and large datasets. The manifold is learned from the training data in the form of an intersection of quadric hypersurfaces—simple but expressive objects. At test time, this manifold can be used to introduce a computationally efficient outlier score for arbitrary new data points and to improve a given similarity metric by incorporating the learned geometric structure into it. } }
Endnote
%0 Conference Paper %T Quadric Hypersurface Intersection for Manifold Learning in Feature Space %A Fedor Pavutnitskiy %A Sergei O. Ivanov %A Evgeniy Abramov %A Viacheslav Borovitskiy %A Artem Klochkov %A Viktor Vyalov %A Anatolii Zaikovskii %A Aleksandr Petiushko %B Proceedings of The 25th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2022 %E Gustau Camps-Valls %E Francisco J. R. Ruiz %E Isabel Valera %F pmlr-v151-pavutnitskiy22a %I PMLR %P 10999--11013 %U https://proceedings.mlr.press/v151/pavutnitskiy22a.html %V 151 %X The knowledge that data lies close to a particular submanifold of the ambient Euclidean space may be useful in a number of ways. For instance, one may want to automatically mark any point far away from the submanifold as an outlier or to use the geometry to come up with a better distance metric. Manifold learning problems are often posed in a very high dimension, e.g. for spaces of images or spaces of words. Today, with deep representation learning on the rise in areas such as computer vision and natural language processing, many problems of this kind may be transformed into problems of moderately high dimension, typically of the order of hundreds. Motivated by this, we propose a manifold learning technique suitable for moderately high dimension and large datasets. The manifold is learned from the training data in the form of an intersection of quadric hypersurfaces—simple but expressive objects. At test time, this manifold can be used to introduce a computationally efficient outlier score for arbitrary new data points and to improve a given similarity metric by incorporating the learned geometric structure into it.
APA
Pavutnitskiy, F., Ivanov, S.O., Abramov, E., Borovitskiy, V., Klochkov, A., Vyalov, V., Zaikovskii, A. & Petiushko, A.. (2022). Quadric Hypersurface Intersection for Manifold Learning in Feature Space . Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 151:10999-11013 Available from https://proceedings.mlr.press/v151/pavutnitskiy22a.html.

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