Measuring dissimilarity with diffeomorphism invariance

Théophile Cantelobre, Carlo Ciliberto, Benjamin Guedj, Alessandro Rudi
Proceedings of the 39th International Conference on Machine Learning, PMLR 162:2572-2596, 2022.

Abstract

Measures of similarity (or dissimilarity) are a key ingredient to many machine learning algorithms. We introduce DID, a pairwise dissimilarity measure applicable to a wide range of data spaces, which leverages the data’s internal structure to be invariant to diffeomorphisms. We prove that DID enjoys properties which make it relevant for theoretical study and practical use. By representing each datum as a function, DID is defined as the solution to an optimization problem in a Reproducing Kernel Hilbert Space and can be expressed in closed-form. In practice, it can be efficiently approximated via Nystr{ö}m sampling. Empirical experiments support the merits of DID.

Cite this Paper

BibTeX
@InProceedings{pmlr-v162-cantelobre22a, title = {Measuring dissimilarity with diffeomorphism invariance}, author = {Cantelobre, Th{\'e}ophile and Ciliberto, Carlo and Guedj, Benjamin and Rudi, Alessandro}, booktitle = {Proceedings of the 39th International Conference on Machine Learning}, pages = {2572--2596}, year = {2022}, editor = {Chaudhuri, Kamalika and Jegelka, Stefanie and Song, Le and Szepesvari, Csaba and Niu, Gang and Sabato, Sivan}, volume = {162}, series = {Proceedings of Machine Learning Research}, month = {17--23 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v162/cantelobre22a/cantelobre22a.pdf}, url = {https://proceedings.mlr.press/v162/cantelobre22a.html}, abstract = {Measures of similarity (or dissimilarity) are a key ingredient to many machine learning algorithms. We introduce DID, a pairwise dissimilarity measure applicable to a wide range of data spaces, which leverages the data’s internal structure to be invariant to diffeomorphisms. We prove that DID enjoys properties which make it relevant for theoretical study and practical use. By representing each datum as a function, DID is defined as the solution to an optimization problem in a Reproducing Kernel Hilbert Space and can be expressed in closed-form. In practice, it can be efficiently approximated via Nystr{ö}m sampling. Empirical experiments support the merits of DID.} }
Endnote
%0 Conference Paper %T Measuring dissimilarity with diffeomorphism invariance %A Théophile Cantelobre %A Carlo Ciliberto %A Benjamin Guedj %A Alessandro Rudi %B Proceedings of the 39th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2022 %E Kamalika Chaudhuri %E Stefanie Jegelka %E Le Song %E Csaba Szepesvari %E Gang Niu %E Sivan Sabato %F pmlr-v162-cantelobre22a %I PMLR %P 2572--2596 %U https://proceedings.mlr.press/v162/cantelobre22a.html %V 162 %X Measures of similarity (or dissimilarity) are a key ingredient to many machine learning algorithms. We introduce DID, a pairwise dissimilarity measure applicable to a wide range of data spaces, which leverages the data’s internal structure to be invariant to diffeomorphisms. We prove that DID enjoys properties which make it relevant for theoretical study and practical use. By representing each datum as a function, DID is defined as the solution to an optimization problem in a Reproducing Kernel Hilbert Space and can be expressed in closed-form. In practice, it can be efficiently approximated via Nystr{ö}m sampling. Empirical experiments support the merits of DID.
APA
Cantelobre, T., Ciliberto, C., Guedj, B. & Rudi, A.. (2022). Measuring dissimilarity with diffeomorphism invariance. Proceedings of the 39th International Conference on Machine Learning, in Proceedings of Machine Learning Research 162:2572-2596 Available from https://proceedings.mlr.press/v162/cantelobre22a.html.

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