Generalized Aitchison Embeddings for Histograms

Tam Le, Marco Cuturi
Proceedings of the 5th Asian Conference on Machine Learning, PMLR 29:293-308, 2013.

Abstract

Learning distances that are specifically designed to compare histograms in the probability simplex has recently attracted the attention of the community. Learning such distances is important because most machine learning problems involve bags of features rather than simple vectors. Ample empirical evidence suggests that the Euclidean distance in general and Mahalanobis metric learning in particular may not be suitable to quantify distances between points in the simplex. We propose in this paper a new contribution to address this problem by generalizing a family of embeddings proposed by Aitchison (1982) to map the probability simplex onto a suitable Euclidean space. We provide algorithms to estimate the parameters of such maps, and show that these algorithms lead to representations that outperform alternative approaches to compare histograms.

Cite this Paper

BibTeX
@InProceedings{pmlr-v29-Le13, title = {Generalized Aitchison Embeddings for Histograms}, author = {Le, Tam and Cuturi, Marco}, booktitle = {Proceedings of the 5th Asian Conference on Machine Learning}, pages = {293--308}, year = {2013}, editor = {Ong, Cheng Soon and Ho, Tu Bao}, volume = {29}, series = {Proceedings of Machine Learning Research}, address = {Australian National University, Canberra, Australia}, month = {13--15 Nov}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v29/Le13.pdf}, url = {https://proceedings.mlr.press/v29/Le13.html}, abstract = {Learning distances that are specifically designed to compare histograms in the probability simplex has recently attracted the attention of the community. Learning such distances is important because most machine learning problems involve bags of features rather than simple vectors. Ample empirical evidence suggests that the Euclidean distance in general and Mahalanobis metric learning in particular may not be suitable to quantify distances between points in the simplex. We propose in this paper a new contribution to address this problem by generalizing a family of embeddings proposed by Aitchison (1982) to map the probability simplex onto a suitable Euclidean space. We provide algorithms to estimate the parameters of such maps, and show that these algorithms lead to representations that outperform alternative approaches to compare histograms.} }
Endnote
%0 Conference Paper %T Generalized Aitchison Embeddings for Histograms %A Tam Le %A Marco Cuturi %B Proceedings of the 5th Asian Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2013 %E Cheng Soon Ong %E Tu Bao Ho %F pmlr-v29-Le13 %I PMLR %P 293--308 %U https://proceedings.mlr.press/v29/Le13.html %V 29 %X Learning distances that are specifically designed to compare histograms in the probability simplex has recently attracted the attention of the community. Learning such distances is important because most machine learning problems involve bags of features rather than simple vectors. Ample empirical evidence suggests that the Euclidean distance in general and Mahalanobis metric learning in particular may not be suitable to quantify distances between points in the simplex. We propose in this paper a new contribution to address this problem by generalizing a family of embeddings proposed by Aitchison (1982) to map the probability simplex onto a suitable Euclidean space. We provide algorithms to estimate the parameters of such maps, and show that these algorithms lead to representations that outperform alternative approaches to compare histograms.
RIS
TY - CPAPER TI - Generalized Aitchison Embeddings for Histograms AU - Tam Le AU - Marco Cuturi BT - Proceedings of the 5th Asian Conference on Machine Learning DA - 2013/10/21 ED - Cheng Soon Ong ED - Tu Bao Ho ID - pmlr-v29-Le13 PB - PMLR DP - Proceedings of Machine Learning Research VL - 29 SP - 293 EP - 308 L1 - http://proceedings.mlr.press/v29/Le13.pdf UR - https://proceedings.mlr.press/v29/Le13.html AB - Learning distances that are specifically designed to compare histograms in the probability simplex has recently attracted the attention of the community. Learning such distances is important because most machine learning problems involve bags of features rather than simple vectors. Ample empirical evidence suggests that the Euclidean distance in general and Mahalanobis metric learning in particular may not be suitable to quantify distances between points in the simplex. We propose in this paper a new contribution to address this problem by generalizing a family of embeddings proposed by Aitchison (1982) to map the probability simplex onto a suitable Euclidean space. We provide algorithms to estimate the parameters of such maps, and show that these algorithms lead to representations that outperform alternative approaches to compare histograms. ER -
APA
Le, T. & Cuturi, M.. (2013). Generalized Aitchison Embeddings for Histograms. Proceedings of the 5th Asian Conference on Machine Learning, in Proceedings of Machine Learning Research 29:293-308 Available from https://proceedings.mlr.press/v29/Le13.html.

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