Algebraic geometric comparison of probability distributions

Franz J. Király, Paul Von Bünau, Frank C. Meinecke, Duncan A J Blythe, Klaus Muller

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

We propose a novel algebraic algorithmic framework for dealing with probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of finding the subspace on which several probability distributions agree. Instead of minimizing an objective function involving the estimated cumulants, we show that by treating the cumulants as elements of the polynomial ring we can directly solve the problem, at a lower computational cost and with higher accuracy. Moreover, the algebraic viewpoint on probability distributions allows us to invoke the theory of algebraic geometry, which we demonstrate in a compact proof for an identifiability criterion.

Original languageEnglish
Pages (from-to)855-903
Number of pages49
JournalJournal of Machine Learning Research
Volume13
Publication statusPublished - 2012 Mar 1

Fingerprint

Cumulants
Probability distributions
Probability Distribution
Unsupervised learning
Unsupervised Learning
Algebraic Geometry
Identifiability
Polynomial ring
Covariance matrix
Computational Cost
High Accuracy
Objective function
Subspace
Polynomials
Geometry
Demonstrate
Costs

Keywords

  • Approximate algebra
  • Computational algebraic geometry
  • Unsupervised Learning

ASJC Scopus subject areas

  • Artificial Intelligence
  • Software
  • Control and Systems Engineering
  • Statistics and Probability

Cite this

Király, F. J., Von Bünau, P., Meinecke, F. C., Blythe, D. A. J., & Muller, K. (2012). Algebraic geometric comparison of probability distributions. Journal of Machine Learning Research, 13, 855-903.

Algebraic geometric comparison of probability distributions. / Király, Franz J.; Von Bünau, Paul; Meinecke, Frank C.; Blythe, Duncan A J; Muller, Klaus.

In: Journal of Machine Learning Research, Vol. 13, 01.03.2012, p. 855-903.

Research output: Contribution to journalArticle

Király, FJ, Von Bünau, P, Meinecke, FC, Blythe, DAJ & Muller, K 2012, 'Algebraic geometric comparison of probability distributions', Journal of Machine Learning Research, vol. 13, pp. 855-903.
Király FJ, Von Bünau P, Meinecke FC, Blythe DAJ, Muller K. Algebraic geometric comparison of probability distributions. Journal of Machine Learning Research. 2012 Mar 1;13:855-903.
Király, Franz J. ; Von Bünau, Paul ; Meinecke, Frank C. ; Blythe, Duncan A J ; Muller, Klaus. / Algebraic geometric comparison of probability distributions. In: Journal of Machine Learning Research. 2012 ; Vol. 13. pp. 855-903.
@article{19d7a38d61da463ea42d35f484bfc1df,
title = "Algebraic geometric comparison of probability distributions",
abstract = "We propose a novel algebraic algorithmic framework for dealing with probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of finding the subspace on which several probability distributions agree. Instead of minimizing an objective function involving the estimated cumulants, we show that by treating the cumulants as elements of the polynomial ring we can directly solve the problem, at a lower computational cost and with higher accuracy. Moreover, the algebraic viewpoint on probability distributions allows us to invoke the theory of algebraic geometry, which we demonstrate in a compact proof for an identifiability criterion.",
keywords = "Approximate algebra, Computational algebraic geometry, Unsupervised Learning",
author = "Kir{\'a}ly, {Franz J.} and {Von B{\"u}nau}, Paul and Meinecke, {Frank C.} and Blythe, {Duncan A J} and Klaus Muller",
year = "2012",
month = "3",
day = "1",
language = "English",
volume = "13",
pages = "855--903",
journal = "Journal of Machine Learning Research",
issn = "1532-4435",
publisher = "Microtome Publishing",

}

TY - JOUR

T1 - Algebraic geometric comparison of probability distributions

AU - Király, Franz J.

AU - Von Bünau, Paul

AU - Meinecke, Frank C.

AU - Blythe, Duncan A J

AU - Muller, Klaus

PY - 2012/3/1

Y1 - 2012/3/1

N2 - We propose a novel algebraic algorithmic framework for dealing with probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of finding the subspace on which several probability distributions agree. Instead of minimizing an objective function involving the estimated cumulants, we show that by treating the cumulants as elements of the polynomial ring we can directly solve the problem, at a lower computational cost and with higher accuracy. Moreover, the algebraic viewpoint on probability distributions allows us to invoke the theory of algebraic geometry, which we demonstrate in a compact proof for an identifiability criterion.

AB - We propose a novel algebraic algorithmic framework for dealing with probability distributions represented by their cumulants such as the mean and covariance matrix. As an example, we consider the unsupervised learning problem of finding the subspace on which several probability distributions agree. Instead of minimizing an objective function involving the estimated cumulants, we show that by treating the cumulants as elements of the polynomial ring we can directly solve the problem, at a lower computational cost and with higher accuracy. Moreover, the algebraic viewpoint on probability distributions allows us to invoke the theory of algebraic geometry, which we demonstrate in a compact proof for an identifiability criterion.

KW - Approximate algebra

KW - Computational algebraic geometry

KW - Unsupervised Learning

UR - http://www.scopus.com/inward/record.url?scp=84859464784&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84859464784&partnerID=8YFLogxK

M3 - Article

VL - 13

SP - 855

EP - 903

JO - Journal of Machine Learning Research

JF - Journal of Machine Learning Research

SN - 1532-4435

ER -