Conformal prediction in manifold learning

Alexander Kuleshov; Alexander Bernstein; Evgeny Burnaev

Conformal prediction in manifold learning

Alexander Kuleshov, Alexander Bernstein, Evgeny Burnaev

Proceedings of the Seventh Workshop on Conformal and Probabilistic Prediction and Applications, PMLR 91:234-253, 2018.

Abstract

The paper presents a geometrically motivated view on conformal prediction applied to nonlinear multi-output regression tasks for obtaining valid measure of accuracy of Manifold Learning Regression algorithms. A considered regression task is to estimate an unknown smooth mapping

$\mathbff$ from

$q$ -dimensional inputs

$\mathbfx ∈\mathbfX$ to

$m$ -dimensional outputs

$\mathbfy = \mathbff(\mathbfx)$ based on training dataset

$\mathbfZ_(n)$ consisting of “input-output” pairs

${Z_i = (\mathbfx_i, \mathbfy_i = \mathbff(\mathbfx_i))^\textrmT, i = 1, 2, ..., n}$ . Manifold Learning Regression (MLR) algorithm solves this task using Manifold learning technique. At first, unknown

$q$ -dimensional Regression manifold

$\mathbfM(\mathbff) = {(\mathbfx,\mathbff(\mathbfx))^\textrmT∈\mathrmR^q+m: \mathbfx ∈\mathbfX ⊂\mathrmR^q}$ , embedded in ambient

$(q+m)$ -dimensional space, is estimated from the training data

$\mathbfZ_(n)$ , sampled from this manifold. The constructed estimator

$\mathbfM_\textMLR$ , which is also

$q$ -dimensional manifold embedded in ambient space

$\textrmR^q+m$ , is close to

$\mathbfM$ in terms of Hausdorff distance. After that, an estimator

$\mathbff_\textMLR$ of the unknown function

$\mathbff$ , mapping arbitrary input

$\mathbfx ∈\mathbfX$ to output

$\mathbff_\textrmMLR(\mathbfx)$ , is constructed as the solution to the equation

$\mathbfM(\mathbff_\textrmMLR) = \mathbfM_\textMLR$ . Conformal prediction allows constructing a prediction region for an unknown output

$\mathbfy = \mathbff(\mathbfx)$ at Out-of-Sample input point

$\mathbfx$ for a given confidence level using given nonconformity measure, characterizing to which extent an example

$Z = (\mathbfx, \mathbfy)^\textrmT$ is different from examples in the known dataset

$\mathbfZ_(n)$ . The paper proposes a new nonconformity measure based on MLR estimator using an analog of Bregman distance.

Cite this Paper

BibTeX


@InProceedings{pmlr-v91-kuleshov18a,
  title = 	 {Conformal prediction in manifold learning},
  author = 	 {Kuleshov, Alexander and Bernstein, Alexander and Burnaev, Evgeny},
  booktitle = 	 {Proceedings of the Seventh Workshop on Conformal and Probabilistic Prediction and Applications},
  pages = 	 {234--253},
  year = 	 {2018},
  editor = 	 {Gammerman, Alex and Vovk, Vladimir and Luo, Zhiyuan and Smirnov, Evgueni and Peeters, Ralf},
  volume = 	 {91},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {11--13 Jun},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v91/kuleshov18a/kuleshov18a.pdf},
  url = 	 {https://proceedings.mlr.press/v91/kuleshov18a.html},
  abstract = 	 {The paper presents a geometrically motivated view on conformal prediction applied to nonlinear multi-output regression tasks for obtaining valid measure of accuracy of Manifold Learning Regression algorithms. A considered regression task is to estimate an unknown smooth mapping $\mathbff$ from $q$-dimensional inputs $\mathbfx ∈\mathbfX$ to $m$-dimensional outputs $\mathbfy = \mathbff(\mathbfx)$ based on training dataset $\mathbfZ_(n)$ consisting of “input-output” pairs ${Z_i = (\mathbfx_i, \mathbfy_i = \mathbff(\mathbfx_i))^\textrmT, i = 1, 2, ..., n}$. Manifold Learning Regression (MLR) algorithm solves this task using Manifold learning technique. At first, unknown $q$-dimensional Regression manifold $\mathbfM(\mathbff) = {(\mathbfx,\mathbff(\mathbfx))^\textrmT∈\mathrmR^q+m: \mathbfx ∈\mathbfX ⊂\mathrmR^q}$, embedded in ambient $(q+m)$-dimensional space, is estimated from the training data $\mathbfZ_(n)$, sampled from this manifold. The constructed estimator $\mathbfM_\textMLR$, which is also $q$-dimensional manifold embedded in ambient space $\textrmR^q+m$, is close to $\mathbfM$ in terms of Hausdorff distance. After that, an estimator $\mathbff_\textMLR$ of the unknown function $\mathbff$, mapping arbitrary input $\mathbfx ∈\mathbfX$ to output $\mathbff_\textrmMLR(\mathbfx)$, is constructed as the solution to the equation $\mathbfM(\mathbff_\textrmMLR) = \mathbfM_\textMLR$. Conformal prediction allows constructing a prediction region for an unknown output $\mathbfy = \mathbff(\mathbfx)$ at Out-of-Sample input point $\mathbfx$ for a given confidence level using given nonconformity measure, characterizing to which extent an example $Z = (\mathbfx, \mathbfy)^\textrmT$ is different from examples in the known dataset $\mathbfZ_(n)$. The paper proposes a new nonconformity measure based on MLR estimator using an analog of Bregman distance.}
}

Endnote

%0 Conference Paper
%T Conformal prediction in manifold learning
%A Alexander Kuleshov
%A Alexander Bernstein
%A Evgeny Burnaev
%B Proceedings of the Seventh Workshop on Conformal and Probabilistic Prediction and Applications
%C Proceedings of Machine Learning Research
%D 2018
%E Alex Gammerman
%E Vladimir Vovk
%E Zhiyuan Luo
%E Evgueni Smirnov
%E Ralf Peeters	
%F pmlr-v91-kuleshov18a
%I PMLR
%P 234--253
%U https://proceedings.mlr.press/v91/kuleshov18a.html
%V 91
%X The paper presents a geometrically motivated view on conformal prediction applied to nonlinear multi-output regression tasks for obtaining valid measure of accuracy of Manifold Learning Regression algorithms. A considered regression task is to estimate an unknown smooth mapping $\mathbff$ from $q$-dimensional inputs $\mathbfx ∈\mathbfX$ to $m$-dimensional outputs $\mathbfy = \mathbff(\mathbfx)$ based on training dataset $\mathbfZ_(n)$ consisting of “input-output” pairs ${Z_i = (\mathbfx_i, \mathbfy_i = \mathbff(\mathbfx_i))^\textrmT, i = 1, 2, ..., n}$. Manifold Learning Regression (MLR) algorithm solves this task using Manifold learning technique. At first, unknown $q$-dimensional Regression manifold $\mathbfM(\mathbff) = {(\mathbfx,\mathbff(\mathbfx))^\textrmT∈\mathrmR^q+m: \mathbfx ∈\mathbfX ⊂\mathrmR^q}$, embedded in ambient $(q+m)$-dimensional space, is estimated from the training data $\mathbfZ_(n)$, sampled from this manifold. The constructed estimator $\mathbfM_\textMLR$, which is also $q$-dimensional manifold embedded in ambient space $\textrmR^q+m$, is close to $\mathbfM$ in terms of Hausdorff distance. After that, an estimator $\mathbff_\textMLR$ of the unknown function $\mathbff$, mapping arbitrary input $\mathbfx ∈\mathbfX$ to output $\mathbff_\textrmMLR(\mathbfx)$, is constructed as the solution to the equation $\mathbfM(\mathbff_\textrmMLR) = \mathbfM_\textMLR$. Conformal prediction allows constructing a prediction region for an unknown output $\mathbfy = \mathbff(\mathbfx)$ at Out-of-Sample input point $\mathbfx$ for a given confidence level using given nonconformity measure, characterizing to which extent an example $Z = (\mathbfx, \mathbfy)^\textrmT$ is different from examples in the known dataset $\mathbfZ_(n)$. The paper proposes a new nonconformity measure based on MLR estimator using an analog of Bregman distance.

APA


Kuleshov, A., Bernstein, A. & Burnaev, E.. (2018). Conformal prediction in manifold learning. Proceedings of the Seventh Workshop on Conformal and Probabilistic Prediction and Applications, in Proceedings of Machine Learning Research 91:234-253 Available from https://proceedings.mlr.press/v91/kuleshov18a.html.

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