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Vector quantile regression and optimal transport, from theory to numerics

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Abstract

In this paper, we first revisit the Koenker and Bassett variational approach to (univariate) quantile regression, emphasizing its link with latent factor representations and correlation maximization problems. We then review the multivariate extension due to Carlier et al. (Ann Statist 44(3):1165–92, 2016,; J Multivariate Anal 161:96–102, 2017) which relates vector quantile regression to an optimal transport problem with mean independence constraints. We introduce an entropic regularization of this problem, implement a gradient descent numerical method and illustrate its feasibility on univariate and bivariate examples.

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Notes

  1. Whenever we write a variable in brackets after a constraint, as \([V_t]\) in (1.1), we mean that this variable plays the role of a multiplier.

  2. It may seem awkward to start with the “dual” formulation before giving out the “primal” one, and the “primal” being the dual to the “dual,” this choice of labeling is pretty arbitrary. However, our choice is motivated by consistency with optimal transport theory, introduced below.

  3. One way to define the nonatomicity of \((\Omega , {\mathcal {F}}, \mathbb {P})\) is by the existence of a uniformly distributed random variable on this space, this somehow ensures that the space is rich enough so that there exists random variables with prescribed law. If, on the contrary, the space is finite for instance only finitely supported probability measures can be realized as the law of such random variables.

  4. In fact for (2.3) to make sense one needs some integrabilty of Y, i.e., \({\mathbb {E}}(\vert Y\vert ) <+\infty \).

  5. if \({\mathbb {E}}(\Vert X\Vert ^2)<+\infty \) then (3.11) amounts to the standard requirement that \({\mathbb {E}}(X X^{\top })\) is nonsingular.

  6. Uniqueness will be discussed later on.

  7. If quantile regression is specified and the pair of functions \((\alpha , \beta )\) is as in definition 3.1, then for every t, \((\alpha (t), \beta (t))\) solves the conditions (3.13). This shows that specification implies quasi-specification.

  8. With a little abuse of notations when a reference number (A) refers to a maximization (minimization) problem, we will simply write \(\sup (A)\) (\(\inf (A) \)) to the denote the value of this optimization problem.

  9. Note the analogy with the fact that in the univariate case the cdf and the quantile of Y are generalized inverse to each other.

  10. In the case where \({\mathbb {E}}(\Vert Y\Vert ^2)<+\infty \), (5.4) is equivalent to minimize \({\mathbb {E}}(\Vert V- Y\Vert ^2)\) among uniformly distributed V’s.

  11. A deep regularity theory initated by Caffarelli (1992) in the 1990’s gives conditions on \(\nu (.\vert x)\) such that this is in fact the case that the optimal transport map is smooth and/or invertible, we refer the interested reader to the textbook of Figalli (2017) for a detailed and recent account of this regularity theory.

  12. here we assume that both X and Y are integrable

  13. Recall that the softmax with regularization parameter \(\varepsilon >0\) of \((\alpha _{1},\ldots ,\alpha _{J})\) is given by \({\mathrm {Softmax}}_{\varepsilon }(\alpha _{1},\ldots \alpha _{J}):=\varepsilon \log (\sum _{j=1}^{J}e^{\frac{\alpha _{j}}{\varepsilon } })\).

  14. Which can be proved either by using the Fenchel-Rockafellar duality theorem or by hand. Indeed, in the primal, there are only finitely many linear constraints and nonnegativity constraints are not binding because of the entropy. The existence of Lagrange multipliers for the equality constraints is then straightforward.

  15. it is even strictly convex once we have chosen normalizations which take into account the two invariances of J explained above.

  16. https://www.openlab.psu.edu/ansur2/

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Funding

Funding from Région Ile-de-France grant is acknowledged. Funding from NSF Grant DMS-1716489 is acknowledged.

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Authors and Affiliations

  1. CEREMADE, UMR CNRS 7534, PSL, Université Paris IX Dauphine, Pl. de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Guillaume Carlier

  2. MOKAPLAN Inria, Paris, France

    Guillaume Carlier

  3. Department of Economics, MIT, 50 Memorial Drive, E52-361B, Cambridge, MA, 02142, USA

    Victor Chernozhukov

  4. DMA, ENS, Paris, France

    Gwendoline De Bie

  5. Economics and Mathematics Departments, New York University, 70 Washington Square South, New York, NY, 10013, USA

    Alfred Galichon

Authors
  1. Guillaume Carlier
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  2. Victor Chernozhukov
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  3. Gwendoline De Bie
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  4. Alfred Galichon
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Correspondence to Alfred Galichon.

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Appendices

Appendix

Proof of Lemma 4.3

Since \(\mathbf {1}_{[0,t]}\in {\mathcal {C}}\), one obviously first has

$$\begin{aligned} \sup _{v\in {\mathcal {C}}}\int _{0}^{1}v(s)q(s)ds\ge \max _{t\in [0,1]}\int _{0}^{t}q(s)ds=\max _{t\in [0,1]}Q(t). \end{aligned}$$

Let us now prove the converse inequality, taking an arbitrary \(v\in { \mathcal {C}}\). We first observe that Q is absolutely continuous and that v is of bounded variation (its derivative in the sense of distributions being a bounded nonpositive measure which we denote by \(\eta \)), integrating by parts and using the definition of \({\mathcal {C}}\) then give:

$$\begin{aligned} \int _{0}^{1}v(s)q(s)ds&=-\int _{0}^{1}Q\eta +v(1^{-})Q(1) \\&\le (\max _{[0,1]}Q)\times (-\eta ([0,1])+v(1^{-})Q(1) \\&=(\max _{[0,1]}Q)(v(0^{+})-v(1^{-}))+v(1^{-})Q(1) \\&=(\max _{[0,1]}Q)v(0^{+})+(Q(1)-\max _{[0,1]}Q)v(1^{-}) \\&\le \max _{[0,1]}Q. \end{aligned}$$

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Carlier, G., Chernozhukov, V., De Bie, G. et al. Vector quantile regression and optimal transport, from theory to numerics. Empir Econ 62, 35–62 (2022). https://doi.org/10.1007/s00181-020-01919-y

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