Regularized geometric quantiles and universal linear distribution functionals

TL;DR

This work introduces regularized geometric quantiles and distribution functions by replacing the singular kernel with a regularizer $oldsymbol{ r}$, yielding $F_P^{oldsymbol{ r}}$ and $Q_P^{oldsymbol{ r}}$ that retain key multivariate quantile properties while removing the need for moment conditions. A universality theorem shows any translation- and orthogonal-equivariant linear distribution functional must arise from such a regularization, yielding a canonical, robust, and computable framework. The authors establish existence, uniqueness, symmetry, and mapping properties, analyze extreme-quantile behavior (including a 'black hole' phenomenon around atoms), and provide thorough empirical-quantile theory with consistency and asymptotic normality via Bahadur representations. Collectively, the results deliver a robust, universally characterized, regularized alternative to geometric quantiles with broad applicability in multivariate statistics and extreme-value analysis.

Abstract

Geometric quantiles are popular location functionals to build rank-based statistical procedures in multivariate settings. They are obtained through the minimization of a non-smooth convex objective function. As a result, the singularity of the directional derivatives leads to numerical instabilities and poor sample properties as well as surprising `phase transitions' from empirical to population distributions. To solve these issues, we introduce a regularized version of geometric distribution functions and quantiles that are provably close to the usual geometric concepts and share their qualitative properties, both in the empirical and continuous case, while allowing for a much broader applicability of asymptotic results without any moment condition. We also show that any linear assignment of probability measures (such as the univariate distribution function), that is also translation- and orthogonal-equivariant, necessarily coincides with one of our regularized geometric distribution functions.

Figures (13)

• Figure 1: Graphical representation of the empirical quantiles $q_{\alpha,u}^\mathfrak{r}$, where $\mathfrak{r}(s)=\mathfrak{r}_{\beta}(s)=1-(1+s)^{-\beta}$, for $\beta=5$ (dashed red curves) and $\beta=10$ (dashed green curves), compared with the classical geometric quantiles $q_{\alpha,u}^{1}$ (full black curves), at levels $\alpha\in \{0.9,0.95,0.99,0.995\}$. The experiments use $n=1000$ data points generated from an equally weighted mixture of the uniform distribution on $[-1,1] \times \{ 0 \}$ and, from left to right, (i) the uniform distribution on $\{ 0 \} \times [-1,1]$, the uniform distribution on $\{ 0 \} \times [-1/2,1/2]$, the uniform distribution on $\{ 0 \} \times [-1/20,1/20]$, and a Dirac mass at the origin. The cross marks the geometric median of the data. The bottom row is a series of zoomed-in versions of the top panels on the square $[-1,1]\times [-1,1]$.
• Figure 2: Graphical representation of the empirical quantiles $q_{\alpha,u}^\mathfrak{r}$, where $\mathfrak{r}(s)=\mathfrak{r}_{\beta}(s)=1-(1+s)^{-\beta}$, for $\beta=5$ (dashed red curves) and $\beta=10$ (dashed green curves), compared with the classical geometric quantiles $q_{\alpha,u}^{1}$ (full black curves), at levels $\alpha\in \{0.9,0.95,0.99,0.995\}$. The experiments use $n=1000$ data points generated from a weighted mixture of the uniform distribution on $[-1,1] \times \{ 0 \}$ and, from left to right, a Dirac mass at (i) $(0,1)$ with weight $1/2$, (ii) $(0,2/3)$ with weight $1/3$, (iii) $(0,1/3)$ with weight $1/6$. The rightmost panels use $n=1000$ data points uniformly generated on $[-1,1] \times \{ 0 \}$. The cross marks the geometric median of the data. The bottom row is a series of zoomed-in versions of the top panels on the square $[-1.5,1.5]\times [-1.5,1.5]$.
• Figure 3: Graphical representation of the empirical quantiles $q_{\alpha,u}^\mathfrak{r}$, where $\mathfrak{r}(s)=\mathfrak{r}_{\beta}(s)=1-(1+s)^{-\beta}$, for $\beta=2$ (dashed blue curves) and $\beta=5$ (dashed red curves), compared with the classical geometric quantiles $q_{\alpha,u}^{1}$ (full black curves), at levels $\alpha\in \{0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8\}$. The data points are the same as in Figure \ref{['fig:extreme1to4']}. The bottom row is a series of zoomed-in versions of the top panels on the square $[-0.3,0.3]\times [-0.3,0.3]$.
• Figure 4: Graphical representation of the empirical quantiles $q_{\alpha,u}^\mathfrak{r}$, where $\mathfrak{r}(s)=\mathfrak{r}_{\beta}(s)=1-(1+s)^{-\beta}$, for $\beta=2$ (dashed blue curves) and $\beta=5$ (dashed red curves), compared with the classical geometric quantiles $q_{\alpha,u}^{1}$ (full black curves), at levels $\alpha\in \{0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8\}$. The data points are the same as in Figure \ref{['fig:extreme5to8']}. The bottom row is a series of zoomed-in versions of the top panels on the square $[-0.3,0.3]\times [-0.3,0.3]$.
• Figure 5: Black holes of classical geometric quantiles corresponding to four uniform distributions on the vertices of triangles in $\mathbb{R}^2$, denoted by $ABC$, with (i) $A(-1,-1/\sqrt{3})$, $B(1,-1/\sqrt{3})$ and $C(0,\sqrt{3}-1/\sqrt{3})$ (equilateral triangle), (ii) $A(-1,-1/3)$, $B(1,-1/3)$ and $C(0,2/3)$ (isosceles triangle), (iii) $A(-1/3,-2/3)$, $B(2/3,-2/3)$ and $C(-1/3,4/3)$ (right-angled triangle) and (iv) $A(0,-1/3)$, $B(1,-1/3)$ and $C(-1,2/3)$ (points in general position). The three black holes are delimited by a dashed circle and centered at points located at the black circular marks.

Theorems & Definitions (55)

• Theorem 1.1
• Theorem 2.5
• Definition 2.6
• Definition 3.1
• Theorem 3.2
• Proposition 3.3
• Proposition 3.4
• Theorem 3.5
• Theorem 3.6
• Proposition 4.1