Singularities in bivariate normal mixtures

TL;DR

This paper studies two-component bivariate normal mixtures through singularity theory, modeling the mixture density as $M_c = c f_1 + (1-c) f_2$ and relating it to the map $F=(f_1,f_2)$. It classifies the product mappings up to $\\mathcal{A}$-equivalence into three types via a generalized distance-squared framework, with the type determined by covariance proportionality and codirectionality of the components. The authors derive explicit normal forms for the singular sets $S(F)$ (hyperbola with a cusp, two intersecting lines, or a line) and establish modality bounds for the mixtures: Type 1 can have up to 3 modes, while Types 2 and 3 have at most 2. The framework provides geometric and topological insights into mixture modality and enables visualization via the image of $F$, linking statistical properties to singularity-theoretic structure.

Abstract

We investigate mappings $F = (f_1, f_2) \colon \mathbb{R}^2 \to \mathbb{R}^2 $ where $ f_1, f_2 $ are bivariate normal densities from the perspective of singularity theory of mappings, motivated by the need to understand properties of two-component bivariate normal mixtures. We show a classification of mappings $ F = (f_1, f_2) $ via $\mathcal{A}$-equivalence and characterize them using statistical notions. Our analysis reveals three distinct types, each with specific geometric properties. Furthermore, we determine the upper bounds for the number of modes in the mixture for each type.

Figures (5)

• Figure 1: The left figure is the graph of two normal densities $f_1, f_2$ with $\hbox{\boldmath $\mu$}_1=(0,0)$, $\hbox{\boldmath $\mu$}_2=(1,1)$, $\Sigma_1=\left(1000.2\right)$, $\Sigma_2=\left(0.2001\right)$. The right figure is the graph of the mixture density $\frac{1}{2} f_1+\frac{1}{2} f_2$ which has three modes (local maximum points).
• Figure 2: The contours of the densities $f_1$ (blue) and $f_2$ (red). The dot lines show the lines going through the mean vectors $\hbox{\boldmath $\mu$}_1$ and $\hbox{\boldmath $\mu$}_2$. The left figure shows a non-codirectional case, and the right figure shows a codirectional case.
• Figure 3: Type 1: The left figure shows the contours of the densities $f_1$ (blue) and $f_2$ (red) which are not codirectional. Here the black curves are the singular set of $F=(f_1,f_2)\colon\mathbb{R}^2\to\mathbb{R}^2$. The right figure shows the image of the mapping $F\colon\mathbb{R}^2\to\mathbb{R}^2$.
• Figure 4: Type 2: The left figure shows the contours of the densities $f_1$ (blue) and $f_2$ (red) which are codirectional. Here the black curves are the singular set of $F=(f_1,f_2)\colon\mathbb{R}^2\to\mathbb{R}^2$. The right figure shows the image of the mapping $F\colon\mathbb{R}^2\to\mathbb{R}^2$.
• Figure 5: Type 3: The left figure shows the contours of the densities $f_1$ (blue) and $f_2$ (red) with the variances $\Sigma_1, \Sigma_2$ being proportional. Here the black curve is the singular set of $F=(f_1,f_2)\colon\mathbb{R}^2\to\mathbb{R}^2$. The right figure shows the image of the mapping $F\colon\mathbb{R}^2\to\mathbb{R}^2$.

Theorems & Definitions (20)

• Remark 1.1
• Proposition 2.1: Ichiki2016
• Theorem 2.2: Theorem 1 and Proposition 2 in Ichiki2016
• Proposition 2.3
• Remark 2.4
• Corollary 3.1
• Remark 3.2
• Definition 3.3
• Example 3.4
• Lemma 3.5