# The existence of maximum likelihood estimate in high-dimensional binary response generalized linear models

## TL;DR

This paper investigates the conditions under which maximum likelihood estimates exist in high-dimensional binary response generalized linear models, revealing a phase transition phenomenon influenced by covariate distribution.

## Contribution

It extends previous work by establishing a phase transition for a broad class of models beyond Gaussian covariates, using geometric and probabilistic tools.

## Key findings

- Existence of MLE exhibits a phase transition depending on model parameters.
- The phase transition is influenced by covariate distribution, not just Gaussianity.
- Simulation studies support theoretical phase transition results.

## Abstract

Motivated by recent works on the high-dimensional logistic regression, we establish that the existence of the maximum likelihood estimate exhibits a phase transition for a wide range of generalized linear models with binary outcome and elliptical covariates. This extends a previous result of Cand\`es and Sur who proved the phase transition for the logistic regression with Gaussian covariates. Our result reveals a rich structure in the phase transition phenomenon, which is simply overlooked by Gaussianity. The main tools for deriving the result are data separation, convex geometry and stochastic approximation. We also conduct simulation studies to corroborate our theoretical findings, and explore other features of the problem.