Convex Optimization · Problem 7.8

Geometric problems — Boyd & Vandenberghe

Mathematics — Optimization · ml-estimation-sign-measurements

Problem

Estimation using sign measurements. We consider the measurement setup

y_i = \mathbf{sign}(a_i^T x + b_i + v_i), \quad i = 1, \ldots, m,

where

x \in \mathbf{R}^n

is the vector to be estimated, and

y_i \in \{-1, 1\}

are the measurements. The vectors

a_i \in \mathbf{R}^n

and scalars

b_i \in \mathbf{R}

are known, and

v_i

are IID noises with a log-concave probability density. (You can assume that

a_i^T x + b_i + v_i = 0

does not occur.) Show that maximum likelihood estimation of

x

is a convex optimization problem.