Convex Optimization · Problem 7.9

Geometric problems — Boyd & Vandenberghe

Mathematics — Optimization · ml-estimation-unknown-sensor-nonlinearity

Problem

Estimation with unknown sensor nonlinearity. We consider the measurement setup

y_i = f(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,

y_i \in \mathbf{R}

are the measurements,

a_i \in \mathbf{R}^n

b_i \in \mathbf{R}

are known, and

v_i

are IID noises with log-concave probability density. The function

f : \mathbf{R} \to \mathbf{R}

, which represents a measurement nonlinearity, is not known. However, it is known that

f'(t) \in [l, u]

for all

t

, where

0 < l < u

are given. Explain how to use convex optimization to find a maximum likelihood estimate of

x

, as well as the function

f

. (This is an infinite-dimensional ML estimation problem, but you can be informal in your approach and explanation.)