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For consistent reconstruction from 1-bit measurements, the measurements are treated as sign constraints that are enforced in the reconstruction to recover the signal.
This paper proposes maximum likelihood estimators which are able to deal with measurement errors on all the signals, at each sampling time (including the initial one) and with intrinsic sign constraints on the parameters.
One of the main obstacles for using the power-shaping approach is to write the dynamics in the required form, since a partial differential equation system submitted to sign constraints has to be solved.
The expression Y Φ x≥0 in (3) means that all entries of the vector Y Φ x are no less than 0. Hence, we can treat the 1-bit measurements as sign constraints that should be enforced in the construction of the signal x of interest.
In another closely related study, [ 23] explicitly considered the situation in which the regressors' slopes are distributed as zero plus a continuous distribution [ 23], also allowed sign constraints on the continuous part of the distribution.
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No sign constraint is imposed on the solution, thus limiting free boundaries may have two-phases.
However, in general, we do not consider necessary to assume this sign constraint.
This is denoted by the following sign constraint: ∀ j : S j ∈ 0, 1 (12).
In what follows, Eq. (3) is referred to as sign constraint or consistency condition, interchangeably.
However, the imposition of a sign constraint on the probability and level effect is a strong empirical assumption within this regression model.4.4
In model (4), the ℓ 1-norm objective function is used to favor sparse solutions, the sign constraint Y Φ x≥0 is used to impose the consistency between the 1-bit measurements and the solution, and the constraint ∥x∥2=1 ensures a nontrivial solution lying on the unit ℓ 2 sphere.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com