[Solved] Measurement with bounded errors. A series of K measurements y1, …,YK E RP, are taken in order to estimate an unknown vector x € RP

Measurement with bounded errors. A series of K measurements y1, …,YK E RP, are taken in order to estimate an unknown vector x € RP. The measurements are related to the unknown vector x by Yi Ax+vi where vi is a measurement noise that satisfies || Villoo <a but is otherwise unknown. (In other words, the entries of v1, …, VK are no larger than a.) The matrix A and the measurement noise norm bound a are known. Let X denote the set of vectors x that are consistent with the observations Yı, …, YK, i.e., the set of x that could have resulted in the measurements made. Is X convex? Now we will examine what happens when the measurements are occasionally in error, i.e., for a few i we have no relation between x and yi. More precisely suppose that Ifault is a subset of /1,…,K), and that Yi = Ax + vị with ||0i||0o = a (as above) for i & Ifault, but for i e Ifault, there is no relation between x and Yi. The set Ifault is the set of times of the faulty measurements. Suppose you know that Ifault has at most J elements, i.e., out of K measurements, at most J are faulty. You do not know Ifault; you know only a bound on its cardinality (size). For what values of J is X, the set of x consistent with the measurements, convex?

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