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数学代写|凸优化代写Convex Optimization代考|EE364A ESTIMATING N-CONVEX FUNCTIONS ON UNIONS OF CONVEX SETS

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数学代写|凸优化代写Convex Optimization代考|ESTIMATING N-CONVEX FUNCTIONS ON UNIONS OF CONVEX SETS

In this section, we apply our testing machinery to the estimation problem as follows.
Given are:

• a simple o.s. $\mathcal{O}=\left(\Omega, \Pi ;\left{p_\mu: \mu \in \mathcal{M}\right} ; \mathcal{F}\right)$,
• a signal space $X \subset \mathbf{R}^n$ along with the affine mapping $x \mapsto A(x): X \rightarrow \mathcal{M}$,
• a real-valued function $f$ on $X$.
Given observation $\omega \sim p_{A\left(x_\right)}$ stemming from unknown signal $x_$ known to belong to $X$, we want to recover $f\left(x_*\right)$.

Our approach imposes severe restrictions on $f$ (satisfied, e.g., when $f$ is linear, or linear-fractional, or is the maximum of several linear functions); as a compensation, we allow for rather “complex” $X$-finite unions of convex sets.

数学代写|凸优化代写Convex Optimization代考|Outline

Though the estimator we develop is, in a nutshell, quite simple, its formal description turns out to be rather involved. ${ }^3$ For this reason we start its presentation with an informal outline, which exposes some simple ideas underlying its construction.
Consider the situation where the signal space $X$ is the $2 \mathrm{D}$ rectangle as presented on the top of Figure 3.2. (a), and let the function to be recovered be $f(x)=x_1$. Thus, “nature” has somehow selected $x=\left[x_1, x_2\right]$ in the rectangle, and we observe a Gaussian random vector with the mean $A(x)$ and known covariance matrix, where $A(\cdot)$ is a given affine mapping. Note that hypotheses $f(x) \geq b$ and $f(x) \leq a$ translate into convex hypotheses on the expectation of the observed Gaussian r.v., so that we can use our hypothesis testing machinery to decide on hypotheses of this type and to localize $f(x)$ in a (hopefully, small) segment by a Bisection-type process. Before describing the process, let us make a terminological agreement. In the sequel we shall use pairwise hypothesis testing in the situation where it may happen that neither of the hypotheses we are deciding upon is true. In this case, we will say that the outcome of a test is correct if the rejected hypothesis indeed is wrong (the accepted hypothesis can be wrong as well, but the latter can happen only in the case when both our hypotheses are wrong).

数学代写|凸优化代写凸优化代考|在凸集的union上估计n -凸函数

• 一个简单的o.s. $\mathcal{O}=\left(\Omega, \Pi ;\left{p_\mu: \mu \in \mathcal{M}\right} ; \mathcal{F}\right)$，
• 一个信号空间$X \subset \mathbf{R}^n$以及仿射映射$x \mapsto A(x): X \rightarrow \mathcal{M}$，
• 一个实值函数$f$在$X$上
已知观测$\omega \sim p_{A\left(x_\right)}$源于未知信号$x_$已知属于$X$，我们想恢复$f\left(x_*\right)$ .

. Outline

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