On polyhedral estimation of signals via indirect observations

Abstract

We consider the problem of recovering linear image of unknown signal belonging to a given convex compact signal set from noisy observation of another linear image of the signal. We develop a simple generic efficiently computable nonlinear in observations "polyhedral" estimate along with computation-friendly techniques for its design and risk analysis. We demonstrate that under favorable circumstances the resulting estimate is provably nearoptimal in the minimax sense, the "favorable circumstances" being less restrictive than the weakest known so far assumptions ensuring near-optimality of estimates which are linear in observations.

Aggregating estimates by convex optimization

“…which is then used to compute N polyhedral estimates z x i .b !/ following the recipe in [27] and [28,Section 5.1.5].…”

“…As such, the problem we consider is that of recovery from noisy indirect observations, the latter being equivalent to estimating univariate function s. /, estimation error being measured in the L 2 -norm on OE 1; 1. We consider two implementations of the recovery procedure; in both implementations we utilize polyhedral estimate of [27] to build pilot estimates z x i . z !/, i D 1; : : : ; N .…”

Aggregating estimates by convex optimization

“…which is then used to compute N polyhedral estimates x i ( ω) following the recipe in [28] and [27, Section 5.1.5]. • Finally, we apply the aggregation routine from Section 3.2 to assemble points…”

“…As such, the problem we consider is that of recovery from noisy indirect observations, the latter being equivalent to estimating univariate function s(•), estimation error being measured in the L 2 -norm on [−1, 1]. We consider two implementations of the recovery procedure; in both implementations we utilize polyhedral estimate of [28] to build pilot estimates x i ( ω), i = 1, ..., N . The first recovery, we denote it x (I) , utilizes the aggregated estimate described in Sections 5.3, 5.4; x (II) is the adaptive estimate of Section 3.3; finally, estimate x (III) is the slightly modified adaptive estimate of Section 3.2 in which, when the set I(ω) of admissible estimates contains more than 1 point, instead of selecting the admissible estimate with the smallest index i, adaptive estimate x is obtained by aggregating admissible points x i , i ∈ I(ω), as the optimal solution to the optimization problem…”