Active MIMO Sensing With Exploration-Exploitation Tradeoff

This paper develops an active sensing framework for designing the transmit and receive beamformers of a multiple-input multiple-output (MIMO) radar system. In the proposed technique, the beamformers are adaptively designed in each sensing stage based on the measurements made in the previous sensing stages. The beamformers are determined by minimizing the Bayesian Cram{é}r-Rao bound (BCRB) for the estimation of the unknown sensing parameters at each stage via Lagrangian dual optimization. To address the exploration-exploitation tradeoff that is inherent to such an adaptive design, this paper proposes two variants of the BCRB optimization problem: an exploration-centric variant, that ensures that multiple orthogonal beamforming directions are probed in each sensing stage, and an exploitation-centric variant, that does not restrict the number of optimal beamformers. Each variant of the optimization problem is solved via an alternating optimization algorithm that alternates between solving for the transmit beamformers and solving for the receive beamformers. The algorithm is shown to converge to a stationary point provided that each optimization problem is solved to global optimality. Moreover, this paper studies each of the two BCRB optimization sub-problems in the Lagrangian dual domain and shows that despite the non-convexity, global optimality is guaranteed provided that certain sufficient conditions hold. The conditions pertain to the multiplicity of the eigenvalues of a specific direction matrix that can be analytically written in terms of the optimal dual variables. These conditions further imply the tightness of the semidefinite relaxation of the optimization problems. Simulation results demonstrate the benefits of the proposed BCRB-based design compared to state-of-the-art adaptive beamforming strategies.

Noisy Computing of the $\mathsf{OR}$ and $\mathsf{MAX}$ Functions

We consider the problem of computing a function of $n$ variables using noisy queries, where each query is incorrect with some fixed and known probability $p \in (0,1/2)$. Specifically, we consider the computation of the $\mathsf{OR}$ function of $n$ bits (where queries correspond to noisy readings of the bits) and the $\mathsf{MAX}$ function of $n$ real numbers (where queries correspond to noisy pairwise comparisons). We show that an expected number of queries of \[ (1 \pm o(1)) \frac{n\log \frac{1}{\delta}}{D_{\mathsf{KL}}(p \| 1-p)} \] is both sufficient and necessary to compute both functions with a vanishing error probability $\delta = o(1)$, where $D_{\mathsf{KL}}(p \| 1-p)$ denotes the Kullback-Leibler divergence between $\mathsf{Bern}(p)$ and $\mathsf{Bern}(1-p)$ distributions. Compared to previous work, our results tighten the dependence on $p$ in both the upper and lower bounds for the two functions.

On the Optimal Bounds for Noisy Computing

We revisit the problem of computing with noisy information considered in Feige et al. 1994, which includes computing the OR function from noisy queries, and computing the MAX, SEARCH and SORT functions from noisy pairwise comparisons. For $K$ given elements, the goal is to correctly recover the desired function with probability at least $1-\delta$ when the outcome of each query is flipped with probability $p$. We consider both the adaptive sampling setting where each query can be adaptively designed based on past outcomes, and the non-adaptive sampling setting where the query cannot depend on past outcomes. The prior work provides tight bounds on the worst-case query complexity in terms of the dependence on $K$. However, the upper and lower bounds do not match in terms of the dependence on $\delta$ and $p$. We improve the lower bounds for all the four functions under both adaptive and non-adaptive query models. Most of our lower bounds match the upper bounds up to constant factors when either $p$ or $\delta$ is bounded away from $0$, while the ratio between the best prior upper and lower bounds goes to infinity when $p\rightarrow 0$ or $p\rightarrow 1/2$. On the other hand, we also provide matching upper and lower bounds for the number of queries in expectation, improving both the upper and lower bounds for the variable-length query model.