The problem of estimation of density functionals like entropy and mutual information has received much attention in the statistics and information theory communities. A large class of estimators of functionals of the probability density suffer from the curse of dimensionality, wherein the mean squared error (MSE) decays increasingly slowly as a function of the sample size T as the dimension d of the samples increases. In particular, the rate is often glacially slow of order O(T−γ/d), where γ > 0 is a rate parameter. Examples of such estimators include kernel density estimators, k-nearest neighbor (k-NN) density estimators, k-NN entropy estimators, intrinsic dimension estimators and other examples. In this paper, we propose a weighted affine combination of an ensemble of such estimators, where optimal weights can be chosen such that the weighted estimator converges at a much faster dimension invariant rate of O(T−1). Furthermore, we show that these optimal weights can be determined by solving a convex optimization problem which can be performed offline and does not require training data. We illustrate the superior performance of our weighted estimator for two important applications: (i) estimating the Panter-Dite distortion-rate factor and (ii) estimating the Shannon entropy for testing the probability distribution of a random sample.

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This paper considers sequential adaptive estimation of sparse signals under a constraint on the total sensing effort. A dynamic programming formulation is derived for the allocation of sensing resources to minimize a cost function related to mean squared estimation error. Allocation policies are developed based on the method of open-loop feedback control. These policies are optimal in the two-stage case and improve monotonically thereafter with the number of stages. Numerical simulations show gains up to several dB as compared to recently proposed adaptive methods, and dramatic gains approaching the oracle limit as compared to non-adaptive estimation.

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Abstract-This paper considers three problems in sparse filter design, the first involving a weighted least-squares constraint on the frequency response, the second a constraint on mean squared error in estimation, and the third a constraint on signalto-noise ratio in detection. The three problems are unified under a single framework based on sparsity maximization under a quadratic performance constraint. Efficient and exact solutions are developed for specific cases in which the matrix in the quadratic constraint is diagonal, block-diagonal, banded, or has low condition number. For the more difficult general case, a low-complexity algorithm based on backward greedy selection is described with emphasis on its efficient implementation. Examples in wireless channel equalization and minimum-variance distortionless-response beamforming show that the backward selection algorithm yields optimally sparse designs in many instances while also highlighting the benefits of sparse design.

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