The exfoliation of acid-exchanged K4Nb6O17 with tetra(n-butyl)ammonium hydroxide in water produces a colloidal suspension of individual sheets, which roll into loosely bound tubular structures. The tubule shape can be made permanent via precipitation of the colloid with alkali cations. Atomic force microscopy and transmission electron micrographs reveal that the tubules have outer diameters ranging from 15 to 30 nm and that they are 0.1 to 1 μm in length. The observed curling tendency, preferential folding, and cleavage angles of the individual sheets are interpreted in terms of the crystal structure of the parent solid, K4Nb6O17. The driving force for tubule formation appears to be relief of strain that is inherent in the asymmetric single sheets. This driving force is absent in bilayer colloids formed early in the exfoliation process, which are found only as flat sheets. Tubules in colloidal suspensions that have been subjected to turbulence have a tendency to unroll into flat sheets on surfaces, indicating that the forces controlling rolling and unrolling are closely balanced.

Location-aware networks are of great importance and interest in both civil and military applications. This paper determines the localization accuracy of an agent, which is equipped with an antenna array and localizes itself using wireless measurements with anchor nodes, in a far-field environment. In view of the Cram\'er-Rao bound, we first derive the localization information for static scenarios and demonstrate that such information is a weighed sum of Fisher information matrices from each anchor-antenna measurement pair. Each matrix can be further decomposed into two parts: a distance part with intensity proportional to the squared baseband effective bandwidth of the transmitted signal and a direction part with intensity associated with the normalized anchor-antenna visual angle. Moreover, in dynamic scenarios, we show that the Doppler shift contributes additional direction information, with intensity determined by the agent velocity and the root mean squared time duration of the transmitted signal. In addition, two measures are proposed to evaluate the localization performance of wireless networks with different anchor-agent and array-antenna geometries, and both formulae and simulations are provided for typical anchor deployments and antenna arrays.Comment: to appear in IEEE Transactions on Information Theor

We consider the problem of estimating functionals of discrete distributions, and focus on tight nonasymptotic analysis of the worst case squared error risk of widely used estimators. We apply concentration inequalities to analyze the random fluctuation of these estimators around their expectations, and the theory of approximation using positive linear operators to analyze the deviation of their expectations from the true functional, namely their \emph{bias}. We characterize the worst case squared error risk incurred by the Maximum Likelihood Estimator (MLE) in estimating the Shannon entropy $H(P) = \sum_{i = 1}^S -p_i \ln p_i$, and $F_\alpha(P) = \sum_{i = 1}^S p_i^\alpha,\alpha>0$, up to multiplicative constants, for any alphabet size $S\leq \infty$ and sample size $n$ for which the risk may vanish. As a corollary, for Shannon entropy estimation, we show that it is necessary and sufficient to have $n \gg S$ observations for the MLE to be consistent. In addition, we establish that it is necessary and sufficient to consider $n \gg S^{1/\alpha}$ samples for the MLE to consistently estimate $F_\alpha(P), 0<\alpha<1$. The minimax rate-optimal estimators for both problems require $S/\ln S$ and $S^{1/\alpha}/\ln S$ samples, which implies that the MLE has a strictly sub-optimal sample complexity. When $1<\alpha<3/2$, we show that the worst-case squared error rate of convergence for the MLE is $n^{-2(\alpha-1)}$ for infinite alphabet size, while the minimax squared error rate is $(n\ln n)^{-2(\alpha-1)}$. When $\alpha\geq 3/2$, the MLE achieves the minimax optimal rate $n^{-1}$ regardless of the alphabet size. As an application of the general theory, we analyze the Dirichlet prior smoothing techniques for Shannon entropy estimation. We show that no matter how we tune the parameters in the Dirichlet prior, this technique cannot achieve the minimax rates in entropy estimation.Comment: 27 pages, 1 figure, published in IEEE Transactions on Information Theor

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We refine the general methodology in [1] for the construction and analysis of essentially minimax estimators for a wide class of functionals of finite dimensional parameters, and elaborate on the case of discrete distributions with support size S comparable with the number of observations n. Specifically, we determine the "smooth" and "non-smooth" regimes based on the confidence set and the smoothness of the functional. In the "non-smooth" regime, we apply an unbiased estimator for a suitable polynomial approximation of the functional. In the "smooth" regime, we construct a general version of the bias-corrected Maximum Likelihood Estimator (MLE) based on Taylor expansion.We apply the general methodology to the problem of estimating the KL divergence between two discrete probability measures P and Q from empirical data in a non-asymptotic and possibly large alphabet setting. We construct minimax rate-optimal estimators for D(P Q) when the likelihood ratio is upper bounded by a constant which may depend on the support size, and show that the performance of the optimal estimator with n samples is essentially that of the MLE with n ln n samples. Our estimator is adaptive in the sense that it does not require the knowledge of the support size nor the upper bound on the likelihood ratio. We show that the general methodology results in minimax rate-optimal estimators for other divergences as well, such as the Hellinger distance and the χ 2 -divergence. Our approach refines the Approximation methodology recently developed for the construction of near minimax estimators of functionals of high-dimensional parameters, such as entropy, Rényi entropy, mutual information and 1 distance in large alphabet settings, and shows that the effective sample size enlargement phenomenon holds significantly more widely than previously established.

We consider the problem of estimating the L1 distance between two discrete probability measures P and Q from empirical data in a nonasymptotic and large alphabet setting. When Q is known and one obtains n samples from P , we show that for every Q, the minimax rate-optimal estimator with n samples achieves performance comparable to that of the maximum likelihood estimator (MLE) with n ln n samples. When both P and Q are unknown, we construct minimax rateoptimal estimators whose worst case performance is essentially that of the known Q case with Q being uniform, implying that Q being uniform is essentially the most difficult case. The effective sample size enlargement phenomenon, identified in Jiao et al. (2015), holds both in the known Q case for every Q and the Q unknown case. However, the construction of optimal estimators for P − Q 1 requires new techniques and insights beyond the approximation-based method of functional estimation in .

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