f-divergence estimation is an important problem in the fields of information theory, machine learning, and statistics. While several divergence estimators exist, relatively few of their convergence rates are known. We derive the MSE convergence rate for a density plug-in estimator of f-divergence. Then by applying the theory of optimally weighted ensemble estimation, we derive a divergence estimator with a convergence rate of O(1 T) that is simple to implement and performs well in high dimensions. We validate our theoretical results with experiments.
@cite_1 proposed a method for estimating @math -divergences by estimating the likelihood ratio of the two densities by solving a convex optimization problem and then plugging it into the divergence formulas. For this method they prove that the minimax convergence rate is parametric ( @math ) when the likelihood ratio is in the bounded H " o lder class @math with @math . This assumption is weaker than ours which requires the densities to be at least @math times differentiable. However, solving the convex problem of @cite_1 is similar in complexity to training the SVM (between @math and @math ) which can be demanding when @math is very large. In contrast, our method of optimally weighted ensemble estimation depends only on simple density plug-in estimates and an offline convex optimization problem. Thus the most computationally demanding step in our approach is the calculation of the @math -nn distances which has complexity no greater than @math .
@cite_1
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We develop and analyze M-estimation methods for divergence functionals and the likelihood ratios of two probability distributions. Our method is based on a nonasymptotic variational characterization of f -divergences, which allows the problem of estimating divergences to be tackled via convex empirical risk optimization. The resulting estimators are simple to implement, requiring only the solution of standard convex programs. We present an analysis of consistency and convergence for these estimators. Given conditions only on the ratios of densities, we show that our estimators can achieve optimal minimax rates for the likelihood ratio and the divergence functionals in certain regimes. We derive an efficient optimization algorithm for computing our estimates, and illustrate their convergence behavior and practical viability by simulations. We consider nonparametric estimation of L2, Renyi-α and Tsallis-α divergences between continuous distributions. Our approach is to construct estimators for particular integral functionals of two densities and translate them into divergence estimators. For the integral functionals, our estimators are based on corrections of a preliminary plug-in estimator. We show that these estimators achieve the parametric convergence rate of n-1 2 when the densities' smoothness, s, are both at least d 4 where d is the dimension. We also derive minimax lower bounds for this problem which confirm that s > d 4 is necessary to achieve the n-1 2 rate of convergence. We validate our theoretical guarantees with a number of simulations. We analyze mixture density approximation and estimation. We form a convex set of density functions by taking the convex hull of a parametric family, e.g. mixtures of the Gaussian location family. A sequence of finite mixture densities is formulated to provide a parsimonious approximation for the target density. If the target density itself is in the convex hull, we show that the approximation error goes to zero with a rate of 1 k, where k is the number of components in the approximation. If the target density is outside of the convex hull, the approximation error is equal to the best achievable error plus a term that goes to zero with a rate of 1 k. A greedy algorithm that introduces one component at each step is shown to achieve such an error rate. Similarly, a greedy estimation algorithm is provided to find such approximation for data from an arbitrary density. This algorithm estimates one mixture component at one time. We prove that such an algorithm achieves a likelihood nearly as good as the MLE (maximum likelihood estimate) over the whole convex hull. And we identify the difference as being bounded by order O(1 k), where k is the number of components in the estimate. Risks of such estimators are shown to be bounded by a sum of approximation error and estimation error. The error terms are identified. An optimal choice of k can be derived by minimizing the risk bound. Acting as a similar role as the bandwidth in non-parainetric density estimation, k controls two error terms in opposite directions. A large k reduces approximation error and increases estimation error. A MDL (minimum description length) principle is derived to provide an estimation method for k. And the estimated k is shown to achieve the risk bound as if we know the best k in advance. A new information projection theory is derived to expand the approximating class to include its information closure. We prove the existence and uniqueness of a f* in the closure of the convex hull C (in a sense we identify), such that D ( fpf* ) = infg∈CD fpg , where Dfpg is the Kullback-Leibler divergence. And log(fk) → log(f*) in L1 (f) for any sequence fk in C with Dfpfk→ infg∈CD fpg. Other characterizing properties of f* are also given. The problem of estimating the phases (angles) of a complex unit-modulus vector @math from their noisy pairwise relative measurements @math , where @math is a complex-valued Gaussian random matrix, is known as phase synchronization. The maximum likelihood estimator (MLE) is a solution to a unit--modulus-constrained quadratic programming problem, which is nonconvex. Existing works have proposed polynomial-time algorithms such as a semidefinite programming (SDP) relaxation or the generalized power method (GPM). Numerical experiments suggest that both of these methods succeed with high probability for @math up to @math , yet existing analyses only confirm this observation for @math up to @math . In this paper, we bridge the gap by proving that the SDP relaxation is tight for @math , and GPM converges to the global optimum under the same regime. Moreover, we establish a linear convergence rate for GPM, and derive a tight...