We study the optimality of the minimax risk of truncated series estimators for symmetric convex polytopes. We show that the optimal truncated series estimator is within @math factor of the optimal if the polytope is defined by @math hyperplanes. This represents the first such bounds towards general convex bodies. In proving our result, we first define a geometric quantity, called the , for lower bounding the minimax risk. We then derive our bounds by establishing a connection between the approximation radius and the Kolmogorov width, the quantity that provides upper bounds for the truncated series estimator. Besides, our proof contains several ingredients which might be of independent interest: 1. The notion of approximation radius depends on the volume of the body. It is an intuitive notion and is flexible to yield strong minimax lower bounds; 2. The connection between the approximation radius and the Kolmogorov width is a consequence of a novel duality relationship on the Kolmogorov width, developed by utilizing some deep results from convex geometry.
On the other hand, the truncated series estimator has a nice geometric interpretation and is related to the classical Kolmogorov width of the underlying space. In addition to its simplicity, @cite_1 shows that it is asymptotically optimal for the classes of orthosymmetric and quadratically convex objects. This includes the class of diagonally stretched @math balls for @math . Present paper shows that the power of truncated series estimators also extend to the family of symmetric convex polytopes, as long as the polytope is defined by @math hyperplanes.
@cite_1
1980999467 2004853061 2333550441 2014689751
Minimum sums of moments or, equivalently, distortion of optimum quantizers play an important role in several branches of mathematics. Fejes Toth's inequality for sums of moments in the plane and Zador's asymptotic formula for minimum distortion in Euclidean d-space are the first precise pertinent results in dimension d⩾2. In this article these results are generalized in the form of asymptotic formulae for minimum sums of moments, resp. distortion of optimum quantizers on Riemannian d-manifolds and normed d-spaces. In addition, we provide geometric and analytic information on the structure of optimum configurations. Our results are then used to obtain information on (i) the minimum distortion of high-resolution vector quantization and optimum quantizers, (ii) the error of best approximation of probability measures by discrete measures and support sets of best approximating discrete measures, (iii) the minimum error of numerical integration formulae for classes of Holder continuous functions and optimum sets of nodes, (iv) best volume approximation of convex bodies by circumscribed convex polytopes and the form of best approximating polytopes, and (v) the minimum isoperimetric quotient of convex polytopes in Minkowski spaces and the form of the minimizing polytopes. Consider estimating the mean of a standard Gaussian shift when that mean is known to lie in an orthosymmetric quadratically convex set in l 2 . The minimax risk among linear estimates is within 25 of the minimax risk among all estimates. The minimax risk among truncated series estimates is within a factor 4.44 of the minimax risk. This implies that the difficulty of estimation ― a statistical quantity ― is measured fairly precisely by the n-widths ― a geometric quantity. If the set is not quadratically convex, as in the case of l p -bodies with p<2, things change appreciably. Minimax linear estimators may be out-performed arbitrarily by nonlinear estimates In this paper, to resolve unclassifiable regions in the support vector machines, we propose fuzzy support vector machines (FSVMs). Using the decision function obtained by training the SVM, for each class, we define a truncated polyhedral pyramidal membership function. Since, for the data in the classifiable regions, the classification results are the same, the generalization ability of the FSVM is the same as or better than that of the SVM. To further improve the generalization ability, we introduce the Bayes theory, assuming that the class distributions are normal, to optimize the bias term of the optimal hyperplane. We evaluate our methods for four benchmark data sets and demonstrate the superiority of the FSVM and Bayes FSVM over the SVM. The method of stable random projections is popular in data stream computations, data mining, information retrieval, and machine learning, for efficiently computing the lα (0 We propose algorithms based on (1) the geometric mean estimator, for all 0 • The general sample complexity bound for α ≠ 1,2. For α = 1, [27] provided a nice argument based on the inverse of Cauchy density about the median, leading to a sample complexity bound, although they did not provide the constants and their proof restricted e to be "small enough." For general α ≠ 1, 2, however, the task becomes much more difficult. [27] provided the "conceptual promise" that the sample complexity bound similar to that for α = 1 should exist for general α, if a "non-uniform algorithm based on t-quantile" could be implemented. Such a conceptual algorithm was only for supporting the arguments in [27], not a real implementation. We consider this is one of the main problems left open in [27]. In this study, we propose a practical algorithm based on the geometric mean estimator and derive the sample complexity bound for all 0 • The practical and optimal algorithm for α = 0+ The l0 norm is an important case. Stable random projections can provide an approximation to the l0 norm using α → 0+. We provide an algorithm based on the harmonic mean estimator, which is simple and statistically optimal. Its tail bounds are sharper than the bounds derived based on the geometric mean. We also discover a (possibly surprising) fact: in boolean data, stable random projections using α = 0+ with the harmonic mean estimator will be about twice as accurate as (l2) normal random projections. Because high-dimensional boolean data are common, we expect this fact will be practically quite useful. • The precise theoretical analysis and practical implications We provide the precise constants in the tail bounds for both the geometric mean and harmonic mean estimators. We also provide the variances (either exact or asymptotic) for the proposed estimators. These results can assist practitioners to choose sample sizes accurately.