Parzen Windows as a nonparametric method has been applied to a variety of density estimation as well as classification problems. Similar to nearest neighbor methods, Parzen Windows does not involve learning. While it converges to true but unknown probability densities in the asymptotic limit, there is a lack of theoretical analysis on its performance with finite samples. In this paper we establ...

In this letter, we evaluate error performance limitations in oversampled subband adaptive lter applications based on an analysis of aliasing in the subband signals. The power spectral density of the minimum error signal is given by the aliased signal components. The presented analysis closely agrees with simulation results.

Assume a standard setting with data D = {(xi, yi)}i=1, where (xi, yi) are sampled iid from the joint distribution p(x, y) on Rd×{±1}. Let H = {h : R 7→ {±1}} be a learning model which produces a hypothesis g ∈ H when given D (we use g for the hypothesis returned by the learning algorithm and h for a generic hypothesis in H). We assume the 0-1 loss, so the in-sample error is ein(h) = 1 2n ∑n i=1...

It’s well known that there is a so-called exponential-type error bound for Gaussian interpolation which is the most powerful error bound hitherto. It’s of the form |f(x) − s(x)| ≤ c1(c2d) c3 d ‖f‖h where f and s are the interpolated and interpolating functions respectively, c1, c2, c3 are positive constants, d is the fill-distance which roughly speaking measures the spacing of the data points, ...

Many problems in applied mathematics require the evaluation of matrix functionals of the form F(A) := u T f(A)u, where A is a large symmetric matrix and u is a vector. Golub and collaborators have described how approximations of such functionals can be computed inexpensively by using the Lanczos algorithm. The present note shows that error bounds for these approximations can be computed essenti...

In this paper, we study properties of rank metric codes in general and maximum rank distance (MRD) codes in particular. For codes with the rank metric, we first establish Gilbert and sphere-packing bounds, and then obtain the asymptotic forms of these two bounds and the Singleton bound. Based on the asymptotic bounds, we observe that asymptotically Gilbert-Varsharmov bound is exceeded by MRD co...

We consider the optimal discrimination of bipartite quantum states and provide an upper bound for maximum success probability local discrimination. also a necessary sufficient condition measurement to realize bound. further establish this be saturated. Finally, we illustrate our results using example.