A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N = 6, ..., 20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases

Empirical Tests for Evaluation of Multirate Filter Bank Parameters

Empirical tests have been developed for evaluating the numerical properties of multirate M-band filter banks represented as N matrices of filter coe#cients. Each test returns a numerically observed estimate ofa1 M vector parameter in which the m element corresponds to the m filter band. These vector valued parameters can be readily converted to scalar valued parameters for comparison of filter bank performance or optimization of filter bank design. However, they are intended primarily for the characterization and verification of filter banks. By characterizing the numerical performance of analytic or algorithmic designs, these tests facilitate the experimental verification of theoretical specifications

A generalization of the Entropy Power Inequality to Bosonic Quantum Systems

In most communication schemes information is transmitted via travelling modes of electromagnetic radiation. These modes are unavoidably subject to environmental noise along any physical transmission medium and the quality of the communication channel strongly depends on the minimum noise achievable at the output. For classical signals such noise can be rigorously quantified in terms of the associated Shannon entropy and it is subject to a fundamental lower bound called entropy power inequality. Electromagnetic fields are however quantum mechanical systems and then, especially in low intensity signals, the quantum nature of the information carrier cannot be neglected and many important results derived within classical information theory require non-trivial extensions to the quantum regime. Here we prove one possible generalization of the Entropy Power Inequality to quantum bosonic systems. The impact of this inequality in quantum information theory is potentially large and some relevant implications are considered in this work

Scale and Translation Invariant Methods for Enhanced Time-Frequency Pattern Recognition

Time-frequency (t-f) analysis has clearly reached a certain maturity. One can now often provide striking visual representations of the joint time-frequency energy representation of signals. However, it has been difficult to take advantage of this rich source of information concerning the signal, especially for multidimensional signals. Properly constructed time-frequency distributions enjoy many desirable properties. Attempts to incorporate t-f analysis results into pattern recognition schemes have not been notably successful to date. Aided by Cohen's scale transform one may construct representations from the t-f results which are highly useful in pattern classification. Such methods can produce two dimensional representations which are invariant to time-shift, frequency-shift and scale changes. In addition, two dimensional objects such as images can be represented in a like manner in a four dimensional form. Even so, remaining extraneous variations often defeat the pattern classification approach. This paper presents a method based on noise subspace concepts. The noise subspace enhancement allows one to separate the desired invariant forms from extraneous variations, yielding much improved classification results. Examples from sound classification are discussed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47350/1/11045_2004_Article_181150.pd