Adaptive Data Depth via Multi-Armed Bandits

Data depth, introduced by Tukey (1975), is an important tool in data science, robust statistics, and computational geometry. One chief barrier to its broader practical utility is that many common measures of depth are computationally intensive, requiring on the order of $n^d$ operations to exactly compute the depth of a single point within a data set of $n$ points in $d$-dimensional space. Often however, we are not directly interested in the absolute depths of the points, but rather in their relative ordering. For example, we may want to find the most central point in a data set (a generalized median), or to identify and remove all outliers (points on the fringe of the data set with low depth). With this observation, we develop a novel and instance-adaptive algorithm for adaptive data depth computation by reducing the problem of exactly computing $n$ depths to an $n$-armed stochastic multi-armed bandit problem which we can efficiently solve. We focus our exposition on simplicial depth, developed by Liu (1990), which has emerged as a promising notion of depth due to its interpretability and asymptotic properties. We provide general instance-dependent theoretical guarantees for our proposed algorithms, which readily extend to many other common measures of data depth including majority depth, Oja depth, and likelihood depth. When specialized to the case where the gaps in the data follow a power law distribution with parameter $\alpha<2$, we show that we can reduce the complexity of identifying the deepest point in the data set (the simplicial median) from $O(n^d)$ to $\tilde{O}(n^{d-(d-1)\alpha/2})$, where $\tilde{O}$ suppresses logarithmic factors. We corroborate our theoretical results with numerical experiments on synthetic data, showing the practical utility of our proposed methods.Comment: Keywords: multi-armed bandits, data depth, adaptivity, large-scale computation, simplicial dept

Electrocardiogram derived respiration during sleep

The aim of this study was quantify the ECG Derived Respiration (EDR) in order to extend the capabilities of ECG-based sleep analysis. We examined our results in normal subjects and in patients with Obstructive Sleep Apnea Syndrome (OSAS) or Central Sleep Apnea. Lead 2 ECG and three measures of respiration (thorax and abdominal effort, and oronasal flow signal) were recorded during sleep studies of 12 normal and 12 OSAS patients. Three parameters, the R-wave amplitude (RWA), R-wave duration (RWD), and QRS area, were extracted from the ECG signal, resulting in time series that displayed a behavior similar to that of the respiration signals. EDR frequency was correlated with directly measured respiratory frequency, and averaged over all subjects. The peak-to-peak value of the EDR signals during the apnea event was compared to the average peak-to-peak of the sleep stage, containing the apnea. 1

All-optical silicon simplified passive modulation

In this paper we present an all-optical silicon based modulator suggested also for high power operation and for pulse picker application being used as part of fiber lasers system. The paper theoretically and experimentally investigates several new and important insights involving the dependence of the relative transmission on the pump pulse energy for different finesse values of the constructed cavity as well as the dependence of the response rate of the device to the pump wavelength due to coexistence of two physical recombination processes: fast surface effect and slow bulk recombination. To adapt the constructed silicon based cavity to be used in lasers applications, we aligned the pump and the signal beams to co-propagate through the device while the usage of a cavity allowed a low power pump to yield a significant extinction ratio at the output of the device

Impedance Matrix Compression Using Adaptively-Constructed Basis Functions

Wavelet expansions have been employed recently in numerical solutions of commonly used frequency-domain integral equations. In this paper we propose a novel method for integrating wavelet-based transforms into existing numerical--solvers. The newly proposed method differs from the presently used ones in two ways. First, the transformation is effected by means of a digital filtering approach. This approach enables a much faster implementation of the transform. It also renders the transform algorithm adaptive and facilitates the derivation of a basis which best suits the problem at hand. Second, the conventional thresholding procedure applied to the impedance-matrix is substituted for by a compression process in which only the significant terms in the expansion of the (yet-unknown) current are retained and subsequently derived. Numerical results for a few TM scattering problems are included to demonstrate the advantages of the proposed method over the presently used ones. 1 Introduction ..

Scattering Analysis Using Fictitious Wavelet Array Sources

In this paper we study the incorporation of wavelet--transforms into the source-model technique (SMT) for efficient analysis of electromagnetic scattering problems. The idea is to divide the discrete sources into groups of arrays with wavelet amplitude distributions. We refer to these array sources as fictitious wavelet array sources. They can be readily formed by applying appropriate wavelet transformations to the original matrix equation obtained based on a conventional SMT solution. The transformed impedance-matrix obtained in this manner is then compressed and thus a substantially smaller matrix equation has to be solved. The conventional as well as the windowed Fourier transform variant of the wavelet transform are considered. The ease with which one can adjust the expansion for resolution of small features and for handling small perturbations in the scatterer geometry is demonstrated. A comparison with a conventional method of moments solution is presented to show the advantages ..

Impedance Matrix Compression (IMC) Using Iteratively Selected Wavelet Basis for MFIE Formulations

In this paper, we present a novel approach to incorporating wavelet expansions in method of moments (MoM) solutions for scattering problems described by a magnetic field integral equation (MFIE) formulation. In this approach, we utilize the fact that when the basis--functions used are wavelet-type functions, only a few terms in a series expansion would be needed to represent the unknown quantity. An iterative procedure is suggested to determine these dominant expansion functions. The new approach combined with the iterative procedure yields a new algorithm which has many advantages over the presently used methods for incorporating wavelets. Numerical results which illustrate the approach are presented. 1 Introduction Wavelet expansions have been employed recently in method of moments solutions of frequencydomain integral equations [1, 2, 3, 4, 5]. In these solutions the unknown quantity of interest (usually the current on the scatterer) is first represented in terms of a set of wavel..

Impedance Matrix Compression (IMC) Using Iteratively Selected Wavelet-Basis

In this paper we present a novel approach for the incorporation of wavelets into the solution of frequencydomain integral equations arising in scattering problems. In this approach, we utilize the fact that when the basis functions used are wavelet-type functions, only a few terms in a series expansion would be needed to represent the unknown quantity. Moreover, an iterative procedure is suggested to determine these dominant expansion functions. The new approach combined with the iterative procedure yields a new algorithm which has many advantages over the presently used methods for incorporating wavelets. Numerical results which illustrate the approach are presented for three scattering problems. I. Introduction W AVELET expansions have been employed recently in numerical solutions of commonly used frequencydomain integral equations [1], [2], [3], [4], [5]. In the conventional approach to the solution of these integral equations [1], the unknown quantity of interest (usually the cur..