6,079 research outputs found
Algorithms for Colourful Simplicial Depth and Medians in the Plane
The colourful simplicial depth of a point x in the plane relative to a
configuration of n points in k colour classes is exactly the number of closed
simplices (triangles) with vertices from 3 different colour classes that
contain x in their convex hull. We consider the problems of efficiently
computing the colourful simplicial depth of a point x, and of finding a point,
called a median, that maximizes colourful simplicial depth.
For computing the colourful simplicial depth of x, our algorithm runs in time
O(n log(n) + k n) in general, and O(kn) if the points are sorted around x. For
finding the colourful median, we get a time of O(n^4). For comparison, the
running times of the best known algorithm for the monochrome version of these
problems are O(n log(n)) in general, improving to O(n) if the points are sorted
around x for monochrome depth, and O(n^4) for finding a monochrome median.Comment: 17 pages, 8 figure
Photon-Efficient Computational 3D and Reflectivity Imaging with Single-Photon Detectors
Capturing depth and reflectivity images at low light levels from active
illumination of a scene has wide-ranging applications. Conventionally, even
with single-photon detectors, hundreds of photon detections are needed at each
pixel to mitigate Poisson noise. We develop a robust method for estimating
depth and reflectivity using on the order of 1 detected photon per pixel
averaged over the scene. Our computational imager combines physically accurate
single-photon counting statistics with exploitation of the spatial correlations
present in real-world reflectivity and 3D structure. Experiments conducted in
the presence of strong background light demonstrate that our computational
imager is able to accurately recover scene depth and reflectivity, while
traditional maximum-likelihood based imaging methods lead to estimates that are
highly noisy. Our framework increases photon efficiency 100-fold over
traditional processing and also improves, somewhat, upon first-photon imaging
under a total acquisition time constraint in raster-scanned operation. Thus our
new imager will be useful for rapid, low-power, and noise-tolerant active
optical imaging, and its fixed dwell time will facilitate parallelization
through use of a detector array.Comment: 11 pages, 8 figure
A Geometric Approach to Covariance Matrix Estimation and its Applications to Radar Problems
A new class of disturbance covariance matrix estimators for radar signal
processing applications is introduced following a geometric paradigm. Each
estimator is associated with a given unitary invariant norm and performs the
sample covariance matrix projection into a specific set of structured
covariance matrices. Regardless of the considered norm, an efficient solution
technique to handle the resulting constrained optimization problem is
developed. Specifically, it is shown that the new family of distribution-free
estimators shares a shrinkagetype form; besides, the eigenvalues estimate just
requires the solution of a one-dimensional convex problem whose objective
function depends on the considered unitary norm. For the two most common norm
instances, i.e., Frobenius and spectral, very efficient algorithms are
developed to solve the aforementioned one-dimensional optimization leading to
almost closed form covariance estimates. At the analysis stage, the performance
of the new estimators is assessed in terms of achievable Signal to Interference
plus Noise Ratio (SINR) both for a spatial and a Doppler processing assuming
different data statistical characterizations. The results show that interesting
SINR improvements with respect to some counterparts available in the open
literature can be achieved especially in training starved regimes.Comment: submitted for journal publicatio
Robust Learning from Bites for Data Mining
Some methods from statistical machine learning and from robust statistics have two drawbacks. Firstly, they are computer-intensive such that they can hardly be used for massive data sets, say with millions of data points. Secondly, robust and non-parametric confidence intervals for the predictions according to the fitted models are often unknown. Here, we propose a simple but general method to overcome these problems in the context of huge data sets. The method is scalable to the memory of the computer, can be distributed on several processors if available, and can help to reduce the computation time substantially. Our main focus is on robust general support vector machines (SVM) based on minimizing regularized risks. The method offers distribution-free confidence intervals for the median of the predictions. The approach can also be helpful to fit robust estimators in parametric models for huge data sets. --Breakdown point,convex risk minimization,data mining,distributed computing,influence function,logistic regression,robustness,scalability
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