27,162 research outputs found
An Interior-Point algorithm for Nonlinear Minimax Problems
We present a primal-dual interior-point method for constrained nonlinear, discrete minimax problems where the objective functions and constraints are not necessarily convex. The algorithm uses two merit functions to ensure progress toward the points satisfying the first-order optimality conditions of the original problem. Convergence properties are described and numerical results provided.Discrete min-max, Constrained nonlinear programming, Primal-dual interior-point methods, Stepsize strategies.
Unsupervised Representation Learning with Minimax Distance Measures
We investigate the use of Minimax distances to extract in a nonparametric way
the features that capture the unknown underlying patterns and structures in the
data. We develop a general-purpose and computationally efficient framework to
employ Minimax distances with many machine learning methods that perform on
numerical data. We study both computing the pairwise Minimax distances for all
pairs of objects and as well as computing the Minimax distances of all the
objects to/from a fixed (test) object.
We first efficiently compute the pairwise Minimax distances between the
objects, using the equivalence of Minimax distances over a graph and over a
minimum spanning tree constructed on that. Then, we perform an embedding of the
pairwise Minimax distances into a new vector space, such that their squared
Euclidean distances in the new space equal to the pairwise Minimax distances in
the original space. We also study the case of having multiple pairwise Minimax
matrices, instead of a single one. Thereby, we propose an embedding via first
summing up the centered matrices and then performing an eigenvalue
decomposition to obtain the relevant features.
In the following, we study computing Minimax distances from a fixed (test)
object which can be used for instance in K-nearest neighbor search. Similar to
the case of all-pair pairwise Minimax distances, we develop an efficient and
general-purpose algorithm that is applicable with any arbitrary base distance
measure. Moreover, we investigate in detail the edges selected by the Minimax
distances and thereby explore the ability of Minimax distances in detecting
outlier objects.
Finally, for each setting, we perform several experiments to demonstrate the
effectiveness of our framework.Comment: 32 page
Design of FIR digital filters for pulse shaping and channel equalization using time-domain optimization
Three algorithms are developed for designing finite impulse response digital filters to be used for pulse shaping and channel equalization. The first is the Minimax algorithm which uses linear programming to design a frequency-sampling filter with a pulse shape that approximates the specification in a minimax sense. Design examples are included which accurately approximate a specified impulse response with a maximum error of 0.03 using only six resonators. The second algorithm is an extension of the Minimax algorithm to design preset equalizers for channels with known impulse responses. Both transversal and frequency-sampling equalizer structures are designed to produce a minimax approximation of a specified channel output waveform. Examples of these designs are compared as to the accuracy of the approximation, the resultant intersymbol interference (ISI), and the required transmitted energy. While the transversal designs are slightly more accurate, the frequency-sampling designs using six resonators have smaller ISI and energy values
Minimax Structured Normal Means Inference
We provide a unified treatment of a broad class of noisy structure recovery
problems, known as structured normal means problems. In this setting, the goal
is to identify, from a finite collection of Gaussian distributions with
different means, the distribution that produced some observed data. Recent work
has studied several special cases including sparse vectors, biclusters, and
graph-based structures. We establish nearly matching upper and lower bounds on
the minimax probability of error for any structured normal means problem, and
we derive an optimality certificate for the maximum likelihood estimator, which
can be applied to many instantiations. We also consider an experimental design
setting, where we generalize our minimax bounds and derive an algorithm for
computing a design strategy with a certain optimality property. We show that
our results give tight minimax bounds for many structure recovery problems and
consider some consequences for interactive sampling
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