In setting up this workshop we had to decide whether we wanted to let you become familiar with the method in a complex setting – non-linear, time-dependent calculations – or restrict ourselves to simpler problems, which allows you to get a better understanding of the method. We have chosen for the second line for several reasons: 1) The method is quite new, so it is hard to find references whic...

How can we search for low dimensional structure in high dimensional data? If the data is mainly confined to a low dimensional subspace, then simple linear methods can be used to discover the subspace and estimate its dimensionality. More generally, though, if the data lies on (or near) a low dimensional submanifold, then its structure may be highly nonlinear, and linear methods are bound to fai...

3 Spectral Method 5 3.1 A Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Toy Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.3 General Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.4 Color agreement problems . . . . . . . . . . . . ....

In this paper we introduce new algorithms for unsupervised learning based on the use of a kernel matrix. All the information required by such algorithms is contained in the eigenvectors of the matrix or of closely related matrices. We use two different but related cost functions, the Alignment and the 'cut cost'. The first one is discussed in a companion paper [3], the second one is based on gr...

Now we recall the statement of the central limit theorem (CLT) and give a proof in the case of IID (independent identically distributed) random variables. The weak law of large numbers says that if Xn is a sequence of IID random variables with E[Xn] = 0, then writing Sn = ∑n−1 k=0 Xk, the time averages 1 n Sn converge to 0 in probability, or equivalently (since the limit is a constant), in dist...

Suppose a certain variable Xt is measured at discrete equally spaced time points t = 1, ... , T and we want to make assertions on the energy distribution of the frequencies in a harmonic analysis. This energy distribution may for example be used to code the data if Xt is a speech signal or to make a prediction if the Xt are economic data. Instead of using a deterministic approach applied scient...

Spectral methods have emerged as powerful computational techniques for simulation of complex, smooth physical phenomena. Among other applications they have contributed to our understanding of turbulence by successfully simulating incompressible turbulent flows, have been extensively used in meteorology and geophysics, and have been recently applied to time domain electromagnetics. Several issue...

We take apart, combine and compare on real and artificial data the features of the four best-known spectral clustering algorithms. We find that the algorithms behave more similarly then expected, especially if the data are near a case called perfect, where three of the algorithms are equivalent.

Harmonic balance (HB) methods are frequency-domain algorithms used for high accuracy computation of the periodic steady-state of circuits. Matrix-implicit Krylov-subspace techniques have made it possible for these methods to simulate large circuits more efficiently. However, the harmonic balance methods are not so efficient in computing steady-state solutions of strongly nonlinear circuits with...

With the rapid growth of the World Wide Web and the capacity of digital data storage, tremendous amount of data are generated daily from business and engineering to the Internet and science. The Internet, financial realtime data, hyperspectral imagery, and DNA microarrays are just a few of the common sources that feed torrential streams of data into scientific and business databases worldwide. ...