In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.

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A tutorial on spectral clustering

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Statistics for Spatial Data.

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

We propose a framework for analyzing and comparing distributions, which we use to construct statistical tests to determine if two samples are drawn from different distributions. Our test statistic is the largest difference in expectations over functions in the unit ball of a reproducing kernel Hilbert space (RKHS), and is called the maximum mean discrepancy (MMD).We present two distribution free tests based on large deviation bounds for the MMD, and a third test based on the asymptotic distribution of this statistic. The MMD can be computed in quadratic time, although efficient linear time approximations are available. Our statistic is an instance of an integral probability metric, and various classical metrics on distributions are obtained when alternative function classes are used in place of an RKHS. We apply our two-sample tests to a variety of problems, including attribute matching for databases using the Hungarian marriage method, where they perform strongly. Excellent performance is also obtained when comparing distributions over graphs, for which these are the first such tests.

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A kernel two-sample test

In ballistic deposition (BD), $(d+1)$-dimensional particles fall sequentially at random towards an initially flat, large but bounded $d$-dimensional surface, and each particle sticks to the first point of contact. For both lattice and continuum BD, a law of large numbers in the thermodynamic limit establishes convergence of the mean height and surface width of the interface to constants $h(t)$ and $w(t)$, respectively, depending on time $t$. We show that $h(t)$ is asymptotically linear in $t$, while $w(t)$ grows at least logarithmically in $t$ when $d=1$. We also give duality results saying that the height above the origin for deposition onto an initially flat surface is equidistributed with the maximum height for deposition onto a surface growing from a single site.

Growth and roughness of the interface for ballistic deposition

Minima of Independent Bessel Processes and of Distances Between Brownian Particles

In the on-line nearest-neighbour graph (ONG), each point after the first in a sequence of points in R^d is joined by an edge to its nearest-neighbour amongst those points that precede it in the sequence. We study the large-sample asymptotic behaviour of the total power-weighted length of the ONG on uniform random points in (0,1)^d. In particular, for d=1 and weight exponent \alpha>1/2, the limiting distribution of the centred total weight is characterized by a distributional fixed-point equation. As an ancillary result, we give exact expressions for the expectation and variance of the standard nearest-neighbour (directed) graph on uniform random points in the unit interval.

/pdf/limit-theory-for-the-random-on-line-nearest-neighbour-graph-412np71aie.pdf

Limit theory for the random on-line nearest-neighbour graph

In the threshold contact process on thed-dimensional integer lattice with ranger, healthy sites become infected at rate λ if they have at least one infectedr-neighbour, and recover at rate 1 We show that the critical valueλc(r) is asymptotic tor−d μc asr→∞, where μc is the critical value of the birth rate μ for a continuum threshold contact process which may be described in terms of an oriented continuous percolation model driven by a Poisson process of rate μ ind+1 dimensions We have bounds of 07320 < μc < 3 ford=1

The threshold contact process: a continuum limit

Various random spatial graphs defined on partially ordered point sets have been treated recently in the literature: these are discussed in the framework of the minimal directed spanning forest. Global distributional limits are discussed, and examples given both of Gaussian and non-Gaussian limiting cases.

Mathew D. Penrose

Papers

Growth and roughness of the interface for ballistic deposition

Minima of Independent Bessel Processes and of Distances Between Brownian Particles

Limit theory for the random on-line nearest-neighbour graph

The threshold contact process: a continuum limit

Random directed and on-line networks