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.

/pdf/a-tutorial-on-spectral-clustering-536pmg9onu.pdf

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.

/pdf/a-kernel-two-sample-test-2mr8hwo4ml.pdf

A kernel two-sample test

The Minimum of a Large Number of Bessel Processes

Quasi-everywhere properties of Brownian level sets and multiple points

We give a general local central limit theorem for the sum of two independent random variables, one of which satisfies a central limit theorem while the other satisfies a local central limit theorem with the same order variance We apply this result to various quantities arising in stochastic geometry, including: size of the largest component for percolation on a box; number of components, number of edges, or number of isolated points, for random geometric graphs; covered volume for germ-grain coverage models; number of accepted points for finite-input random sequential adsorption; sum of nearest-neighbour distances for a random sample from a continuous multidimensional distribution

/pdf/local-central-limit-theorems-in-stochastic-geometry-59yetknt7b.pdf

Local Central Limit Theorems in Stochastic Geometry

Consider a graph on randomly scattered points in an arbitrary space, with two points $x,y$ connected with probability $\phi(x,y)$. Suppose the number of points is large but the mean number of isolated points is $O(1)$. We give general criteria for the latter to be approximately Poisson distributed. More generally, we consider the number of vertices of fixed degree, the number of components of fixed order, and the number of edges. We use a general result on Poisson approximation by Stein's method for a set of points selected from a Poisson point process. This method also gives a good Poisson approximation for U-statistics of a Poisson process.

Inhomogeneous random graphs, isolated vertices, and Poisson approximation

Consider a Boolean model in $\R^d$ with balls of random, bounded radii with distribution $F_0$, centered at the points of a Poisson process of intensity $t>0$. The capacity functional of the infinite cluster $Z_\infty$ is given by $\theta_L(t) = \BP\{ Z_\infty\cap L\ne\emptyset\}$, defined for each compact $L\subset\R^d$. 
We prove for any fixed $L$ and $F_0$ that $\theta_L(t)$ is infinitely differentiable in $t$, except at the critical value $t_c$; we give a Margulis-Russo type formula for the derivatives. More generally, allowing the distribution $F_0$ to vary and viewing $\theta_L$ as a function of the measure $F:=tF_0$, we show that it is infinitely differentiable in all directions with respect to the measure $F$ in the supercritical region of the cone of positive measures on a bounded interval. 
We also prove that $\theta_L(\cdot)$ grows at least linearly at the critical value. This implies that the critical exponent known as $\beta$ is at most 1 (if it exists) for this model. Along the way, we extend a result of H.~Tanemura (1993), on regularity of the supercritical Boolean model in $d \geq 3$ with fixed-radius balls, to the case with bounded random radii.

Mathew D. Penrose

Papers

The Minimum of a Large Number of Bessel Processes

Quasi-everywhere properties of Brownian level sets and multiple points

Local Central Limit Theorems in Stochastic Geometry

Inhomogeneous random graphs, isolated vertices, and Poisson approximation

On the capacity functional of the infinite cluster of a Boolean model