Convergence and Energy Landscape for Cheeger Cut Clustering

This paper provides both theoretical and algorithmic results for the l 1-relaxation of the Cheeger cut problem. The l2- relaxation, known as spectral clustering, only loosely relates to the Cheeger cut; however, it is convex and leads to a simple optimization problem. The l1-relaxation, in contrast, is non-convex but is provably equivalent to the original problem. The l1-relaxation therefore trades convexity for exactness, yielding improved clustering results at the cost of a more challenging optimization. The first challenge is understanding convergence of algorithms. This paper provides the first complete proof of convergence for algorithms that minimize the l1-relaxation. The second challenge entails comprehending the l1-energy landscape, i.e. the set of possible points to which an algorithm might converge. We show that l 1-algorithms can get trapped in local minima that are not globally optimal and we provide a classification theorem to interpret these local minima. This classification gives meaning to these suboptimal solutions and helps to explain, in terms of graph structure, when the l1-relaxation provides the solution of the original Cheeger cut problem

Spectral Analysis of the Supreme Court

The focus of this paper is the linear algebraic framework in which the spectral analysis of voting data like that above is carried out. As we will show, this framework can be used to pinpoint voting coalitions in small voting bodies like the United States Supreme Court. Our goal is to show how simple ideas from linear algebra can come together to say something interesting about voting. And what could be more simple than where our story begins— with counting

Ring patterns and their bifurcations in a nonlocal model of biological swarms

In this paper we study the pattern formation of a kinematic aggregation model for biological swarming in two dimensions. The swarm is represented by particles and the dynamics are driven by a gradient flow of a non-local interaction potential which has a local repulsion long range attraction structure. We leverage a co-dimension one formulation of the continuum gradient flow to characterize the stability of ring solutions for general interaction kernels. In the regime of long-wave instability we show that the resulting ground state is as a low mode bifurcation away from the ring and use weakly nonlinear analysis to provide conditions for when this bifurcation is a pitchfork. In the regime of short-wave instabilities we show that the rings break up into fully 2D ground states in the large particle limit. We analyze the dependence on the stability of a ring on the number of particles and provide examples of complex multi-ring bifurcation behavior as the number of particles increases. We are also able to provide a solution for the “designer potential” problem in 2D. Finally, we characterize the stability of the rotating rings in the second order kinetic swarming model

Stability and clustering of self-similar solutions of aggregation equations

In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρt = ∇ · (ρ∇K ∗ ρ) in Rd, d ≥ 2, where K(r) = r γ /γ with γ> 2. It was previously observed 1 that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for γ> 2. In this paper we compute the stability of the similarity solution and show that the collapsing shell solution is stable for 2 < γ < 4. For γ> 4, we show that the shell solution is always unstable and destabilizes into clusters that form a simplex which we observe to be the long time attractor. We then classify the stability of these simplex solutions and prove that two dimensional (in-)stability implies n dimensional (in-)stability