An Adaptive Total Variation Algorithm for Computing the Balanced Cut of a Graph

We propose an adaptive version of the total variation algorithm proposed in [3] for computing the balanced cut of a graph. The algorithm from [3] used a sequence of inner total variation minimizations to guarantee descent of the balanced cut energy as well as convergence of the algorithm. In practice the total variation minimization step is never solved exactly. Instead, an accuracy parameter is specified and the total variation minimization terminates once this level of accuracy is reached. The choice of this parameter can vastly impact both the computational time of the overall algorithm as well as the accuracy of the result. Moreover, since the total variation minimization step is not solved exactly, the algorithm is not guarantied to be monotonic. In the present work we introduce a new adaptive stopping condition for the total variation minimization that guarantees monotonicity. This results in an algorithm that is actually monotonic in practice and is also significantly faster than previous, non-adaptive algorithms

Multiclass Total Variation Clustering

Ideas from the image processing literature have recently motivated a new set of clustering algorithms that rely on the concept of total variation. While these algorithms perform well for bi-partitioning tasks, their recursive extensions yield unimpressive results for multiclass clustering tasks. This paper presents a general framework for multiclass total variation clustering that does not rely on recursion. The results greatly outperform previous total variation algorithms and compare well with state-of-the-art NMF approaches

Convergence of a Steepest Descent Algorithm for Ratio Cut Clustering

Unsupervised clustering of scattered, noisy and high-dimensional data points is an important and difficult problem. Tight continuous relaxations of balanced cut problems have recently been shown to provide excellent clustering results. In this paper, we present an explicit-implicit gradient flow scheme for the relaxed ratio cut problem, and prove that the algorithm converges to a critical point of the energy. We also show the efficiency of the proposed algorithm on the two moons dataset

Local circuits targeting parvalbumin-containing interneurons in layer IV of rat barrel cortex

Interactions between inhibitory interneurons and excitatory spiny neurons and also other inhibitory cells represent fundamental network properties which cause the so-called thalamo-cortical response transformation and account for the well-known receptive field differences of cortical layer IV versus thalamic neurons. We investigated the currently largely unknown morphological basis of these interactions utilizing acute slice preparations of barrel cortex in P19-21 rats. Layer IV spiny (spiny stellate, star pyramidal and pyramidal) neurons or inhibitory (basket and bitufted) interneurons were electrophysiologically characterized and intracellularly biocytin-labeled. In the same slice, we stained parvalbumin-immunoreactive (PV-ir) interneurons as putative target cells after which the tissue was subjected to confocal image acquisition. Parallel experiments confirmed the existence of synaptic contacts in these types of connection by correlated light and electron microscopy. The axons of the filled neurons differentially targeted barrel PV-ir interneurons: (1) The relative number of all contacted PV-ir cells within the axonal sphere was 5–17% for spiny (n = 10), 32 and 58% for basket (n = 2) and 12 and 13% for bitufted (n = 2) cells. (2) The preferential subcellular site which was contacted on PV-ir target cells was somatic for four and dendritic for five spiny cells; for basket cells, there was a somatic and for bitufted cells a dendritic preference in each examined case. (3) The highest number of contacts on a single PV-ir cell was 9 (4 somatic and 5 dendritic) for spiny neurons, 15 (10 somatic and 5 dendritic) for basket cells and 4 (1 somatic and 3 dendritic) for bitufted cells. These patterns suggest a cell type-dependent communication within layer IV microcircuits in which PV-ir interneurons provide not only feed-forward but also feedback inhibition thus triggering the thalamo-cortical response transformation

Communications in Mathematical Physics manuscript No. (will be inserted by the editor) Well-Posedness Theory for Aggregation Sheets

Abstract: In this paper, we consider distribution solutions to the aggregation equation ρt + div(ρu) = 0, u = −∇V ∗ ρ in R d where the density ρ concentrates on a co-dimension one manifold. We show that an evolution equation for the manifold itself completely determines the dynamics of such solutions. We refer to such solutions aggregation sheets. When the equation for the sheet is linearly well-posed, we show that the fully non-linear evolution is also well-posed locally in time for the class of bi-Lipschitz surfaces. Moreover, we show that if the initial sheet is C 1 then the solution itself remains C 1 as long as it remains Lipschitz. Lastly, we provide conditions on the kernel g(s) = − dV ds that guarantee the solution remains a bi-Lipschitz surface globally in time, and construct explicit solutions that either collapse or blow up in finite time when these conditions fail. 1. Background Systems with a large number of pairwise interacting particles pervade many disciplines, ranging from models of self-assembly processes in physics and chemistry [21,22,23,29] to models for biological swarming [1,8,16,28] to algorithms for the cooperative control of autonomous vehicles [32]. A simple example of these models employs a first order system of ordinary differential equations for the positions xi(t) ∈ R d of N particles, dxi d