Circular object arrangement using spherical embeddings

Abstract

We consider the problem of recovering a circular arrangement of data instances with respect to some proximity measure, such that nearby instances are more similar. Applications of this problem, also referred to as circular seriation, can be found in various disciplines such as genome sequencing, data visualization and exploratory data analysis. Circular seriation can be expressed as a quadratic assignment problem, which is in general an intractable problem. Spectral-based approaches can be used to find approximate solutions, but are shown to perform well only for a specific class of data matrices. We propose a bilevel optimization framework where we employ a spherical embedding approach together with a spectral method for circular ordering in order to recover circular arrangements of the embedded data. Experiments on real and synthetic datasets demonstrate the competitive performance of the proposed method.

An Optimal Algorithm for Strict Circular Seriation

“…In this example we consider Q-node α = ((0, 1, 2), (6, 5, (3, 4)), (7, (8, 9, 10))) taking values over X = [11]. There are three Q-trees, which are consecutive in α, which are T 1 = (0, 1, 2), T 2 = (6, 5, (3, 4)) and T 3 = (7, (8,9,10)). An example of a possible outcome of Consecutive Orientation (T 1 , T 2 , T 3 ) would be that the root of T 2 is reversed and fixed into α.…”

“…In this case, the matrix representation of the pairwise dissimilarities is symmetric, with entries that increase monotonically starting from the diagonal along each row until they reach a maximum and then decrease monotonically, when the columns are wrapped around (see Figure 1). Matrices of this form are called circular Robinson [10,12] in contrast to linear Robinson dissimilarities, where the entries are monotone non-decreasing along rows and columns when moving toward the diagonal [15].…”

“…The approach proposed for circular seriation is an instance of the quadratic assignment problem, which is NP-hard. A recent work following a similar line is [10]. The authors propose an optimization framework where they employ a spherical embedding together with a spectral method for circular ordering in order to recover circular arrangements of the embedded objects.…”

A simple and optimal algorithm for strict circular seriation

“…For origins of circular seriation see the papers [10,12,11] and for recent applications of circular seriation in planar tomographic reconstruction, gene expression, DNA replication and 3D conformation, see the papers [7,17,16]. For a spectral approach to circular seriation, see the papers [8,9,21]. At the difference of the classical seriation, where the notion of Robinson dissimilarity is a well-established standard, in circular seriation there are several non-equivalent notions of circular Robinson dissimilarities.…”