168 research outputs found
Convex Relaxations for Permutation Problems
Seriation seeks to reconstruct a linear order between variables using
unsorted, pairwise similarity information. It has direct applications in
archeology and shotgun gene sequencing for example. We write seriation as an
optimization problem by proving the equivalence between the seriation and
combinatorial 2-SUM problems on similarity matrices (2-SUM is a quadratic
minimization problem over permutations). The seriation problem can be solved
exactly by a spectral algorithm in the noiseless case and we derive several
convex relaxations for 2-SUM to improve the robustness of seriation solutions
in noisy settings. These convex relaxations also allow us to impose structural
constraints on the solution, hence solve semi-supervised seriation problems. We
derive new approximation bounds for some of these relaxations and present
numerical experiments on archeological data, Markov chains and DNA assembly
from shotgun gene sequencing data.Comment: Final journal version, a few typos and references fixe
The quadratic assignment problem is easy for Robinsonian matrices with Toeplitz structure
We present a new polynomially solvable case of the Quadratic Assignment
Problem in Koopmans-Beckman form , by showing that the identity
permutation is optimal when and are respectively a Robinson similarity
and dissimilarity matrix and one of or is a Toeplitz matrix. A Robinson
(dis)similarity matrix is a symmetric matrix whose entries (increase) decrease
monotonically along rows and columns when moving away from the diagonal, and
such matrices arise in the classical seriation problem.Comment: 15 pages, 2 figure
A graph-spectral approach to shape-from-shading
In this paper, we explore how graph-spectral methods can be used to develop a new shape-from-shading algorithm. We characterize the field of surface normals using a weight matrix whose elements are computed from the sectional curvature between different image locations and penalize large changes in surface normal direction. Modeling the blocks of the weight matrix as distinct surface patches, we use a graph seriation method to find a surface integration path that maximizes the sum of curvature-dependent weights and that can be used for the purposes of height reconstruction. To smooth the reconstructed surface, we fit quadrics to the height data for each patch. The smoothed surface normal directions are updated ensuring compliance with Lambert's law. The processes of height recovery and surface normal adjustment are interleaved and iterated until a stable surface is obtained. We provide results on synthetic and real-world imagery
An Optimal Algorithm for Strict Circular Seriation
We study the problem of circular seriation, where we are given a matrix of
pairwise dissimilarities between objects, and the goal is to find a {\em
circular order} of the objects in a manner that is consistent with their
dissimilarity. This problem is a generalization of the classical {\em linear
seriation} problem where the goal is to find a {\em linear order}, and for
which optimal algorithms are known. Our contributions can be
summarized as follows. First, we introduce {\em circular Robinson matrices} as
the natural class of dissimilarity matrices for the circular seriation problem.
Second, for the case of {\em strict circular Robinson dissimilarity matrices}
we provide an optimal algorithm for the circular seriation
problem. Finally, we propose a statistical model to analyze the well-posedness
of the circular seriation problem for large . In particular, we establish
rates on the distance between any circular ordering found
by solving the circular seriation problem to the underlying order of the model,
in the Kendall-tau metric.Comment: 27 pages, 5 figure
Graph edit distance from spectral seriation
This paper is concerned with computing graph edit distance. One of the criticisms that can be leveled at existing methods for computing graph edit distance is that they lack some of the formality and rigor of the computation of string edit distance. Hence, our aim is to convert graphs to string sequences so that string matching techniques can be used. To do this, we use a graph spectral seriation method to convert the adjacency matrix into a string or sequence order. We show how the serial ordering can be established using the leading eigenvector of the graph adjacency matrix. We pose the problem of graph-matching as a maximum a posteriori probability (MAP) alignment of the seriation sequences for pairs of graphs. This treatment leads to an expression in which the edit cost is the negative logarithm of the a posteriori sequence alignment probability. We compute the edit distance by finding the sequence of string edit operations which minimizes the cost of the path traversing the edit lattice. The edit costs are determined by the components of the leading eigenvectors of the adjacency matrix and by the edge densities of the graphs being matched. We demonstrate the utility of the edit distance on a number of graph clustering problems
Numerical Linear Algebra applications in Archaeology: the seriation and the photometric stereo problems
The aim of this thesis is to explore the application of Numerical Linear Algebra to Archaeology. An ordering problem called the seriation problem, used for dating findings and/or artifacts deposits, is analysed in terms of graph theory. In particular, a Matlab implementation of an algorithm for spectral seriation, based on the use of the Fiedler vector of the Laplacian matrix associated with the problem, is presented. We consider bipartite graphs for describing the seriation problem, since the interrelationship between the units (i.e. archaeological sites) to be reordered, can be described in terms of these graphs. In our archaeological metaphor of seriation, the two disjoint nodes sets into which the vertices of a bipartite graph can be divided, represent the excavation sites and the artifacts found inside
them.
Since it is a difficult task to determine the closest bipartite network to a given one, we describe how a starting network can be approximated by a bipartite one by solving a sequence of fairly simple optimization problems.
Another numerical problem related to Archaeology is the 3D reconstruction of the shape of an object from a set of digital pictures. In particular, the Photometric Stereo (PS) photographic technique is considered
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