Based on a non-rigorous formalism called the “cavity method”, physicists have put forward intriguing predictions on phase transitions in discrete structures. One of the most remarkable ones is that in problems such as random k-SAT or random graph k-coloring, very shortly before the threshold for the existence of solutions there occurs another phase transition called condensation [Krzakala et al...

We reformulate the cavity approximation (CA), a class of algorithms recently introduced for improving the Bethe approximation estimates of marginals in graphical models. In our formulation, which allows for the treatment of multivalued variables, a further generalization to factor graphs with arbitrary order of interaction factors is explicitly carried out, and a message passing algorithm that ...

We study the problem of finding shortest paths in the plane among h convex obstacles, where the path is allowed to pass through (violate) up to k obstacles, for k ≤ h. Equivalently, the problem is to find shortest paths that become obstacle-free if k obstacles are removed from the input. Given a fixed source point s, we show how to construct a map, called a shortest k-path map, so that all dest...

A classification is obtained of Borel measurable, GL(n) covariant, symmetric matrix valued valuations on the space of n-dimensional convex polytopes. The only ones turn out to be the moment matrix corresponding to the classical Legendre ellipsoid and the matrix corresponding to the ellipsoid recently discovered by Lutwak, Yang, and Zhang. A classical concept from mechanics is the Legendre ellip...

In this paper we present a comparison of multiple cluster algorithms and their suitability for clustering text data. The clustering is based on similarities only, employing the Kolmogorov complexity as a similiarity measure. This motivates the set of considered clustering algorithms which take into account the similarity between objects exclusively. Compared cluster algorithms are Median kMeans...

Boolean factor analysis is the task of decomposing a binary matrix to the Boolean product of two binary factors. This unsupervised data-analysis approach is desirable due to its interpretability, but hard to perform due its NP-hardness. A closely related problem is low-rank Boolean matrix completion from noisy observations. We treat these problems as maximum a posteriori inference problems, and...

X-irradiation of single crystals of Tp–GeH3 (Tp: triptycene) led to the trapping of the radical Tp– GeH2. The angular variations of the resulting EPR spectra were recorded at 300 and 77 K. The drastic temperature dependence of the spectra was caused by both a strong anisotropy of the g-tensor and a rotation of the GeH2 moiety around the C–Ge bond. The determination of the EPR tensors as well as...

In this paper we prove lower bounds on randomized mul-tiparty communication complexity, both in the blackboardmodel (where each message is written on a blackboard for allplayers to see) and (mainly) in the message-passing model,where messages are sent player-to-player. We introduce anew technique for proving such bounds, called symmetriza-tion, which is natural, intuitiv...

The Message Passing Interface 1.1 (MPI 1.1) standard defines a library of message-passing functions for parallel and distributed computing. We have developed a new software tool called C++2MPI which can automatically generate MPI derived datatypes for a specified C++ class. C++2MPI can generate data types for derived classes, for partially and fully-specialized templated classes, and for classe...

We study the problem of reconstructing low-rank matrices from their noisy observations. We formulate the problem in the Bayesian framework, which allows us to exploit structural properties of matrices in addition to low-rankedness, such as sparsity. We propose an efficient approximate message passing algorithm, derived from the belief propagation algorithm, to perform the Bayesian inference for...