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A deterministic attitude estimation problem for a rigid body in an attitude dependent potential field with bounded measurement errors is studied. An attitude estimation scheme that does not use generalized coordinate representations of the…
We study the problem of computationally efficient robust estimation of the covariance/scatter matrix of elliptical distributions -- that is, affine transformations of spherically symmetric distributions -- under the strong contamination…
We explore ways in which the covariance ellipsoid ${\cal B}=\{v \in \mathbb{R}^d : \mathbb{E} <X,v>^2 \leq 1\}$ of a centred random vector $X$ in $\mathbb{R}^d$ can be approximated by a simple set. The data one is given for constructing the…
A variational approach to finite connectivity spin-glass-like models is developed and applied to describe the structure of optimal solutions in random satisfiability problems. Our variational scheme accurately reproduces the known replica…
In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing:…
Semiclassical approximations often involve the use of stationary phase approximations. This method can be applied when $\hbar$ is small in comparison to relevant actions or action differences in the corresponding classical system. In many…
We study the Vitale zonoid (a convex body associated to a probability distribution) associated to a non--centered Gaussian vector. This defines a family of convex bodies, that contains and generalizes ellipsoids, which we call Gaussian…
This study presents a generalised least squares based method for fitting polygons and ellipses to data points. The method is based on a trigonometric fitness function that approximates a unit shape accurately, making it applicable to…
Attitude estimation is often a prerequisite for control of the attitude or orientation of mechanical systems. Current attitude estimation algorithms use coordinate representations for the group of rigid body orientations. All coordinate…
Since the early 2000s physicists have developed an ingenious but non-rigorous formalism called the cavity method to put forward precise conjectures on phase transitions in random problems [Mezard, Parisi, Zecchina: Science 2002]. The cavity…
The (2+p)-Satisfiability (SAT) problem interpolates between different classes of complexity theory and is believed to be of basic interest in understanding the onset of typical case complexity in random combinatorics. In this paper, a…
We present a novel method for deciding whether a given n-dimensional ellipsoid contains another one (possibly with a different center). This method consists in constructing a particular concave function and deciding whether it has any value…
We revisit the replica method for analyzing inference and learning in parametric models, considering situations where the data-generating distribution is unknown or analytically intractable. Instead of assuming idealized distributions to…
We obtain an asymptotic H\"older estimate for functions satisfying a dynamic programming principle arising from a so-called ellipsoid process. By the ellipsoid process we mean a generalization of the random walk where the next step in the…
A satisfiability (SAT-UNSAT) transition takes place for many optimization problems when the number of constraints, graphically represented by links between variables nodes, is brought above some threshold. If the network of constraints is…
We establish the satisfiability threshold for random $k$-SAT for all $k\ge k_0$, with $k_0$ an absolute constant. That is, there exists a limiting density $\alpha_*(k)$ such that a random $k$-SAT formula of clause density $\alpha$ is with…
We study and solve some variations of the random K-satisfiability problem - balanced K-SAT and biased random K-SAT - on a regular tree, using techniques we have developed earlier(arXiv:1110.2065). In both these problems, as well as…
We consider versions of the FIND algorithm where the pivot element used is the median of a subset chosen uniformly at random from the data. For the median selection we assume that subsamples of size asymptotic to $c \cdot n^\alpha$ are…
This paper describes an approach for fitting an immersed submanifold of a finite-dimensional Euclidean space to random samples. The reconstruction mapping from the ambient space to the desired submanifold is implemented as a composition of…
The problem of fitting concentric ellipses is a vital problem in image processing, pattern recognition, and astronomy. Several methods have been developed but all address very special cases. In this paper, this problem has been investigated…