Several statistics-based detectors, based on unimodal matrix models, for determining the number of sources in a field are designed. A new variance ratio statistic is proposed, and its asymptotic distribution analyzed. The detector shown to outperform alternatives. It that further improvements achievable via optimally selected rotations. Numerical experiments demonstrate performance gains our de...

Bayesian nonparametric inference for unimodal and multimodal random probability measures on a finite dimensional Euclidean space is examined. After a short discussion on several concepts of multivatiate unimodality, we introduce and study a new class of nonparametric prior distributions on the subspace of random multivariate multimodal distributions. This class in a way generalizes the very res...

We consider stochastic bandit problems with a continuum set of arms and where the expected re-ward is a continuous and unimodal function of the arm. No further assumption is made regarding thesmoothness and the structure of the expected reward function. We propose Stochastic Pentachotomy(SP), an algorithm for which we derive finite-time regret upper bounds. In particular, we sho...

The Fibonacci recurrence of the critical orbit appeared in the work of Branner and Hubbard on complex cubic polynomials [BH, §12] and in Yoccoz's work [Yl, Y2] on quadratic ones, as the "worst" pattern of recurrence. On the other hand, a real quadratic Fibonacci map was suggested by Hofbauer and Keller [HK] as a possible candidate for a map having a "wild" attractor (that is, a set A which is t...

This paper deals with optimal learning and provides a uniied viewpoint of most signiicant results in the eld. The focus is on the problem of local minima in the cost function that is likely to aaect more or less any learning algorithm. We give some intriguing links between optimal learning and the computational complexity of loading problems. We exhibit a computational model such that the solut...

We review recent results that lead to a very precise understanding of the dynamics of typical unimodal maps from the statistical point of view. We also describe the (generalized) renormalization approach to the study of the statistical properties of typical unimodal maps.

This paper gives optimal algorithms for determining realvalued univariate unimodal regressions, that is, for determining the optimal regression which is increasing and then decreasing. Such regressions arise in a wide variety of applications. They are a form of shape-constrained nonparametric regression, closely related to isotonic regression. For the L2 metric our algorithm requires only (n) t...

We describe up to conjugacy all unimodal and bimodal maps that are chaotic, by giving necessary and sufficient conditions for unimodal and bimodal maps of slopes ±s to be transitive.

The complexity of formal languages which are generated by S-unimodal systems on interval covers is studied. It is shown that there exist S-unimodal systems with nonrecursive languages, those with recursive but not context-sensitive languages, and those with context-sensitive but not regular languages. It is also shown that S-unimodal systems with regular languages include all systems with finit...