Given noisy observations Xi = θi + Zi, i = 1, . . . , n, with noise Zi iid ∼ N(0, σ), we wish to recover the signal θ with small mean-squared error. We consider the Minimum Kolmogorov Complexity Estimator (MKCE), defined roughly as the n-vector θ̂(X) solving the problem min Y K(Y ) subject to ‖X − Y ‖2l2n ≤ σ 2 · n, where K(Y ) denotes the length of the shortest computer program that can compute...

We present a method for hierarchical clustering of data called mutual information clustering (MIC) algorithm. It uses mutual information (MI) as a similarity measure and exploits its grouping property: The MI between three objects X, Y, and Z is equal to the sum of the MI between X and Y , plus the MI between Z and the combined object (XY ). We use this both in the Shannon (probabilistic) versi...

We propose a way to construct consistent generalizations of the Cram er-von Mises test on the line, and the Watson U 2 n test on the circle, based on classes of partitions invariant under various group actions. The framework developed extends our earlier work generalizing the Kolmogorov-Smirnov and Kuiper goodness-oft tests, and provides a conceptually unifying description. Many goodness-oft te...

A new approach to the Kolmogorov-Arnold-Moser theory concermng the existence of invariant ton having prescnbed frequencies îs presented It îs based on the Lagrangian formahsm m configuration space instead of the Hamiltonian formahsm in phase space used in earher approaches In particular, the construction of the invariant ton avoids the composition of infinitely many coordinate transformations T...

We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain new rates of convergence with respect to the Wasserstein and Kolmogorov metrics for the theorem of Rényi on random sums and generalizations of it, hitting times for Markov chains, and to obtain a new rate for the clas...