Characterization of trace spaces on regular trees via dyadic norms

Dyadic Classification Trees via Structural Risk Minimization

Classification trees are one of the most popular types of classifiers, with ease of implementation and interpretation being among their attractive features. Despite the widespread use of classification trees, theoretical analysis of their performance is scarce. In this paper, we show that a new family of classification trees, called dyadic classification trees (DCTs), are near optimal (in a min...

Norms on Quotient Spaces

Dyadic Decision Trees

These notes introduce a new kind of classifier called a dyadic decision tree (DDT). We also introduce a discrimination rule for learning a DDT that achieves the optimal rate of convergence, ER(ĥn) − R∗ = O(n−1/d), for the box-counting class, which was defined in the previous set of notes. This improves on the rate of ER(ĥn)−R = O(n−1/(d+2)) for the histogram sieve estimator from the previous no...

Invasion percolation on regular trees

We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behaviour of its r-point function for any r ≥ 2 and of its volume both at a given height and below a given height. In addition, we derive scaling estimates for simple random walk on the cluster starting from the root. We find that while the power laws of the scaling...

Reconstruction Thresholds on Regular Trees

We consider the model of broadcasting on a tree, with binary state space, on the infinite rooted tree k in which each node has k children. The root of the tree takes a random value 0 or 1, and then each node passes a value independently to each of its children according to a 2 2 transition matrix P. We say that reconstruction is possible if the values at the dth level of the tree contain non-va...