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The flexibility of these methods is often limited bytherequirement thatthecutsbeaxisaligned.

Space partitioning methods such as random forests and the Mondrian process are powerful machine learning methods for multi-dimensional and relational data, and are based on recursively cutting a domain. The flexibility of these methods is often limited by the requirement that the cuts be axis aligned. The Ostomachion process and the self-consistent binary space partitioning-tree process were recently introduced as generalizations of the Mondrian process for space partitioning with non-axis aligned cuts in the plane. Motivated by the need for a multi-dimensional partitioning tree with non-axis aligned cuts, we propose the Random Tessellation Process, a framework that includes the Mondrian process as a special case. We derive a sequential Monte Carlo algorithm for inference, and provide random forest methods. Our methods are self-consistent and can relax axis-aligned constraints, allowing complex inter-dimensional dependence to be captured. We present a simulation study and analyze gene expression data of brain tissue, showing improved accuracies over other methods.

Indeed, the fact that this parameter is fixed hinders the statistical consistency of the original procedure.

By tailoring a multi-dimensional space (or multi-dimensional array) into a number of rectangular regions, the partition model can fit data using these "blocks" such that the data within each block

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The paper describes a Bayesian model for online learning in the context of random forests models for supervised classification. The main contribution of the paper is the formulation of a novel prior on binary rooted trees that relies on the Mondrian process. An additional novelty of the paper is the use of hierarchical normalized stable processes as priors for the probabilities of the different classes at each terminal node. The paper is well written and the formulation novel.

Neural Information Processing Systems http://nips.cc/

Ensembles of randomized decision trees, usually referred to as random forests, are widely used for classification and regression tasks in machine learning and statistics. Random forests achieve competitive predictive performance and are computationally efficient to train and test, making them excellent candidates for real-world prediction tasks. The most popular random forest variants (such as Breiman's random forest and extremely randomized trees) operate on batches of training data. Online methods are now in greater demand. Existing online random forests, however, require more training data than their batch counterpart to achieve comparable predictive performance. In this work, we use Mondrian processes (Roy and Teh, 2009) to construct ensembles of random decision trees we call Mondrian forests. Mondrian forests can be grown in an incremental/online fashion and remarkably, the distribution of online Mondrian forests is the same as that of batch Mondrian forests. Mondrian forests achieve competitive predictive performance comparable with existing online random forests and periodically retrained batch random forests, while being more than an order of magnitude faster, thus representing a better computation vs accuracy tradeoff.

First proposed by Rahimi and Recht, random features are used to decrease the computational cost of kernel machines in large-scale problems. The Mondrian kernel is one such example of a fast random feature approximation of the Laplace kernel, generated by a computationally efficient hierarchical random partition of the input space known as the Mondrian process. In this work, we study a variation of this random feature map by using uniformly randomly rotated Mondrian processes to approximate a kernel that is invariant under rotations. We obtain a closed-form expression for this isotropic kernel, as well as a uniform convergence rate of the uniformly rotated Mondrian kernel to this limit. To this end, we utilize techniques from the theory of stationary random tessellations in stochastic geometry and prove a new result on the geometry of the typical cell of the superposition of uniformly random rotations of Mondrian tessellations. Finally, we test the empirical performance of this random feature map on both synthetic and real-world datasets, demonstrating its improved performance over the Mondrian kernel on a debiased dataset.

Summary and Contributions: The Baxter Permutation Process provides a generalization of the Mondrian process so that an arbitrary tiling with hyperrectables is supported. In particular, the Mondrian process (MP) is a Bayesian nonparametric version of a decision tree. Since MP is defined by a tree, each cut must extend to the edges of the hyperrectangle being cut. This means that the first cut must extend from -infinity to infinity, and the second cut (should it be perpendicular) must extend from infinity to the level of the first cut (or, if it's parallel, again from -infinity to infinity). This extensive cut nature is not incredibly terrible, but could lead to a lack of local modelling.