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**1 - 4**of**4**### Discovering and Visualizing Hierarchy in the Data

"... Abstract—How to extract useful insights from data in a human perceivable manner is always a challenge when the dimension and amount of the data is large. Often, the data can be organized according to certain hierarchical structure that are stemmed either from data collection process or from the info ..."

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Abstract—How to extract useful insights from data in a human perceivable manner is always a challenge when the dimension and amount of the data is large. Often, the data can be organized according to certain hierarchical structure that are stemmed either from data collection process or from the information and phenomena carried by the data itself. The current study attempts to discover and visualize these underlying hierarchies. Regarding each observation as a draw from a (hypothetical) multidimensional joint density, our first goal is to approximate this unknown density with a piecewise constant function over the binary partitioned sample space; our non-parametric approach makes no assumptions on the form of the density, such as assuming that it is Multivariate Gaussian, or that it is a mixture of a small number of Gaussians. Given the piecewise constant density function and its corresponding partitions of the sample space, our second goal is to construct a connected graph and build up a tree representation of the data from sub-level sets. To demonstrate that our method is a general data mining and visualization tool which can provide “multi-resolution ” summaries and reveal different levels of information of the data, we apply it to two real data sets from different fields.

### LOCAL EXTREMA IN QUANTUM CHAOS

"... Abstract. We numerically investigate both the number and the spatial distribution of local ex-trema of ’chaotic ’ Laplacian eigenfunctions on two-dimensional manifolds and demonstrate two new universality phenomena. Blum, Gnutzmann & Smilansky have numerically demonstrated that the k−th eigenfun ..."

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Abstract. We numerically investigate both the number and the spatial distribution of local ex-trema of ’chaotic ’ Laplacian eigenfunctions on two-dimensional manifolds and demonstrate two new universality phenomena. Blum, Gnutzmann & Smilansky have numerically demonstrated that the k−th eigenfunction has typically ∼ 0.06k nodal domains – we give numerical evidence that it typically has ∼ σ · k local extrema, where σ = 0.58 ± 0.02 is a universal constant. Using the discrepancy as a measure of quality of distribution, we show that the local extrema are more regularly spread than a regular grid. 1.

### 1Density Estimation via Adaptive Partition and Discrepancy Control

"... Abstract—Given iid samples from some unknown continuous density on hyper-rectangle [0, 1]d, we attempt to learn a piecewise constant function that approximates this underlying density nonparametrically. Our density estimate is defined on a binary split of [0, 1]d and built up sequentially according ..."

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Abstract—Given iid samples from some unknown continuous density on hyper-rectangle [0, 1]d, we attempt to learn a piecewise constant function that approximates this underlying density nonparametrically. Our density estimate is defined on a binary split of [0, 1]d and built up sequentially according to discrepancy criteria; the key ingredient is to control the discrepancy adaptively in each sub-rectangle to achieve overall bound. We prove that the estimate, even though simple as it appears, preserves most of the estimation power. By exploiting its structure, it can be directly applied to some important pattern recognition tasks such as mode seeking and density landscape exploration, we demonstrate its applicability through simulations and examples.