Sparse probability density function estimation using the minimum integrated square error

We develop a new sparse kernel density estimator using a forward constrained regression framework, within which the nonnegative and summing-to-unity constraints of the mixing weights can easily be satisfied. Our main contribution is to derive a recursive algorithm to select significant kernels one at time based on the minimum integrated square error (MISE) criterion for both the selection of kernels and the estimation of mixing weights. The proposed approach is simple to implement and the associated computational cost is very low. Specifically, the complexity of our algorithm is in the order of the number of training data N, which is much lower than the order of N2 offered by the best existing sparse kernel density estimators. Numerical examples are employed to demonstrate that the proposed approach is effective in constructing sparse kernel density estimators with comparable accuracy to those of the classical Parzen window estimate and other existing sparse kernel density estimators.

Chemistry and structure by design: ordered CuNi(CN)4 sheets with copper(II) in a square-planar environment

Layered copper–nickel cyanide, CuNi(CN)4, a 2-D negative thermal expansion material, is one of a series of copper(II)-containing cyanides derived from Ni(CN)2. In CuNi(CN)4, unlike in Ni(CN)2, the cyanide groups are ordered generating square-planar Ni(CN)4 and Cu(NC)4 units. The adoption of square-planar geometry by Cu(II) in an extended solid is very unusual.