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David Zimmermann

Publications and source records attributed to David Zimmermann.

4 recordsLinked to original sources
arXiv2019

Asymmetric Impurity Functions, Class Weighting, and Optimal Splits for Binary Classification Trees

We investigate how asymmetrizing an impurity function affects the choice of optimal node splits when growing a decision tree for binary classification. In particular, we relax the usual axioms of an impurity function and show how skewing an impurity function biases the optimal splits to isolate points of a particular class when splitting a node. We give a rigorous definition of this notion, then give a necessary and sufficient condition for such a bias to hold. We also show that the technique of class weighting is equivalent to applying a specific transformation to the impurity function, and tie all these notions together for a class of impurity functions that includes the entropy and Gini impurity. We also briefly discuss cost-insensitive impurity functions and give a characterization of such functions.

cs.LG↗
arXiv2014

Logarithmic Sobolev inequalities for mollified compactly supported measures

We show that the convolution of a compactly supported measure on $\mathbb{R}$ with a Gaussian measure satisfies a logarithmic Sobolev inequality (LSI). We use this result to give a new proof of a classical result in random matrix theory that states that, under certain hypotheses, the empirical law of eigenvalues of a sequence of random real symmetric matrices converges weakly in probability to its mean. We then examine the optimal constants in the LSIs for the convolved measures in terms of the variance of the convolving Gaussian. We conclude with partial results on the extension of our main theorem to higher dimensions.

math.PR↗
arXiv2014

Random Matrices with Log-Range Correlations, and Log-Sobolev Inequalities

Let $X_N$ be a symmetric $N\times N$ random matrix whose $\sqrt{N}$-scaled centered entries are uniformly square integrable. We prove that if the entries of $X_N$ can be partitioned into independent subsets each of size $o(\log N)$, then the empirical eigenvalue distribution of $X_N$ converges weakly to its mean in probability. This significantly extends the best previously known results on convergence of eigenvalues for matrices with correlated entries (where the partition subsets are blocks and of size $O(1)$.) we prove this result be developing a new log-Sobolev inequality, generalizing the first author's introduction of mollified log-Sobolev inequalities: we show that if $\mathbf{Y}$ is a bounded random vector and $\mathbf{Z}$ is a standard normal random vector independent from $\mathbf{Y}$, then the law of $\mathbf{Y}+t\mathbf{Z}$ satisfies a log-Sobolev inequality for all $t>0$, and we give bounds on the optimal log-Sobolev constant.

math.FA↗