Fast Algorithms for Constructing Maximum Entropy Summary Trees

Karloff? and Shirley recently proposed summary trees as a new way to visualize large rooted trees (Eurovis 2013) and gave algorithms for generating a maximum-entropy k-node summary tree of an input n-node rooted tree. However, the algorithm generating optimal summary trees was only pseudo-polynomial (and worked only for integral weights); the authors left open existence of a olynomial-time algorithm. In addition, the authors provided an additive approximation algorithm and a greedy heuristic, both working on real weights. This paper shows how to construct maximum entropy k-node summary trees in time O(k^2 n + n log n) for real weights (indeed, as small as the time bound for the greedy heuristic given previously); how to speed up the approximation algorithm so that it runs in time O(n + (k^4/eps?) log(k/eps?)), and how to speed up the greedy algorithm so as to run in time O(kn + n log n). Altogether, these results make summary trees a much more practical tool than before.Comment: 17 pages, 4 figures. Extended version of paper appearing in ICALP 201

Modelling of selection and mating decisions in tree breeding programs

Hardwood trees from the temperate forests of southern Australia are an important source of timber for high quality paper. Two species in particular, Eucalyptus globulus and Eucalyptus nitens are well suited to this purpose and are now widely grown in commercial plantations. These plantations have been established by professional tree breeders using seedlings derived originally from broadly based collection of seed in natural forests. To increase productivity it is desirable to select trees that grow quickly and give high yields of top quality timber. Nevertheless it is important to maintain genetic diversity in the breeding population and thereby retain a robust capacity to adapt to changing environmental factors. In this article we formulate a number of related mathematical models for the selection and mating processes and discuss the consequences of these models. We recommend a relatively simple scheme which can be implemented on an IBM compatible PC using standard algorithms

Model Extraction Warning in MLaaS Paradigm

Cloud vendors are increasingly offering machine learning services as part of their platform and services portfolios. These services enable the deployment of machine learning models on the cloud that are offered on a pay-per-query basis to application developers and end users. However recent work has shown that the hosted models are susceptible to extraction attacks. Adversaries may launch queries to steal the model and compromise future query payments or privacy of the training data. In this work, we present a cloud-based extraction monitor that can quantify the extraction status of models by observing the query and response streams of both individual and colluding adversarial users. We present a novel technique that uses information gain to measure the model learning rate by users with increasing number of queries. Additionally, we present an alternate technique that maintains intelligent query summaries to measure the learning rate relative to the coverage of the input feature space in the presence of collusion. Both these approaches have low computational overhead and can easily be offered as services to model owners to warn them of possible extraction attacks from adversaries. We present performance results for these approaches for decision tree models deployed on BigML MLaaS platform, using open source datasets and different adversarial attack strategies

Variable Selection Bias in Classification Trees Based on Imprecise Probabilities

Classification trees based on imprecise probabilities provide an advancement of classical classification trees. The Gini Index is the default splitting criterion in classical classification trees, while in classification trees based on imprecise probabilities, an extension of the Shannon entropy has been introduced as the splitting criterion. However, the use of these empirical entropy measures as split selection criteria can lead to a bias in variable selection, such that variables are preferred for features other than their information content. This bias is not eliminated by the imprecise probability approach. The source of variable selection bias for the estimated Shannon entropy, as well as possible corrections, are outlined. The variable selection performance of the biased and corrected estimators are evaluated in a simulation study. Additional results from research on variable selection bias in classical classification trees are incorporated, implying further investigation of alternative split selection criteria in classification trees based on imprecise probabilities

Localizing the Latent Structure Canonical Uncertainty: Entropy Profiles for Hidden Markov Models

This report addresses state inference for hidden Markov models. These models rely on unobserved states, which often have a meaningful interpretation. This makes it necessary to develop diagnostic tools for quantification of state uncertainty. The entropy of the state sequence that explains an observed sequence for a given hidden Markov chain model can be considered as the canonical measure of state sequence uncertainty. This canonical measure of state sequence uncertainty is not reflected by the classic multivariate state profiles computed by the smoothing algorithm, which summarizes the possible state sequences. Here, we introduce a new type of profiles which have the following properties: (i) these profiles of conditional entropies are a decomposition of the canonical measure of state sequence uncertainty along the sequence and makes it possible to localize this uncertainty, (ii) these profiles are univariate and thus remain easily interpretable on tree structures. We show how to extend the smoothing algorithms for hidden Markov chain and tree models to compute these entropy profiles efficiently.Comment: Submitted to Journal of Machine Learning Research; No RR-7896 (2012