Efficient Identification of Equivalences in Dynamic Graphs and Pedigree Structures

We propose a new framework for designing test and query functions for complex structures that vary across a given parameter such as genetic marker position. The operations we are interested in include equality testing, set operations, isolating unique states, duplication counting, or finding equivalence classes under identifiability constraints. A motivating application is locating equivalence classes in identity-by-descent (IBD) graphs, graph structures in pedigree analysis that change over genetic marker location. The nodes of these graphs are unlabeled and identified only by their connecting edges, a constraint easily handled by our approach. The general framework introduced is powerful enough to build a range of testing functions for IBD graphs, dynamic populations, and other structures using a minimal set of operations. The theoretical and algorithmic properties of our approach are analyzed and proved. Computational results on several simulations demonstrate the effectiveness of our approach.Comment: Code for paper available at http://www.stat.washington.edu/~hoytak/code/hashreduc

Matching in the Pi-Calculus

We study whether, in the pi-calculus, the match prefix-a conditional operator testing two names for (syntactic) equality-is expressible via the other operators. Previously, Carbone and Maffeis proved that matching is not expressible this way under rather strong requirements (preservation and reflection of observables). Later on, Gorla developed a by now widely-tested set of criteria for encodings that allows much more freedom (e.g. instead of direct translations of observables it allows comparison of calculi with respect to reachability of successful states). In this paper, we offer a considerably stronger separation result on the non-expressibility of matching using only Gorla's relaxed requirements.Comment: In Proceedings EXPRESS/SOS 2014, arXiv:1408.127

Graphical description of the action of Clifford operators on stabilizer states

We introduce a graphical representation of stabilizer states and translate the action of Clifford operators on stabilizer states into graph operations on the corresponding stabilizer-state graphs. Our stabilizer graphs are constructed of solid and hollow nodes, with (undirected) edges between nodes and with loops and signs attached to individual nodes. We find that local Clifford transformations are completely described in terms of local complementation on nodes and along edges, loop complementation, and change of node type or sign. Additionally, we show that a small set of equivalence rules generates all graphs corresponding to a given stabilizer state; we do this by constructing an efficient procedure for testing the equality of any two stabilizer graphs.Comment: 14 pages, 8 figures. Version 2 contains significant changes. Submitted to PR

Matching in the Pi-Calculus (Technical Report)

We study whether, in the pi-calculus, the match prefix---a conditional operator testing two names for (syntactic) equality---is expressible via the other operators. Previously, Carbone and Maffeis proved that matching is not expressible this way under rather strong requirements (preservation and reflection of observables). Later on, Gorla developed a by now widely-tested set of criteria for encodings that allows much more freedom (e.g. instead of direct translations of observables it allows comparison of calculi with respect to reachability of successful states). In this paper, we offer a considerably stronger separation result on the non-expressibility of matching using only Gorla's relaxed requirements.Comment: This report extends a paper in EXPRESS/SOS'14 and provides the missing proof

A U-statistic estimator for the variance of resampling-based error estimators

We revisit resampling procedures for error estimation in binary classification in terms of U-statistics. In particular, we exploit the fact that the error rate estimator involving all learning-testing splits is a U-statistic. Therefore, several standard theorems on properties of U-statistics apply. In particular, it has minimal variance among all unbiased estimators and is asymptotically normally distributed. Moreover, there is an unbiased estimator for this minimal variance if the total sample size is at least the double learning set size plus two. In this case, we exhibit such an estimator which is another U-statistic. It enjoys, again, various optimality properties and yields an asymptotically exact hypothesis test of the equality of error rates when two learning algorithms are compared. Our statements apply to any deterministic learning algorithms under weak non-degeneracy assumptions. In an application to tuning parameter choice in lasso regression on a gene expression data set, the test does not reject the null hypothesis of equal rates between two different parameters

A unified approach to testing for and against a set of linear inequality constraints in the product multinomial setting

AbstractA problem that is frequently encountered in statistics concerns testing for equality of multiple probability vectors corresponding to independent multinomials against an alternative they are not equal. In applications where an assumption of some type of stochastic ordering is reasonable, it is desirable to test for equality against this more restrictive alternative. Similar problems have been considered heretofore using the likelihood ratio approach. This paper aims to generalize the existing results and provide a unified technique for testing for and against a set of linear inequality constraints placed upon on any r(r≥1) probability vectors corresponding to r independent multinomials. The paper shows how to compute the maximum likelihood estimates under all hypotheses of interest and obtains the limiting distributions of the likelihood ratio test statistics. These limiting distributions are of chi bar square type and the expression of the weighting values is given. To illustrate our theoretical results, we use a real life data set to test against second-order stochastic ordering

The Communication Complexity of Set Intersection and Multiple Equality Testing

In this paper we explore fundamental problems in randomized communication complexity such as computing Set Intersection on sets of size $k$ and Equality Testing between vectors of length $k$. Sa\u{g}lam and Tardos and Brody et al. showed that for these types of problems, one can achieve optimal communication volume of $O(k)$ bits, with a randomized protocol that takes $O(\log^* k)$ rounds. Aside from rounds and communication volume, there is a \emph{third} parameter of interest, namely the \emph{error probability} $p_{\mathrm{err}}$. It is straightforward to show that protocols for Set Intersection or Equality Testing need to send $\Omega(k + \log p_{\mathrm{err}}^{-1})$ bits. Is it possible to simultaneously achieve optimality in all three parameters, namely $O(k + \log p_{\mathrm{err}}^{-1})$ communication and $O(\log^* k)$ rounds? In this paper we prove that there is no universally optimal algorithm, and complement the existing round-communication tradeoffs with a new tradeoff between rounds, communication, and probability of error. In particular: 1. Any protocol for solving Multiple Equality Testing in $r$ rounds with failure probability $2^{-E}$ has communication volume $\Omega(Ek^{1/r})$. 2. There exists a protocol for solving Multiple Equality Testing in $r + \log^*(k/E)$ rounds with $O(k + rEk^{1/r})$ communication, thereby essentially matching our lower bound and that of Sa\u{g}lam and Tardos. Our original motivation for considering $p_{\mathrm{err}}$ as an independent parameter came from the problem of enumerating triangles in distributed ($\textsf{CONGEST}$) networks having maximum degree $\Delta$. We prove that this problem can be solved in $O(\Delta/\log n + \log\log \Delta)$ time with high probability $1-1/\operatorname{poly}(n)$.Comment: 44 page