arXiv subjects

Ngoc Mai Tran

Publications and source records attributed to Ngoc Mai Tran.

26 records · Page 2Linked to original sources

arXiv2017

Classification on Large Networks: A Quantitative Bound via Motifs and Graphons

When each data point is a large graph, graph statistics such as densities of certain subgraphs (motifs) can be used as feature vectors for machine learning. While intuitive, motif counts are expensive to compute and difficult to work with theoretically. Via graphon theory, we give an explicit quantitative bound for the ability of motif homomorphisms to distinguish large networks under both generative and sampling noise. Furthermore, we give similar bounds for the graph spectrum and connect it to homomorphism densities of cycles. This results in an easily computable classifier on graph data with theoretical performance guarantee. Our method yields competitive results on classification tasks for the autoimmune disease Lupus Erythematosus.

cs.LG↗

arXiv2016

Enumerating Polytropes

Polytropes are both ordinary and tropical polytopes. We show that tropical types of polytropes in $\mathbb{TP}^{n-1}$ are in bijection with cones of a certain Gröbner fan $\mathcal{GF}_n$ in $\mathbb{R}^{n^2 - n}$ restricted to a small cone called the polytrope region. These in turn are indexed by compatible sets of bipartite and triangle binomials. Geometrically, on the polytrope region, $\mathcal{GF}_n$ is the refinement of two fans: the fan of linearity of the polytrope map appeared in \cite{tran.combi}, and the bipartite binomial fan. This gives two algorithms for enumerating tropical types of polytropes: one via a general Gröbner fan software such as \textsf{gfan}, and another via checking compatibility of systems of bipartite and triangle binomials. We use these algorithms to compute types of full-dimensional polytropes for $n = 4$, and maximal polytropes for $n = 5$.

math.CO↗

arXiv2015

Antibiotics Time Machine is NP-hard

The antibiotics time machine is an optimization question posed by Mira \latin{et al.} on the design of antibiotic treatment plans to minimize antibiotic resistance. The problem is a variation of the Markov decision process. These authors asked if the problem can be solved efficiently. In this paper, we show that this problem is NP-hard in general.

math.OC↗

arXiv2014

Zeros of random tropical polynomials, random polytopes and stick-breaking

For $i = 0, 1, \ldots, n$, let $C_i$ be independent and identically distributed random variables with distribution $F$ with support $(0,\infty)$. The number of zeros of the random tropical polynomials $\mathcal{T}f_n(x) = \min_{i=1,\ldots,n}(C_i + ix)$ is also the number of faces of the lower convex hull of the $n+1$ random points $(i,C_i)$ in $\mathbb{R}^2$. We show that this number, $Z_n$, satisfies a central limit theorem when $F$ has polynomial decay near $0$. Specifically, if $F$ near $0$ behaves like a $gamma(a,1)$ distribution for some $a > 0$, then $Z_n$ has the same asymptotics as the number of renewals on the interval $[0,\log(n)/a]$ of a renewal process with inter-arrival distribution $-\log(Beta(a,2))$. Our proof draws on connections between random partitions, renewal theory and random polytopes. In particular, we obtain generalizations and simple proofs of the central limit theorem for the number of vertices of the convex hull of $n$ uniform random points in a square. Our work leads to many open problems in stochastic tropical geometry, the study of functionals and intersections of random tropical varieties.

math.PR↗

arXiv2014

Principal Component Analysis in an Asymmetric Norm

Principal component analysis (PCA) is a widely used dimension reduction tool in the analysis of many kind of high-dimensional data. It is used in signal processing, mechanical engineering, psychometrics, and other fields under different names. It still bears the same mathematical idea: the decomposition of variation of a high dimensional object into uncorrelated factors or components. However, in many of the above applications, one is interested in capturing the tail variables of the data rather than variation around the mean. Such applications include weather related event curves, expected shortfalls, and speeding analysis among others. These are all high dimensional tail objects which one would like to study in a PCA fashion. The tail character though requires to do the dimension reduction in an asymmetric norm rather than the classical $L_2$-type orthogonal projection. We develop an analogue of PCA in an asymmetric norm. These norms cover both quantiles and expectiles, another tail event measure. The difficulty is that there is no natural basis, no `principal components', to the $k$-dimensional subspace found. We propose two definitions of principal components and provide algorithms based on iterative least squares. We prove upper bounds on their convergence times, and compare their performances in a simulation study. We apply the algorithms to a Chinese weather dataset with a view to weather derivative pricing

stat.ME↗

arXiv2012

HodgeRank is the limit of Perron Rank

We study the map which takes an elementwise positive matrix to the k-th root of the principal eigenvector of its k-th Hadamard power. We show that as $k$ tends to 0 one recovers the row geometric mean vector and discuss the geometric significance of this convergence. In the context of pairwise comparison ranking, our result states that HodgeRank is the limit of Perron Rank, thereby providing a novel mathematical link between two important pairwise ranking methods.

stat.ME↗

arXiv2012

Combinatorial Types of Tropical Eigenvectors

The map which takes a square matrix to its tropical eigenvalue-eigenvector pair is piecewise linear. We determine the cones of linearity of this map. They are simplicial but they do not form a fan. Motivated by statistical ranking, we also study the restriction of that cone decomposition to the subspace of skew-symmetric matrices.

math.CO↗

arXiv2011

Pairwise ranking: choice of method can produce arbitrarily different rank order

We examine three methods for ranking by pairwise comparison: Principal Eigenvector, HodgeRank and Tropical Eigenvector. It is shown that the choice of method can produce arbitrarily different rank order.To be precise, for any two of the three methods, and for any pair of rankings of at least four items, there exists a comparison matrix for the items such that the rankings found by the two methods are the prescribed ones. We discuss the implications of this result in practice, study the geometry of the methods, and state some open problems.

stat.ME↗