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Ngoc Mai Tran

Publications and source records attributed to Ngoc Mai Tran.

At least 19 recordsLinked to original sources
arXiv2024

Minimal Representations of Tropical Rational Functions

This paper studies the following question: given a piecewise-linear function, find its minimal algebraic representation as a tropical rational signomial. We put forward two different notions of minimality, one based on monomial length, the other based on factorization length. We show that in dimension one, both notions coincide, but this is not true in dimensions two or more. We prove uniqueness of the minimal representation for dimension one and certain subclasses of piecewise-linear functions in dimension two. As a proof step, we obtain counting formulas and lower bounds for the number of regions in an arrangement of tropical hypersurfaces, giving a small extension for a result by Montúfar, Ren and Zhang. As an equivalent formulation, it gives a lower bound on the number of vertices in a regular mixed subdivision of a Minkowski sum, giving a small extension for Adiprasito's Lower Bound Theorem for Minkowski sums.

math.CO↗
arXiv2023

Estimating a Directed Tree for Extremes

We propose a new method to estimate a root-directed spanning tree from extreme data. A prominent example is a river network, to be discovered from extreme flow measured at a set of stations. Our new algorithm utilizes qualitative aspects of a max-linear Bayesian network, which has been designed for modelling causality in extremes. The algorithm estimates bivariate scores and returns a root-directed spanning tree. It performs extremely well on benchmark data and new data. We prove that the new estimator is consistent under a max-linear Bayesian network model with noise. We also assess its strengths and limitations in a small simulation study.

stat.ML↗
arXiv2023

Minimax Rates for High-Dimensional Random Tessellation Forests

Random forests are a popular class of algorithms used for regression and classification. The algorithm introduced by Breiman in 2001 and many of its variants are ensembles of randomized decision trees built from axis-aligned partitions of the feature space. One such variant, called Mondrian forests, was proposed to handle the online setting and is the first class of random forests for which minimax rates were obtained in arbitrary dimension. However, the restriction to axis-aligned splits fails to capture dependencies between features, and random forests that use oblique splits have shown improved empirical performance for many tasks. In this work, we show that a large class of random forests with general split directions also achieve minimax optimal convergence rates in arbitrary dimension. This class includes STIT forests, a generalization of Mondrian forests to arbitrary split directions, as well as random forests derived from Poisson hyperplane tessellations. These are the first results showing that random forest variants with oblique splits can obtain minimax optimality in arbitrary dimension. Our proof technique relies on the novel application of the theory of stationary random tessellations in stochastic geometry to statistical learning theory.

math.ST↗
arXiv2022

A Generalized Muirhead Inequality and Symmetric Sums of Nonnegative Circuits

Circuit polynomials are a certificate of nonnegativity for real polynomials, which can be derived via a generalization of the classical inequality of arithmetic and geometric means. In this article, we show that similarly nonnegativity of symmetric real polynomials can be certified via a generalization of the classical Muirhead inequality. Moreover, we show that a nonnegative symmetric polynomial admits a decomposition into sums of nonnegative circuit polynomials if and only if it satisfies said generalized Muirhead condition. The latter re-proves a result by Moustrou, Naumann, Riener, Theobald, and Verdure for the case of the symmetric group in a shortened and more elementary way.

math.AG↗
arXiv2022

Predicting the Future of AI with AI: High-quality link prediction in an exponentially growing knowledge network

A tool that could suggest new personalized research directions and ideas by taking insights from the scientific literature could significantly accelerate the progress of science. A field that might benefit from such an approach is artificial intelligence (AI) research, where the number of scientific publications has been growing exponentially over the last years, making it challenging for human researchers to keep track of the progress. Here, we use AI techniques to predict the future research directions of AI itself. We develop a new graph-based benchmark based on real-world data -- the Science4Cast benchmark, which aims to predict the future state of an evolving semantic network of AI. For that, we use more than 100,000 research papers and build up a knowledge network with more than 64,000 concept nodes. We then present ten diverse methods to tackle this task, ranging from pure statistical to pure learning methods. Surprisingly, the most powerful methods use a carefully curated set of network features, rather than an end-to-end AI approach. It indicates a great potential that can be unleashed for purely ML approaches without human knowledge. Ultimately, better predictions of new future research directions will be a crucial component of more advanced research suggestion tools.

cs.AI↗
arXiv2022

OpenEMS: an open-source Package for Two-Stage Stochastic and Robust Optimization for Ambulance Location and Routing with Applications to Austin-Travis County EMS Data

Emergency Medical Systems (EMS) provide crucial pre-hospital care and transportation. Faster EMS response time provides quicker pre-hospital care and thus increases survival rate. We reduce response time by providing optimal ambulance stationing and routing decisions by solving two stage stochastic and robust linear programs. Although operational research on ambulance systems is decades old, there is little open-source code and consistency in simulations. We begin to bridge this gap by publishing OpenEMS, in collaboration with the Austin-Travis County EMS (ATCEMS) in Texas, an end-to-end pipeline to optimize ambulance strategic decisions. It includes data handling, optimization, and a calibrated simulation. We hope this open source framework will foster future research with and for EMS. Finally, we provide a detailed case study on the city of Austin, Texas. We find that optimal stationing would increase response time by 88.02 seconds. Further, we design optimal strategies in the case where Austin EMS must permanently add or remove one ambulance from their fleet.

stat.AP↗
arXiv2021

Improving random walk rankings with feature selection and imputation

The Science4cast Competition consists of predicting new links in a semantic network, with each node representing a concept and each edge representing a link proposed by a paper relating two concepts. This network contains information from 1994-2017, with a discretization of days (which represents the publication date of the underlying papers). Team Hash Brown's final submission, \emph{ee5a}, achieved a score of 0.92738 on the test set. Our team's score ranks \emph{second place}, 0.01 below the winner's score. This paper details our model, its intuition, and the performance of its variations in the test set.

cs.SI↗
arXiv2021

Competitive equilibrium always exists for combinatorial auctions with graphical pricing schemes

We show that a competitive equilibrium always exists in combinatorial auctions with anonymous graphical valuations and pricing, using discrete geometry. This is an intuitive and easy-to-construct class of valuations that can model both complementarity and substitutes, and to our knowledge, it is the first class besides gross substitutes that have guaranteed competitive equilibrium. We prove through counter-examples that our result is tight, and we give explicit algorithms for constructive competitive pricing vectors. We also give extensions to multi-unit combinatorial auctions (also known as product-mix auctions). Combined with theorems on graphical valuations and pricing equilibrium of Candogan, Ozdagar and Parrilo, our results indicate that quadratic pricing is a highly practical method to run combinatorial auctions.

math.CO↗
arXiv2021

Predicting Covid-19 EMS Incidents from Daily Hospitalization Trends

Introduction: The aim of our retrospective study was to quantify the impact of Covid-19 on the temporal distribution of Emergency Medical Services (EMS) demand in Travis County, Austin, Texas and propose a robust model to forecast Covid-19 EMS incidents. Methods: We analyzed the temporal distribution of EMS calls in the Austin-Travis County area between January 1st, 2019, and December 31st, 2020. Change point detection was performed to identify critical dates marking changes in EMS call distributions, and time series regression was applied for forecasting Covid-19 EMS incidents. Results: Two critical dates marked the impact of Covid-19 on the distribution of EMS calls: March 17th, when the daily number of non-pandemic EMS incidents dropped significantly, and May 13th, by which the daily number of EMS calls climbed back to 75% of the number in pre-Covid-19 time. The new daily count of the hospitalization of Covid-19 patients alone proves a powerful predictor of the number of pandemic EMS calls, with an r2 value equal to 0.85. In particular, for every 2.5 cases where EMS takes a Covid-19 patient to a hospital, one person is admitted. Conclusion: The mean daily number of non-pandemic EMS demand was significantly less than the period before Covid-19 pandemic. The number of EMS calls for Covid-19 symptoms can be predicted from the daily new hospitalization of Covid-19 patients. These findings may be of interest to EMS departments as they plan for future pandemics, including the ability to predict pandemic-related calls in an effort to adjust a targeted response.

stat.AP↗
arXiv2021

Statistics and tropicalization of local field Gaussian measures

This paper aims to lay the foundations for statistics over local fields, such as the field of $p$-adic numbers. Over such fields, we give characterizations for maximum likelihood estimation and conditional independence for multivariate Gaussian distributions. We also give a bijection between the tropicalization of such Gaussian measures in dimension 2 and supermodular functions on the 2-dimensional discrete cube. Finally, we introduce the Bruhat-Tits building as a parameter space for Gaussian distributions and discuss their connections to conditional independence statements as an open problem.

math.ST↗
arXiv2020

The Finite Matroid-Based Valuation Conjecture is False

The matroid-based valuation conjecture of Ostrovsky and Paes Leme states that all gross substitutes valuations on $n$ items can be produced from merging and endowments of weighted ranks of matroids defined on at most $m(n)$ items. We show that if $m(n) = n$, then this statement holds for $n \leq 3$ and fails for all $n \geq 4$. In particular, the set of gross substitutes valuations on $n \geq 4$ items is strictly larger than the set of matroid based valuations defined on the ground set $[n]$. Our proof uses matroid theory and discrete convex analysis to explicitly construct a large family of counter-examples. It indicates that merging and endowment by themselves are poor operations to generate gross substitutes valuations. We also connect the general MBV conjecture and related questions to long-standing open problems in matroid theory, and conclude with open questions at the intersection of this field and economics.

math.CO↗
arXiv2020

Clustering with Fast, Automated and Reproducible assessment applied to longitudinal neural tracking

Across many areas, from neural tracking to database entity resolution, manual assessment of clusters by human experts presents a bottleneck in rapid development of scalable and specialized clustering methods. To solve this problem we develop C-FAR, a novel method for Fast, Automated and Reproducible assessment of multiple hierarchical clustering algorithms simultaneously. Our algorithm takes any number of hierarchical clustering trees as input, then strategically queries pairs for human feedback, and outputs an optimal clustering among those nominated by these trees. While it is applicable to large dataset in any domain that utilizes pairwise comparisons for assessment, our flagship application is the cluster aggregation step in spike-sorting, the task of assigning waveforms (spikes) in recordings to neurons. On simulated data of 96 neurons under adverse conditions, including drifting and 25\% blackout, our algorithm produces near-perfect tracking relative to the ground truth. Our runtime scales linearly in the number of input trees, making it a competitive computational tool. These results indicate that C-FAR is highly suitable as a model selection and assessment tool in clustering tasks.

stat.ML↗
arXiv2019

Linear and Rational Factorization of Tropical Polynomials

Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables. Special families of these polynomials have appeared in economics, discrete convex analysis, and combinatorics. Our theorems rely on an intrinsic characterization of regular mixed subdivisions of integral polytopes, and lead to many open problems of interest in discrete geometry.

math.CO↗
arXiv2019

Two-player incentive compatible outcome functions are affine maximizers

In mechanism design, for a given type space, there may be incentive compatible outcome functions which are not affine maximizers. Using tools from linear algebra and tropical geometry, we prove that for two-player games on a discrete type space, any given outcome function can be turned into an affine maximizer through a nontrivial perturbation of the type space. Furthermore, our theorems are the strongest possible in this setup.

math.CO↗
arXiv2018

Tropical Geometry and Mechanism Design

We develop a novel framework to construct and analyze finite valued, multidimensional mechanisms using tropical convex geometry. We geometrically characterize incentive compatibility using cells in the tropical convex hull of the type set. These cells are the sets of incentive compatible payments and form tropical simplices, spanned by generating payments whose number equals the dimension of the simplex. The analysis of the collection of incentive compatible mechanisms via tropical simplices and their generating payments facilitates the use of geometric techniques. We use this view to derive a new geometric characterization of revenue equivalence but also show how to handle multidimensional mechanisms in the absence of revenue equivalence.

cs.GT↗
arXiv2018

Geometry and algorithms for upper triangular tropical matrix identities

We provide geometric methods and algorithms to verify, construct and enumerate pairs of words (of specified length over a fixed $m$-letter alphabet) that form identities in the semigroup $\ut{n}$ of $n\times n$ upper triangular tropical matrices. In the case $n=2$ these identities are precisely those satisfied by the bicyclic monoid, whilst in the case $n=3$ they form a subset of the identities which hold in the plactic monoid of rank $3$. To each word we associate a signature sequence of lattice polytopes, and show that two words form an identity for $\ut{n}$ if and only if their signatures are equal. Our algorithms are thus based on polyhedral computations and achieve optimal complexity in some cases. For $n=m=2$ we prove a Structural Theorem, which allows us to quickly enumerate the pairs of words of fixed length which form identities for $\ut{2}$. This allows us to recover a short proof of Adjan's theorem on minimal length identities for the bicyclic monoid, and to construct minimal length identities for $\ut{3}$, providing counterexamples to a conjecture of Izhakian in this case. We conclude with six conjectures at the intersection of semigroup theory, probability and combinatorics, obtained through analysing the outputs of our algorithms.

math.CO↗
arXiv2017

Product-Mix Auctions and Tropical Geometry

In a recent and ongoing work, Baldwin and Klemperer explored a connection between tropical geometry and economics. They gave a sufficient condition for the existence of competitive equilibrium in product-mix auctions of indivisible goods. This result, which we call the Unimodularity Theorem, can also be traced back to the work of Danilov, Koshevoy, and Murota in discrete convex analysis. We give a new proof of the Unimodularity Theorem via the classical unimodularity theorem in integer programming. We give a unified treatment of these results via tropical geometry and formulate a new sufficient condition for competitive equilibrium when there are only two types of product. Generalizations of our theorem in higher dimensions are equivalent to various forms of the Oda conjecture in algebraic geometry.

cs.GT↗