Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS156: Tropical geometry in statistics
Time:
Wednesday, 10/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F013
53 seats, 74m^2

Presentations
10:00am - 12:00pm

Tropical geometry in statistics

Chair(s): Carlos Améndola (TU Munich), Anthea Monod (Columbia University), Ruriko Yoshida (Naval Postgraduate School)

Classically, statistics is the branch of mathematics that deals with data. The challenges of modern data demand the development of new statistical methods to handle them. Modern data collection technology brings not only “big data” that are extremely high dimensional, but additionally, they are made up of complex structures, which can be prohibitive to the Euclidean setting of classical statistics. Tropical geometry defines and studies piecewise linear structures in an algebraic framework that, if interpreted appro- priately, is amenable to modern data structures and challenges. This session focuses on leveraging the potential of tropical geometry to reinterpret classical statistics and enhance the utility of statistical methodology in the face of modern data challenges. Specifically, we seek to adapt the linearizing properties of the tropical semiring to statistical settings that rely on principles of linear algebra and optimization. These encompass fundamental descriptive and inferential statistics, such as the computation of Fréchet means, principal component analysis, linear regression, and hypothesis testing. This is a very new direction of research with potential for wide-reaching applications from biology to economics, and it is our hope to bring together researchers to develop and advance the interaction between tropical geometry and statistics.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

Tropical principal component analysis

Leon Zhang
UC Berkeley

We introduce a notion of principal component analysis in the setting of tropical geometry. We also describe some results on the containment of a Stiefel linear space within a larger tropical linear space and apply them to our setting of tropical principal component analysis.

 

Tropical Foundations for Probability and Statistics on Phylogenetic Tree Spaces

Bo Lin
Georgia Tech

A geometric approach to phylogenetic tree space was first introduced by Billera, Holmes, and Vogtmann. We reinterpret the tree space via tropical geometry and introduce a novel framework for the statistical analysis of phylogenetic trees: the palm tree space, which represents phylogenetic trees as points in a space endowed with the tropical metric. We show that the palm tree space possesses a variety of properties that allow for the definition of probability measures, and thus expectations, variances, and other fundamental statistical quantities. In addition, they lead to increased computational efficiency. Our approach provides a new, tropical basis for a statistical treatment of evolutionary biological processes represented by phylogenetic trees. This is a joint work with Anthea Monod (Columbia University, USA) and Ruriko Yoshida (Naval Postgraduate School, USA).

 

Tropical Gaussians

Ngoc Tran
University of Texas, Austin

There is a growing need for a systematic study of probability distributions in tropical settings. Over the classical algebra, the Gaussian measure is arguably the most important distribution to both theoretical probability and applied statistics. In this work, we review the existing analogues of the Gaussian measure in the tropical semiring and outline various research directions.

 

Tropical hardware for data intensive applications: DNA sequence alignment to machine learning

Advait Madhavan
University of Maryland College Park, NIST

New encodings are being explored to allay the energy efficiency concerns, that fundamentally limit performance of data intensive, modern computing systems. One such brain inspired encoding, known as Race Logic, encodes information in the arrival time of signals. With such an encoding, conventional-computing gates such as OR, AND and delay gates perform MIN, MAX and addition-by-constant operations respectively. Hence with we end up with elegant hardware implementations for the fundamental operations of tropical algebra. This allows tropical operations to be easily expressed with temporally-coded hardware, which allows data intensive problems to be solved with low latency, low energy computer architectures. One such architecture is a DNA sequence alignment engine which calculates the edit distance between two input sequences. Our architecture physically implements the dynamic programming nature of tropical graph traversal methods, on a programmable edit-graph. The other architecture describes a programmable methodology of mapping various decision tree based forests to MIN/MAX gates and is used for high throughput in-sensor image classification. The main message of this talk is to stress on the symbiosis between tropical algebra and computing hardware communities, which we believe can lead to development of compact, energy efficient, computing hardware for new classes of complex optimization problems.