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
Location: Unitobler, F011
30 seats, 59m^2
Date: Tuesday, 09/Jul/2019
10:00am - 12:00pmMS177, part 1: Algebraic and combinatorial phylogenetics
Unitobler, F011 
 
10:00am - 12:00pm

Algebraic and combinatorial phylogenetics

Chair(s): Marta Casanellas (Universitat Politècnica de Catalunya), Jane Coons (North Carolina State University), Seth Sullivant (North Carolina State University)

Since late eighties, algebraic tools have been present in phylogenetic theory and have been crucial in understanding the limitations of models and methods and in proposing improvements to the existing tools. In this session we intend to present some of the most recent work in this area.

 

(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)

 

An Introduction to Algebraic and Combinatorial Phylogenetics

Jane Coons
North Carolina State University

The purpose of this talk is to provide participants with the necessary background information to attend the subsequent talks in this session. As such, we will discuss preliminary definitions and key results in the field of mathematical phylogenetics. In particular, we will discuss results concerning tree and network combinatorics and introduce some important phylogenetic models. We will also describe the algebraic methods that have been used to understand these models.

 

Inferring species networks from gene trees

Elizabeth S. Allman, Hector Baños, John Rhodes
University of Alaska Fairbanks

Phylogenetic trees for different genes from the same taxa often differ from one another, with incomplete lineage sorting and hybridization considered to be two of the most important biological reasons underlying this discordance. Inferring a hybridization network that shows species relationships from a set of gene trees is made difficult by the confounding of these two sources of conflicting signal. We present a new algorithm for this inference problem, under the Network Multispecies Coalescent model of these processes on a level-1 network. Building on a number of combinatorial insights, the topological species network estimator is statistically consistent with reasonable running time for moderate size data sets. Analyses of several simulated and empirical datasets indicate its practical value.

 

Algebraic versus semi-algebraic conditions for phylogenetic varieties

Marina Garrote-López
BGSMath and Universitat Politècnica de Catalunya

It is common to model evolution adopting a parametric statistical model which allows to define a joint probability distribution at the leaves of phylogenetic trees. When these models are algebraic, one is able to deduce polynomial relationships between these probabilities, and the study of these polynomials and the geometry of the algebraic varieties that arise from them can be used to reconstruct phylogenetic trees. However, not all points in these algebraic varieties have biological sense. In this talk, we would like to discuss the importance of studying the subset of these varieties with biological sense and explore the extent to which restricting to these subsets can provide insight into existent methods of phylogenetic reconstruction. One of our main focuses is to understand and describe these subsets of points that come from positive parameters. We are interested in the algebraic and semi-algebraic conditions that describe them and in knowing which of these conditions are relevant for topology inference. The projection into these subsets can be seen as an optimization problem and can be solved using nonlinear programming algorithms. As these algorithms do not guarantee a global solution, we use a different approach that allows us to find a global optimum. Numerical algebraic geometry and computational algebra play a fundamental role here. We will show some results on trees evolving under groups-based models and, in particular, we will explore the long branch attraction phenomenon.

 

Trait evolution on two gene trees

James Degnan
The University of New Mexico

Models of trait evolution use a phylogenetic tree to determine the correlation structure for traits sampled from a set of species. Typically, the phylogenetic tree is estimated from genetic data from many loci, and a single tree is used to model the trait evolution, for example by assuming that the mean trait value follows a Brownian motion on the tree. Here, we model trait evolution by assuming that there are two genetic loci influencing the trait. In this case separate evolutionary trees (called gene trees) can occur for the two loci. We model the correlation structure as arising from a linear combination of Brownian motions on the two trees, and develop a model to estimate the proportion of trait evolution contributed by each gene.

 
3:00pm - 5:00pmRoom free
Unitobler, F011 

Date: Wednesday, 10/Jul/2019
10:00am - 12:00pmMS177, part 2: Algebraic and combinatorial phylogenetics
Unitobler, F011 
 
10:00am - 12:00pm

Algebraic and combinatorial phylogenetics

Chair(s): Marta Casanellas (Universitat Politècnica de Catalunya), Jane Coons (North Carolina State University), Seth Sullivant (North Carolina State University)

Since late eighties, algebraic tools have been present in phylogenetic theory and have been crucial in understanding the limitations of models and methods and in proposing improvements to the existing tools. In this session we intend to present some of the most recent work in this area.

 

(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)

 

Weighting the Coalescent

Joseph Rusinko
Hobart and William Smith Colleges

Under the coalescent model, the dominant quartet should match the topology on the species tree. However, in practice we only have a finite sample of gene trees from which to estimate the dominant quartet. We introduce a quartet weighting system which enables accurate species tree reconstruction when combined with a quartet amalgamation algorithm such as MaxCut. The weighting system also provides a mechanism for determining which data should be included in their analysis.

 

Identifiability of 2-tree mixtures for the Kimura 3ST model

Jesús Fernández-Sánchez1, Marta Casanellas1, Alessandro Oneto2
1Universitat Politècnica de Catalunya, 2BGSMath and Universitat Politècnica de Catalunya

The inference of evolutionary trees from molecular sequence data relies on modeling site substitutions by a Markov process on a phylogenetic tree, or by a mixture of such processes in a number of trees (not necessarily distinct). The identifiability of the parameters of the models is a crucial feature for this inference process to be consistent. From an algebraic geometry perspective, the unmixed substitution models can be described in terms of some algebraic varieties associated to the tree topologies while the mixtures of these models correspond to the join of these varieties (secant varieties, when the trees considered are the same).

The identifiability of the 2-tree mixtures (mixed models obtained from two tree topologies) under the so-called group-based models with 4 states has been deeply studied and, in particular, Allman et al. 2011 proved that under the JC and K2P models, it is possible to distinguish unmixed models from mixtures obtained from two trees. A key point in the proof is the existence of linear constraints that allow us to distinguish between different tree topologies. Unfortunately, such linear equations do not exist for the more general K3P model.

In this talk, we will recall some general facts on mixed models and state some results of our joint work with Marta Casanellas and Alessandro Oneto. In particular, we will present some advances related to the generic identifiability of tree parameters under the K3P model.

 

Markov association schemes

Jeremy Sumner
University of Tasmania

This work concerns a compelling example of the mathematics of phylogenetics leading to a novel algebraic/combinatorial structure. The motivation for this work comes from a simple model of aminoacyl-tRNA synthetase (aaRS) evolution devised by Julia Shore (UTAS) and Peter Wills (U Auckland). Starting with a proposed rooted tree describing the specialization of aaRS through evolution of the genetic code, their model produces a space of symmetric Markov rate matrices that form a commutative algebra under matrix multiplication. We refer to each of these as a `tree-algebra'.

From their construction, one most naturally expects that the tree-algebras occur as special instances of association schemes (which are well-studied in algebraic combinatorics). However, this is incorrect as one finds that a tree-algebra corresponds to an association scheme only in a highly degenerate case. In fact, further study has revealed that both the tree-algebras and association schemes can be conceived of as occurring as special cases of a novel class of combinatorial structures, which we (possibly imperfectly) refer to as `Markov association schemes'.

In this talk, I will describe our attempts thus far to characterize Markov association schemes. In particular, I will present two natural binary operations of `sum' and `product' on the class of schemes and show that the tree algebras arise precisely from repeatedly applying the sum operation to the trivial scheme.

 

Existence of maximally probable ranked gene tree topologies with a matching unranked topology

Filippo Disanto1, Pasquale Miglionico2, Guido Narduzzi2
1University of Pisa, 2Scuola Normale Superiore, Pisa

A ranked gene tree topology is a labeled gene tree topology together with a temporal ordering (a ranking) of its coalescence events. A species tree is a labeled species tree topology considered with a set of lengths for its branches that naturally induces a ranking of the coalescence events present in the tree. Disregarding the ordering of the internal nodes of a ranked tree yields a leaf labeled tree topology, which is the unranked topology of the tree. When exactly one gene copy is sampled for each species, we consider ranked gene tree topologies realized in a ranked species tree under the multispecies coalescent model, and study the unranked topology of the ranked gene tree topologies with the largest conditional probability. We show that among the ranked gene tree topologies that are maximally probable, there is always at least one whose unranked topology matches that of the species tree. We also show that not all of the maximally probable ranked gene tree topologies have a concordant unranked topology.

 
3:00pm - 5:00pmMS199, part 1: Applications of topology in neuroscience
Unitobler, F011 
 
3:00pm - 5:00pm

Applications of topology in neuroscience

Chair(s): Kathryn Hess Bellwald (Laboratory for topology and neuroscience, EPFL, Switzerland), Ran Levi (University of Aberdeen, UK)

Research at the interface of topology and neuroscience is growing rapidly and has produced many remarkable results in the past five years. In this minisymposium, speakers will present a wide and exciting array of current applications of topology in neuroscience, including classification and synthesis of neuron morphologies, analysis of synaptic plasticity, and diagnosis of traumatic brain injuries.

 

(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)

 

Understanding neuronal shapes with algebraic topology

Lida Kanari
Blue Brain Project, EPFl, Switzerland

The morphological diversity of neurons supports the complex information-processing capabilities of biological neuronal networks. A major challenge in neuroscience has been to reliably describe neuronal shapes with universal morphometrics that generalize across cell types and species. Inspired by algebraic topology, we have conceived a topological descriptor of trees that couples the topology of their complex arborization with the geometric features of its structure, retaining more information than traditional morphometrics. The topological morphology descriptor (TMD) has proved to be very powerful in categorizing neurons into concrete groups based on morphological grounds. The TMD algorithm has lead to the discovery of two distinct classes of pyramidal cells in the human cortex, and the identification of robust groups for rodent cortical neurons.

 

Computing homotopy types of directed flag complexes

Dejan Govc
University of Aberdeen, UK

I will present some techniques from elementary algebraic topology that can in some cases be used to completely determine the homotopy types of directed flag complexes (or more generally, ordered simplicial complexes). These involve simplicial collapses, heuristic homology computations and coning operations. Then I will explain how to use these techniques to classify the homotopy types for certain large families of tournaments. Finally, I will show how to compute the homotopy type for the C. Elegans connectome.

 

Applications of persistent homology to stroke therapy

Philip Egger
Hummel Lab, EPFL. Switzerland

Stroke has recently been called “the epidemic of the 21st century." Today, 1.5 million new strokes occur each year in Europe alone, and this number is expected to increase by a factor of 1.5 to 2 by the year 2050. Despite advances in treatment, only a small proportion of patients recover enough to re-enter normal life.

The brain is a highly networked organ; accordingly, many brain diseases, including stroke, are increasingly understood as network disorders. Stroke lesions cause impairment by disabling nearby nodes and edges in the structural network, which in turn affects the functional network and the corresponding behavior. Likewise, it is hypothesized that recovery from stroke can be expressed in terms of reorganization of both functional and structural networks.

The use of graph theory-based metrics to study brain networks is well established. Some metrics, such as degree or betweenness centrality, capture local characteristics; others, such as connection density or the small-world index, capture global characteristics of a network. In recent years, algebraic topology has become increasingly prominent for its ability to integrate local network characteristics to a global notion of shape.

We will present evidence that in vivo structural brain networks possess significantly more cavities than random networks with the same degree distribution, building on in silico evidence from Hess et al that information flow is organized by topological invariants. We will also attempt to use persistent homology to distinguish between two groups of patients: those who, within 3 months poststroke, recover roughly 70% of lost motor function (called “fitters") and those who do not (called “non-fitters").

 

Neural decoding using TDA

Erik Rybakken
NTNU, Norway

Neural decoding is the process of determining which stimuli are driving the activity of neurons. For instance, head direction neurons fire depending on which direction the animal is looking. Determining this relationship, however, can be a tedious proces, where the researcher would have to track and process all kinds of stimuli that might be relevant for the neural activity.

I will demonstrate how we can use topological methods to decodethe relevant features from neural recordings alone, without having to observe the external stimuli or the behavior of the animal at all. In particular, we have decoded the head direction of mice with high accuracy, without even knowing that the neurons were coding for head direction. This demonstrates that topologi-cal methods provide a useful tool in neural decoding that could possibly be used to discover new types of neurons.

 

Date: Thursday, 11/Jul/2019
10:00am - 12:00pmMS126, part 1: Euclidean distance geometry and its applications
Unitobler, F011 
 
10:00am - 12:00pm

Euclidean distance geometry and its applications

Chair(s): Kaie Kubjas (Sorbonne Université)

Given a natural number d and a weighted graph G=(V,E), the fundamental problem in Euclidean distance geometry is to determine whether there exists a realization of the graph G in Rd such that distances between pairs of points are equal to the corresponding edge weights. This problem naturally arises in many applications that require recovering locations of objects from the distances between these objects. Usually, measurements of the distances are noisy and there can be missing data. Examples of applications are sensor network localization, molecular conformation, genome reconstruction, robotics and data visualization. Algebraic varieties and semialgebraic sets naturally come up in Euclidean distance geometry, since distances between objects are given by polynomials. Hence questions about uniqueness and finiteness of realizations are often algebraic in nature, whereas realizations are found using semidefinite or nonconvex optimization methods. The goal of this minisymposium is to present theory and applications of Euclidean distance geometry, and connect researchers working in Euclidean distance geometry with applied algebraic geometers.

 

(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)

 

Isometries in Euclidean, Homogeneous, and Conformal Spaces

Carlile Lavor
University of Campinas, Brazil

It is known, from linear algebra, that isometries in Euclidean Spaces are described by orthogonal transformations up to translations. We will discuss what happens when isometries are considered in Homogeneous and Conformal Spaces.
 

Auxetic deformations of triply periodic minimal surfaces

Ciprian S. Borcea
Rider University, USA

The notion of one-parameter auxetic deformation, introduced previously for periodic frameworks, can be used in the context of triply periodic minimal surfaces. We exhibit auxetic paths in several new families of triply periodic minimal surfaces of low genus.

 

Voronoi Cells of Varieties

Maddie Weinstein
University of California, Berkeley, USA

Every real algebraic variety determines a Voronoi decomposition of its ambient Euclidean space. Each Voronoi cell is a convex semialgebraic set in the normal space of the variety at a point. We compute the algebraic boundaries of these Voronoi cells. Using intersection theory, we give a formula for the degrees of the algebraic boundaries of Voronoi cells of curves and surfaces. We discuss an application to low-rank matrix approximation. This is joint work with Diego Cifuentes, Kristian Ranestad, and Bernd Sturmfels.

 

Critical points of the Hamming and taxicab distance functions

Jonathan Hauenstein
University of Notre Dame, USA

Minimizing the Euclidean distance from a given point to the solution set of a given system of polynomial equations can be accomplished via critical point techniques. This talk will explore extending critical point techniques to minimization with respect to the Hamming distance and taxicab distance. Numerical algebraic geometric techniques are derived for computing a finite set of real points satisfying the polynomial equations which contains a global minimizer. Several examples will be used to demonstrate the new techniques. This is joint work with D. Brake, N. Daleo, and S. Sherman.

 
3:00pm - 5:00pmMS166, part 2: Computational aspects of finite groups and their representations
Unitobler, F011 
 
3:00pm - 5:00pm

Computational aspects of finite groups and their representations

Chair(s): Armin Jamshidpey (University of Waterloo, Canada), Eric Schost (University of Waterloo, Canada), Mark Giesbrecht (University of Waterloo, Canada)

The theory of finite groups and their representations is not only an interesting topic for mathematicians but also provides powerful tools in solving problems in science. New computational tools are making this even more feasible. To name a few, one may find applications in physics, coding theory and cryptography. On the other hand representation theory is useful in different areas of mathematics such as algebraic geometry and algebraic topology. Due to this wide range of applications, new algorithmic methods are being developed to study finite groups and their representations from a computational perspective.

Recent developments in computer algebra systems and more specifically computational linear algebra, provide tools for developments in computational aspects of finite groups and their representations. The aim of this minisymposium is to gather experts in the area to discuss the recent achievements and potential new directions.

 

(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)

 

Calculations with Symplectic Hypergeometric Groups

Alexander Hulpke
Colorado State University

We study monodromy groups of hypergeometric equations that can be considered as matrix groups generated by the companion matrices of pimitive pairs of integral polynomials. It is known that these groups preseve a nondegenerate symplectic form and are Zariski-dense in the respective symplectic group, and much recent work has concentrated on the question whether particular groups are arithmetic, that is have finite index in the symplectic group.
Using matrix group algorithms on congruence images, we are able to calculate indices of the arithmetic closures of these groups, The information obtained also enables us to prove arithmeticity in some cases through a coset enumeration.
This is joint work with Dane Flannery (Galway) and Alla Detinko (St Andrews).

 

Algorithmic factorization of noncommutative polynomials

Viktor Levandovskyy
RWTH Acchen University

We are interested in factorizing polynomials over non-commutative rings. Let us start with a field K and a finitely presented associative K-algebra A, which is a domain.

There are at least two distinct notions of a factorization of polynomials over A. One of them originates from the ring theory (N. Jacobson, P.M. Cohn) and uses a weak notion of association relation (called left or right similarity), what is at the same time hard to approach algorithmically. On the contrary, in applications we'd like to use the classical association relation, i.e. when two elements differ by a factor, which is nonzero central unit.

I will present long-seeked conditions for a given algebra A to be a finite factorization domain, i.e. a domain, where every nonunit has at most finite number of factorizations. Over such domains a factorization procedure thus becomes into an algorithm. Examples, bounds and counterexamples will be given. Over the well-known class of ubiquitous G-algebras (a.k.a. PBW a.k.a. algebras of solvable type), we provide a factorization algorithm, its' smarter graded-driven version for graded algebras and a factorizing Groebner algorithm. All of these are implemented in Singular:Plural (www.singular.uni-kl.de). We view the factorizing Groebner algorithm as the only general possibility to obtain a weaker analogon to the primary decomposition from the commutative algebra.
Recent complexity results and applications of the mentioned algorithms will be presented.

 

Finite groups of Lie type and computer algebra

Meinolf Geck
Universität Stuttgart

The classification of finite simple groups highlights the importance of studying the class of groups in the title. These are defined in terms of algebraic groups over algebraically closed fields of positive characteristic. We discuss a few recent examples where computer algebra methods have played a significant role in developing and establishing new results.

 

Classification of regular parametrized one-relation operads

Murray Bremner
University of Saskatchewan

I will discuss an application of representation theory of symmetric groups to algebraic operads and nonassociative algebra. J.-L. Loday introduced parametrized one-relation operads (POROs): symmetric operads generated by one binary operation subject to one relation showing how to reassociate a left-normed product into to a linear combination of right-normed products:

(ab)c =Σσ in S_3 xσ aσ ( bσ cσ ) ( xσ in Q ).

For some values of xσ, the operad is regular: for all n its homogeneous component of degree n is isomorphic to the regular representation of Sn. (Equivalently, the corresponding free algebra on a vector space V is isomorphic as a graded vector space to the tensor algebra of V.) The familiar examples of regular POROs are those governing associative, Poisson, Leibniz, and Zinbiel algebras. We use computer algebra based on a constructive version of representation theory of symmetric groups to classify all regular POROs. We show that in addition to the above four operads, the only other example is the nilpotent operad. This is joint work with Vladimir Dotsenko of Trinity College Dublin.

 
5:15pm - 6:30pmSIAGA meeting for corresponding and associate editors
Unitobler, F011 

Date: Friday, 12/Jul/2019
10:00am - 12:00pmMS126, part 2: Euclidean distance geometry and its applications
Unitobler, F011 
 
10:00am - 12:00pm

Euclidean distance geometry and its applications

Chair(s): Kaie Kubjas (Sorbonne Université)

Given a natural number d and a weighted graph G=(V,E), the fundamental problem in Euclidean distance geometry is to determine whether there exists a realization of the graph G in Rd such that distances between pairs of points are equal to the corresponding edge weights. This problem naturally arises in many applications that require recovering locations of objects from the distances between these objects. Usually, measurements of the distances are noisy and there can be missing data. Examples of applications are sensor network localization, molecular conformation, genome reconstruction, robotics and data visualization. Algebraic varieties and semialgebraic sets naturally come up in Euclidean distance geometry, since distances between objects are given by polynomials. Hence questions about uniqueness and finiteness of realizations are often algebraic in nature, whereas realizations are found using semidefinite or nonconvex optimization methods. The goal of this minisymposium is to present theory and applications of Euclidean distance geometry, and connect researchers working in Euclidean distance geometry with applied algebraic geometers.

 

(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)

 

Rigidity theory and algebraic matroids

Jessica Sidman
Mount Holyoke College, USA

Consider a framework consisting of fixed length bars attached at flexible joints. The central question in rigidity theory is to determine if the resulting framework is rigid or flexible. The minimally locally rigid graphs in dimension d are the bases of a matroid which can be realized as a linear matroid in joint coordinates associated to the classical "rigidity matrix" or as the algebraic matroid in (squared) distance coordinates associated to the Cayley-Menger variety. In this talk we focus on what the algebraic matroid can tell us about stresses and finite motions. This is joint work with Zvi Rosen, Louis Theran, and Cynthia Vinzant.

 

Periodic framework enhancements

Ileana Streinu
Smith College, USA

A (periodic) bar-and-joint framework is a geometric (periodic) graph whose vertices are mapped to points in R^d (for a fixed dimension d) and its edges to straight-line segments between them. The framework’s configuration space consists in all the placements of the same graph which retain the edge lengths (and the abstract periodicity). We enhance the (periodic) bar-and-joint framework structure to include faces of higher dimensions and study several scenarios that preserve or alter in a controlled manner the dimension of the original configuration space.

 

Barvinok's Naive Algorithm in Distance Geometry

Leo Liberti1, Ky Vu2
1CNRS and Ecole Polytechnique, France, 2Chinese University of Hong Kong, P.R. China

In 1997, A. Barvinok gave a probabilistic algorithm to derive a feasible solution of a quadratically (equation) constrained problem from its semidefinite relaxation. We generalize this algorithm to handle matrix (instead of vector) variables and to two-sided inequalities, and derive a heuristic for the distance geometry problem. We showcase its computational performance on a set of instances related to protein conformation.

 

Mathematics of 3D genome reconstruction in diploid organisms

Kaie Kubjas
Sorbonne Université, France

The 3D organization of the genome plays an important role for gene regulation. Chromosome conformation capture techniques allow one to measure the number of contacts between genomic loci that are nearby in the 3D space. In this talk, we study the problem of reconstructing the 3D organization of the genome from whole genome contact frequencies in diploid organisms, i.e. organisms that contain two indistinguishable copies of each genomic locus. In particular, we study the identifiability of the 3D organization of the genome and optimization methods for reconstructing it. This talk is based on joint work with Anastasiya Belyaeva, Lawrence Sun and Caroline Uhler.

 
3:00pm - 5:00pmMS186, part 1: Algebraic vision
Unitobler, F011 
 
3:00pm - 5:00pm

Algebraic vision

Chair(s): Max David Lieblich (University of Washington, United States of America), Tomas Pajdla (Czech Technical University in Prague), Matthew Trager (Courant Institute of Mathematical Sciences at NYU)

There has been a burst of recent activity focused on the applications of modern abstract and numerical algebraic geometry to problems in computer vision, ranging from highly-optimized Gröbner-basis techniques, to homotopy continuation methods, to Ulrich sheaves and Chow forms, to functorial moduli theory. We will discuss this recent progress, with a focus on multiview geometry, both in theory and in practice.

 

(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)

 

"Real" Algebraic Vision

Sameer Agarwal
Google

Computer Vision is rarely done over the complex numbers. On the other hand, it does often involve semi-algebraic sets, non-genericity and finite precision arithmetic. Dealing with these complications leads to a variety of interesting and hard mathematical questions. I will talk about a few of my favorite ones.

 

A geometric construction of the essential variety

Lucas Van Meter
University of Washington

We give a new construction of the essential variety using a geometric definition of calibration. This construction recovers classical results about the essential variety while also yielding a new 2-1 cover that is strongly related to previous work of Kileel-Fløystad-Ottaviani.

 

Classification of Point-Line Minimal Problems in Complete Multi-View Visibility

Timothy Duff1, Kathlén Kohn2, Anton Leykin1, Tomas Pajdla3
1Georgia Tech, 2University of Oslo, 3CIIRC, CTU Prague

We present a complete classification of all minimal problems for generic arrangements of points and lines completely observed by calibrated perspective cameras. We show that there are only 30 minimal problems in total, no problems exist for more than 6 cameras, for more than 5 points, and for more than 6 lines. For all minimal problems discovered, we present their algebraic degrees, i.e. the number of solutions, which measure their intrinsic difficulty. Our classification shows that there are many interesting new minimal problems. Our results also show how exactly the difficulty of problems grows with the number of views. Importantly, we discovered several new minimal problems with small degrees that might be practical in image matching and 3D reconstruction.

 

Date: Saturday, 13/Jul/2019
10:00am - 12:00pmMS186, part 2: Algebraic vision
Unitobler, F011 
 
10:00am - 12:00pm

Algebraic vision

Chair(s): Max David Lieblich (University of Washington, United States of America), Tomas Pajdla (Czech Technical University in Prague), Matthew Trager (Courant Institute of Mathematical Sciences at NYU)

There has been a burst of recent activity focused on the applications of modern abstract and numerical algebraic geometry to problems in computer vision, ranging from highly-optimized Gröbner-basis techniques, to homotopy continuation methods, to Ulrich sheaves and Chow forms, to functorial moduli theory. We will discuss this recent progress, with a focus on multiview geometry, both in theory and in practice.

 

(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)

 

Solving for camera configurations from pairs

Brian Osserman
University of California, Davis

We study the question of when a configuration of multiple cameras can be recovered when one has information about a subset of the pairs (given for instance as a collection of fundamental matrices). We find the minimal number of pairs which can suffice, and analyze more generally what sorts of conditions can ensure either that a given set of pairs does or does not suffice to determine the configuration. This is joint work with Matthew Trager.

 

Ideals of the Multiview Variety

Andrew Pryhuber
University of Washington

The multiview variety of an arrangement of cameras is the Zariski closure of the images of world points in the cameras. The prime vanishing ideal of this complex projective variety is called the multiview ideal. We show that the bifocal and trifocal polynomials from the cameras generate the multiview ideal when the foci are distinct. In the computer vision literature, many sets of (determinantal) polynomials have been proposed to describe the multiview variety. While the ideals of these polynomials are all contained in the multiview ideal, we show that none of them coincide with the multiview ideal. We establish precise algebraic relationships between the multiview ideal and these various determinantal ideals. When the camera foci are non-coplanar, we prove that the ideal of bifocal polynomials saturate to give the multiview ideal. Finally, we prove that all the ideals we consider coincide when dehomogenized to cut out the space of finite images.

 

Estimation under group action and fast polynomial solvers, with applications to cryo-EM

Joe Kileel
Princeton

In many applied contexts, the task is to estimate latent variables from noisy observations involving unknown rotations. One challenging example comes from cryo-electron microscopy (cryo-EM), recognized by the 2017 Nobel Prize in Chemistry, where the objective is to estimate a 3D molecule from highly noisy 2D projection images taken from unknown viewing directions.

In this talk, we introduce an abstract framework for statistical estimation under noisy group actions. We prove, for this class of problems, sample complexity relates to invariant rings and secant varieties, while method-of-moments is sample-efficient. In special cases, we find a computationally-efficient algorithm for inverting moments, using tensor decomposition and polynomial solving.

In particular, for one model of cryo-EM we present a polynomial solver for ~10000 variables running in ~2 minutes. Further, we develop a new variant of the power method for symmetric tensor decomposition, e.g. decomposing random 15^6 symmetric tensors of rank 450 in ~45 seconds. Our principled approach is validated on a real cryo-EM dataset, in the context of ab initio modeling.

Joint work with Amit Singer’s group and Afonso Bandeira’s group.

 
3:00pm - 5:00pmRoom free
Unitobler, F011