# Applied Algebra Seminar Fall 2021

**When**: 11am Central, Thursdays.

**Where**: Virtual: Zoom link (The passcode: Bucky )

**List**: to join email mathaas+join@g-groups.wisc.edu and subscribe to the google group.

**Contact**: Julia Lindberg (Lead) and Jose Israel Rodriguez

**Remark**: This seminar is held virtually on the first and third Thursday of the month.

**Inclusiveness**: We also welcome participants with applied algebra interests from other institutions.

## Contents

## Fall 2021 Schedule

date | speaker | title | host(s) |
---|---|---|---|

September 16 | Jeroen Zuiddam (University of Amsterdam) | Geometric rank of tensors and applications | Virtual |

October 7 | Timo de Wolff (TU Braunschweig) | Certificates of Nonnegativity and Maximal Mediated Sets | Virtual |

October 21 | Jesús A. De Loera (UC Davis) | A Combinatorial Geometry View of Inference and Learning Problems | Virtual |

November 4 | Kevin Shu (Georgia Tech) | Nonnegative Quadratics over Stanley Reisner Varieties | Virtual |

November 18 | Angélica Torres (KTH Stockholm) | The line multi-view variety in computer vision | Virtual |

December 2 | Papri Dey (Missouri) | Hyperbolic Polynomials, and Helton-Vinnikov curves | Virtual |

## Spring 2021 Schedule

date | speaker | title | host(s) |
---|---|---|---|

February 25 | Greg Blekherman (Georgia Tech) | Locally Positive Semidefinite Matrices | Virtual |

March 25 | James Saunderson (Monash University) | Lifting For Simplicity: Concise Descriptions of Convex Sets | Virtual |

April 22 (At the new 11am time!) | Hamza Fawzi (Cambridge) | Semidefinite representations, and the set of separable states | Virtual |

May 13 (At the new 11am time!) | Anna Seigal (University of Oxford) | Invariant Theory for Maximum Likelihood Estimation | Virtual |

## Spring 2020 Schedule

date | speaker | title | host(s) |
---|---|---|---|

February 20 | Carla Michini (UW Madison) | Short simplex paths in lattice polytopes | Local |

March 5 | Alisha Zachariah (UW Madison) | Efficient Estimation of a Sparse Delay-Doopler Channel | Local |

March 19 | Spring Break | ||

March 26 | (Seminar on Hiatus because of Covid-19) |

## Abstracts

### Papri Dey

**Hyperbolic Polynomials, and Helton-Vinnikov curves**

In this talk, I shall give a panoramic view of my research work. I shall introduce the notion of hyperbolic polynomials and discuss an algebraic method to test hyperbolicity of a multivariate polynomial w.r.t some fixed point via sum-of-squares relaxation, proposed in my research work. An important class of hyperbolic polynomials are definite determinantal polynomials. Helton-Vinnikov curves in the projective plane are cut-out by hyperbolic polynomials in three variables. This leads to the computational problem of explicitly producing a symmetric positive definite linear determinantal representation for a given curve. I shall focus on two approaches to this problem proposed in my research work: an algebraic approach via solving polynomial equations, and a geometric-combinatorial approach via scalar product representations of the coefficients and its connection with polytope membership problem. The algorithms to solve determinantal representation problems are implemented in Macaulay2 as a software package DeterminantalRepresentations. m2.

### Angélica Torres

**The line multi-view variety in computer vision**

Given an arrangement of cameras, the line multi-view variety is the Zariski closure of the image of world lines in the cameras. This is a similar definition to the multi-view variety studied extensively in computer vision. The goal of this talk is to present a description of the ideal of the line multi-view variety, and discuss some geometric properties of this variety. This is joint work with Felix Rydell, Elima Shehu, Paul Breiding, and Kathlén Kohn.

### Kevin Shu

**Nonnegative Quadratics over Stanley Reisner Varieties**

Nonnegative polynomials are of fundamental interest in the field of real algebraic geometry. We will discuss the convex geometry of nonnegative polynomials over an interesting class of algebraic varieties known as Stanley-Reisner varieties. Stanley-Reisner varieties are fairly simple, as they are unions of coordinate planes, but they have surprising connections to algebraic topology that we can use to obtain a detailed understanding of the convex geometry of these nonnegative quadratics. We will be particularly interested in the extreme nonnegative quadratic forms over such varieties. Though this will not be emphasized in the talk, this subject also has some direct applications to optimization theory.

### Jesús A. De Loera

**A Combinatorial Geometry View of Inference and Learning Problems**

In statistical inference we wish to learn the properties or parameters of a distribution through sufficiently many samples. A famous example is logistic regression, a popular non-linear model in multivariate statistics and supervised learning. Users often rely on optimizing of maximum likelihood estimation, but how much training data do we need, as a function of the dimension of the covariates of the data, before we expect an MLE to exist with high probability? Similarly, for unsupervised learning and non-parametric statistics, one wishes to uncover the shape and patterns from samples of a measure or measures. We use only the intrinsic geometry and topology of the sample. A famous example of this type of method is the $k$-means clustering algorithm. A fascinating challenge is to explain the variability of behavior of $k$-means algorithms with distinct random initializations and the shapes of the clusters.

In this talk we present new stochastic combinatorial theorems, that give bounds on the probability of existence of maximum likelihood estimators in multinomial logistic regression and also relate to the variability of clustering initializations. Along the way we will see fascinating connections to the coupon collector problem, topological data analysis, and to the computation of Tukey centerpoints of data clouds (a high-dimensional generalization of median). This is joint work with T. Hogan, R. D. Oliveros, E. Jaramillo-Rodriguez, and A. Torres-Hernandez.

### Timo de Wolff

**Certificates of Nonnegativity and Maximal Mediated Sets**

In science and engineering, we regularly face polynomial optimization problems, that is: minimize a real, multivariate polynomial under polynomial constraints. Solving these problems is essentially equivalent to certifying of nonnegativity of real polynomials -- a key problem in real algebraic geometry since the 19th century. Since this problem is notoriously hard to solve, one is interested in certificates that imply nonnegativity and are easier to check than nonnegativity itself. In particular, a polynomial is nonnegative if it is a sums of squares (SOS) of other polynomials. In 2014, Iliman and I introduced a new certificate of nonnegativity based on sums of nonnegative circuit polynomials (SONC), which I have developed further since then both in theory and practice joint with different coauthors.

A canonical question is: When is a nonnegative circuit polynomial also a sums of squares. It turns out that this question is purely combinatorial and is governed by sets of particular (integral) lattice points called maximal mediated sets, which where first introduced by Reznick. In this talk, I will give a brief introduction to nonnegativity, SOS, and SONC, and then discuss recent results on maximal mediated sets.

### Jeroen Zuiddam

**Geometric rank of tensors and applications**

There are several important problems in computational complexity and extremal combinatorics that can naturally be phrased in terms of tensors. This includes the matrix multiplication problem, the cap set problem and the sunflower problem. We present a new geometric method in this area, called Geometric Rank (Kopparty–Moshkovitz–Zuiddam, 2020). Geometric Rank is a natural extension of matrix rank to tensors. As an application, we use Geometric Rank to solve a problem of Strassen about matrix multiplication tensors.

### Anna Seigal

**Invariant Theory for Maximum Likelihood Estimation**

Maximum likelihood estimation is an optimization problem over a statistical model, to obtain the parameters that best fit observed data. I will focus on two settings: log-linear models and Gaussian group models. I will describe connections between maximum likelihood estimation and notions of stability from invariant theory. This talk is based on joint work with Carlos Améndola, Kathlén Kohn and Philipp Reichenbach.

### Hamza Fawzi

**Semidefinite representations, and the set of separable states**

The convex set of separable states plays a fundamental role in quantum information theory as it corresponds to the set of non-entangled states. In this talk I will discuss the question of semidefinite programming representations for this convex set. Using connections with nonnegative polynomials and sums of squares I will show that except in the low-dimensions, this semialgebraic set has no semidefinite representation. As a consequence this set furnishes a counter-example to the Helton-Nie conjecture. The proof I will present is elementary and relies only on basic results about semialgebraic sets and functions.

### James Saunderson

**Lifting For Simplicity: Concise Descriptions of Convex Sets**

This talk will give a selective overview of the theory and applications of lifts of convex sets. A lift of a convex set is a higher-dimensional convex set that projects onto the original set. Many interesting convex sets have lifts that are dramatically simpler to describe than the original set. Finding such simple lifts has significant algorithmic implications, particularly for associated optimization problems. We will consider both the classical case of polyhedral lifts, which are described by linear inequalities, as well as spectrahedral lifts, which are defined by linear matrix inequalities. The talk will include discussion of ways to construct lifts, ways to find obstructions to the existence of lifts, and a number of interesting examples from a variety of mathematical contexts. (Based on joint work with H. Fawzi, J. Gouveia, P. Parrilo, and R. Thomas).

### Greg Blekherman

**Locally Positive Semidefinite Matrices**

The cone of positive semidefinite matrices plays a prominent role in optimization, and many hard computational problems have well-performing semidefinite relaxations. In practice, enforcing the constraint that a large matrix is positive semidefinite can be expensive. We introduce the cone of k-locally posiitive semidefinite matrices, which consists of matrices all of whose k by k principal submatrices are positive semidefinite. We consider the distance between the cones of positive and locally positive semidefinite matrices, and possible eigenvalues of locally positive semidefinite matrices. Hyperbolic polynomials play a role in some of the proofs. Joint work with Santanu Dey, Marco Molinaro, Kevin Shu and Shengding Sun.

### Carla Michini

**Short simplex paths in lattice polytopes**

We consider the problem of optimizing a linear function over a lattice polytope P contained in [0,k]^n and defined via m linear inequalities. We design a simplex algorithm that, given an initial vertex, reaches an optimal vertex by tracing a path along the edges of P of length at most O(n^6 k log k). The length of this path is independent on m and is the best possible up to a polynomial function, since it is only polynomially far from the worst case diameter. The number of arithmetic operations needed to compute the next vertex in the path is polynomial in n, m and log k. If k is polynomially bounded by n and m, the algorithm runs in strongly polynomial time. This is a joint work with Alberto Del Pia.

### Alisha Zachariah

**Efficiently Estimating a Sparse Delay-Doppler Channel**
** **

Multiple wireless sensing tasks, e.g., radar detection for driver safety, involve estimating the ”channel” or relationship between signal transmitted and received. In this talk, I will focus on a certain type of channel known as the delay-doppler channel. This channel model starts to be applicable in high frequency carrier settings, which are increasingly common with recent developments in mmWave technology. Moreover, in this setting, both the channel model and existing technologies are amenable to working with signals of large bandwidth, and using such signals is a standard approach to achieving high resolution channel estimation. However, when high resolution is desirable, this approach creates a tension with the desire for efficiency because, in particular, it immediately implies that the signals in play live in a space of very high dimension N (e.g., ~10^6 in some applications), as per the Shannon-Nyquist sampling theorem.

To address this, I will propose a randomized algorithm for channel estimation in the k-sparse setting (e.g., k objects in radar detection), with sampling and space complexity both on the order of k(log N)^2, and arithmetic complexity on the order of k(log N)^3+k^2, for N sufficiently large.

While this algorithm seems to be extremely efficient -- to the best of our knowledge, the first of this nature in terms of complexity -- it is just a simple combination of three ingredients, two of which are well-known and widely used, namely digital chirp signals and discrete Gaussian filter functions, and the third being recent developments in Sparse Fast Fourier Transform algorithms.

## Related events to note

date | event/title | location/speaker | info |
---|---|---|---|

Postponed | Applied Algebra Days 4 - Tensors | Several talks on tensors | |

1:30pm, 2nd Thursday of the month | Informal Seminar: Algebra in Statistics and Computation | Virtual | |

3:30pm | SIAM Student Chapter | Virtual | |

10am, 2nd Tuesday of the month | SIAM SAGA | Virtual: Recordings | Registration needed once. |

10am, Most Tuesdays | Nonlinear algebra seminar online | Virtual: Recordings | Registration required once |

Biweekly Mondays | Algebraic Statistics Online Seminar (ASOS) | Virtual: Recordings | Mailing list sign-up for Zoom-links |

January 26th-29th, 2021 | Sanya Workshop on Algebraic Geometry and Machine Learning | Virtual: Recordings | |

July 29-30, 2021 | Real algebraic geometry and convex geometry Conference | TBD: TU Braunschweig, Germany or Online |