# Applied Algebra Seminar Spring 2021

**When**: 11am, Thursdays

**Where**: Virtual: https://uwmadison.zoom.us/j/93664753217

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

**Contact**: Shamgar Gurevich, Jose Israel Rodriguez, Julia Lindberg (Lead)

**Remark**: This seminar is held on the fourth Thursday of the month and other select Thursdays.

## Contents

## 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

### 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 |