Events

Sum of squares of positive eigenvalues

The spectral Turán theorem says that if a graph has largest eigenvalue $\lambda_1$, $m$ edges and clique number $\omega$, then $\lambda_1^2 \leq 2m (1-\frac{1}{\omega})$. This result implies the classical Turán bound $m \leq (1-\frac{1}{\omega})\frac{n^2}{2}$.
In this talk, we present the proof of the Wocjan, Elphick and Anekstein conjecture in which, in the spectral Turán bound, the square of the first eigenvalue is replaced by the sum of the squares of the positive eigenvalues and the clique number is replaced by the vector chromatic number. 
We will also present recent progress towards a conjecture by Bollobás and Nikiforov in which, in the spectral Turán bound, the square of the first eigenvalue is replaced by the sum of the squares of the two largest eigenvalues. This is joint work with Gabriel Coutinho and Shengtong Zhang.

A strongly polynomial algorithm for linear programs with at most two non-zero entries per row or column.

Abstract:We give a strongly polynomial algorithm for minimum cost generalized flow, and hence for optimizing any linear program with at most two non-zero entries per row, or at most two non-zero entries per column. Primal and dual feasibility were shown by Megiddo (SICOMP '83) and Végh (MOR '17) respectively. Our result can be viewed as progress towards understanding whether all linear programs can be solved in strongly polynomial time, also referred to as Smale's 9th problem. Our approach is based on the recent primal-dual interior point method (IPM) due to Allamigeon, Dadush, Loho, Natura and Végh (FOCS '22). The number of iterations needed by the IPM is bounded, up to a polynomial factor in the number of inequalities, by the straight line complexity of the central path. Roughly speaking, this is the minimum number of pieces of any piecewise linear curve that multiplicatively approximates the central path. As our main contribution, we show that the straight line complexity of any minimum cost generalized flow instance is polynomial in the number of arcs and vertices. By applying a reduction of Hochbaum (ORL '04), the same bound applies to any linear program with at most two non-zeros per column or per row. To be able to run the IPM, one requires a suitable initial point. For this purpose, we develop a novel multistage approach, where each stage can be solved in strongly polynomial time given the result of the previous stage. Beyond this, substantial work is needed to ensure that the bit complexity of each iterate remains bounded during the execution of the algorithm. For this purpose, we show that one can maintain a representation of the iterates as a low complexity convex combination of vertices. Our approach is black-box and can be applied to any log-barrier path following method. 

Bento Natura is an Assistant Professor in Industrial Engineering and Operations Research (IEOR) at Columbia University. He spent two years as a Postdoctoral Fellow at Georgia Tech, Brown University, and UC Berkeley. Prior to that, he obtained his PhD in Mathematics from the London School of Economics.

His research interests are focused on the areas of algorithms, optimization, and game theory, with a special emphasis on the theory of linear programming.

Graph Property Testing using the Container Method

Abstract: The Graph and Hypergraph Container Methods have recently been used to obtain multiple striking results across different areas of mathematics. In this talk, we will see how the graph container method is particularly well-suited for the study of some fundamental problems in graph property testing.

The main problem we will discuss in the talk is the Independent Set Testing problem introduced by Goldreich, Goldwasser, and Ron (1998). In this problem, we are given oracle access to a graph on $n$ vertices that either (i) contains an independent set on $\rho n$ vertices, or (ii) is $\epsilon$-far from the property in the sense that at least $\epsilon n^2$ edges must be removed from the graph to make it have an independent set of this size. We will introduce a new container lemma for the latter class of graphs and we will show how this lemma can be used to obtain a near-optimal solution to the Independent Set Testing problem. We will also discuss how variants and extensions of the new container lemma can be used to prove a variety of other results in property testing.

This is joint work with Cameron Seth.

Title: Purifying arbitrarily noisy quantum states

Abstract: Quantum state purification is the task of recovering a nearly pure copy of an unknown pure quantum state using multiple noisy copies of the state. We derive an efficient purification procedure based on the swap test for qudits of any dimension, starting with any initial error parameter. For constant initial error parameter and dimension, our procedure has sample complexity asymptotically optimal in the final error parameter, and almost matches the known optimal protocol for qubits. Our protocol has a simple recursive structure that can be applied when the states are provided one at a time in a streaming fashion, requiring only a small quantum memory to implement.  Joint work with Andrew Childs, Honghao Fu, Zhi Li, Maris Ozols, Vedang Vyas. 

Title: Rotation-invariant web bases from hourglass plabic graphs and symmetrized six-vertex configurations

Abstract: Many combinatorial objects with strikingly good enumerative formulae also have remarkable dynamical behavior and underlying algebraic structure. In this talk, we consider the promotion action on certain rectangular tableaux and explain its small, predictable order by reinterpreting promotion as a rotation in disguise. We show how the search for this visual explanation of combinatorial dynamics led to the solution of an algebraic problem involving web bases and a generalization of the six-vertex model of statistical physics. We find that this framework includes several unexpected combinatorial objects of interest: alternating sign matrices, plane partitions, and their symmetry classes (joint with Ashleigh Adams). This talk is based primarily on joint work with Christian Gaetz, Stephan Pfannerer, Oliver Pechenik, and Joshua P. Swanson.

Title: Concrete analysis of a few aspects of lattice-based cryptography

Abstract: A seminal 2013 paper by Lyubashevsky, Peikert, and Regev proposed using ideal lattices as a foundation for post-quantum cryptography, supported by a polynomial-time security reduction from the approximate Shortest Independent Vectors Problem (SIVP) to the Decision Learning With Errors (DLWE) problem in ideal lattices. In our concrete analysis of this multi-step reduction, we find that the reduction’s tightness gap is so significant that it undermines any meaningful security guarantees. Additionally, we have concerns about the feasibility of the quantum aspect of the reduction in the near future. Moreover, when making the reduction concrete, the approximation factor for the SIVP problem turns out to be much larger than anticipated, suggesting that the approximate SIVP problem may not be hard for the proposed cryptosystem parameters.

Title: Problem Decomposition in Optimization:  Algorithmic Advances Beyond ADMM

Abstract:

Decomposition schemes like those coming from ADMM typically start by posing a separable-type problem in the Fenchel duality format.  They then pass to an augmented Lagrangian, which however can interfere with the separability and cause a slow-down.  Progressive decoupling offers a more flexible approach which can utilize augmented Lagrangians while maintaining decomposability.  Based on a variable metric extension of the proximal point algorithm that's applied in a twisted sort of way, progressive decoupling benefits from stopping criteria which can guarantee convergence despite inexact minimization in each iteration.   The convergence is generically at a linear rate, and for convex problems, is global. But the method also works for nonconvex problems when initiated close enough to a point that satisfies a natural extension of the strong sufficient condition for local optimality known from nonlinear programming.

This talk is held as part of the 26^th Annual Midwest Optimization Meeting (“MOM26”).

Title: Proximal methods for nonsmooth and nonconvex fractional programs: when sparse optimization meets fractional programs

Abstract:Nonsmooth and nonconvex fractional programs are ubiquitous and also highly challenging. It includes the composite optimization problems studied extensively lately, and encompasses many important modern optimization problems arising from diverse areas such as the recent proposed scale invariant sparse signal reconstruction problem in signal processing, the robust Sharpe ratio optimization problems in finance and the sparse generalized eigenvalue problem in discrimination analysis.

In this talk, we will introduce extrapolated proximal methods for solving nonsmooth and nonconvex fractional programs and analyse their convergence behaviour. Interestingly, we will show that the proposed algorithm exhibits linear convergence for the scale invariant sparse signal reconstruction model,  and the sparse generalized eigenvalue problem with either cardinality regularization or sparsity constraints. This is achieved by identifying the explicit desingularization function of the Kurdyka-Ł ojasiewicz inequality for the merit function of the fractional optimization models. Finally, if time permits, we will present some preliminary encouraging numerical results for the proposed methods for sparse signal reconstruction and sparse Fisher discriminant analysis

The talk is based on joint work with R.I. Bo ̧t, M. Dao, T.K. Pong and P. Yu.

Title: Soft Condorcet Optimization

Abstract:

A common way to drive the progress of AI models and agents is to compare their performance on standardized benchmarks. This often involves aggregating individual performances across a potentially wide variety of tasks and benchmarks and many of the leaderboards that draw greatest attention are Elo-based. 

In this paper, we describe a novel ranking scheme inspired by social choice frameworks, called Soft Condorcet Optimization (SCO), to compute the optimal ranking of agents: the one that makes the fewest mistakes in predicting the agent comparisons in the evaluation data. This optimal ranking is the maximum likelihood estimate when evaluation data (which we view as votes) are interpreted as noisy samples from a ground truth ranking, a solution to Condorcet's original voting system criteria and inherits desirable social-choice inspired properties since SCO ratings are maximal for Condorcet winners when they exist, which we show is not necessarily true for the classical rating system Elo.

We propose three optimization algorithms to compute SCO ratings and evaluate their empirical performance across a variety of synthetic and real-world datasets, to illustrate different properties.

With Marc Lanctot, Ian Gemp, Quentin Berthet, Yoram Bachrach, Manfred Diaz, Roberto-Rafael Maura-Rivero,  Anna Koop, and Doina Precup