The maximum 2D subarray polytope: facet-inducing inequalities and polyhedral computations

Given a matrix with real-valued entries, the maximum 2D subarray problem consists in finding a rectangular submatrix with consecutive rows and columns maximizing the sum of its entries. In this work we start a polyhedral study of an integer programming formulation for this problem.We thus define the 2D subarray polytope, explore conditions ensuring the validity of linear inequalities, and provide several families of facet-inducing inequalities. We also report com- putational experiments assessing the reduction of the dual bound for the linear relaxation achieved by these families of inequalities.Este documento es una versión del artículo publicado en Applied Mathematics 323, 286-301

Supporting Scientific Analytics under Data Uncertainty and Query Uncertainty

Data management is becoming increasingly important in many applications, in particular, in large scientific databases where (1) data can be naturally modeled by continuous random variables, and (2) queries can involve complex predicates and/or be difficult for users to express explicitly. My thesis work aims to provide efficient support to both the data uncertainty and the query uncertainty . When data is uncertain, an important class of queries requires query answers to be returned if their existence probabilities pass a threshold. I start with optimizing such threshold query processing for continuous uncertain data in the relational model by (i) expediting selections by reducing dimensionality of integration and using faster filters, (ii) expediting joins using new indexes on uncertain data, and (iii) optimizing a query plan using a dynamic, per-tuple based approach. Evaluation results using real-world data and benchmark queries show the accuracy and efficiency of my techniques and the dynamic query planning has over 50% performance gains in most cases over a state-of-the-art threshold query optimizer and is very close to the optimal planning in all cases. Next I address uncertain data management in the array model, which has gained popu- larity for scientific data processing recently due to performance benefits. I define the formal semantics of array operations on uncertain data involving both value uncertainty within individual tuples and position uncertainty regarding where a tuple should belong in an array given uncertain dimension attributes, and propose a suite of storage and evaluation strategies for array operators, with a focus on a novel scheme that bounds the overhead of querying by strategically placing a few replicas of the tuples with large variances. Evaluation results show that for common workloads, my best-performing techniques outperform baselines up to 1 to 2 orders of magnitude while incurring only small storage overhead. Finally, to bridge the increasing gap between the fast growth of data and the limited human ability to comprehend data and help the user retrieve high-value content from data more effectively, I propose to build interactive data exploration as a new database service, using an approach called “explore-by-example”. To build an effective system, my work is grounded in a rigorous SVM-based active learning framework and focuses on the following three problems: (i) accuracy-based and convergence-based stopping criteria, (ii) expediting example acquisition in each iteration, and (iii) expediting the final result retrieval. Evaluation results using real-world data and query patterns show that my system significantly outperforms state-of-the-art systems in accuracy (18x accuracy improvement for 4-dimensional workloads) while achieving desired efficiency for interactive exploration (2 to 5 seconds per iteration)

COMBINATORIAL ASPECTS OF EXCEDANCES AND THE FROBENIUS COMPLEX

In this dissertation we study the excedance permutation statistic. We start by extending the classical excedance statistic of the symmetric group to the affine symmetric group eSn and determine the generating function of its distribution. The proof involves enumerating lattice points in a skew version of the root polytope of type A. Next we study the excedance set statistic on the symmetric group by defining a related algebra which we call the excedance algebra. A combinatorial interpretation of expansions from this algebra is provided. The second half of this dissertation deals with the topology of the Frobenius complex, that is the order complex of a poset whose definition was motivated by the classical Frobenius problem. We determine the homotopy type of the Frobenius complex in certain cases using discrete Morse theory. We end with an enumeration of Q-factorial posets. Open questions and directions for future research are located at the end of each chapter

Boletín de Investigación Ditelliano n°4

La Universidad Torcuato Di Tella es una universidad de investigación cuya misión es contribuir al avance del conocimiento, y a la búsqueda de la excelencia intelectual, científica y artística dentro de un marco de libertad de expresión, pluralismo y diversidad de enfoques. El Boletín de Investigación Ditelliano refleja la actividad de investigación de nuestro cuerpo docente. Incluye las publicaciones, los premios, subsidios y becas de investigación obtenidos, y las participaciones en seminarios y conferencias argentinas e internacionales. El Boletín tiene dos ediciones anuales. Con el objetivo de difundir a una audiencia internacional la investigación desarrollada por nuestro cuerpo académico regular, este año, además, lanzamos el Boletín de Investigación en su versión en inglés. Este Boletín es un trabajo conjunto de la Secretaría Académica de la Universidad y el área de Apoyo a la Investigación de la Biblioteca. Cada publicación reflejada en este Boletín contribuye a la actualización de nuestro Repositorio Digital y a la adquisición de los nuevos libros de ditellianos publicados en cada semestre.Universidad Torcuato Di Tella is a research University whose mission is to contribute to the advancement of knowledge, the pursuit of intellectual, scientific, and artistic excellence within a framework of freedom of expression, pluralism, and diversity of approaches. The Research Newsletter reflects the research activity of our faculty. It includes publications, awards, research grants and scholarships obtained, as well as participation in Argentine and international seminars and conferences. The newsletter has two annual editions. With the aim of disseminating the research developed by our regular academic staff to an international audience, this year we are also launching the newsletter in its English version. This newsletter is a joint effort of the Academic Secretariat of the University and the Research Support area of the Library. Each publication reflected in this newsletter contributes to the update of our Digital Repository and the acquisition of new books by Torcuato Di Tella authors published each semester

Enumeration of max-pooling responses with generalized permutohedra

We investigate the combinatorics of max-pooling layers, which are functions that downsample input arrays by taking the maximum over shifted windows of input coordinates, and which are commonly used in convolutional neural networks. We obtain results on the number of linearity regions of these functions by equivalently counting the number of vertices of certain Minkowski sums of simplices. We characterize the faces of such polytopes and obtain generating functions and closed formulas for the number of vertices and facets in a 1D max-pooling layer depending on the size of the pooling windows and stride, and for the number of vertices in a special case of 2D max-pooling.Comment: 35 pages, 11 figures, 4 tables. V2: Improved exposition, added computations in Section 4, and expanded analysis of dat

Boletín de Investigación Ditelliano n°3

Compartimos con ustedes la 3.ª edición del Boletín de Investigación Ditelliano con la idea de articular y potenciar nuestros esfuerzos en investigación. Creemos que este Boletín es el espacio propicio para compartir entre colegas y, por eso, agradecemos a todos los que se suman a participar en este espacio. Este trabajo se realiza junto con el área de Apoyo a la Investigación de la Biblioteca. Es por ello que cada publicación da lugar a la actualización de nuestro Repositorio Digital y a la adquisición de los nuevos libros de ditellianos publicados en cada semestre

Combinatorics

Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session

ADVANCES IN QUANTUM PARAMETER ESTIMATION AND OTHER TOPICS

The first half of this thesis deals with the problem of parameter estimation in a quantum system. In quantum parameter estimation theory, the quantum Fisher information is usually considered as the ultimate precision limit, beyond which no further improvement is possible due to the inherent stochasticity of quantum measurements. On the other hand, in this thesis we will show that a better precision is achievable than predicted by a quantum Fisher information analysis. This is true if some regularity assumptions about the underlying quantum parametric model are relaxed. In such situations, the quantum Fisher information does not completely capture the best possible performance of quantum measurements, and a different approach must be followed. In the second part of the thesis, we will focus on some applications of orthogonal array theory to two notable quantum information problems: the problem of multipartite entanglement classification and the quantum marginal problem. Introduced by Rao in 1947, orthogonal arrays have been usefully applied to different fields, from cryptography and coding theory to the statistical design of experiments, software testing and quality control. Remarkably, orthogonal arrays have also found application in quantum information and, in particular, in the study of quantum entanglement. We will employ tools from orthogonal array theory to study a toy version of the multipartite entanglement classification problem. Finally, we will show how orthogonal arrays can be employed to build constructive solutions to low-dimensional quantum marginal problems

Spectral Aspects of Cocliques in Graphs

This thesis considers spectral approaches to finding maximum cocliques in graphs. We focus on the relation between the eigenspaces of a graph and the size and location of its maximum cocliques. Our main result concerns the computational problem of finding the size of a maximum coclique in a graph. This problem is known to be NP-Hard for general graphs. Recently, Codenotti et al. showed that computing the size of a maximum coclique is still NP-Hard if we restrict to the class of circulant graphs. We take an alternative approach to this result using quotient graphs and coding theory. We apply our method to show that computing the size of a maximum coclique is NP-Hard for the class of Cayley graphs for the groups $\mathbb{Z}_p^n$ where $p$ is any fixed prime. Cocliques are closely related to equitable partitions of a graph, and to parallel faces of the eigenpolytopes of a graph. We develop this connection and give a relation between the existence of quadratic polynomials that vanish on the vertices of an eigenpolytope of a graph, and the existence of elements in the null space of the Veronese matrix. This gives a us a tool for finding equitable partitions of a graph, and proving the non-existence of equitable partitions. For distance-regular graphs we exploit the algebraic structure of association schemes to derive an explicit formula for the rank of the Veronese matrix. We apply this machinery to show that there are strongly regular graphs whose $\tau$-eigenpolytopes are not prismoids. We also present several partial results on cocliques and graph spectra. We develop a linear programming approach to the problem of finding weightings of the adjacency matrix of a graph that meets the inertia bound with equality, and apply our technique to various families of Cayley graphs. Towards characterizing the maximum cocliques of the folded-cube graphs, we find a class of large facets of the least eigenpolytope of a folded cube, and show how they correspond to the structure of the graph. Finally, we consider equitable partitions with additional structural constraints, namely that both parts are convex subgraphs. We show that Latin square graphs cannot be partitioned into a coclique and a convex subgraph