Related papers: Multi-scale representation of integer sets: applic…
Hierarchical classification addresses the problem of classifying items into a hierarchy of classes. An important issue in hierarchical classification is the evaluation of different classification algorithms, which is complicated by the…
An important problem in multi-label classification is to capture label patterns or underlying structures that have an impact on such patterns. This paper addresses one such problem, namely how to exploit hierarchical structures over labels.…
The use of complex networks as a modern approach to understanding the world and its dynamics is well-established in literature. The adjacency matrix, which provides a one-to-one representation of a complex network, can also yield several…
We propose a hierarchical architecture for efficiently computing high-quality solutions to structured mixed-integer programs (MIPs). To reduce computational effort, our approach decouples the original problem into a higher level problem and…
Prime numbers have fascinated mathematicians since antiquity, with ongoing efforts to uncover both their properties and ever-larger examples. While giant primes rarely aid cryptography, they find use in areas such as locally decodable…
Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. "Structure" can be understood as symmetry and a range of symmetries are expressed by hierarchy. Such symmetries directly…
Let p(n) be the nth prime and p(p(n)) be the nth prime-indexed prime (PIP). The process of taking prime-indexed subsequences of primes can be iterated, and the number of such iterations is the prime-index order. We report empirical evidence…
This paper aims for set-to-hypergraph prediction, where the goal is to infer the set of relations for a given set of entities. This is a common abstraction for applications in particle physics, biological systems, and combinatorial…
We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\eps…
We present a detailed proof of the prime number theorem suitable for a typical undergraduate- or graduate-level complex analysis course. Our presentation is particularly useful for any instructor who seeks to use the prime number theorem…
Cutting planes (cuts) play an important role in solving mixed-integer linear programs (MILPs), which formulate many important real-world applications. Cut selection heavily depends on (P1) which cuts to prefer and (P2) how many cuts to…
When studying multi-body protein complexes, biochemists use computational tools that can suggest hundreds or thousands of their possible spatial configurations. However, it is not feasible to experimentally verify more than only a very…
In this paper, we present an alternative approach to multiple inheritance for typed feature structures. In our approach, a feature structure can be associated with several types coming from different hierarchies (dimensions). In case of…
Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat's theorem. For an integer $N$, the test tries to find an…
Recent years have been marked with the fast-pace diversification and increasing ubiquity of machine learning applications. Yet, a firm theoretical understanding of the surprising efficiency of neural networks to learn from high-dimensional…
Graph classification aims to categorize graphs based on their structural and attribute features, with applications in diverse fields such as social network analysis and bioinformatics. Among the methods proposed to solve this task, those…
The First Hilbert problem is studied in this paper by applying two instruments: a new methodology distinguishing between mathematical objects and mathematical languages used to describe these objects; and a new numeral system allowing one…
Counting the number of prime numbers up to a certain natural number and describing the asymptotic behavior of such a counting function has been studied by famous mathematicians like Gauss, Legendre, Dirichlet, and Euler. The prime number…
In this thesis, we develop various techniques for working with sets in machine learning. Each input or output is not an image or a sequence, but a set: an unordered collection of multiple objects, each object described by a feature vector.…
We develop a new, powerful method for counting elements in a multiset. As a first application, we use this algorithm to study the number of occurrences of patterns in a permutation. For patterns of length 3 there are two Wilf classes, and…