Lattice-based and topological representations of binary relations with an application to music

International audienceFormal concept analysis associates a lattice of formal concepts to a binary relation. The structure of the relation can then be described in terms of lattice theory. On the other hand Q-analysis associates a simplicial complex to a binary relation and studies its properties using topological methods. This paper investigates which mathematical invariants studied in one approach can be captured in the other. Our main result is that all homotopy invariant properties of the simplicial complex can be recovered from the structure of the concept lattice. This not only clarifies the relationships between two frameworks widely used in symbolic data analysis but also offers an effective new method to establish homotopy equivalence in the context of Q-analysis. As a musical application, we will investigate Olivier Messiaen's modes of limited transposition. We will use our theoretical result to show that the simplicial complex associated to a maximal mode with m transpositions is homotopy equivalent to the (m − 2)–dimensional sphere

From music to mathematics and backwards: introducing algebra, topology and category theory into computational musicology

International audienceDespite a long historical relationship between mathematics and music, the interest of mathematicians is a recent phenomenon. In contrast to statistical methods and signal-based approaches currently employed in MIR (Music Information Research), the research project described in this paper stresses the necessity of introducing a structural multidisciplinary approach into computational musicology making use of advanced mathematics. It is based on the interplay between three main mathematical disciplines: algebra, topology and category theory. It therefore opens promising perspectives on important prevailing challenges, such as the automatic classification of musical styles or the solution of open mathematical conjectures, asking for new collaborations between mathematicians, computer scientists, musicologists, and composers. Music can in fact occupy a strategic place in the development of mathematics since music-theoretical constructions can be used to solve open mathematical problems. The SMIR project also differs from traditional applications of mathematics to music in aiming to build bridges between different musical genres, ranging from contemporary art music to popular music, including rock, pop, jazz and chanson. Beyond its academic ambition, the project carries an important societal dimension stressing the cultural component of 'mathemusical' research, that naturally resonates with the underlying philosophy of the “Imagine Maths”conference series. The article describes for a general public some of the most promising interdisciplinary research lines of this project

Fuzzy Intervals for Designing Structural Signature: An Application to Graphic Symbol Recognition

Revised selected papers from Eighth IAPR International Workshop on Graphics RECognition (GREC) 2009.The motivation behind our work is to present a new methodology for symbol recognition. The proposed method employs a structural approach for representing visual associations in symbols and a statistical classifier for recognition. We vectorize a graphic symbol, encode its topological and geometrical information by an attributed relational graph and compute a signature from this structural graph. We have addressed the sensitivity of structural representations to noise, by using data adapted fuzzy intervals. The joint probability distribution of signatures is encoded by a Bayesian network, which serves as a mechanism for pruning irrelevant features and choosing a subset of interesting features from structural signatures of underlying symbol set. The Bayesian network is deployed in a supervised learning scenario for recognizing query symbols. The method has been evaluated for robustness against degradations & deformations on pre-segmented 2D linear architectural & electronic symbols from GREC databases, and for its recognition abilities on symbols with context noise i.e. cropped symbols

Pattern Recognition

A wealth of advanced pattern recognition algorithms are emerging from the interdiscipline between technologies of effective visual features and the human-brain cognition process. Effective visual features are made possible through the rapid developments in appropriate sensor equipments, novel filter designs, and viable information processing architectures. While the understanding of human-brain cognition process broadens the way in which the computer can perform pattern recognition tasks. The present book is intended to collect representative researches around the globe focusing on low-level vision, filter design, features and image descriptors, data mining and analysis, and biologically inspired algorithms. The 27 chapters coved in this book disclose recent advances and new ideas in promoting the techniques, technology and applications of pattern recognition

The bifiltration of a relation and extended Dowker duality

We explain how homotopical information of two composeable relations can be organized in two simplicial categories that augment the relations row and column complexes. We show that both of these categories realize to weakly equivalent spaces, thereby extending Dowker's duality theorem. We also prove a functorial version of this result. Specializing the above construction a bifiltration of Dowker complexes that coherently incorporates the total weights of a relation's row and column complex into one single object is introduced. This construction is motivated by challenges in data analysis that necessitate the simultaneous study of a data matrix rows and columns. To illustrate the applicability of our constructions for solving those challenges we give an appropriate reconstruction result

Topological Deep Learning: Going Beyond Graph Data

Topological deep learning is a rapidly growing field that pertains to the development of deep learning models for data supported on topological domains such as simplicial complexes, cell complexes, and hypergraphs, which generalize many domains encountered in scientific computations. In this paper, we present a unifying deep learning framework built upon a richer data structure that includes widely adopted topological domains. Specifically, we first introduce combinatorial complexes, a novel type of topological domain. Combinatorial complexes can be seen as generalizations of graphs that maintain certain desirable properties. Similar to hypergraphs, combinatorial complexes impose no constraints on the set of relations. In addition, combinatorial complexes permit the construction of hierarchical higher-order relations, analogous to those found in simplicial and cell complexes. Thus, combinatorial complexes generalize and combine useful traits of both hypergraphs and cell complexes, which have emerged as two promising abstractions that facilitate the generalization of graph neural networks to topological spaces. Second, building upon combinatorial complexes and their rich combinatorial and algebraic structure, we develop a general class of message-passing combinatorial complex neural networks (CCNNs), focusing primarily on attention-based CCNNs. We characterize permutation and orientation equivariances of CCNNs, and discuss pooling and unpooling operations within CCNNs in detail. Third, we evaluate the performance of CCNNs on tasks related to mesh shape analysis and graph learning. Our experiments demonstrate that CCNNs have competitive performance as compared to state-of-the-art deep learning models specifically tailored to the same tasks. Our findings demonstrate the advantages of incorporating higher-order relations into deep learning models in different applications