43 research outputs found
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Mathematical Foundations for a Compositional Account of the Bayesian Brain
This dissertation reports some first steps towards a compositional account of
active inference and the Bayesian brain. Specifically, we use the tools of
contemporary applied category theory to supply functorial semantics for
approximate inference. To do so, we define on the `syntactic' side the new
notion of Bayesian lens and show that Bayesian updating composes according to
the compositional lens pattern. Using Bayesian lenses, and inspired by
compositional game theory, we define fibrations of statistical games and
classify various problems of statistical inference as corresponding sections:
the chain rule of the relative entropy is formalized as a strict section, while
maximum likelihood estimation and the free energy give lax sections. In the
process, we introduce a new notion of `copy-composition'.
On the `semantic' side, we present a new formalization of general open
dynamical systems (particularly: deterministic, stochastic, and random; and
discrete- and continuous-time) as certain coalgebras of polynomial functors,
which we show collect into monoidal opindexed categories (or, alternatively,
into algebras for multicategories of generalized polynomial functors). We use
these opindexed categories to define monoidal bicategories of cilia: dynamical
systems which control lenses, and which supply the target for our functorial
semantics. Accordingly, we construct functors which explain the bidirectional
compositional structure of predictive coding neural circuits under the free
energy principle, thereby giving a formal mathematical underpinning to the
bidirectionality observed in the cortex. Along the way, we explain how to
compose rate-coded neural circuits using an algebra for a multicategory of
linear circuit diagrams, showing subsequently that this is subsumed by lenses
and polynomial functors.Comment: DPhil thesis; as submitted. Main change from v1: improved treatment
of statistical games. A number of errors also fixed, and some presentation
improved. Comments most welcom
Bifurcation analysis of the Topp model
In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao
Classification and homological invariants of compact quantum groups of combinatorial type
Compact quantum groups can be found by solving certain combinatorics problems, as first shown by Banica and Speicher. Any system of partitions of finite sets which is closed under reflection and two kinds of concatenation gives rise to a quantum subgroup of the free orthogonal quantum group. Later Freslon, Tarrago and Weber extended this construction to colored partitions. Only recently, Mančinska and Roberson generalized this from finite sets to finite graphs. The present thesis contributes to the classification programs for quantum groups induced by two-colored partitions in Chapter 1 and those induced by uncolored graphs in Chapter 2. While these constructions produce numerous quantum groups, little is known about which of those are actually new and not isomorphic to others. In an effort to elucidate this, Chapter 3 shows that any such quantum group interpolating the unitary group and the free unitary quantum group can be written as a quotient of a wreath graph product of one of the two. Another way of making distinctions between such quantum groups of combinatorial type is to study quantum group invariants, such as cohomology. Chapter 4 computes the first order with trivial coefficients for the discrete duals of all of Tarrago and Weber’s quantum groups. For a handful of those Chapter 5 computes the L²-Betti numbers following Bichon, Kyed and Raum’s method. Chapter 6 proposes a common categorial framework covering all the aforementioned constructions for the first time.Durch das Lösen gewisser Kombinatorikrätsel lassen sich kompakte Quantengruppen finden, wie von Banica und Speicher gezeigt. Jede Sammlung von Partitionen endlicher Mengen, die unter Spiegelung und zwei Arten Konkatenierung abgeschlossen ist, ergibt eine Unterquantengruppe der freien orthogonalen Quantengruppe. Freslon, Tarrago und Weber erweiterten dies auf “gefärbte Partionen”. Erst kürzlich ersetzten Mancinska und Roberson die endlichen Mengen durch endliche Graphen. Die Dissertation trägt zu zwei entsprechenden Klassifikationsvorhaben bei: zweifarbige Partitionen in Kapitel 1, ungefärbte Graphen in Kapitel 2. Zwar ergeben sich viele Quantengruppen. Doch ist nur wenig darüber bekannt, welche davon tatsächlich neu sind. Um dieser Frage nachzugehen, wird in Kapitel 3 bewiesen, dass jede solche Quantengruppe zwischen der unitären Gruppe und der freien unitären Quantengruppe Quotient eines Kranzgraphprodukts einer dieser beiden ist. Eine andere Möglichkeit, solche Quantengruppen kombinatorischen Typs von einander zu unterscheiden bieten Invarianten wie Kohomologie. Von letzterer, mit trivialen Koeffizienten, wird in Kapitel 4 die erste Ordnung berechnet, und zwar für die diskreten Dualen aller von Tarrago und Webers Quantengruppen. Für eine handvoll davon werden in Kapitel 5 noch nach der Methode von Bichon, Kyed und Raum die L2-Betti-Zahlen bestimmt. Kapitel 6 enthält den Vorschlag eines gemeinsamen Rahmens für erstmals alle zuvor genannten Konstruktionen von Quantengruppen.Deutsche Forschungsgemeinschaft, SFB-TRR 195, IRTG scholarshi
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science
Topos and Stacks of Deep Neural Networks
Every known artificial deep neural network (DNN) corresponds to an object in
a canonical Grothendieck's topos; its learning dynamic corresponds to a flow of
morphisms in this topos. Invariance structures in the layers (like CNNs or
LSTMs) correspond to Giraud's stacks. This invariance is supposed to be
responsible of the generalization property, that is extrapolation from learning
data under constraints. The fibers represent pre-semantic categories (Culioli,
Thom), over which artificial languages are defined, with internal logics,
intuitionist, classical or linear (Girard). Semantic functioning of a network
is its ability to express theories in such a language for answering questions
in output about input data. Quantities and spaces of semantic information are
defined by analogy with the homological interpretation of Shannon's entropy
(P.Baudot and D.B. 2015). They generalize the measures found by Carnap and
Bar-Hillel (1952). Amazingly, the above semantical structures are classified by
geometric fibrant objects in a closed model category of Quillen, then they give
rise to homotopical invariants of DNNs and of their semantic functioning.
Intentional type theories (Martin-Loef) organize these objects and fibrations
between them. Information contents and exchanges are analyzed by Grothendieck's
derivators
The Foundation of Pattern Structures and their Applications
This thesis is divided into a theoretical part, aimed at developing statements around the newly introduced concept of pattern morphisms, and a practical part, where we present use cases of pattern structures.
A first insight of our work clarifies the facts on projections of pattern structures. We discovered that a projection of a pattern structure does not always lead again to a pattern structure.
A solution to this problem, and one of the most important points of this thesis, is the introduction of pattern morphisms in Chapter4. Pattern morphisms make it possible to describe relationships between pattern structures, and thus enable a deeper understanding of pattern structures in general. They also provide the means to describe projections of pattern structures that lead to pattern structures again. In Chapter5 and Chapter6, we looked at the impact of morphisms between pattern structures on concept lattices and on their representations and thus clarified the theoretical background of existing research in this field.
The application part reveals that random forests can be described through pattern structures, which constitutes another central achievement of our work.
In order to demonstrate the practical relevance of our findings, we included a use case where this finding is used to build an algorithm that solves a real world classification problem of red wines. The prediction accuracy of the random forest is better, but the high interpretability makes our algorithm valuable.
Another approach to the red wine classification problem is presented in Chapter 8, where, starting from an elementary pattern structure, we built a classification model that yielded good results
On the Existence of Right Adjoints for Surjective Mappings between Fuzzy Structures
Abstract. We continue our study of the characterization of existence of adjunctions (isotone Galois connections) whose codomain is insufficiently structured. This paper focuses on the fuzzy case in which we have a fuzzy ordering ρA on A and a surjective mapping f : A, ≈A → B, ≈B compatible with respect to the fuzzy equivalences ≈A and ≈B. Specifically, the problem is to find a fuzzy ordering ρB and a compatible mapping g : B, ≈B → A, ≈A such that the pair (f, g) is a fuzzy adjunction