arXiv subjects
Publications and source records attributed to Alexander Kurz.
Drawing on well-known results from the theory of canonical extensions and the theory of categories enriched over a quantale, we define canonical extensions of quantale-enriched categories and establish their basic properties.
Deep Neural Networks have shown promising classification performance when predicting certain biomarkers from Whole Slide Images in digital pathology. However, the calibration of the networks' output probabilities is often not evaluated. Communicating uncertainty by providing reliable confidence scores is of high relevance in the medical context. In this work, we compare three neural network architectures that combine feature representations on patch-level to a slide-level prediction with respect to their classification performance and evaluate their calibration. As slide-level classification task, we choose the prediction of Microsatellite Instability from Colorectal Cancer tissue sections. We observe that Transformers lead to good results in terms of classification performance and calibration. When evaluating the classification performance on a separate dataset, we observe that Transformers generalize best. The investigation of reliability diagrams provides additional insights to the Expected Calibration Error metric and we observe that especially Transformers push the output probabilities to extreme values, which results in overconfident predictions.
We study varieties generated by semi-primal lattice-expansions by means of category theory. We provide a new proof of the Keimel-Werner topological duality for such varieties and, using similar methods, establish its discrete version. We describe multiple adjunctions between the variety of Boolean algebras and the variety generated by a semi-primal lattice-expansion, both on the topological side and explicitly algebraic. In particular, we show that the Boolean skeleton functor has two adjoints, both defined by taking certain Boolean powers, and we identify properties of these adjunctions which fully characterize semi-primality of an algebra. Lastly, we give a new characterization of canonical extensions of algebras in semi-primal varieties in terms of their Boolean skeletons.
We study many-valued coalgebraic logics with semi-primal algebras of truth-degrees. We provide a systematic way to lift endofunctors defined on the variety of Boolean algebras to endofunctors on the variety generated by a semi-primal algebra. We show that this can be extended to a technique to lift classical coalgebraic logics to many-valued ones, and that (one-step) completeness and expressivity are preserved under this lifting. For specific classes of endofunctors, we also describe how to obtain an axiomatization of the lifted many-valued logic directly from an axiomatization of the original classical one. In particular, we apply all of these techniques to classical modal logic.
In the past years, deep learning has seen an increase in usage in the domain of histopathological applications. However, while these approaches have shown great potential, in high-risk environments deep learning models need to be able to judge their uncertainty and be able to reject inputs when there is a significant chance of misclassification. In this work, we conduct a rigorous evaluation of the most commonly used uncertainty and robustness methods for the classification of Whole Slide Images, with a focus on the task of selective classification, where the model should reject the classification in situations in which it is uncertain. We conduct our experiments on tile-level under the aspects of domain shift and label noise, as well as on slide-level. In our experiments, we compare Deep Ensembles, Monte-Carlo Dropout, Stochastic Variational Inference, Test-Time Data Augmentation as well as ensembles of the latter approaches. We observe that ensembles of methods generally lead to better uncertainty estimates as well as an increased robustness towards domain shifts and label noise, while contrary to results from classical computer vision benchmarks no systematic gain of the other methods can be shown. Across methods, a rejection of the most uncertain samples reliably leads to a significant increase in classification accuracy on both in-distribution as well as out-of-distribution data. Furthermore, we conduct experiments comparing these methods under varying conditions of label noise. Lastly, we publish our code framework to facilitate further research on uncertainty estimation on histopathological data.
We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This leads us to define nominal PROPs and nominal monoidal theories. We show that the categories of ordinary PROPs and nominal PROPs are equivalent. This equivalence is then extended to symmetric monoidal theories and nominal monoidal theories, which allows us to transfer completeness results between ordinary and nominal calculi for string diagrams.
The familiar adjunction between ordered sets and completely distributive lattices can be extended to generalised metric spaces, that is, categories enriched over a quantale (a lattice of "truth values"), via an appropriate distributive law between the "down-set" monad and the "up-set" monad on the category of quantale-enriched categories. If the underlying lattice of the quantale is completely distributive, and if powers distribute over non-empty joins in the quantale, then this distributive law can be concretely formulated in terms of operations, equations and choice functions, similar to the familiar distributive law of lattices.
Accurate and reliable localization is a fundamental requirement for autonomous vehicles to use map information in higher-level tasks such as navigation or planning. In this paper, we present a novel approach to vehicle localization in dense semantic maps, including vectorized high-definition maps or 3D meshes, using semantic segmentation from a monocular camera. We formulate the localization task as a direct image alignment problem on semantic images, which allows our approach to robustly track the vehicle pose in semantically labeled maps by aligning virtual camera views rendered from the map to sequences of semantically segmented camera images. In contrast to existing visual localization approaches, the system does not require additional keypoint features, handcrafted localization landmark extractors or expensive LiDAR sensors. We demonstrate the wide applicability of our method on a diverse set of semantic mesh maps generated from stereo or LiDAR as well as manually annotated HD maps and show that it achieves reliable and accurate localization in real-time.
We show how Stone duality can be extended from maps to relations. This is achieved by working order enriched and defining a relation from A to B as both an order-preserving function from the opposite of A times B to the 2-element chain and as a subobject of A times B. We show that dual adjunctions and equivalences between regular categories, taken in a suitably order enriched sense, extend to (framed bi)categories of relations.
We introduce a proper display calculus for first-order logic, of which we prove soundness, completeness, conservativity, subformula property and cut elimination via a Belnap-style metatheorem. All inference rules are closed under uniform substitution and are without side conditions.
We define a new logic-induced notion of bisimulation (called $ρ$-bisimulation) for coalgebraic modal logics given by a logical connection, and investigate its properties. We show that it is structural in the sense that it is defined only in terms of the coalgebra structure and the one-step modal semantics and, moreover, can be characterised by a form of relation lifting. Furthermore we compare $ρ$-bisimulations to several well-known equivalence notions, and we prove that the collection of bisimulations between two models often forms a complete lattice. The main technical result is a Hennessy-Milner type theorem which states that, under certain conditions, logical equivalence implies $ρ$-bisimilarity. In particular, the latter does \emph{not} rely on a duality between functors $\mathsf{T}$ (the type of the coalgebras) and $\mathsf{L}$ (which gives the logic), nor on properties of the logical connection $ρ$.
Choreographic approaches to message-passing applications can be regarded as an instance of the model-driven development principles. Choreographies specify interactions among distributed participants coordinating among themselves with message-passing at two levels of abstractions. A global view of the application is specified with a model that abstracts away from asynchrony while a local view of the application specifies the communication pattern of each participant. Noteworthy, the latter view can typically be algorithmically obtained by projection of the global view. A crucial element of this approach is to verify the so-called well-formed conditions on global views so that its projections realise a sound communication protocol. We introduce a novel local model, group interface automata, to represent the local view of choreographies and propose a new method to verify the well-formedness of global choreographies. We rely on a recently proposed semantics of global views formalised in terms of pomsets.
We introduce nominal string diagrams as, string diagrams internal in the category of nominal sets. This requires us to take nominal sets as a monoidal category, not with the cartesian product, but with the separated product. To this end, we develop the beginnings of a theory of monoidal categories internal in a symmetric monoidal category. As an instance, we obtain a notion of a nominal PROP as a PROP internal in nominal sets. A 2-dimensional calculus of simultaneous substitutions is an application.
For a commutative quantale $\mathcal{V}$, the category $\mathcal{V}-cat$ can be perceived as a category of generalised metric spaces and non-expanding maps. We show that any type constructor $T$ (formalised as an endofunctor on sets) can be extended in a canonical way to a type constructor $T_{\mathcal{V}}$ on $\mathcal{V}-cat$. The proof yields methods of explicitly calculating the extension in concrete examples, which cover well-known notions such as the Pompeiu-Hausdorff metric as well as new ones. Conceptually, this allows us to to solve the same recursive domain equation $X\cong TX$ in different categories (such as sets and metric spaces) and we study how their solutions (that is, the final coalgebras) are related via change of base. Mathematically, the heart of the matter is to show that, for any commutative quantale $\mathcal{V}$, the `discrete' functor $D:\mathsf{Set}\to \mathcal{V}-cat$ from sets to categories enriched over $\mathcal{V}$ is $\mathcal{V}-cat$-dense and has a density presentation that allows us to compute left-Kan extensions along $D$.
We present positive coalgebraic logic in full generality, and show how to obtain a positive coalgebraic logic from a boolean one. On the model side this involves canonically computing an endofunctor $T': Pos\to Pos$ from an endofunctor $T: Set\to Set$, in a procedure previously defined by the second author et alii called posetification. On the syntax side, it involves canonically computing a syntax-building functor $L': DL\to DL$ from a syntax-building functor $L: BA\to BA$, in a dual procedure which we call positivication. These operations are interesting in their own right and we explicitly compute posetifications and positivications in the case of several modal logics. We show how the semantics of a boolean coalgebraic logic can be canonically lifted to define a semantics for its positive fragment, and that weak completeness transfers from the boolean case to the positive case.
The present paper provides an analysis of the existing proof systems for dynamic epistemic logic from the viewpoint of proof-theoretic semantics. Dynamic epistemic logic is one of the best known members of a family of logical systems which have been successfully applied to diverse scientific disciplines, but the proof theoretic treatment of which presents many difficulties. After an illustration of the proof-theoretic semantic principles most relevant to the treatment of logical connectives, we turn to illustrating the main features of display calculi, a proof-theoretic paradigm which has been successfully employed to give a proof-theoretic semantic account of modal and substructural logics. Then, we review some of the most significant proposals of proof systems for dynamic epistemic logics, and we critically reflect on them in the light of the previously introduced proof-theoretic semantic principles. The contributions of the present paper include a generalisation of Belnap's cut elimination metatheorem for display calculi, and a revised version of the display-style calculus D.EAK. We verify that the revised version satisfies the previously mentioned proof-theoretic semantic principles, and show that it enjoys cut elimination as a consequence of the generalised metatheorem.
In the present paper, we introduce a multi-type display calculus for dynamic epistemic logic, which we refer to as Dynamic Calculus. The display-approach is suitable to modularly chart the space of dynamic epistemic logics on weaker-than-classical propositional base. The presence of types endows the language of the Dynamic Calculus with additional expressivity, allows for a smooth proof-theoretic treatment, and paves the way towards a general methodology for the design of proof systems for the generality of dynamic logics, and certainly beyond dynamic epistemic logic. We prove that the Dynamic Calculus adequately captures Baltag-Moss-Solecki's dynamic epistemic logic, and enjoys Belnap-style cut elimination.
We introduce a multi-type display calculus for Propositional Dynamic Logic (PDL). This calculus is complete w.r.t. PDL, and enjoys Belnap-style cut-elimination and subformula property.