arXiv subjects

Moritz Schönherr

Publications and source records attributed to Moritz Schönherr.

7 recordsLinked to original sources

arXiv2026

Neuro-Relational Programs: Unifying Queries and Neural Computation over Structured Data

The conventional approach to deep learning over relational databases applies neural models, such as Graph Neural Networks (GNNs), to a graph representation of the database. Recent approaches instead operate on databases directly, associating tuples with embeddings and extending query mechanisms to jointly process embeddings and relational content. Inspired by these developments, we introduce Neuro-Relational Programs (NRPs), a declarative query language for relational databases whose facts carry numeric vector embeddings. NRPs extend Datalog-style rules with operations that combine, aggregate, and transform embeddings, thereby interleaving relational reasoning and learnable neural components within a single formalism. This yields a general approach to neural computation over relational data: an NRP can be read both as a query plan with trainable components and as a neural architecture with relational structure built in. Natural syntactic fragments of NRPs recover existing architectures and query formalisms. Zero-ary NRPs correspond to non-adaptive query algorithms; monadic NRPs generalize GNN-style message passing and precisely capture Deep Homomorphism Networks, a connection that we extend to frontier-guarded NRPs over databases with row-ids. We characterize the expressive power of unrestricted NRPs with ReLU-FFN transformations by FOCQ, an extension of first-order logic with counting interpreted over real-weighted structures, yielding a precise connection with uniform TC$^0$ over ordered databases. Together, these results establish NRPs as a broad declarative framework for querying and neural computation over relational data.

cs.DB↗

arXiv2026

Expressive Power of Deep Homomorphism Networks over Relational Databases

The expressive limitations of message-passing Graph Neural Networks (GNNs) have motivated a wide range of more powerful graph learning architectures. We advocate Deep Homomorphism Networks (DHNs) as a model particularly well-suited for learning over relational databases, due to their close connection to important fragments of SQL such as conjunctive queries. We study the precise expressive power of DHNs by relating them to various natural fragments and extensions of first-order logic (FO). For DHNs with max, sum, and mean aggregations, we establish connections to the unary negation fragment (UNFO) and to the extensions of UNFO with counting quantifiers and with ratio quantifiers. We further relate sum-aggregation DHNs to the unary quantifier alternation fragment of FO and to an extension of FO with expressive counting. Through the classical correspondence between FO and SQL, these results also illuminate the relation between DHNs and SQL. They also enable us to study the decidability of two fundamental static analysis problems for DHNs, the emptiness problem and the subsumption problem. Finally, we confirm through experiments that the established differences in expressive power are reflected in the performance on suitable prediction tasks.

cs.DB↗

arXiv2026

Density Measures

The paper treats density measures as typical examples of finitely additive measures in $\mathbb{R}^n$. We study their structure and derive basic properties. In addition, estimates for related integrals are provided. The results are applied to the precise representative of general integrable functions and then they are specialized to functions of bounded variation. Moreover, a new representation of the generalized gradients in the sense of Clarke is given for the finite dimensional case.

math.AP↗

arXiv2025

Logical Characterizations of GNNs with Mean Aggregation

We study the expressive power of graph neural networks (GNNs) with mean as the aggregation function, with the following results. In the non-uniform setting, such GNNs have exactly the same expressive power as ratio modal logic, which has modal operators expressing that at least a certain ratio of the successors of a vertex satisfies a specified property. In the uniform setting, the expressive power relative to MSO is exactly that of modal logic, and thus identical to the (absolute) expressive power of GNNs with max aggregation. The proof, however, depends on constructions that are not satisfactory from a practical perspective. This leads us to making the natural assumptions that combination functions are continuous and classification functions are thresholds. The resulting class of GNNs with mean aggregation turns out to be much less expressive: relative to MSO and in the uniform setting, it has the same expressive power as alternation-free modal logic. This is in contrast to the expressive power of GNNs with max and sum aggregation, which is not affected by these assumptions.

cs.AI↗

arXiv2022

A theory of traces and the divergence theorem

We introduce a general approach to traces that we consider as linear continuous functionals on some function space where we focus on some special choices for that space. This leads to an integral calculus for the computation of the precise representative of an integrable function and of the trace of a Sobolev or BV function. For integrable vector fields with distributional divergence being a measure, we also obtain Gauss-Green formulas on arbitrary Borel sets. It turns out that a second boundary integral is needed in general. The advantage of the integral calculus is that neither a normal field nor a trace function on the boundary is needed. The Gauss-Green formulas are also available for Sobolev and BV functions. Finally, for any open set the existence of a weak solution of a boundary value problem is shown as application of the trace theory.

math.AP↗

arXiv2017

Pure Measures, Density Measures and the Dual of L-infinity

Measures play an important role in the characterisation of various function spaces. In this paper, the structure of density measures will be investigated. These are elements of the dual of the space of essentially bounded func- tions. The main results presented here are a more precise representation of the dual of the space of essentially bounded functions, leading to the notion of pure measures, and the definition and analysis of density measures which constitute a large class of such measures. It is shown that density measures have applications in the context of traces. In particular, new and meaningful examples of pure measures are given on Rn, in contrast to common examples in the literature, which are usually constructed on N.

math.MG↗

arXiv2017

A General Theorem of Gauß Using Pure Measures

This paper shows that finitely additive measures occur naturally in very general Divergence Theorems. The main results are two such theorems. The first proves the existence of pure normal measures for sets of finite perime- ter, which yield a Gauß formula for essentially bounded vector fields having divergence measure. The second extends a result of Silhavy [19] on normal traces. In particular, it is shown that a Gauß Theorem for unbounded vector fields having divergence measure necessitates the use of pure measures acting on the gradient of the scalar field. All of these measures are shown to have their core on the boundary of the domain of integration.

math.AP↗