Computing the Minimum Error Distance of Graphs in 0(n3) Time with Precedence Graph Grammars

AbstractIn the field of Model-Driven Engineering, Triple Graph Grammars (TGGs) play an important role as a rule-based means of implementing consistency management. From a declarative specification of a consistency relation, several operations including forward and backward transformations, (concurrent) synchronisation, and consistency checks can be automatically derived. For TGGs to be applicable in realistic application scenarios, expressiveness in terms of supported language features is very important. A TGG tool is schema compliant if it can take domain constraints, such as multiplicity constraints in a meta-model, into account when performing consistency management tasks. To guarantee schema compliance, most TGG tools allow application conditions to be attached as necessary to relevant rules. This strategy is problematic for at least two reasons: First, ensuring compliance to a sufficiently expressive schema for all previously mentioned derived operations is still an open challenge; to the best of our knowledge, all existing TGG tools only support a very restricted subset of application conditions. Second, it is conceptually demanding for the user to indirectly specify domain constraints as application conditions, especially because this has to be completely revisited every time the TGG or domain constraint is changed. While domain constraints can in theory be automatically transformed to obtain the required set of application conditions, this has only been successfully transferred to TGGs for a very limited subset of domain constraints. To address these limitations, this paper proposes a search-based strategy for achieving schema compliance. We show that all correctness and completeness properties, previously proven in a setting without domain constraints, still hold when schema compliance is to be additionally guaranteed. An implementation and experimental evaluation are provided to support our claim of practical applicability.

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Approximated least-squares solutions of a generalized Sylvester-transpose matrix equation via gradient-descent iterative algorithm

Advances in Difference Equations ◽

10.1186/s13662-021-03427-4 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Adisorn Kittisopaporn ◽

Pattrawut Chansangiam

Keyword(s):

Least Squares ◽

Iterative Algorithm ◽

Gradient Descent ◽

Main Idea ◽

Full Column Rank ◽

Minimum Error ◽

Matrix Coefficients ◽

Special Cases ◽

Efficiency And Effectiveness ◽

Associated Matrix

AbstractThis paper proposes an effective gradient-descent iterative algorithm for solving a generalized Sylvester-transpose equation with rectangular matrix coefficients. The algorithm is applicable for the equation and its interesting special cases when the associated matrix has full column-rank. The main idea of the algorithm is to have a minimum error at each iteration. The algorithm produces a sequence of approximated solutions converging to either the unique solution, or the unique least-squares solution when the problem has no solution. The convergence analysis points out that the algorithm converges fast for a small condition number of the associated matrix. Numerical examples demonstrate the efficiency and effectiveness of the algorithm compared to renowned and recent iterative methods.

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