Structured versus Decorated Cospans

Baez, John C.; Courser, Kenny; Vasilakopoulou, Christina

doi:10.48550/arxiv.2101.09363

Cited by 2 publications

(5 citation statements)

References 34 publications

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“…What we want is to attach to each morphism some sort of parameter. This corresponds to the concept of a decoration, see [Fon15,BCV21]. In those manuscripts, the authors use the decorations to model electrical circuits and Petri Nets.…”

Section: Decorated Algorithmsmentioning

confidence: 99%

Exploring explainable AI: category theory insights into machine learning algorithms

Fabregat-Hernández,

Palanca,

Botti

2023

Mach. Learn.: Sci. Technol.

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Explainable artificial intelligence (XAI) is a growing field that aims to increase the transparency and interpretability of machine learning (ML) models. The aim of this work is to use the categorical properties of learning algorithms in conjunction with the categorical perspective of the information in the datasets to give a framework for explainability. In order to achieve this, we are going to define the enriched categories, with decorated morphisms, L e a r n L e a r n , P a r a P a r a and M N e t M N e t of learners, parameterized functions, and neural networks over metric spaces respectively. The main idea is to encode information from the dataset via categorical methods, see how it propagates, and lastly, interpret the results thanks again to categorical (metric) information. This means that we can attach numerical (computable) information via enrichment to the structural information of the category. With this, we can translate theoretical information into parameters that are easily understandable. We will make use of different categories of enrichment to keep track of different kinds of information. That is, to see how differences in attributes of the data are modified by the algorithm to result in differences in the output to achieve better separation. In that way, the categorical framework gives us an algorithm to interpret what the learning algorithm is doing. Furthermore, since it is designed with generality in mind, it should be applicable in various different contexts. There are three main properties of category theory that help with the interpretability of ML models: formality, the existence of universal properties, and compositionality. The last property offers a way to combine smaller, simpler models that are easily understood to build larger ones. This is achieved by formally representing the structure of ML algorithms and information contained in the model. Finally, universal properties are a cornerstone of category theory. They help us characterize an object, not by its attributes, but by how it interacts with other objects. Thus, we can formally characterize an algorithm by how it interacts with the data. The main advantage of the framework is that it can unify under the same language different techniques used in XAI. Thus, using the same language and concepts we can describe a myriad of techniques and properties of ML algorithms, streamlining their explanation and making them easier to generalize and extrapolate.

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Section: Decorated Algorithmsmentioning

confidence: 99%

Exploring explainable AI: category theory insights into machine learning algorithms

Fabregat-Hernández,

Palanca,

Botti

2023

Mach. Learn.: Sci. Technol.

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“…Alternatively, we can compose stock-flow diagrams with any number of interfaces using an operad. Both approaches let us describe the differential equation semantics for open stock-flow diagrams following a paradigm already explored for open Petri nets with rates [5,4]. We describe that paradigm here.…”

Section: The Mathematics Of Stock-flow Diagramsmentioning

confidence: 99%

“…In fact, our semantics is more general than suggested above: we describe a map from open stock-flow diagrams to open dynamical systems. Our strategy for defining this semantics closely follows the strategy already used for open Petri nets with rates [3,5,4] and implemented for epidemiological models using AlgebraicJulia [18].…”

Section: Open Dynamical Systemsmentioning

confidence: 99%

“…While the mathematics described in Section 3 uses decorated cospans, we can also describe open stock-flow diagrams using structured cospans [3,4]. A structured cospan is a diagram of the form…”

Section: Composition Using Structured Cospansmentioning

confidence: 99%

“…The AlgebraicJulia ecosystem of software implements compositionality and functorial semantics in a thorough-going way [1]. Decorated and structured cospans are broad mathematical frameworks for turning "closed" system descriptions into "open" ones that can be composed along their boundaries [10,3,4]. One part of AlgebraicJulia, called Catlab [22], provides a generic interface for working with such cospans, among other categorical abstractions.…”

Section: Introductionmentioning

confidence: 99%

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Compositional Modeling with Stock and Flow Diagrams

Baez¹,

Li²,

Libkind³

et al. 2022

Preprint

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Stock and flow diagrams are widely used in epidemiology to model the dynamics of populations. Although tools already exist for building these diagrams and simulating the systems they describe, we have created a new package called StockFlow, part of the AlgebraicJulia ecosystem, which uses ideas from category theory to overcome notable limitations of existing software. Compositionality is provided by the theory of decorated cospans: stock and flow diagrams can composed to form larger ones in an intuitive way formalized by the operad of undirected wiring diagrams. Our approach also cleanly separates the syntax of stock and flow diagrams from the semantics they can be assigned. We consider semantics in ordinary differential equations, although others are possible. As an example, we explain code in StockFlow that implements a simplified version of a COVID-19 model used in Canada.

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Structured versus Decorated Cospans

Cited by 2 publications

References 34 publications

Exploring explainable AI: category theory insights into machine learning algorithms

Exploring explainable AI: category theory insights into machine learning algorithms

Compositional Modeling with Stock and Flow Diagrams

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