Metric Geometry

Burago, Yuri; Shoenthal, David

doi:10.1007/978-3-662-08966-8_1

Cited by 6 publications

(4 citation statements)

References 1 publication

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“…is the computational cost of the random sampling of Sj [24], O(D) is the computational cost for the computation of the distance d [25] and O(s j ) is the computational cost of the research of the minimum of the vector of distances relative to the j−th random sampling ( Sj ) [26]. So, considering all m classes, the computational cost of the counterfactuals search, O (SCF), can be estimated in…”

Section: A Numerical Solutionmentioning

confidence: 99%

Multiclass Counterfactual Explanations Using Support Vector Data Description

Carlevaro,

Lenatti,

Paglialonga

et al. 2024

IEEE Trans. Artif. Intell.

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Explainability has become crucial in Artificial Intelligence studies and, as the complexity of the model increases, so does the complexity of its explanation. However, the higher the complexity of the problem, the higher the amount of information it may provide, and this information can be exploited to generate a more precise explanation of how the model works. One of the most valuable ways to recover such input-output relation is to extract counterfactual explanations that allow us to find minimal changes from an observation to another one belonging to a different class. In this article, we propose a novel methodology to extract multiple counterfactual explanations (MUCH, MUlti Counterfactual via Halton sampling) from an original Multi-Class Support Vector Data Description algorithm. To evaluate the performance of the proposed method, we extracted a set of counterfactual explanations from three state-of-the-art datasets achieving satisfactory results that pave the way to a range of real-world applications.Impact Statement-When a system is analyzed by Artificial Intelligence, the inherent models are posed to the attention of domain experts, thus delegating further possible actions. Counterfactual explanations, on the other hand, directly suggest actuation on the system. Counterfactual control still remains under experts' supervision, but the system improves its level of autonomy. The long-term goal is to make the AI model aware of how to affect the environment properly (both in terms of performance and safety). Examples may include: manoeuvering of autonomous cars, clinical diagnosis, and finance. The proposed approach generalizes counterfactuals intelligibility and control to the multi-class case. The validation over practical scenarios (e.g., the FIFA dataset) corroborates both control precision and quality of counterfactual explanations, thus increasing the readiness level of the approach.

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Section: A Numerical Solutionmentioning

confidence: 99%

Multiclass Counterfactual Explanations Using Support Vector Data Description

Carlevaro,

Lenatti,

Paglialonga

et al. 2024

IEEE Trans. Artif. Intell.

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“…We refer to [17] and [18] for further details on metric geometry, and on geodesic spaces in particular. From now on we will assume that X is a compact path-connected space, endowed with intrinsic metric; in other words, X is a compact geodesic space.…”

Section: Intrinsic Metric and Geodesic Spacesmentioning

confidence: 99%

The game of Cops and Robber on geodesic spaces

Mohar¹

2022

Preprint

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The game of Cops and Robber is traditionally played on a finite graph. The purpose of this paper is to introduce and analyse the game that is played on an arbitrary geodesic space (a compact, path-connected space endowed with intrinsic metric). It is shown that the game played on metric graphs is essentially the same as the discrete game played on abstract graphs and that for every compact geodesic surface there is an integer c such that c cops can win the game against one robber, and c only depends on the genus g of the surface. It is shown that c = 3 for orientable surfaces of genus 0 or 1 and nonorientable surfaces of crosscap number 1 or 2 (with any number of boundary components) and that c = O(g) and that c = Ω( √ g) when the genus g is larger. The main motivation for discussing this game is to view the cop number (the minimum number of cops needed to catch the robber) as a new geometric invariant describing how complex is the geodesic space.

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“…To prove (8), we just argue that if the robber, when using strategy s, is able to keep distance v from the cops for any cops' strategy q, he is also able to keep the same distance by using his strategy s ′ . Suppose not.…”

Section: Choice Of Agility Functionsmentioning

confidence: 99%

“…In this article we discuss the game of cops and robbers on arbitrary geodesic spaces (see Section 2 or [7,8] for definitions). We come up with a version of the game that is somewhat different from the game version in [5], but preserves all the beauty and power of discrete games played on graphs.…”

Section: Introductionmentioning

confidence: 99%

Min-max theorem for the game of Cops and Robber on geodesic spaces

Mohar¹

2021

Preprint

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The game of Cops and Robber is traditionally played on a finite graph. The purpose of this note is to introduce and analyze the game that is played on an arbitrary geodesic space. The game is defined in such a way that it preserves the beauty and power of discrete games played on graphs and also keeps the specialties of the pursuit-evasion games played on polyhedral complexes. It is shown that the game can be approximated by finite games of discrete type and as a consequence a min-max theorem is obtained.

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Metric Geometry

Cited by 6 publications

References 1 publication

Multiclass Counterfactual Explanations Using Support Vector Data Description

Multiclass Counterfactual Explanations Using Support Vector Data Description

The game of Cops and Robber on geodesic spaces

Min-max theorem for the game of Cops and Robber on geodesic spaces

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