A Categorical Semantics of Fuzzy Concepts in Conceptual Spaces

• URL: http://arxiv.org/abs/2110.05985v1
• Date: Tue, 12 Oct 2021 13:22:27 GMT
• Title: A Categorical Semantics of Fuzzy Concepts in Conceptual Spaces
• Authors: Sean Tull
• Abstract summary: We define a symmetric monocave category fuzzy concepts and fuzzy conceptual reasoning within G"ardenfors' spaces. We propose log-concave functions, showing that fuzzy concepts are the most general choice satisfying a criterion.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We define a symmetric monoidal category modelling fuzzy concepts and fuzzy conceptual reasoning within G\"ardenfors' framework of conceptual (convex) spaces. We propose log-concave functions as models of fuzzy concepts, showing that these are the most general choice satisfying a criterion due to G\"ardenfors and which are well-behaved compositionally. We then generalise these to define the category of log-concave probabilistic channels between convex spaces, which allows one to model fuzzy reasoning with noisy inputs, and provides a novel example of a Markov category.

Related papers

• FaCT: Faithful Concept Traces for Explaining Neural Network Decisions [56.796533084868884]
  Deep networks have shown remarkable performance across a wide range of tasks, yet getting a global concept-level understanding of how they function remains a key challenge.<n>We put emphasis on the faithfulness of concept-based explanations and propose a new model with model-inherent mechanistic concept-explanations.<n>Our concepts are shared across classes and, from any layer, their contribution to the logit and their input-visualization can be faithfully traced.
  arXiv  Detail & Related papers  (2025-10-29T13:35:46Z)
• The Origins of Representation Manifolds in Large Language Models [52.68554895844062]
  We show that cosine similarity in representation space may encode the intrinsic geometry of a feature through shortest, on-manifold paths.<n>The critical assumptions and predictions of the theory are validated on text embeddings and token activations of large language models.
  arXiv  Detail & Related papers  (2025-05-23T13:31:22Z)
• Analogical Reasoning Within a Conceptual Hyperspace [1.1049608786515839]
  We propose an approach to analogical inference that marries the neuro-symbolic computational power of hyperdimensional computing with Conceptual Spaces Theory. We present preliminary proof-of-concept experimental results within a toy domain and describe how it can perform category-based and property-based analogical reasoning.
  arXiv  Detail & Related papers  (2024-11-13T15:20:14Z)
• Non-monotonic Extensions to Formal Concept Analysis via Object Preferences [3.4873708563238277]
  We introduce a non-monotonic conditional between sets of attributes, which assumes a preference over the set of objects. We show that this conditional gives rise to a consequence relation consistent with the postulates for non-monotonicty. This notion of typical concepts is a further introduction of KLM-style typicality into Formal Concept Analysis.
  arXiv  Detail & Related papers  (2024-10-05T15:01:00Z)
• A Canonicalization Perspective on Invariant and Equivariant Learning [54.44572887716977]
  We introduce a canonicalization perspective that provides an essential and complete view of the design of frames. We show that there exists an inherent connection between frames and canonical forms. We design novel frames for eigenvectors that are strictly superior to existing methods.
  arXiv  Detail & Related papers  (2024-05-28T17:22:15Z)
• An Axiomatic Approach to Model-Agnostic Concept Explanations [67.84000759813435]
  We propose an approach to concept explanations that satisfy three natural axioms: linearity, recursivity, and similarity. We then establish connections with previous concept explanation methods, offering insight into their varying semantic meanings.
  arXiv  Detail & Related papers  (2024-01-12T20:53:35Z)
• Involutive Markov categories and the quantum de Finetti theorem [1.90365714903665]
  We show that involutive Markov categories are simpler to work with than Parzygnat's quantum categories. We prove a quantum de Finetti theorem for both the minimal and the maximal C*-tensor norms.
  arXiv  Detail & Related papers  (2023-12-15T10:26:19Z)
• A Geometric Notion of Causal Probing [85.49839090913515]
  The linear subspace hypothesis states that, in a language model's representation space, all information about a concept such as verbal number is encoded in a linear subspace. We give a set of intrinsic criteria which characterize an ideal linear concept subspace. We find that, for at least one concept across two languages models, the concept subspace can be used to manipulate the concept value of the generated word with precision.
  arXiv  Detail & Related papers  (2023-07-27T17:57:57Z)
• Concept Activation Regions: A Generalized Framework For Concept-Based Explanations [95.94432031144716]
  Existing methods assume that the examples illustrating a concept are mapped in a fixed direction of the deep neural network's latent space. In this work, we propose allowing concept examples to be scattered across different clusters in the DNN's latent space. This concept activation region (CAR) formalism yields global concept-based explanations and local concept-based feature importance.
  arXiv  Detail & Related papers  (2022-09-22T17:59:03Z)
• Concept Gradient: Concept-based Interpretation Without Linear Assumption [77.96338722483226]
  Concept Activation Vector (CAV) relies on learning a linear relation between some latent representation of a given model and concepts. We proposed Concept Gradient (CG), extending concept-based interpretation beyond linear concept functions. We demonstrated CG outperforms CAV in both toy examples and real world datasets.
  arXiv  Detail & Related papers  (2022-08-31T17:06:46Z)
• Automatic Concept Extraction for Concept Bottleneck-based Video Classification [58.11884357803544]
  We present an automatic Concept Discovery and Extraction module that rigorously composes a necessary and sufficient set of concept abstractions for concept-based video classification. Our method elicits inherent complex concept abstractions in natural language to generalize concept-bottleneck methods to complex tasks.
  arXiv  Detail & Related papers  (2022-06-21T06:22:35Z)
• The Conceptual VAE [7.15767183672057]
  We present a new model of concepts, based on the framework of variational autoencoders. The model is inspired by, and closely related to, the Beta-VAE model of concepts. We show how the model can be used as a concept classifier, and how it can be adapted to learn from fewer labels per instance.
  arXiv  Detail & Related papers  (2022-03-21T17:27:28Z)
• Human-Centered Concept Explanations for Neural Networks [47.71169918421306]
  We introduce concept explanations including the class of Concept Activation Vectors (CAV) We then discuss approaches to automatically extract concepts, and approaches to address some of their caveats. Finally, we discuss some case studies that showcase the utility of such concept-based explanations in synthetic settings and real world applications.
  arXiv  Detail & Related papers  (2022-02-25T01:27:31Z)
• A conditional, a fuzzy and a probabilistic interpretation of self-organising maps [0.0]
  We establish a link between preferential semantics for description logics and self-organising maps. We show that a concept-wise multipreference semantics can be used to provide a logical interpretation of SOMs.
  arXiv  Detail & Related papers  (2021-03-11T18:31:00Z)
• Reasoning about Typicality and Probabilities in Preferential Description Logics [0.15749416770494704]
  We describe preferential Description Logics of typicality, a nonmonotonic extension of standard Description Logics by means of a typicality operator T. This extension is based on a minimal model semantics corresponding to a notion of rational closure. We consider other strengthening of the rational closure semantics and construction to avoid the so-called blocking of property inheritance problem.
  arXiv  Detail & Related papers  (2020-04-20T14:50:31Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.