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Towards Grounding Conceptual Spaces in Neural Representations

Lucas Bechberger and Kai-Uwe Kühnberger https://www.lucas-bechberger.de

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The Different Layers of Representation

Symbolic Layer Subsymbolic Layer [0.42; -1.337] ∀ x:apple( x)⇒red(x) Formal Logics Sensor Values, Machine Learning

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Conceptual Layer Geometric Representation

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Dimensions & Domains

• Quality dimensions
• Different ways stimuli are judged to be similar or different
• Interpretable by a human
• E.g., temperature, weight, brightness, pitch
• Domain
• Set of dimensions that inherently belong together
• Color: hue, saturation, and brightness
• Distance in this space is inversely related to similarity
• Geometric betweenness represents semantic betweenness
• Concepts: regions in this space

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Example: The Color Domain

https://en.wikipedia.org/wiki/HSL_and_HSV#/media/File:HSL_color_solid_dblcone_chroma_gray.png

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Towards A Thorough Implementation

Symbolic Layer Subsymbolic Layer Conceptual Layer

Manually define regions Manually define dimensions

representation learning

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InfoGAN – Architecture

• Information Maximizing Generative Adversarial Networks
• X. Chen et al., “InfoGAN: Interpretable Representation Learning by Information Maximizing

Generative Adversarial Nets”, Advances in Neural Information Processing Systems, 2016

D

z c

G

G(z) x c ?

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InfoGAN – MNIST Results

• X. Chen et al., “InfoGAN: Interpretable Representation Learning by Information Maximizing

Generative Adversarial Nets”, Advances in Neural Information Processing Systems, 2016

• Three latent variables
• Categorical (10 classes)
• Continuous (uniform)
• Continuous (uniform)

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Domains and InfoGAN

Domains in CS framework

• Interpretable dimensions
• Distance-based notion of

semantic similarity

• Geometric betweenness

represents semantic betweenness Latent Space of InfoGAN

• Tends to be the case
• Smoothness assumption
• Interpolations in latent

space describe a meaningful morph

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Our Proposal

• Use InfoGAN to learn dimensions of “difficult” domains
• For starters: shape domain
• Hyperparameters
• Number of latent variables

→ as few as possible, as much as necessary

• Type of latent variables

→ continuous (i.e., uniform or Gaussian)

• Make sure that we learn new dimensions
• Pre-select training data (if possible)
• Additional term in loss function: correlation to other dimensions

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First Preliminary Results

• Data set of right-angled triangles, rectangles, and ellipses
• 2 continuous varibles (uniform distribution), 500 epochs

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Towards A Thorough Implementation

Symbolic Layer Subsymbolic Layer Conceptual Layer

Manually define regions Manually define dimensions

clustering representation learning

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Overall Envisioned System

apple banana color domain shape domain

ANN HSB

Symbolic layer Conceptual layer Subsymbolic layer

Thank you for your attention!

Questions? Comments? Discussions? https://www.lucas-bechberger.de