The highly in uential framework of conceptual spaces provides a geometric way of representing knowledge. It aims at bridging the gap between symbolic and subsymbolic processing. Instances are represented by points in a high-dimensional space and concepts are represented by convex regions in this space. In this paper, we present our approach towards grounding the dimensions of a conceptual space in latent spaces learned by an InfoGAN from unlabeled data.
The cognitive framework of conceptual spaces [
The framework of conceptual spaces has been highly in uential in the last
15 years within cognitive science and cognitive linguistics [
Although this framework provides means for representing concepts, it does not consider the question of how these concepts can be learned from mostly unlabeled data. Moreover, the framework assumes that the dimensions spanning the conceptual space are already given a priori. In practical applications of the framework, they thus often need to be handcrafted by a human expert.
In this paper, we argue that by using neural networks, one can automatically
extract the dimensions of a conceptual space from unlabeled data. We propose
that latent spaces learned by an InfoGAN [
The remainder of this paper is structured as follows: Section 2 presents the framework of conceptual spaces and Section 3 introduces the InfoGAN framework. In Section 4, we present our idea of combining these two frameworks.
Section 5 gives an illustrative example and Section 6 concludes the paper. 2
A conceptual space [
The overall conceptual space is de ned as the product space of all dimensions. Distance within the overall conceptual space is measured by the Manhattan metric of the intra-domain distances. The similarity of two points in a conceptual space is inversely related to their distance { the closer two instances are in the conceptual space, the more similar they are considered to be.
The framework distinguishes properties like \red", \round", and \sweet" from full- eshed concepts like \apple" or \dog": Properties are represented as regions within individual domains (e.g., color, shape, taste), whereas full- eshed concepts span multiple domains. Reasoning within a conceptual space can be done based on geometric relationships (e.g., betweenness and similarity) and geometric operations (e.g., intersection or projection).
Recently, Balkenius & Gardenfors [
Within the research area of neural networks, there has been some substantial
work on learning compressed representations of a given feature space. Bengio et
al. [
The GAN framework (depicted in the left part of Figure 1) consists of two networks, the generator and the discriminator. The generator is fed with a lowdimensional vector of noise values. Its task is to create high-dimensional data vectors that have a similar distribution as real data vectors taken from an unlabeled training set. The discriminator receives a data vector that was either created by the generator or taken from the training set. Its task is to distinguish real inputs from generated inputs. Although the discriminator is trained on a classi cation task, the overall system works in an unsupervised way. The overall architecture can be interpreted as a two-player game: The generator tries to fool the discriminator by creating realistic inputs and the discriminator tries to avoid being fooled by the generator. When the GAN framework converges, the discriminator is expected to make predictions only at chance level and the generator is expected to create realistic data vectors. Although the overall framework works quite well, the dimensions of the input noise vector are usually not interpretable.
Chen et al. [
For some domains of a conceptual space, a dimensional representation is already available. For instance, the color domain can be represented by the threedimensional HSB space. For other domains, it is however quite unclear how to represent them based on a handful of dimensions. One prominent example is the shape domain: To the best of our knowledge, there are no widely accepted dimensional models for describing shapes.
We propose to use the InfoGAN framework in order to learn such a dimensional representation based on an unlabeled data set: Each of the latent variables can be interpreted as one dimension of the given domain of interest. For instance, the latent variables learned on a data set of shapes can be interpreted as dimensions of the shape domain. Three important properties of domains in a conceptual space are the following: interpretable dimensions, a distance-based notion of similarity, and a geometric way of describing semantic betweenness. We think that the latent space of an InfoGAN is a good candidate for representing a domain of a conceptual space, because it ful lls all of the above requirements:
As described before, Chen et al. [
Moreover, the smoothness assumption used in representation learning (cf. [
Finally, Radford et al. [
There are two important hyperparameters to the approach of grounding domains in InfoGANs: The number of latent variables (i.e., the dimensionality of the learned domain) and the type of distribution used for the latent variables (e.g., uniform vs. Gaussian). Note that one would probably aim for the lowestdimensional representation that still describes the domain su ciently well.
Finally, we would like to address a critical aspect of this proposal: How can one make sure that the representation learned by the neural network only represents information from the target domain (e.g., shape) and not anything related to other domains (e.g., color)? In our opinion, there are two complementary methods to \steer" the network towards the desired representation:
The rst option consists of selecting only such inputs for the training set that do not exhibit major di erences with respect to other domains. For instance, a training set for the shape domain should only include images of shapes that have the same color (e.g., black shape on white ground). If there is only very small variance in the data set with respect to other domains (e.g., color), the network is quite unlikely to incorporate this information into its latent representation.
The second option concerns modi cations of the network's loss function: One could for instance introduce an additional term into the loss function which measures the correlation between the learned latent representation and dimensions from other (already de ned) domains. This would cause a stronger error signal if the network starts to re-discover already known dimensions from other domains and therefore drive the network away from learning redundant representations.
A simple proof of concept implementation for the shape domain could be
based on a data set of simple 2D shapes (circles, triangles, rectangles, etc.) in
various orientations and locations. For a more thorough experiment, one could
for instance use ShapeNet3 [
Figure 2 illustrates a simpli ed example of our envisioned overall system. Here, we consider only two domains: color and shape. Color can be represented by the HSB space using the three dimensions hue, saturation and brightness. This is an example for a hard-coded domain. The representation of the shape domain, however, needs to be learned. The arti cial neural network depicted in Figure
2 corresponds to the discriminator of an InfoGAN trained on a data set of shapes.
Let us consider two example concepts: The concept of an apple can be described by the \red" region in the color domain and the \round" region in the shape domain. The concept of a banana can be represented by the \yellow" region in the color domain and the \cylindric" region in the shape domain.
If the system makes a new observation (e.g., an apple as depticted in Figure 2), it will convert this observation into a point in the conceptual space. For the color domain, this is done by a hard-coded conversion to the HSB color space. For the shape domain, the observation is fed into the discriminator and its latent representation is extracted, resulting in the coordinates for the shape domain. Now in order to classify this observation, the system needs to check whether the resulting data point is contained in any of the de ned regions. If the data point is an element of the apple region in both domains (which is the case in our example), this observation should be classi ed as an apple. If the data point is an element of the banana region, the object should be classi ed as a banana.
Based on a new observation, the existing concepts can also be updated: If the observation was classi ed as an apple, but it is not close to the center of the apple region in one of the domains, this region might be enlarged or moved a bit, such that the observed instance is better matched by the concept description. If the observation does not match any of the given concepts at all, even a new concept might be created. This means that concepts cannot only be applied for classi cation, but they can also be learned and updated. Note that this can take place without explicit label information, i.e., in an unsupervised way. Our overall reserach goal is to develop a clustering algorithm that can take care of incrementally updating the regions in such a conceptual space.
Please note that the updates considered above only concern the connections between the conceptual and the symbolic layer. The connections between the subsymbolic and the conceptual layer remain xed. The neural network thus only serves as a preprocessing step in our approach: It is trained before the overall system is used and remains unchanged afterwards. Simultaneous updates of both the neural network and the concept description might be desirable, but would probably introduce a great amount of additional complexity. 6
In this paper, we outlined how neural representations can be used to ground the domains of a conceptual space in perception. This is especially useful for domains like shape, where handcrafting a dimensional representation is di cult. We argued that the latent representations learned by an InfoGAN have suitable properties for being combined with the conceptual spaces framework. In future work, we will implement the proposed idea by giving a neural grounding to the domain of simple 2D shapes. Furthermore, we will devise a clustering algorithm for discovering and updating conceptual representations in a conceptual space.