In the context of analogies, similarity refers to the extent to which two things are alike or have shared characteristics. When making an analogy, the goal is often to show that two seemingly unrelated things are actually similar in some way, and the strength of the analogy depends on the degree of similarity between the things being compared.
For example, if we are trying to explain the concept of photosynthesis to a student who is familiar with the process of digestion, we might use the analogy that "photosynthesis is to plants as digestion is to animals," highlighting the shared characteristic of both processes being ways that organisms obtain energy from their environments. In this case, the degree of similarity between photosynthesis and digestion is fairly high, as both processes involve the conversion of energy from one form to another.
On the other hand, if we were to use the analogy that "photosynthesis is to plants as breathing is to animals," the degree of similarity would be lower, as photosynthesis and breathing serve different functions and involve different processes.
Overall, the concept of similarity is central to the idea of analogy-making, as it allows us to draw connections between things that may not be immediately obvious and to better understand complex ideas by relating them to things that are more familiar to us.
I asked ChatGPT the following: Explain to me what is analogy mathematically? and how are you able to make analogies from latent representations?
An analogy mathematically is a way of expressing a relationship between three or more elements, where one element is to another as a third element is to a fourth. For example, the statement "A is to B as C is to D" is an analogy, where A is related to B in the same way that C is related to D.
In the context of machine learning and language processing, a latent representation is a way of representing data in a reduced-dimensionality space. Latent representations can be used to make analogies by identifying the relationships between different elements in the data. For example, a machine learning model can be trained to identify the relationship between a word and its synonyms in a latent representation. Once the model has learned these relationships, it can use them to make analogies by identifying related words based on the relationships in the latent representation.
In summary, an analogy mathematically is a comparison of two relationships and with machine learning, we can use latent representations to identify the relationships between different elements in data, making it possible to make analogies.