Understanding how vector differences encode transformations and relationships between concepts is a fundamental insight in working with latent spaces of large language models (LLMs). When you subtract one vector from another in latent space, the result is not just a mathematical operation—it often represents a meaningful semantic transformation. For example, consider the well-known analogy: king - man + woman ≈ queen. Here, the difference between the vectors for "king" and "man" captures the royal aspect associated with masculinity. When you add the vector for "woman", you are effectively transferring the royal concept to femininity, resulting in a vector close to "queen". This demonstrates that vector arithmetic can capture complex relationships and analogies between words and concepts.

From a geometric perspective, you can think of these transformations as translations along specific directions in latent space. Each direction can correspond to a particular semantic feature or relationship. For instance, the direction from "man" to "woman" might encode gender, while the direction from "Paris" to "France" might encode the capital-to-country relationship. When you move along these directions by adding or subtracting vectors, you are navigating the latent space in a way that mirrors real-world conceptual changes.

To summarize the main points about the role of vector arithmetic in capturing semantic relationships, consider these key insights:

• Vector subtraction in latent space often encodes specific semantic transformations;
• Directions in latent space correspond to interpretable relationships, such as gender or royalty;
• Adding or subtracting vectors allows you to perform analogical reasoning within the latent space;
• The ability of LLMs to capture these relationships underpins their effectiveness in understanding and generating language;
• Vector arithmetic provides a geometric framework for interpreting and manipulating meaning in high-dimensional representations.