How can one experiment to determine which distance metric yields the best retrieval quality for a given task (e.g., trying both cosine and Euclidean and comparing recall/precision of results)?

To determine which distance metric (e.g., cosine similarity or Euclidean distance) yields the best retrieval quality for a task, you can design an experiment that compares their performance using standard evaluation metrics like precision and recall. Start by defining a clear task, such as retrieving relevant documents for a query, and prepare a labeled dataset where each query has a known set of relevant results. For example, if you’re working with text embeddings, use a dataset like MS MARCO or a custom corpus with precomputed vector representations. Split your data into queries and a retrieval corpus, and ensure you have ground-truth relevance labels (e.g., which documents are relevant to each query).

Next, implement retrieval using both metrics. For cosine similarity, normalize the vectors to unit length and compute the dot product between query and document vectors. For Euclidean distance, calculate the straight-line distance between vectors. Use a library like FAISS or Annoy to efficiently retrieve the top-k results for each query using both methods. After retrieval, compute precision (the fraction of retrieved results that are relevant) and recall (the fraction of all relevant results retrieved) for each metric. For example, if a query has 5 relevant documents and the top-10 results using cosine similarity include 4 of them, the precision is 0.4 and recall is 0.8. Repeat this for all queries and average the results to get overall metrics for each distance method.

Finally, analyze the results to determine which metric performs better. Compare the average precision and recall scores, and consider using statistical tests (e.g., paired t-tests) to verify if differences are significant. For instance, you might find that cosine similarity outperforms Euclidean distance in high-dimensional spaces (common in text embeddings) because it focuses on vector angles rather than magnitudes. However, Euclidean distance could be more effective in low-dimensional or normalized data where magnitude differences matter. If the results are close, test on edge cases—for example, see how each metric handles queries with ambiguous terms or documents with varying lengths. Document your findings and iterate if needed, adjusting parameters like vector normalization or the number of retrieved results (k) to refine the comparison.