Embedding

An embedding maps an object such as a word, sentence, document, image, or user item into a vector of numbers. Useful embedding spaces place objects with related meanings or functions near one another according to a chosen similarity measure.

Embedding models are trained so that useful relationships are reflected in vector geometry. Applications often compare vectors with cosine similarity, dot product, or distance metrics. Sentence and document embeddings are widely used to index large text collections.

Embeddings power semantic search, clustering, recommendation, duplicate detection, and many RAG pipelines. Their quality depends on the training objective and the domain in which they are used.

See also Vector Search, Tokenization, and Retrieval-Augmented Generation (RAG).

These coordinates are fictional teaching vectors, not outputs from a real embedding model. Query q=(3, 4) and document a=(0, 5) each have norm 5. Their dot product is 20, so cosine similarity is 20 / (5 × 5) = 0.8. Document b=(4, -3) has a dot product of zero with q and therefore cosine similarity zero. Under this metric, a ranks closer than b. The score is not an 80% probability of correctness. A zero-norm vector needs separate handling because the denominator would be zero.

Separate candidate ranking from factual correctness. Eligible for return and ineligible for return can share many words while expressing opposite policies. Check negation, product identifiers and effective dates in the actual source before treating a nearby candidate as evidence.

Record the model, tokenizer, dimension, normalization and query/document input formats together. Equal dimensions do not make vectors from different models interchangeable. Changes to input segmentation or chunk boundaries can also change retrieval behavior.

Build a separate new document index and evaluate it with the new query encoder on the same held-out questions. Switch encoder and index together, retaining the previous bundle for recovery. Check that source updates and deletions reach derived vectors and source versions. Recomputing coordinates over stale text does not make the underlying evidence current.

For a fictional AI product manual, separate Korean questions, English questions, cross-language equivalents, product codes, negation and date conditions. The question whether X-20 works offline paired with a source saying X-20 requires an internet connection tests both language alignment and a critical condition. Record duplicate candidates that come from the same original document.

Measure whether the correct source appears within the top k candidates and whether returned sources meet the actual constraints. Separate threshold tuning from final evaluation. A universal cosine threshold of 0.8 across models and languages is not justified by the worked example; compare false matches and missed evidence on the intended workload.

Sentence-BERT ↗ studies separately encoded sentence vectors compared with cosine similarity.^[2] Dense Passage Retrieval ↗ evaluates a dual-encoder approach to passage retrieval for open-domain question answering.^[4] Their findings are bounded to the evaluated models, tasks and data, not all Korean material or current systems.

When a RAG answer fails, check source availability, retained conditions and encoder/index consistency first. If correct evidence was retrieved but the generator ignored an exception, replacing embeddings alone does not address that failure.

Sentence-BERT ↗, section 3, describes shared-weight siamese encoders and triplet training. Its pooling comparison includes CLS, mean and max operations, with mean as the default. Distinguish classification/regression training objectives from sentence comparison at inference.^[2] Results on the studied similarity tasks do not guarantee relevance or factual correctness for every Korean retrieval workload.

This is a mathematical explanation, not measured research performance. Only for unit-length vectors u and v does squared Euclidean distance satisfy ||u-v||² = 2 - 2·cos(u,v). A fictional unit-vector pair with cosine 0.25 has squared distance 1.5. Under this condition, descending cosine and ascending squared distance give the same ranking; do not extend that equivalence to unnormalized vectors. A zero vector cannot be normalized to unit length. The transformation does not produce correctness probabilities. Record index distance settings with input preprocessing.

This is an ORBiS Tree evaluation design. Separate direct answer evidence, background relevance and irrelevance for each question, retaining source sentences and versions. The same AI product code with different usage conditions need not supply direct evidence. Have another reviewer independently label a subset and preserve disagreement reasons. Separate unstable labels from retrieval failures before attributing an improvement to a new model. Manage evaluation composition and source changes together.