Blog comment on Distance Metrics in Vector Search

[giscus[bot]]

Blog comment on Distance Metrics in Vector Search

Learn about why you need distance metrics in vector search and the metrics implemented in Weaviate (Cosine, Dot Product, L2-Squared, Manhattan, and Hamming).

https://weaviate.io/blog/distance-metrics-in-vector-search

[Vodyanoy17]

Dot Produc:"It keeps the intuition that a smaller distance value means they are similar." is not true
AB= 13, but AA= 16+1=17 ,

[patmejia]

Weaviate's Distance Metrics $-(\vec{A} \cdot \vec{B})$ vs. Traditional Dot Product's Similarity Measurement:

Negative of the dot product: "It keeps the intuition that a smaller distance value means they are similar." - Weaviate's Docs is correct within the context of Weaviate's distance metrics or vector search.

1. Euclidean Geometry:

Similarity Measurement: In Euclidean geometry, the dot product measures the alignment and magnitude of vectors, with higher values indicating greater similarity. Larger magnitudes increase the dot product when vectors are aligned or at an acute angle, but larger angles, especially over $> 90^\circ$, can result in low or negative values regardless of magnitude.

Example:
For vectors $\vec{A} = [4, 0, 1]$ and $\vec{B} = [3, 0, 1]$,
the dot product $\vec{A} \cdot \vec{B} = 13$ suggests moderate similarity.
$\vec{A} \cdot \vec{A} = 17$ indicates a maximum similarity.

2. Weaviate's Approach:

Distance Metric: Weaviate introduces the negative of the dot product as a distance metric, where more negative values signal higher similarity: $-(\vec{A} \cdot \vec{B})$.

Example:
For vectors $\vec{A}$ and $\vec{B}$, the distance metric in Weaviate,
$-(\vec{A} \cdot \vec{B}) = -13$ indicates high similarity.
$-(\vec{A} \cdot \vec{A}) = -17$ implies even greater similarity or proximity.

In summary, Weaviate's distance metrics prioritize similarity differently than raw dot product, with lower (more negative) values indicating greater similarity or closeness.

3. Weaviate's Docs:

Credit:
Distance Metric | https://weaviate.io/developers/weaviate/config-refs/distances