feat: Add doc for binary vector (#52)

* feat: Add a separate doc for bvector

Signed-off-by: Ce Gao <cegao@tensorchord.ai>

* chore: Update

Signed-off-by: Ce Gao <cegao@tensorchord.ai>

---------

Signed-off-by: Ce Gao <cegao@tensorchord.ai>

.vitepress/config.mts

@@ -66,6 +66,7 @@ export default defineConfig({
           items: [
             { text: 'Indexing', link: '/usage/indexing' },
             { text: 'Search', link: '/usage/search' },
+            { text: 'Vector Types', link: '/usage/vector-types' },
             { text: 'Monitoring', link: '/usage/monitoring' },
             { text: 'Quantization', link: '/usage/quantization' },
             { text: 'Compatibility', link: '/usage/compatibility' },

src/usage/vector-types.md

@@ -0,0 +1,47 @@
+# Vector Types
+
+We have seen `vector` type in the previous section. In this section, we will show other vector types.
+
+## `bvector` binary vector
+
+The `bvector` type is a binary vector type in pgvecto.rs. It represents a binary vector, which is a vector where each component can take on two possible values, typically 0 and 1. 
+
+Here's an example of creating a table with a bvector column and inserting values:
+
+```sql {3}
+CREATE TABLE items (
+  id bigserial PRIMARY KEY,
+  embedding bvector(3) NOT NULL
+);
+
+INSERT INTO items (embedding) VALUES ('[1,0,1]'), ('[0,1,0]');
+```
+
+We support three operators to calculate the distance between two `bvector` values.
+
+- `<->` (`bvector_l2_ops`): squared Euclidean distance, defined as $\Sigma (x_i - y_i) ^ 2$. The Hamming distance is equivalent to the squared Euclidean distance for binary vectors.
+- `<#>` (`bvector_dot_ops`): negative dot product, defined as $- \Sigma x_iy_i$.
+- `<=>` (`bvector_cos_ops`): cosine distance, defined as $1 - \frac{\Sigma x_iy_i}{\sqrt{\Sigma x_i^2 \Sigma y_i^2}}$.
+- `<~>` (`bvector_jaccard_ops`): Jaccard distance, defined as $1 - \frac{|X\cap Y|}{|X\cup Y|}$.
+
+```sql
+
+Index can be created on `bvector` type as well.
+
+```sql
+CREATE INDEX bvector ON items USING vectors (embedding bvector_l2_ops);
+
+SELECT * FROM items ORDER BY embedding <-> '[1,0,1]' LIMIT 5;
+```
+
+### Performance
+
+The `bvector` type is optimized for storage and performance. It uses a bit-packed representation to store the binary vector. The distance calculation is also optimized for binary vectors.
+
+Here are some performance benchmarks for the `bvector` type. We use the [dbpedia-entities-openai3-text-embedding-3-large-3072-1M](https://huggingface.co/datasets/Qdrant/dbpedia-entities-openai3-text-embedding-3-large-3072-1M) dataset for the benchmark. The VM is n2-standard-8 (8 vCPUs, 32 GB memory) on Google Cloud.
+
+We upsert 1M binary vectors into the table and then run a KNN query for each embedding. It only takes about 600MB memory to index 1M binary vectors.
+
+![bvector](./images/bvector.png)
+
+We can see that the `bvector`'s accuracy is not as good as the `vector` type, but it exceeds 95%  if we adopt adaptive retrieval.