diff --git a/_search-plugins/knn/approximate-knn.md b/_search-plugins/knn/approximate-knn.md index f8921033e0..26277b8588 100644 --- a/_search-plugins/knn/approximate-knn.md +++ b/_search-plugins/knn/approximate-knn.md @@ -402,7 +402,7 @@ Not every method supports each of these spaces. Be sure to check out [the method | `l1` | $$ d(\mathbf{x}, \mathbf{y}) = \sum_{i=1}^n \lvert x_i - y_i \rvert $$ | $$ score = {1 \over {1 + d} } $$ | | `l2` | $$ d(\mathbf{x}, \mathbf{y}) = \sum_{i=1}^n (x_i - y_i)^2 $$ | $$ score = {1 \over 1 + d } $$ | | `linf` | $$ d(\mathbf{x}, \mathbf{y}) = max(\lvert x_i - y_i \rvert) $$ | $$ score = {1 \over 1 + d } $$ | -| `cosinesimil` | $$ d(\mathbf{x}, \mathbf{y}) = 1 - cos { \theta } = 1 - {\mathbf{x} \cdot \mathbf{y} \over \lVert \mathbf{x}\rVert \cdot \lVert \mathbf{y}\rVert}$$$$ = 1 - {\sum_{i=1}^n x_i y_i \over \sqrt{\sum_{i=1}^n x_i^2} \cdot \sqrt{\sum_{i=1}^n y_i^2}}$$,
where $$\lVert \mathbf{x}\rVert$$ and $$\lVert \mathbf{y}\rVert$$ represent the norms of vectors $$\mathbf{x}$$ and $$\mathbf{y}$$, respectively. | **NMSLIB** and **Faiss**:
$$ score = {1 \over 1 + d } $$

**Lucene**:
$$ score = {2 - d \over 2}$$ | +| `cosinesimil` | $$ d(\mathbf{x}, \mathbf{y}) = 1 - cos { \theta } = 1 - {\mathbf{x} \cdot \mathbf{y} \over \lVert \mathbf{x}\rVert \cdot \lVert \mathbf{y}\rVert}$$$$ = 1 - {\sum_{i=1}^n x_i y_i \over \sqrt{\sum_{i=1}^n x_i^2} \cdot \sqrt{\sum_{i=1}^n y_i^2}}$$,
where $$\lVert \mathbf{x}\rVert$$ and $$\lVert \mathbf{y}\rVert$$ represent the norms of vectors $$\mathbf{x}$$ and $$\mathbf{y}$$, respectively. | $$ score = {2 - d \over 2} $$ | | `innerproduct` (supported for Lucene in OpenSearch version 2.13 and later) | **NMSLIB** and **Faiss**:
$$ d(\mathbf{x}, \mathbf{y}) = - {\mathbf{x} \cdot \mathbf{y}} = - \sum_{i=1}^n x_i y_i $$

**Lucene**:
$$ d(\mathbf{x}, \mathbf{y}) = {\mathbf{x} \cdot \mathbf{y}} = \sum_{i=1}^n x_i y_i $$ | **NMSLIB** and **Faiss**:
$$ \text{If} d \ge 0, score = {1 \over 1 + d }$$
$$\text{If} d < 0, score = −d + 1$$

**Lucene:**
$$ \text{If} d > 0, score = d + 1 $$
$$\text{If} d \le 0, score = {1 \over 1 + (-1 \cdot d) }$$ | | `hamming` (supported for binary vectors in OpenSearch version 2.16 and later) | $$ d(\mathbf{x}, \mathbf{y}) = \text{countSetBits}(\mathbf{x} \oplus \mathbf{y})$$ | $$ score = {1 \over 1 + d } $$ | diff --git a/_search-plugins/knn/knn-score-script.md b/_search-plugins/knn/knn-score-script.md index a184de2d3d..45e7d1a67c 100644 --- a/_search-plugins/knn/knn-score-script.md +++ b/_search-plugins/knn/knn-score-script.md @@ -293,7 +293,7 @@ A _space_ corresponds to the function used to measure the distance between two p | `l1` | $$ d(\mathbf{x}, \mathbf{y}) = \sum_{i=1}^n \lvert x_i - y_i \rvert $$ | $$ score = {1 \over {1 + d} } $$ | | `l2` | $$ d(\mathbf{x}, \mathbf{y}) = \sum_{i=1}^n (x_i - y_i)^2 $$ | $$ score = {1 \over 1 + d } $$ | | `linf` | $$ d(\mathbf{x}, \mathbf{y}) = max(\lvert x_i - y_i \rvert) $$ | $$ score = {1 \over 1 + d } $$ | -| `cosinesimil` | $$ d(\mathbf{x}, \mathbf{y}) = 1 - cos { \theta } = 1 - {\mathbf{x} \cdot \mathbf{y} \over \lVert \mathbf{x}\rVert \cdot \lVert \mathbf{y}\rVert}$$$$ = 1 - {\sum_{i=1}^n x_i y_i \over \sqrt{\sum_{i=1}^n x_i^2} \cdot \sqrt{\sum_{i=1}^n y_i^2}}$$,
where $$\lVert \mathbf{x}\rVert$$ and $$\lVert \mathbf{y}\rVert$$ represent the norms of vectors $$\mathbf{x}$$ and $$\mathbf{y}$$, respectively. | $$ score = 2 - d $$ | +| `cosinesimil` | $$ d(\mathbf{x}, \mathbf{y}) = 1 - cos { \theta } = 1 - {\mathbf{x} \cdot \mathbf{y} \over \lVert \mathbf{x}\rVert \cdot \lVert \mathbf{y}\rVert}$$$$ = 1 - {\sum_{i=1}^n x_i y_i \over \sqrt{\sum_{i=1}^n x_i^2} \cdot \sqrt{\sum_{i=1}^n y_i^2}}$$,
where $$\lVert \mathbf{x}\rVert$$ and $$\lVert \mathbf{y}\rVert$$ represent the norms of vectors $$\mathbf{x}$$ and $$\mathbf{y}$$, respectively. | $$ score = {2 - d \over 2 } $$ | | `innerproduct` (supported for Lucene in OpenSearch version 2.13 and later) | $$ d(\mathbf{x}, \mathbf{y}) = - {\mathbf{x} \cdot \mathbf{y}} = - \sum_{i=1}^n x_i y_i $$ | $$ \text{If} d \ge 0, score = {1 \over 1 + d }$$
$$\text{If} d < 0, score = −d + 1$$ | | `hammingbit` (supported for binary and long vectors)

`hamming` (supported for binary vectors in OpenSearch version 2.16 and later) | $$ d(\mathbf{x}, \mathbf{y}) = \text{countSetBits}(\mathbf{x} \oplus \mathbf{y})$$ | $$ score = {1 \over 1 + d } $$ |