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Spaces

In vector search, a space defines how the distance (or similarity) between two vectors is calculated. The choice of space affects how nearest neighbors are determined during search operations.

Distance calculation

A space defines the function used to measure the distance between two points in order to determine the k-nearest neighbors. In k-NN search, a lower score equates to a closer and better result. This is the opposite of how OpenSearch scores results, where a higher score equates to a better result. OpenSearch supports the following spaces.

Not every method/engine combination supports each of the spaces. For a list of supported spaces, see the section for a specific engine in the method documentation.

Space type Search type Distance function (\(d\) ) OpenSearch score
l1 Approximate, exact \(d(\mathbf{x}, \mathbf{y}) = \sum_{i=1}^n \lvert x_i - y_i \rvert\) \(score = {1 \over {1 + d} }\)
l2 Approximate, exact \(d(\mathbf{x}, \mathbf{y}) = \sum_{i=1}^n (x_i - y_i)^2\) \(score = {1 \over 1 + d }\)
linf Approximate, exact \(d(\mathbf{x}, \mathbf{y}) = max(\lvert x_i - y_i \rvert)\) \(score = {1 \over 1 + d }\)
cosinesimil Approximate, exact \(d(\mathbf{x}, \mathbf{y}) = 1 - cos { \theta } = 1 - {\mathbf{x} \cdot \mathbf{y} \over \lVert \mathbf{x}\rVert \cdot \lVert \mathbf{y}\rVert}\)\(= 1 - {\sum_{i=1}^n x_i y_i \over \sqrt{\sum_{i=1}^n x_i^2} \cdot \sqrt{\sum_{i=1}^n y_i^2}}\),
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) Approximate 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:
\(\text{If} d > 0, score = d + 1\)
\(\text{If} d \le 0, score = {1 \over 1 + (-1 \cdot d) }\)
innerproduct (supported for Lucene in OpenSearch version 2.13 and later) Exact \(d(\mathbf{x}, \mathbf{y}) = - {\mathbf{x} \cdot \mathbf{y}} = - \sum_{i=1}^n x_i y_i\) \(\text{If} d \ge 0, score = {1 \over 1 + d }\)
\(\text{If} d < 0, score = −d + 1\)
hamming (supported for binary vectors in OpenSearch version 2.16 and later) Approximate, exact \(d(\mathbf{x}, \mathbf{y}) = \text{countSetBits}(\mathbf{x} \oplus \mathbf{y})\) \(score = {1 \over 1 + d }\)
hammingbit (supported for binary and long vectors) Exact \(d(\mathbf{x}, \mathbf{y}) = \text{countSetBits}(\mathbf{x} \oplus \mathbf{y})\) \(score = {1 \over 1 + d }\)

The cosine similarity formula does not include the 1 - prefix. However, because similarity search libraries equate lower scores with closer results, they return 1 - cosineSimilarity for the cosine similarity space—this is why 1 - is included in the distance function.

With cosine similarity, it is not valid to pass a zero vector ([0, 0, ...]) as input. This is because the magnitude of such a vector is 0, which raises a divide by 0 exception in the corresponding formula. Requests containing the zero vector will be rejected, and a corresponding exception will be thrown.

The hamming space type is supported for binary vectors in OpenSearch version 2.16 and later. For more information, see Binary k-NN vectors.

Specifying the space type

The space type is specified when creating an index.

You can specify the space type at the top level of the field mapping:

PUT /test-index
{
  "settings": {
    "index": {
      "knn": true
    }
  },
  "mappings": {
    "properties": {
      "my_vector1": {
        "type": "knn_vector",
        "dimension": 3,
        "space_type": "l2"
      }
    }
  }
}

Alternatively, you can specify the space type within the method object if defining a method:

PUT test-index
{
  "settings": {
    "index": {
      "knn": true,
      "knn.algo_param.ef_search": 100
    }
  },
  "mappings": {
    "properties": {
      "my_vector1": {
        "type": "knn_vector",
        "dimension": 1024,
        "method": {
          "name": "hnsw",
          "space_type": "l2",
          "engine": "nmslib",
          "parameters": {
            "ef_construction": 128,
            "m": 24
          }
        }
      }
    }
  }
}

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