# Approximate Nearest Neighbor Search Indexes [experimental]

Nearest neighborhood search is the problem of finding the M closest points for a given point in an N-dimensional vector space. The most
straightforward approach to solve this problem is a brute force search where the distance between all points in the vector space and the
reference point is computed. This method guarantees perfect accuracy, but it is usually too slow for practical applications. Thus, nearest
neighborhood search problems are often solved with [approximative algorithms](https://github.com/erikbern/ann-benchmarks). Approximative
nearest neighborhood search techniques, in conjunction with [embedding
methods](https://cloud.google.com/architecture/overview-extracting-and-serving-feature-embeddings-for-machine-learning) allow to search huge
amounts of media (pictures, songs, articles, etc.) in milliseconds.

Blogs:
- [Vector Search with ClickHouse - Part 1](https://clickhouse.com/blog/vector-search-clickhouse-p1)
- [Vector Search with ClickHouse - Part 2](https://clickhouse.com/blog/vector-search-clickhouse-p2)


In terms of SQL, the nearest neighborhood problem can be expressed as follows:

``` sql
SELECT *
FROM table
ORDER BY Distance(vectors, Point)
LIMIT N
```

`vectors` contains N-dimensional values of type [Array(Float32)](../../../sql-reference/data-types/array.md) or Array(Float64), for example
embeddings. Function `Distance` computes the distance between two vectors. Often, the Euclidean (L2) distance is chosen as distance function
but [other distance functions](/docs/en/sql-reference/functions/distance-functions.md) are also possible. `Point` is the reference point,
e.g. `(0.17, 0.33, ...)`, and `N` limits the number of search results.

This query returns the top-`N` closest points to the reference point. Parameter `N` limits the number of returned values which is useful for
situations where `MaxDistance` is difficult to determine in advance.

With brute force search, the query is expensive (linear in the number of points) because the distance between all points in `vectors` and
`Point` must be computed. To speed this process up, Approximate Nearest Neighbor Search Indexes (ANN indexes) store a compact representation
of the search space (using clustering, search trees, etc.) which allows to compute an approximate answer much quicker (in sub-linear time).

# Creating and Using Vector Similarity Indexes

Syntax to create a vector similarity index over an [Array(Float32)](../../../sql-reference/data-types/array.md) column:

```sql
CREATE TABLE table
(
  id Int64,
  vectors Array(Float32),
  INDEX index_name vectors TYPE vector_similarity(method, distance_function[, quantization, connectivity, expansion_add, expansion_search]) [GRANULARITY N]
)
ENGINE = MergeTree
ORDER BY id;
```

Parameters:
- `method`: Supports currently only `hnsw`.
- `distance_function`: either `L2Distance` (the [Euclidean distance](https://en.wikipedia.org/wiki/Euclidean_distance) - the length of a
  line between two points in Euclidean space), or `cosineDistance` (the [cosine
  distance](https://en.wikipedia.org/wiki/Cosine_similarity#Cosine_distance)- the angle between two non-zero vectors).
- `quantization`: either `f32`, `f16`, or `i8` for storing the vector with reduced precision (optional, default: `f32`)
- `m`: the number of neighbors per graph node (optional, default: 16)
- `ef_construction`: (optional, default: 128)
- `ef_search`: (optional, default: 64)

Value 0 for parameters `m`, `ef_construction`, and `ef_search` refers to the default value.

Example:

```sql
CREATE TABLE table
(
  id Int64,
  vectors Array(Float32),
  INDEX idx vectors TYPE vector_similarity('hnsw', 'L2Distance') -- Alternative syntax: TYPE vector_similarity(hnsw, L2Distance)
)
ENGINE = MergeTree
ORDER BY id;
```

Vector similarity indexes are based on the [USearch library](https://github.com/unum-cloud/usearch), which implements the [HNSW
algorithm](https://arxiv.org/abs/1603.09320), i.e., a hierarchical graph where each point represents a vector and the edges represent
similarity. Such hierarchical structures can be very efficient on large collections. They may often fetch 0.05% or less data from the
overall dataset, while still providing 99% recall. This is especially useful when working with high-dimensional vectors, that are expensive
to load and compare. The library also has several hardware-specific SIMD optimizations to accelerate further distance computations on modern
Arm (NEON and SVE) and x86 (AVX2 and AVX-512) CPUs and OS-specific optimizations to allow efficient navigation around immutable persistent
files, without loading them into RAM.

USearch indexes are currently experimental, to use them you first need to `SET allow_experimental_vector_similarity_index = 1`.

Vector similarity indexes currently support two distance functions:
- `L2Distance`, also called Euclidean distance, is the length of a line segment between two points in Euclidean space
  ([Wikipedia](https://en.wikipedia.org/wiki/Euclidean_distance)).
- `cosineDistance`, also called cosine similarity, is the cosine of the angle between two (non-zero) vectors
  ([Wikipedia](https://en.wikipedia.org/wiki/Cosine_similarity)).

Vector similarity indexes allows storing the vectors in reduced precision formats. Supported scalar kinds are `f64`, `f32`, `f16` or `i8`.
If no scalar kind was specified during index creation, `f16` is used as default.

For normalized data, `L2Distance` is usually a better choice, otherwise `cosineDistance` is recommended to compensate for scale. If no
distance function was specified during index creation, `L2Distance` is used as default.

:::note
All arrays must have same length. To avoid errors, you can use a
[CONSTRAINT](/docs/en/sql-reference/statements/create/table.md#constraints), for example, `CONSTRAINT constraint_name_1 CHECK
length(vectors) = 256`. Also, empty `Arrays` and unspecified `Array` values in INSERT statements (i.e. default values) are not supported.
:::

:::note
The vector similarity index currently does not work with per-table, non-default `index_granularity` settings (see
[here](https://github.com/ClickHouse/ClickHouse/pull/51325#issuecomment-1605920475)). If necessary, the value must be changed in config.xml.
:::

ANN indexes are built during column insertion and merge. As a result, `INSERT` and `OPTIMIZE` statements will be slower than for ordinary
tables. ANNIndexes are ideally used only with immutable or rarely changed data, respectively when are far more read requests than write
requests.

ANN indexes support these queries:

  ``` sql
  SELECT *
  FROM table
  [WHERE ...]
  ORDER BY Distance(vectors, Point)
  LIMIT N
  ```

:::tip
To avoid writing out large vectors, you can use [query
parameters](/docs/en/interfaces/cli.md#queries-with-parameters-cli-queries-with-parameters), e.g.

```bash
clickhouse-client --param_vec='hello' --query="SELECT * FROM table WHERE L2Distance(vectors, {vec: Array(Float32)}) < 1.0"
```
:::

**Restrictions**: Approximate algorithms used to determine the nearest neighbors require a limit, hence queries without `LIMIT` clause
cannot utilize ANN indexes. Also, ANN indexes are only used if the query has a `LIMIT` value smaller than setting
`max_limit_for_ann_queries` (default: 1 million rows). This is a safeguard to prevent large memory allocations by external libraries for
approximate neighbor search.

**Differences to Skip Indexes** Similar to regular [skip indexes](https://clickhouse.com/docs/en/optimize/skipping-indexes), ANN indexes are
constructed over granules and each indexed block consists of `GRANULARITY = [N]`-many granules (`[N]` = 1 by default for normal skip
indexes). For example, if the primary index granularity of the table is 8192 (setting `index_granularity = 8192`) and `GRANULARITY = 2`,
then each indexed block will contain 16384 rows. However, data structures and algorithms for approximate neighborhood search (usually
provided by external libraries) are inherently row-oriented. They store a compact representation of a set of rows and also return rows for
ANN queries. This causes some rather unintuitive differences in the way ANN indexes behave compared to normal skip indexes.

When a user defines an ANN index on a column, ClickHouse internally creates an ANN "sub-index" for each index block. The sub-index is "local"
in the sense that it only knows about the rows of its containing index block. In the previous example and assuming that a column has 65536
rows, we obtain four index blocks (spanning eight granules) and an ANN sub-index for each index block. A sub-index is theoretically able to
return the rows with the N closest points within its index block directly. However, since ClickHouse loads data from disk to memory at the
granularity of granules, sub-indexes extrapolate matching rows to granule granularity. This is different from regular skip indexes which
skip data at the granularity of index blocks.

The `GRANULARITY` parameter determines how many ANN sub-indexes are created. Bigger `GRANULARITY` values mean fewer but larger ANN
sub-indexes, up to the point where a column (or a column's data part) has only a single sub-index. In that case, the sub-index has a
"global" view of all column rows and can directly return all granules of the column (part) with relevant rows (there are at most 
`LIMIT [N]`-many such granules). In a second step, ClickHouse will load these granules and identify the actually best rows by performing a
brute-force distance calculation over all rows of the granules. With a small `GRANULARITY` value, each of the sub-indexes returns up to
`LIMIT N`-many granules. As a result, more granules need to be loaded and post-filtered. Note that the search accuracy is with both cases
equally good, only the processing performance differs. It is generally recommended to use a large `GRANULARITY` for ANN indexes and fall
back to a smaller `GRANULARITY` values only in case of problems like excessive memory consumption of the ANN structures. If no `GRANULARITY`
was specified for ANN indexes, the default value is 100 million.