Distance

L1Distance

Calculates the distance between two points (the elements of the vectors are the coordinates) in L1 space (1-norm (taxicab geometry distance)).

Syntax

L1Distance(vector1, vector2)

Arguments

vector1 — First vector. Tuple(T) or Array(T)
vector2 — Second vector. Tuple(T) or Array(T)

Returned value

Returns the 1-norm distance. UInt32 or Float64

Examples

Basic usage

SELECT L1Distance((1, 2), (2, 3))

┌─L1Distance((1, 2), (2, 3))─┐
│                          2 │
└────────────────────────────┘

Introduced in version 21.11.

L1Norm

Calculates the sum of absolute elements of a vector.

Syntax

L1Norm(vector)

Arguments

vector — Vector or tuple of numeric values. Array(T) or Tuple(T)

Returned value

Returns the L1-norm or taxicab geometry distance. UInt* or Float* or Decimal

Examples

Basic usage

SELECT L1Norm((1, 2))

┌─L1Norm((1, 2))─┐
│              3 │
└────────────────┘

Introduced in version 21.11.

L1Normalize

Calculates the unit vector of a given vector (the elements of the tuple are the coordinates) in L1 space (taxicab geometry).

Syntax

L1Normalize(tuple)

Arguments

tuple — A tuple of numeric values. Tuple(T)

Returned value

Returns the unit vector. Tuple(Float64)

Examples

Basic usage

SELECT L1Normalize((1, 2))

┌─L1Normalize((1, 2))─────────────────────┐
│ (0.3333333333333333,0.6666666666666666) │
└─────────────────────────────────────────┘

Introduced in version 21.11.

L2Distance

Calculates the distance between two points (the elements of the vectors are the coordinates) in Euclidean space (Euclidean distance).

Syntax

L2Distance(vector1, vector2)

Arguments

vector1 — First vector. Tuple(T) or Array(T)
vector2 — Second vector. Tuple(T) or Array(T)

Returned value

Returns the 2-norm distance. Float64

Examples

Basic usage

SELECT L2Distance((1, 2), (2, 3))

┌─L2Distance((1, 2), (2, 3))─┐
│         1.4142135623730951 │
└────────────────────────────┘

Introduced in version 21.11.

L2DistanceTransposed

Calculates the approximate distance between two points (the values of the vectors are the coordinates) in Euclidean space (Euclidean distance).

Syntax

L2DistanceTransposed(vector1, vector2, p)

Arguments

vectors — Vectors. QBit(T, UInt64)
reference — Reference vector. Array(T)
p — Number of bits from each vector element to use in the distance calculation (1 to element bit-width). The quantization level controls the precision-speed trade-off. Using fewer bits results in faster I/O and calculations with reduced accuracy, while using more bits increases accuracy at the cost of performance. UInt

Returned value

Returns the approximate 2-norm distance. Float64

Examples

Basic usage

CREATE TABLE qbit (id UInt32, vec QBit(Float64, 2)) ENGINE = Memory;
INSERT INTO qbit VALUES (1, [0, 1]);
SELECT L2DistanceTransposed(vec, array(1, 2), 16) FROM qbit;

┌─L2DistanceTransposed([0, 1], [1, 2], 16)─┐
│                       1.4142135623730951 │
└──────────────────────────────────────────┘

Introduced in version 25.10.

L2Norm

Calculates the square root of the sum of the squares of the vector elements.

Syntax

L2Norm(vector)

Arguments

vector — Vector or tuple of numeric values. Tuple(T) or Array(T)

Returned value

Returns the L2-norm or Euclidean distance. UInt* or Float*

Examples

Basic usage

SELECT L2Norm((1, 2))

┌───L2Norm((1, 2))─┐
│ 2.23606797749979 │
└──────────────────┘

Introduced in version 21.11.

L2Normalize

Calculates the unit vector of a given vector (the elements of the tuple are the coordinates) in Euclidean space (using Euclidean distance).

Syntax

L2Normalize(tuple)

Arguments

tuple — A tuple of numeric values. Tuple(T)

Returned value

Returns the unit vector. Tuple(Float64)

Examples

Basic usage

SELECT L2Normalize((3, 4))

┌─L2Normalize((3, 4))─┐
│ (0.6,0.8)           │
└─────────────────────┘

Introduced in version 21.11.

L2SquaredDistance

Calculates the sum of the squares of the difference between the corresponding elements of two vectors.

Syntax

L2SquaredDistance(vector1, vector2)

Arguments

vector1 — First vector. Tuple(T) or Array(T)
vector2 — Second vector. Tuple(T) or Array(T)

Returned value

Returns the sum of the squares of the difference between the corresponding elements of two vectors. Float64

Examples

Basic usage

SELECT L2SquaredDistance([1, 2, 3], [0, 0, 0])

┌─L2SquaredDis⋯ [0, 0, 0])─┐
│                       14 │
└──────────────────────────┘

Introduced in version 22.7.

L2SquaredNorm

Calculates the square root of the sum of the squares of the vector elements (the L2Norm) squared.

Syntax

L2SquaredNorm(vector)

Arguments

vector — Vector or tuple of numeric values. Array(T) or Tuple(T)

Returned value

Returns the L2-norm squared. UInt* or Float* or Decimal

Examples

Basic usage

SELECT L2SquaredNorm((1, 2))

┌─L2SquaredNorm((1, 2))─┐
│                     5 │
└───────────────────────┘

Introduced in version 22.7.

LinfDistance

Calculates the distance between two points (the elements of the vectors are the coordinates) in L_{inf} space (maximum norm).

Syntax

LinfDistance(vector1, vector2)

Arguments

vector1 — First vector. Tuple(T) or Array(T)
vector2 — Second vector. Tuple(T) or Array(T)

Returned value

Returns the Infinity-norm distance. Float64

Examples

Basic usage

SELECT LinfDistance((1, 2), (2, 3))

┌─LinfDistance((1, 2), (2, 3))─┐
│                            1 │
└──────────────────────────────┘

Introduced in version 21.11.

LinfNorm

Calculates the maximum of absolute elements of a vector.

Syntax

LinfNorm(vector)

Arguments

vector — Vector or tuple of numeric values. Array(T) or Tuple(T)

Returned value

Returns the Linf-norm or the maximum absolute value. Float64

Examples

Basic usage

SELECT LinfNorm((1, -2))

┌─LinfNorm((1, -2))─┐
│                 2 │
└───────────────────┘

Introduced in version 21.11.

LinfNormalize

Calculates the unit vector of a given vector (the elements of the tuple are the coordinates) in L_{inf} space (using maximum norm).

Syntax

LinfNormalize(tuple)

Arguments

tuple — A tuple of numeric values. Tuple(T)

Returned value

Returns the unit vector. Tuple(Float64)

Examples

Basic usage

SELECT LinfNormalize((3, 4))

┌─LinfNormalize((3, 4))─┐
│ (0.75,1)              │
└───────────────────────┘

Introduced in version 21.11.

LpDistance

Calculates the distance between two points (the elements of the vectors are the coordinates) in Lp space (p-norm distance).

Syntax

LpDistance(vector1, vector2, p)

Arguments

vector1 — First vector. Tuple(T) or Array(T)
vector2 — Second vector. Tuple(T) or Array(T)
p — The power. Possible values: real number from [1; inf). UInt* or Float*

Returned value

Returns the p-norm distance. Float64

Examples

Basic usage

SELECT LpDistance((1, 2), (2, 3), 3)

┌─LpDistance((1, 2), (2, 3), 3)─┐
│            1.2599210498948732 │
└───────────────────────────────┘

Introduced in version 21.11.

LpNorm

Calculates the p-norm of a vector, which is the p-th root of the sum of the p-th powers of the absolute elements of its elements.

Special cases:

When p=1, it's equivalent to L1Norm (Manhattan distance).
When p=2, it's equivalent to L2Norm (Euclidean distance).
When p=∞, it's equivalent to LinfNorm (maximum norm).

Syntax

LpNorm(vector, p)

Arguments

vector — Vector or tuple of numeric values. Tuple(T) or Array(T)
p — The power. Possible values are real numbers in the range [1; inf). UInt* or Float*

Returned value

Returns the Lp-norm. Float64

Examples

Basic usage

SELECT LpNorm((1, -2), 2)

┌─LpNorm((1, -2), 2)─┐
│   2.23606797749979 │
└────────────────────┘

Introduced in version 21.11.

LpNormalize

Calculates the unit vector of a given vector (the elements of the tuple are the coordinates) in Lp space (using p-norm).

Syntax

LpNormalize(tuple, p)

Arguments

tuple — A tuple of numeric values. Tuple(T)
p — The power. Possible values are any number in the range range from [1; inf). UInt* or Float*

Returned value

Returns the unit vector. Tuple(Float64)

Examples

Usage example

SELECT LpNormalize((3, 4), 5)

┌─LpNormalize((3, 4), 5)──────────────────┐
│ (0.7187302630182624,0.9583070173576831) │
└─────────────────────────────────────────┘

Introduced in version 21.11.

cosineDistance

Calculates the cosine distance between two vectors (the elements of the tuples are the coordinates). The smaller the returned value is, the more similar are the vectors.

Syntax

cosineDistance(vector1, vector2)

Arguments

vector1 — First tuple. Tuple(T) or Array(T)
vector2 — Second tuple. Tuple(T) or Array(T)

Returned value

Returns the cosine of the angle between two vectors subtracted from one. Float64

Examples

Basic usage

SELECT cosineDistance((1, 2), (2, 3));

┌─cosineDistance((1, 2), (2, 3))─┐
│           0.007722123286332261 │
└────────────────────────────────┘

Introduced in version 1.1.

cosineDistanceTransposed

Calculates the approximate cosine distance between two points (the values of the vectors are the coordinates). The smaller the returned value is, the more similar are the vectors.

Syntax

cosineDistanceTransposed(vector1, vector2, p)

Arguments

vectors — Vectors. QBit(T, UInt64)
reference — Reference vector. Array(T)
p — Number of bits from each vector element to use in the distance calculation (1 to element bit-width). The quantization level controls the precision-speed trade-off. Using fewer bits results in faster I/O and calculations with reduced accuracy, while using more bits increases accuracy at the cost of performance. UInt

Returned value

Returns the approximate cosine of the angle between two vectors subtracted from one. Float64

Examples

Basic usage

CREATE TABLE qbit (id UInt32, vec QBit(Float64, 2)) ENGINE = Memory;
INSERT INTO qbit VALUES (1, [0, 1]);
SELECT cosineDistanceTransposed(vec, array(1, 2), 16) FROM qbit;

┌─cosineDistanceTransposed([0, 1], [1, 2], 16)─┐
│                          0.10557281085638826 │
└──────────────────────────────────────────────┘

Introduced in version 26.1.

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