View Source Scholar.Metrics.Distance (Scholar v0.3.0)
Distance metrics between multi-dimensional tensors. They all support distance calculations between any subset of axes.
Summary
Functions
Chebyshev or $L_{\infty}$ distance.
Cosine distance.
Standard euclidean distance ($L_{2}$ distance).
Hamming distance.
Hamming distance in weighted version.
Manhattan, Taxicab, or $L_{1}$ distance.
Minkowski distance.
Pairwise cosine distance. It is equivalent to Scholar.Metrics.Distance.pairwise_euclidean(x, x)
Pairwise cosine distance.
Pairwise euclidean distance. It is equivalent to Scholar.Metrics.Distance.pairwise_euclidean(x, x)
Pairwise euclidean distance.
Computes the pairwise Minkowski distance.
Pairwise squared euclidean distance. It is equivalent to Scholar.Metrics.Distance.pairwise_squared_euclidean(x, x)
Pairwise squared euclidean distance.
Squared euclidean distance.
Functions
Chebyshev or $L_{\infty}$ distance.
$$ D(x, y) = \max_i |x_i - y_i| $$
Options
:axes- Axes to calculate the distance over. By default the distance is calculated between the whole tensors.
Examples
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([3, 2])
iex> Scholar.Metrics.Distance.chebyshev(x, y)
#Nx.Tensor<
f32
2.0
>
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([1, 2])
iex> Scholar.Metrics.Distance.chebyshev(x, y)
#Nx.Tensor<
f32
0.0
>
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([1, 2, 3])
iex> Scholar.Metrics.Distance.chebyshev(x, y)
** (ArgumentError) tensors must be broadcast compatible, got tensors with shapes {2} and {3}
iex> x = Nx.tensor([[1, 2, 5], [3, 4, 3]])
iex> y = Nx.tensor([[8, 3, 1], [2, 5, 2]])
iex> Scholar.Metrics.Distance.chebyshev(x, y, axes: [1])
#Nx.Tensor<
f32[2]
[7.0, 1.0]
>
iex> x = Nx.tensor([[6, 2, 9], [2, 5, 3]])
iex> y = Nx.tensor([[8, 3, 1]])
iex> Scholar.Metrics.Distance.chebyshev(x, y)
#Nx.Tensor<
f32
8.0
>Cosine distance.
$$ D(u, v) = 1 - \frac{u \cdot v}{\|u\|_2 \|v\|_2} $$
Options
:axes- Axes to calculate the distance over. By default the distance is calculated between the whole tensors.
Examples
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([5, 2])
iex> Scholar.Metrics.Distance.cosine(x, y)
#Nx.Tensor<
f32
0.25259071588516235
>
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([1, 2, 3])
iex> Scholar.Metrics.Distance.cosine(x, y)
** (ArgumentError) tensors must be broadcast compatible, got tensors with shapes {2} and {3}
iex> x = Nx.tensor([[1, 2, 3], [0, 0, 0], [5, 2, 4]])
iex> y = Nx.tensor([[1, 5, 2], [2, 4, 1], [0, 0, 0]])
iex> Scholar.Metrics.Distance.cosine(x, y, axes: [1])
#Nx.Tensor<
f32[3]
[0.1704850196838379, 1.0, 1.0]
>
iex> x = Nx.tensor([[6, 2, 9], [2, 5, 3]])
iex> y = Nx.tensor([[8, 3, 1]])
iex> Scholar.Metrics.Distance.cosine(x, y)
#Nx.Tensor<
f32
0.10575336217880249
>Standard euclidean distance ($L_{2}$ distance).
$$ D(x, y) = \sqrt{\sum_i (x_i - y_i)^2} $$
Options
:axes- Axes to calculate the distance over. By default the distance is calculated between the whole tensors.
Examples
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([3, 2])
iex> Scholar.Metrics.Distance.euclidean(x, y)
#Nx.Tensor<
f32
2.0
>
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([1, 2])
iex> Scholar.Metrics.Distance.euclidean(x, y)
#Nx.Tensor<
f32
0.0
>
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([1, 2, 3])
iex> Scholar.Metrics.Distance.euclidean(x, y)
** (ArgumentError) tensors must be broadcast compatible, got tensors with shapes {2} and {3}
iex> x = Nx.tensor([[1, 2, 5], [3, 4, 3]])
iex> y = Nx.tensor([[8, 3, 1], [2, 5, 2]])
iex> Scholar.Metrics.Distance.euclidean(x, y, axes: [0])
#Nx.Tensor<
f32[3]
[7.071067810058594, 1.4142135381698608, 4.123105525970459]
>
iex> x = Nx.tensor([[6, 2, 9], [2, 5, 3]])
iex> y = Nx.tensor([[8, 3, 1]])
iex> Scholar.Metrics.Distance.euclidean(x, y)
#Nx.Tensor<
f32
10.630146026611328
>Hamming distance.
$$ hamming(x, y) = \frac{\left \lvert x\_{i, j, ...} \neq y\_{i, j, ...}\right \rvert}{\left \lvert x\_{i, j, ...}\right \rvert} $$ where $i, j, ...$ are the aggregation axes
Options
:axes- Axes to calculate the distance over. By default the distance is calculated between the whole tensors.
Examples
iex> x = Nx.tensor([1, 0, 0])
iex> y = Nx.tensor([0, 1, 0])
iex> Scholar.Metrics.Distance.hamming(x, y)
#Nx.Tensor<
f32
0.6666666865348816
>
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([1, 2, 3])
iex> Scholar.Metrics.Distance.hamming(x, y)
** (ArgumentError) tensors must be broadcast compatible, got tensors with shapes {2} and {3}
iex> x = Nx.tensor([[1, 2, 3], [0, 0, 0], [5, 2, 4]])
iex> y = Nx.tensor([[1, 5, 2], [2, 4, 1], [0, 0, 0]])
iex> Scholar.Metrics.Distance.hamming(x, y, axes: [1])
#Nx.Tensor<
f32[3]
[0.6666666865348816, 1.0, 1.0]
>
iex> x = Nx.tensor([[6, 2, 9], [2, 5, 3]])
iex> y = Nx.tensor([[8, 3, 1]])
iex> Scholar.Metrics.Distance.hamming(x, y)
#Nx.Tensor<
f32
1.0
>Hamming distance in weighted version.
Options
:axes- Axes to calculate the distance over. By default the distance is calculated between the whole tensors.
Examples
iex> x = Nx.tensor([1, 0, 0])
iex> y = Nx.tensor([0, 1, 0])
iex> weights = Nx.tensor([1, 0.5, 0.5])
iex> Scholar.Metrics.Distance.hamming(x, y, weights)
#Nx.Tensor<
f32
0.75
>
iex> x = Nx.tensor([[6, 2, 9], [2, 5, 3]])
iex> y = Nx.tensor([[8, 3, 1]])
iex> weights = Nx.tensor([1, 0.5, 0.5])
iex> Scholar.Metrics.Distance.hamming(x, y, weights, axes: [1])
#Nx.Tensor<
f32[2]
[1.0, 1.0]
>Manhattan, Taxicab, or $L_{1}$ distance.
$$ D(x, y) = \sum_i |x_i - y_i| $$
Options
:axes- Axes to calculate the distance over. By default the distance is calculated between the whole tensors.
Examples
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([3, 2])
iex> Scholar.Metrics.Distance.manhattan(x, y)
#Nx.Tensor<
f32
2.0
>
iex> x = Nx.tensor([1.0, 2.0])
iex> y = Nx.tensor([1, 2])
iex> Scholar.Metrics.Distance.manhattan(x, y)
#Nx.Tensor<
f32
0.0
>
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([1, 2, 3])
iex> Scholar.Metrics.Distance.manhattan(x, y)
** (ArgumentError) tensors must be broadcast compatible, got tensors with shapes {2} and {3}
iex> x = Nx.tensor([[1, 2, 5], [3, 4, 3]])
iex> y = Nx.tensor([[8, 3, 1], [2, 5, 2]])
iex> Scholar.Metrics.Distance.manhattan(x, y, axes: [0])
#Nx.Tensor<
f32[3]
[8.0, 2.0, 5.0]
>
iex> x = Nx.tensor([[6, 2, 9], [2, 5, 3]])
iex> y = Nx.tensor([[8, 3, 1]])
iex> Scholar.Metrics.Distance.manhattan(x, y)
#Nx.Tensor<
f32
21.0
>Minkowski distance.
$$ D(x, y) = \left(\sum_i |x_i - y_i|^p\right)^{\frac{1}{p}} $$
Options
:axes- Axes to calculate the distance over. By default the distance is calculated between the whole tensors.:p- A positive parameter of Minkowski distance or :infinity (then Chebyshev metric computed). The default value is2.0.
Examples
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([5, 2])
iex> Scholar.Metrics.Distance.minkowski(x, y)
#Nx.Tensor<
f32
4.0
>
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([1, 2])
iex> Scholar.Metrics.Distance.minkowski(x, y)
#Nx.Tensor<
f32
0.0
>
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([1, 2, 3])
iex> Scholar.Metrics.Distance.minkowski(x, y)
** (ArgumentError) tensors must be broadcast compatible, got tensors with shapes {2} and {3}
iex> x = Nx.tensor([[1, 2, 5], [3, 4, 3]])
iex> y = Nx.tensor([[8, 3, 1], [2, 5, 2]])
iex> Scholar.Metrics.Distance.minkowski(x, y, p: 2.5, axes: [0])
#Nx.Tensor<
f32[3]
[7.021548271179199, 1.3195079565048218, 4.049539089202881]
>
iex> x = Nx.tensor([[6, 2, 9], [2, 5, 3]])
iex> y = Nx.tensor([[8, 3, 1]])
iex> Scholar.Metrics.Distance.minkowski(x, y, p: 2.5)
#Nx.Tensor<
f32
9.621805191040039
>Pairwise cosine distance. It is equivalent to Scholar.Metrics.Distance.pairwise_euclidean(x, x)
Examples
iex> x = Nx.iota({6, 6})
iex> Scholar.Metrics.Distance.pairwise_cosine(x)
#Nx.Tensor<
f32[6][6]
[
[0.0, 0.0793418288230896, 0.1139642596244812, 0.13029760122299194, 0.1397092342376709, 0.14581435918807983],
[0.0793418288230896, 0.0, 0.0032819509506225586, 0.006624102592468262, 0.008954286575317383, 0.01060718297958374],
[0.1139642596244812, 0.0032819509506225586, 0.0, 5.82277774810791e-4, 0.0013980269432067871, 0.0020949840545654297],
[0.13029760122299194, 0.006624102592468262, 5.82277774810791e-4, 0.0, 1.7595291137695312e-4, 4.686713218688965e-4],
[0.1397092342376709, 0.008954286575317383, 0.0013980269432067871, 1.7595291137695312e-4, 0.0, 7.027387619018555e-5],
[0.14581435918807983, 0.01060718297958374, 0.0020949840545654297, 4.686713218688965e-4, 7.027387619018555e-5, 0.0]
]
>Pairwise cosine distance.
Examples
iex> x = Nx.iota({6, 6})
iex> y = Nx.reverse(x)
iex> Scholar.Metrics.Distance.pairwise_cosine(x, y)
#Nx.Tensor<
f32[6][6]
[
[0.2050153613090515, 0.21226388216018677, 0.22395789623260498, 0.24592703580856323, 0.30156970024108887, 0.6363636255264282],
[0.03128105401992798, 0.03429150581359863, 0.039331674575805664, 0.049365341663360596, 0.07760530710220337, 0.30156970024108887],
[0.014371514320373535, 0.01644366979598999, 0.020004630088806152, 0.02736520767211914, 0.049365341663360596, 0.24592703580856323],
[0.0091819167137146, 0.010854601860046387, 0.013785064220428467, 0.020004630088806152, 0.039331674575805664, 0.22395789623260498],
[0.006820023059844971, 0.008272230625152588, 0.010854601860046387, 0.01644366979598999, 0.03429150581359863, 0.21226388216018677],
[0.005507469177246094, 0.006820023059844971, 0.0091819167137146, 0.014371514320373535, 0.03128105401992798, 0.2050153613090515]
]
>Pairwise euclidean distance. It is equivalent to Scholar.Metrics.Distance.pairwise_euclidean(x, x)
Examples
iex> x = Nx.iota({6, 6})
iex> Scholar.Metrics.Distance.pairwise_euclidean(x)
#Nx.Tensor<
f32[6][6]
[
[0.0, 14.696938514709473, 29.393877029418945, 44.090816497802734, 58.78775405883789, 73.48469543457031],
[14.696938514709473, 0.0, 14.696938514709473, 29.393877029418945, 44.090816497802734, 58.78775405883789],
[29.393877029418945, 14.696938514709473, 0.0, 14.696938514709473, 29.393877029418945, 44.090816497802734],
[44.090816497802734, 29.393877029418945, 14.696938514709473, 0.0, 14.696938514709473, 29.393877029418945],
[58.78775405883789, 44.090816497802734, 29.393877029418945, 14.696938514709473, 0.0, 14.696938514709473],
[73.48469543457031, 58.78775405883789, 44.090816497802734, 29.393877029418945, 14.696938514709473, 0.0]
]
>Pairwise euclidean distance.
Examples
iex> x = Nx.iota({2, 3})
iex> y = Nx.reverse(x)
iex> Scholar.Metrics.Distance.pairwise_euclidean(x, y)
#Nx.Tensor<
f32[2][2]
[
[5.916079998016357, 2.8284270763397217],
[2.8284270763397217, 5.916079998016357]
]
>Computes the pairwise Minkowski distance.
Examples
iex> x = Nx.iota({2, 3})
iex> y = Nx.reverse(x)
iex> Scholar.Metrics.Distance.pairwise_minkowski(x, y)
#Nx.Tensor<
f32[2][2]
[
[5.916079998016357, 2.8284270763397217],
[2.8284270763397217, 5.916079998016357]
]
>Pairwise squared euclidean distance. It is equivalent to Scholar.Metrics.Distance.pairwise_squared_euclidean(x, x)
Examples
iex> x = Nx.iota({6, 6})
iex> Scholar.Metrics.Distance.pairwise_squared_euclidean(x)
#Nx.Tensor<
s64[6][6]
[
[0, 216, 864, 1944, 3456, 5400],
[216, 0, 216, 864, 1944, 3456],
[864, 216, 0, 216, 864, 1944],
[1944, 864, 216, 0, 216, 864],
[3456, 1944, 864, 216, 0, 216],
[5400, 3456, 1944, 864, 216, 0]
]
>Pairwise squared euclidean distance.
Examples
iex> x = Nx.iota({6, 6})
iex> y = Nx.reverse(x)
iex> Scholar.Metrics.Distance.pairwise_squared_euclidean(x, y)
#Nx.Tensor<
s64[6][6]
[
[5470, 3526, 2014, 934, 286, 70],
[3526, 2014, 934, 286, 70, 286],
[2014, 934, 286, 70, 286, 934],
[934, 286, 70, 286, 934, 2014],
[286, 70, 286, 934, 2014, 3526],
[70, 286, 934, 2014, 3526, 5470]
]
>Squared euclidean distance.
$$ D(x, y) = \sum_i (x_i - y_i)^2 $$
Options
:axes- Axes to calculate the distance over. By default the distance is calculated between the whole tensors.
Examples
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([3, 2])
iex> Scholar.Metrics.Distance.squared_euclidean(x, y)
#Nx.Tensor<
f32
4.0
>
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([1.0, 2.0])
iex> Scholar.Metrics.Distance.squared_euclidean(x, y)
#Nx.Tensor<
f32
0.0
>
iex> x = Nx.tensor([1, 2])
iex> y = Nx.tensor([1, 2, 3])
iex> Scholar.Metrics.Distance.squared_euclidean(x, y)
** (ArgumentError) tensors must be broadcast compatible, got tensors with shapes {2} and {3}
iex> x = Nx.tensor([[1, 2, 5], [3, 4, 3]])
iex> y = Nx.tensor([[8, 3, 1], [2, 5, 2]])
iex> Scholar.Metrics.Distance.squared_euclidean(x, y, axes: [0])
#Nx.Tensor<
f32[3]
[50.0, 2.0, 17.0]
>
iex> x = Nx.tensor([[6, 2, 9], [2, 5, 3]])
iex> y = Nx.tensor([[8, 3, 1]])
iex> Scholar.Metrics.Distance.squared_euclidean(x, y)
#Nx.Tensor<
f32
113.0
>