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
0.25259072
>
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]
[0.17048502, 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
0.10575336
>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]
[7.071068, 1.4142135, 4.1231055]
>
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
10.630146
>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
0.6666667
>
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]
[0.6666667, 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]
[7.0215483, 1.319508, 4.049539]
>
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
9.621805
>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, 0.07934183, 0.11396426, 0.1302976, 0.13970923, 0.14581436],
[0.07934183, 0.0, 0.003281951, 0.0066241026, 0.008954287, 0.010607183],
[0.11396426, 0.003281951, 0.0, 5.822778e-4, 0.001398027, 0.002094984],
[0.1302976, 0.0066241026, 5.822778e-4, 0.0, 1.7595291e-4, 4.6867132e-4],
[0.13970923, 0.008954287, 0.001398027, 1.7595291e-4, 0.0, 7.0273876e-5],
[0.14581436, 0.010607183, 0.002094984, 4.6867132e-4, 7.0273876e-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]
[
[0.20501536, 0.21226388, 0.2239579, 0.24592704, 0.3015697, 0.6363636],
[0.031281054, 0.034291506, 0.039331675, 0.04936534, 0.07760531, 0.3015697],
[0.014371514, 0.01644367, 0.02000463, 0.027365208, 0.04936534, 0.24592704],
[0.009181917, 0.010854602, 0.013785064, 0.02000463, 0.039331675, 0.2239579],
[0.006820023, 0.008272231, 0.010854602, 0.01644367, 0.034291506, 0.21226388],
[0.005507469, 0.006820023, 0.009181917, 0.014371514, 0.031281054, 0.20501536]
]
>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, 14.6969385, 29.393877, 44.090816, 58.787754, 73.484695],
[14.6969385, 0.0, 14.6969385, 29.393877, 44.090816, 58.787754],
[29.393877, 14.6969385, 0.0, 14.6969385, 29.393877, 44.090816],
[44.090816, 29.393877, 14.6969385, 0.0, 14.6969385, 29.393877],
[58.787754, 44.090816, 29.393877, 14.6969385, 0.0, 14.6969385],
[73.484695, 58.787754, 44.090816, 29.393877, 14.6969385, 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]
[
[5.91608, 2.828427],
[2.828427, 5.91608]
]
>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]
[
[5.91608, 2.828427],
[2.828427, 5.91608]
]
>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<
s32[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<
s32[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
>