[Tensor] The dimensions or axes of a tensor in Python(Numpy, PyTorch)
Theory3D Tensor and Axes
When dealing with a 3D tensor, such as the one we had in the previous example, it can be conceptualized as a collection of 2D matrices or blocks, stacked along a new dimension. The axes represent the following:
[*] First Component (axis=0) - Batch Size / Number of Blocks:
[*]Direction along the batch size or the stack of matrices.
[*]When you have a 3D tensor, this first dimension essentially tells you how many blocks (matrices) you have.
[*]This axis essentially indexes the individual blocks in your collection.
[*]For example, if the shape is (2, 2, 3), the value 2 in the first position tells you there are 2 blocks.
[*] Second Component (axis=1) - Rows of Each Block (Matrix):
[*]Represents the rows within each matrix or block.
[*]When referring to axis 1, you are looking along the rows within each individual block.
[*]For each block (or matrix), you have multiple rows. In the example of shape (2, 2, 3), the second 2 refers to the fact that each block has 2 rows.
[*] Third Component (axis=2) - Columns of Each Block (Matrix):
[*]Represents the columns within each row of a matrix.
[*]When referring to axis 2, you are looking at the columns within each row of a matrix.
[*]In the example of shape (2, 2, 3), the 3 refers to the fact that each row has 3 columns.
Conceptual Visualization
Consider the example we used:
arr_3d = np.array([
[[ 1, 2, 3], # Block 0, Row 0
[ 4, 5, 6]], # Block 0, Row 1
[[ 7, 8, 9], # Block 1, Row 0
]# Block 1, Row 1
]) The shape of this array is (2, 2, 3), which can be described as follows:
[*]2 Blocks (first dimension) → We have 2 blocks or matrices.
[*]2 Rows per block (second dimension) → Each block has 2 rows.
[*]3 Columns per row (third dimension) → Each row has 3 columns.
Axis Summary:
[*] axis = 0: Refers to the batch size or block level.
[*]Example: Summing along this axis means combining all the corresponding elements from each block.
[*] axis = 1: Refers to the rows within each block.
[*]Example: Summing along this axis means summing across rows within each block.
[*] axis = 2: Refers to the columns within each row.
[*]Example: Summing along this axis means summing across columns within each row.
Practical Meaning
[*]The first axis (axis=0) typically represents a batch when you're dealing with data processing. For instance, in machine learning models, this is commonly the batch size—how many examples are being processed in parallel.
[*]The second axis (axis=1) represents the rows of each block (matrix). In data like images, this could represent the height of the image (number of rows of pixels).
[*]The third axis (axis=2) represents the columns of each block (matrix). For images, this could represent the width of the image (number of columns of pixels).https://i-blog.csdnimg.cn/direct/0f11889113cd4b4e90f2a5b1300590f4.png
As is shown in the above picture, dimension#3 is the axis 0 of a tensor in python
dimension#1 is the axis 1 of a tensor in python
dimension#2 is the axis 2 of a tensor in python
Examples
Consider the following 3D array:
import numpy as np
# Create a 3D array arr_3d = np.array([
[[ 1, 2, 3],
[ 4, 5, 6]],
[[ 7, 8, 9],
]
])
# The shape of arr_3d is (2, 2, 3) # It has 2 blocks, each with 2 rows and 3 columns. This 3D array can be visualized as having two 2x3 matrices (blocks).
.sum(0) - Summing along the First Axis (Combining Blocks)
[*]axis=0 means we are summing along the first axis, which corresponds to stacking the blocks on top of each other and summing along that dimension.
result_0 = arr_3d.sum(0)
print("Sum along axis 0:\n", result_0)
# Output: # [[ 8 10 12] # ] Explanation:
[*]Summing along axis 0 means adding the corresponding elements from each of the two blocks:
[*] = (first row of the result).
[*] = (second row of the result).
.sum(1) - Summing along the Second Axis (Combining Rows within Each Block)
[*]axis=1 means we are summing along the second axis, which corresponds to summing across the rows within each block.
result_1 = arr_3d.sum(1)
print("Sum along axis 1:\n", result_1)
# Output: # [[ 5 7 9] # ] Explanation:
[*]Summing along axis 1 means adding the rows together within each block:
[*]For the first block ([, ]):
[*] = .
[*]For the second block ([, ]):
[*] = .
.sum(2) - Summing along the Third Axis (Summing Across Columns)
[*]axis=2 means we are summing along the third axis, which corresponds to summing across the columns within each row.
result_2 = arr_3d.sum(2)
print("Sum along axis 2:\n", result_2)
# Output: # [[ 6 15] # [ 24 33]] Explanation:
[*]Summing along axis 2 means summing across the columns within each row:
[*]For the first block:
[*]First row: 1 + 2 + 3 = 6
[*]Second row: 4 + 5 + 6 = 15
[*]For the second block:
[*]First row: 7 + 8 + 9 = 24
[*]Second row: 10 + 11 + 12 = 33
.sum(-1) - Summing along the Last Axis (Equivalent to .sum(2) in a 3D Array)
[*]axis=-1 always refers to the last axis. In this case, the last axis is axis 2, which corresponds to summing across the columns within each row. Therefore, .sum(-1) will give the same result as .sum(2).
result_minus_1 = arr_3d.sum(-1)
print("Sum along axis -1:\n", result_minus_1)
# Output: # [[ 6 15] # [ 24 33]] Explanation:
[*]Since axis=-1 refers to the last axis, which is axis 2 in this case, it produces the same output as .sum(2):
[*]For the first block:
[*]First row: 1 + 2 + 3 = 6
[*]Second row: 4 + 5 + 6 = 15
[*]For the second block:
[*]First row: 7 + 8 + 9 = 24
[*]Second row: 10 + 11 + 12 = 33
Summary Table
MethodDescriptionOutput.sum(0)Sum across the first axis (combine blocks)[, ].sum(1)Sum across the second axis (combine rows)[, ].sum(2)Sum across the third axis (sum columns)[, ].sum(-1)Sum across the last axis (equivalent to .sum(2) for 3D tensor)[, ] These examples illustrate how different axes are used to reduce the tensor along specific dimensions. Summing along a particular axis effectively removes that dimension and produces a new array with fewer dimensions.
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