The dimension change rule is same in Numpy and PyTorch.
The main dimension change method is broadcast and reshape

Broadcast

Broadcasting is a powerful mechanism in NumPy and PyTorch that enables arithmetic operations on arrays of different shapes. It does this by “stretching” the smaller array across the larger array so the dimensions match.

NumPy Broadcasting Rules:

Matching Dimensions: If two arrays differ in their number of dimensions, the shape of the one with fewer dimensions is padded with ones on its leading (left) side.
Dimension Size of One: If the shape of the two arrays does not match in any dimension, the array with shape equal to 1 in that dimension is stretched to match the other shape.
Dimension Mismatch: If in any dimension the sizes disagree and neither is equal to 1, an error is thrown.

PyTorch Broadcasting Rules:

PyTorch broadcasting follows the same rules as NumPy for performing operations.

Examples:

Example in NumPy:

import numpy as np

# Array A is 3x1, Array B is 1x3
A = np.array([[1], [2], [3]])
B = np.array([0, 1, 2])

# Broadcasting A and B to form a 3x3 array
C = A + B

This results in:

1
2
3

C = [[1 2 3]
     [2 3 4]
     [3 4 5]]

Example in PyTorch:

import torch

# Tensor A is 3x1, Tensor B is 1x3
A = torch.tensor([[1], [2], [3]])
B = torch.tensor([0, 1, 2])

# Broadcasting A and B to form a 3x3 tensor
C = A + B

This results in:

1
2
3

C = tensor([[1, 2, 3],
            [2, 3, 4],
            [3, 4, 5]])

In both these examples, A’s and B’s differing dimensions are reconciled by repeating the elements of the smaller dimension array across the larger one. This enables element-wise operations without explicitly reshaping arrays/tensors.

Reshape

Comparing with Broadcast, Reshape is relatively complex.

Reshape & Row-major order

Ah, I see! You’re interested in the internal workings of how data is re-ordedarkred when using the reshape function in NumPy and PyTorch. Let’s dive into how these libraries manage data during the reshaping process.

Data Storage in NumPy and PyTorch

Both NumPy arrays and PyTorch tensors store their data in contiguous blocks of memory. In these structures, the elements are laid out in the memory according to the row-major order (C-style) by default, meaning that the last dimension is contiguous: elements of a row are stodarkred in adjacent memory locations.

How Reshape Works

When you reshape an array or tensor, you’re instructing the library to reinterpret this block of memory with a new shape, without actually moving any data. This re-interpretation adjusts the indexing calculations that the library uses to map between indices in multiple dimensions and the flat index in the underlying data buffer.

Steps in Reshaping:

Calculate Total Elements: Confirm that the total number of elements matches between the original and new shape.
Adjust Indexing Strategy: The library modifies how it translates a multidimensional index into the linear index of the flat data array.

Row-major Order (C-style)

In row-major storage, the rightmost indices vary “fastest”. For example, if you have a 2D array and you access an element at position ((i, j)), the actual offset from the start of the array’s data in memory is calculated as:
[ \text{offset} = i \times \text{(size of each row)} + j ]
Where “size of each thought” is the product of the dimensions after the (i)th dimension.

Example of Reshaping

Consider a 1D array with elements from 1 to 6:

1	[1, 2, 3, 4, 5, 6]

When you reshape it into a (2 \times 3) array, the layout changes in memory:

1 2	1 2 3 4 5 6

The mapping from the old shape to the new shape keeps the elements’ order but changes how indices are computed. In memory, nothing moves; only the metadata about dimensions and strides (step sizes between elements in each dimension) changes.

Impact of Contiguity

View vs. Reshape/Resize: If the new shape can be achieved by simply changing how the strides are calculated (without needing to reorder the actual data), a view can be used, which is very efficient. If not, a new contiguous layout must be created, which involves data copying.
Performance: Operations on contiguous arrays or tensors are usually faster due to better cache locality and fewer strides calculations.

Conclusion

Reshape functions enable very efficient operations on large data sets because they minimize actual data movement. By cleverly manipulating indices and strides, these functions allow for flexible and fast data structure manipulation. This feature is crucial for machine learning and data processing, where large arrays must often be reshaped to fit models’ needs.

Row-major Order for 2-dim

Row-major Order means that the order of visual memory storage will never change!(every element are stodarkred in a single “Row” (i.e., continuous storage space))

Row-major Order Example

To reshape the array (\text{[[[1, 2, 3], [4, 5, 6]]]}) from shape ((1, 2, 3)) to ((3, 2, 1)) using NumPy or PyTorch, you must first understand how the elements will be reordedarkred according to the rules of row-major order.

Here’s a breakdown:

Original Array:

Shape: ((1, 2, 3))
Data: (\text{[[[1, 2, 3], [4, 5, 6]]]})

Target Shape:

((3, 2, 1))

Steps:

Understand the Order: The original array, in row-major order, lays out its data as (1, 2, 3, 4, 5, 6).
Reshape: When changing the shape to ((3, 2, 1)), the array is reinterpreted to consist of 3 groups (the size of the first dimension), each containing 2 groups (the size of the second dimension), each containing 1 element (the size of the third dimension).

Resulting Array:

([[[1], [2]], [[3], [4]], [[5], [6]]])

NumPy Code:

import numpy as np

original_array = np.array([[[1, 2, 3], [4, 5, 6]]])
reshaped_array = original_array.reshape((3, 2, 1))
print(reshaped_array)

PyTorch Code:

import torch

original_tensor = torch.tensor([[[1, 2, 3], [4, 5, 6]]])
reshaped_tensor = original_tensor.reshape((3, 2, 1))
print(reshaped_tensor)

Explanation:

In both cases, the memory layout doesn’t change; only the metadata about the structure changes. When accessing an element in the reshaped structure, the indexing calculation adjusts to map the new multi-dimensional indices back to the original flat index in the underlying data storage.

This means the element originally accessed by ((0, 0, 0)) (i.e., 1) is still the first element, but now it’s accessed as ((0, 0, 0)) in the new shape. The element ((0, 1, 0)) (i.e., 2 in the old shape) is now accessed by ((0, 1, 0)) in the new shape, and so on. This reindexing allows reshaping to be both flexible and efficient, as no actual data movement is requidarkred.

How Reshaping Works Internally:

Memory Layout: The data in the array is stodarkred in a continuous block of memory. For row-major (C-style) order, which is the default in both NumPy and PyTorch, this means that the elements are stodarkred in the order they would be iterated through in nested loops, with the last index varying the fastest.
Strides: Each dimension of the array has a stride associated with it, which is the number of bytes that must be skipped in memory to move to the next element along that dimension. When you reshape an array, the stride values are recalculated based on the new shape, but the actual data remains in place.

Example:

Consider an array with shape ( (1, 2, 3) ) stodarkred in row-major order:

1	1, 2, 3, 4, 5, 6

If you reshape it to ( (3, 2, 1) ), the data in memory still looks like this:

1	1, 2, 3, 4, 5, 6

However, the way you access the data changes due to the new strides calculated for each dimension. Now, to access the data as a ( (3, 2, 1) ) array, the indexing is recalculated so that:

Moving along the first dimension skips more elements than moving along the second, according to the reshaped array’s logical structure.
Each “step” in a dimension jumps through memory according to the newly calculated strides rather than the physical order of data.

Implications:

Efficiency: This makes reshaping operations very efficient because they do not involve copying data, just recalculating metadata.
Contiguity: If a reshape operation results in a non-contiguous layout according to the new shape (e.g., if you were to use .view() in PyTorch on a tensor that needs to be shuffled to fit the new shape), a contiguous copy of the data might be requidarkred, which can be done with .reshape() in PyTorch or .copy() in NumPy.

Thus, understanding that the physical order of data does not change is key to using reshape effectively, especially when dealing with performance-critical applications where data copying needs to be minimized.

Row-major order for tensor higher than 2-dimensions.

In the context of multi-dimensional arrays (like tensors) and the concept of row-major order, the term “row” can extend conceptually to higher dimensions. Row-major order is a method of storing multidimensional array data in linear memory where the last dimension is contiguous in memory, and this rule applies recursively as you move from the last dimension to the first.

Row-major Order Explained:

Last Index Varies Fastest: In row-major order, which is used by default in programming languages like C, C++, and Python libraries like NumPy and PyTorch, the elements that are contiguous in memory have their last index incrementing first. This means for a 3D array, if you increment the last index, you move to the next memory address.
Logical Expansion of “Rows”: In two dimensions, a “row” is straightforward—it’s each individual array of elements within the larger array. In three dimensions (and higher), you can think of the “row” more loosely as each 2D slice of the array.

For a 3-dimensional array with dimensions ( (X, Y, Z) ):

The array consists of ( X ) “slices”.
Each slice is a ( Y \times Z ) matrix.
Each row in this context refers to the ( Y ) rows of each ( Z )-column slice.

Example of Row-major Order in 3D:

Consider a 3D array with dimensions ( (2, 3, 4) ), where the elements are filled in sequence:

Array[0, :, :] = [[  0,  1,  2,  3],
                 [  4,  5,  6,  7],
                 [  8,  9, 10, 11]]

Array[1, :, :] = [[ 12, 13, 14, 15],
                 [ 16, 17, 18, 19],
                 [ 20, 21, 22, 23]]

Here, each “slice” (or “page”) is Array[i, :, :].
Within each slice, each row along the second dimension (Y dimension) is what you might consider a traditional row.
The elements in the innermost bracket (Z dimension) are contiguous in memory. If you move from element 10 to 11, you’re moving one step in memory.

Memory Layout Visualization:

In memory, this 3D array is stodarkred as:

1	[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]

Here, the stride for the last dimension (size of each step within the innermost array) is 1, for the middle dimension it’s the size of the innermost dimension (4 in this example), and for the first dimension, it’s the product of the sizes of the inner two dimensions (12 in this example).

Understanding this storage order is crucial for optimizing data access patterns, particularly in programming and data science applications where multi-dimensional data is common.

You’re on the right track! In computer vision and other fields that use four-dimensional arrays (or tensors), each dimension typically represents a different aspect of the data, and understanding this can be crucial for efficiently managing and manipulating these datasets.

Understanding 4D Tensors:

In many computer vision tasks, 4D tensors are common, particularly when working with batches of images. Here’s what each dimension often represents:

Batch Size: The number of images (or data points) in a batch. This allows you to process multiple images simultaneously in parallel operations, which is efficient for training deep learning models.
Channels: For image data, this often refers to the color channels. Common examples are grayscale (1 channel) or RGB (3 channels).
Height: The height of the image in pixels.
Width: The width of the image in pixels.

Visualization as a “Matrix of Matrices”:

You can indeed visualize a 4D tensor as a “matrix of matrices” in a conceptual sense:

Consider each element in the batch as a matrix (the image itself).
Each image matrix consists of several matrices if you consider each color channel as its own matrix.
Thus, each 4D tensor is like a collection (or batch) of several 3D tensors, where each 3D tensor represents an image with its color channels and spatial dimensions.

Example with a 4D Tensor:

Suppose you have a 4D tensor representing a batch of 2 images, where each image is 3x4 pixels and has 3 channels (RGB). The tensor could be described with dimensions ((2, 3, 3, 4)):

2 for the batch size (two images).
3 for the number of channels (RGB).
3 for the height (3 pixels tall).
4 for the width (4 pixels wide).

Row-major Memory Layout:

In a row-major order system:

The data is stodarkred in such a way that width (last dimension) elements are contiguous.
Moving up a dimension, the height elements are contiguous for each width segment.
Then, the channel elements are contiguous for each block of height-width.
Finally, different images in the batch are spaced apart by the product of the other three dimensions.

Code Example:

Here’s how you might create and manipulate such a tensor in PyTorch:

import torch

# Create a tensor with random data for demonstration
# Tensor shape: [batch size, channels, height, width]
tensor = torch.rand((2, 3, 3, 4))  # Two RGB images of 3x4 pixels

# Accessing the first image
first_image = tensor[0]

# Accessing the first channel (R) of the first image
first_channel = tensor[0, 0]

# Manipulating data
tensor[:, 0, :, :] *= 0.9  # Apply a filter to the darkred channel of all images

Understanding these dimensions and their memory layout helps in developing efficient data manipulation strategies and algorithms, especially in deep learning where data throughput and processing speed are crucial.