Tensor Explained with Python Numpy Examples

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In this post, you will learn about the concepts of Tensor using Python Numpy examples with the help of simple explanation. When starting to learn deep learning, you must get a good understanding of the data structure namely tensor as it is used widely as the basic data structure in frameworks such as tensorflow, PyTorch, Keras etc.

Simply speaking, Tensor is a container of data. The data can be numerical or character. However, mostly, tensors hold numbers or numerical dataTensors can be represented as an array data structure. In this post, you will learn about how to express tensor as 1D, 2D, 3D Numpy array. Before going ahead and see the examples of 1D, 2D and 3D tensors, lets understand the key features of tensors: 

  • Rank or Number of Axes: Number of axes of a tensor is also termed as the rank of the tensor. Simply speaking, the axes or rank of the tensor represented as an array is number of indices that will be required to access a specific value in the multi-dimensional array aka tensor. Accessing a specific element in a tensor is also called as tensor slicing. The first axis of the tensor is also called as a sample axis.
  • Shape: Tuple of integers representing the dimensions that the tensor have along each axes.

Before getting into the details, lets look at the diagram given below which represents 0D, 1D, 2D and 3D tensors. 0D tensor is a scalar data or simply a number.

Fig 1. Pictorial representation of a Tensor

Here are some examples of 2D, 3D, 4D and 5D tensors:

  • Vector data— 2D tensors of shape (samples, features)
  • Timeseries data or sequence data— 3D tensors of shape (samples, timesteps, features)
  • Images— 4D tensors of shape (samples, height, width, channels or samples, channels, height, width)
  • Video— 5D tensors of shape (samples, frames, height, width, channels or samples, frames, channels, height, width)

1D Tensor (Vector) Example

A one-dimensional array also called as vector can be termed as a 1D-tensor. 1D tensor represents the tensor of rank/axes as 1. Take an example of an array representing the tensor using the following 1D Numpy array:

#
# Tensor having rank / axes = 1 and dimension along the axis as 3
#
x = np.array([56, 183, 1])
#
# Dimension = 1
# Shape = 1,
#
x.ndim, x.shape

In the above example, the axes or rank of the tensor x is 1. The axes of the tensor can be printed using ndim command invoked on Numpy array. In order to access elements such as 56, 183 and 1, all one needs to do is use x[0], x[1], x[2] respectively. Note that just one indices is used. Printing x.ndim, x.shape will print the following: (1, (3,)). This represents the fact that the tensor x has the axes or rank has 1 and there are three dimension on that axes.

2D Tensor (Matrix) Example

A two-dimensional array also called as matrix can be termed as a 2D-tensor. 2D tensor represents the 2-dimensional array which is nothing but a matrix. As like matrices, the two-dimensional array can be represented as a set of rows and columns. Here is the code for 2D numpy array representing 2D tensor:

#
# Tensor having rank / axes = 2 and dimension along each axis as 3
#
x = np.array([[56, 183, 1],
             [78, 178, 2],
             [50, 165, 0]])
#
# Dimension = 2
# Shape = 3, 3
#
x.ndim, x.shape

Printing x.ndim, x.shape will print the following: (2, (3,3)). This represents the fact that the tensor x has the axes or rank has 2 and there are three dimension on each of the axes. n order to access elements such as 56, 178 and 50, one would require to use the following respectively: x[0][0], x[1][1], x[2][0]. Note that you will need two indices to access a particular number.

3D Tensor Example

A three-dimensional array can be termed as a 3D-Tensor. Here is the code representing the 3D tensor:

#
# Tensor having rank / axes = 3 and dimension along the axis as 3
#
x = np.array([[[56, 183, 1],
               [65, 164, 0]],
              [[85, 176, 1],
               [44, 164, 0]]])
#
# Dimension = 1
# Shape = 1,
#
x.ndim, x.shape

Printing x.ndim, x.shape will print the following: (2, (3,3)). This represents the fact that the tensor x has the axes or rank has 2 and there are three dimension on each of the axes. n order to access elements such as 56, 178 and 50, one would require to use the following respectively: x[0][0], x[1][1], x[2][0]. Note that you will need two indices to access a particular number.

Understanding Tensor using MNIST Image Example

Here is an explanation of tensor axes / rank and shape using Keras and MNIST dataset.

from keras.dataset import mnist
#
# Load the training and test images MNIST Dataset
#
(train_images, train_labels), (test_images, test_labels) = mnist.load_data()
#
# Accessing the tensor rank / axes and shape
#
print(train_images.ndim, train_images.shape)

Executing the above would print the information about the tensor rank / axes and the shape – 3 (60000, 28, 28). It represents the fact that training images is stored in a 3D tensor of axes 3 and having shape representing 60,000 matrices of 28×28 integers.

Conclusions

Here is what you learned about tensors with the help of simple Python Numpy code samples.

  • Tensor can be represented as a multi-dimensional array.
  • Numpy np.array can be used to create tensor of different dimensions such as 1D, 2D, 3D etc.
  • A vector is 1D tensor, a matrix is a 2D tensor. 0D tensor is a scalar or a numerical value.
  • Accessing a specific value of tensor is also called as tensor slicing.
  • Two key attributes of tensors include A. rank or axes of tensor B. Shape of the tensor
  • ndim and shape when invoked on Numpy array gives the axes / rank and shape of the tensor respectively.
Ajitesh Kumar
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