Tensors
- [Rice University - Brandon Martin]
- Overview
In mathematics, a tensor is a mathematical object that describes the relationship between sets of algebraic objects related to a vector space. Tensors can map between different objects such as vectors, scalars, and other tensors.
Tensors are a generalization of scalars and vectors. A scalar is a zero rank tensor, and a vector is a first rank tensor.
Tensors can be used to describe physical properties. For example, a tensor can describe the relationship between one vector and another. If you start with one vector, such as a force, and mathematically apply it to a tensor, then you get another vector. That vector might be, for example, the stress caused by the force.
The word tensor comes from the Latin word tendere meaning "to stretch". Tensors provide a mathematical framework for solving physics problems in areas such as elasticity, fluid mechanics, and general relativity.
In essence, tensors are a powerful mathematical tool for representing and working with multi-dimensional data, and their recursive nature allows for flexible and efficient computation and manipulation across various scientific and computational domains.
Please refer to the following for more details:
- Wikipedia: Tensor
- Understanding Tensors
Tensors are mathematical objects that generalize vectors and matrices to n-dimensional arrays. They are fundamental in various fields, including physics, computer science, and machine learning, where they represent and manipulate data across multiple dimensions.
In essence, tensors are containers for data, and their dimensionality (number of axes or dimensions) determines their rank.
1. Tensors as N-dimensional Arrays:
- A tensor can be thought of as an extension of vectors (1D arrays) and matrices (2D arrays) to higher dimensions.
- For example, a scalar (a single number) is a 0-dimensional tensor, a vector is a 1-dimensional tensor, and a matrix is a 2-dimensional tensor.
- Tensors are used to represent data with more than two dimensions, like color images (3D tensors) or video data (4D tensors).
2. Recursive Representation:
- The concept of tensors can be recursively applied. A tensor of any rank can be built up from tensors of lower rank.
- This means that a 3D tensor can be seen as a collection of 2D tensors (matrices), which in turn can be seen as collections of 1D tensors (vectors), and so on.
- This recursive nature allows for a hierarchical and structured way to represent and manipulate complex data.
3. Tensors in Different Fields:
- Physics and Differential Geometry: Tensors are defined in a way that ensures their properties remain consistent under coordinate transformations. This is crucial for describing physical laws and properties that are independent of the chosen coordinate system.
- Computer Science and Machine Learning: In these contexts, tensors are primarily used as data structures to represent and process data. They are the fundamental building blocks for many algorithms, especially in deep learning.
- Tensor Networks: In fields like quantum physics and applied mathematics, tensor networks are used to represent and manipulate tensors efficiently, especially for high-dimensional data.
4. Key Characteristics:
- Shape: The shape of a tensor defines the number of elements along each dimension.
- Rank: The rank of a tensor is determined by the number of dimensions.
- Data Type: Tensors typically store data of a single type (e.g., integers, floating-point numbers).
5. Relationship to Arrays:
- While tensors are often implemented as n-dimensional arrays in programming languages, they are more than just arrays.
- Tensors have a mathematical definition and properties that are independent of their implementation as arrays.
- Arrays are the concrete data structures used to represent tensors in code, and libraries like NumPy (in Python) are commonly used for array manipulation.
- The Tensor: The Fundamental Data Structure of ML
Tensors are containers for data, particularly in the context of machine learning (ML) and deep learning (DL).
Think of tensors as specialized containers that efficiently organize and manipulate data for ML tasks, especially those involving complex and high-dimensional information.
- Multi-dimensional Arrays: Tensors are essentially multi-dimensional arrays, which means they can store data in various formats, such as scalars (0D), vectors (1D), matrices (2D), and higher-dimensional arrays (3D and beyond).
- Data Representation: This structure is essential for representing complex data like images, audio, and text, which often require multiple dimensions to capture their intricacies. For example, a color image can be represented as a 3D tensor with dimensions for height, width, and color channels.
- Fundamental Data Structure: Tensors serve as the fundamental data structure in popular machine learning frameworks like TensorFlow and PyTorch.
- Efficient Computation: Tensors are optimized for efficient numerical computations, especially on GPUs, which are crucial for speeding up the training and inference processes in deep learning models.
- Data Flow: In frameworks like TensorFlow, tensors represent the data flow through a computation graph, enabling the definition, optimization, and efficient evaluation of mathematical expressions involving these multi-dimensional arrays.
[More to come ...]
