Nonlinear Mapping (NLM)

TL;DR

Transforms input into output using nonlinear functions to model relationships that linear functions cannot.
Common in machine learning and image processing; examples include neural networks and nonlinear filters such as the median filter.
Offers greater representational power than linear models but can be harder to interpret, more sensitive to noise, and more computationally demanding.

Definition

Nonlinear mapping (NLM) is a mathematical technique that involves the use of nonlinear functions to transform input data into output data.

Explanation

NLM applies nonlinear functions to input data so the resulting output can capture complex relationships and interactions that linear functions cannot represent. In multilayer systems such as neural networks, inputs pass through successive processing units (neurons) that apply nonlinear functions to extract features and make predictions. In image processing, nonlinear filters apply nonlinear operations to pixel neighborhoods to enhance or modify appearance.

Common nonlinear functions used in NLM include sigmoidal functions (with an “S” shaped curve), ReLU (Rectified Linear Unit) functions, and polynomial functions. The choice of function depends on the application and the data characteristics.

Advantages of NLM include the ability to model complex, nonlinear relationships and often produce more accurate predictions or aesthetically pleasing image results compared with linear approaches. Challenges include reduced interpretability relative to linear models, greater sensitivity to noise or small input variations, and increased computational cost for training and evaluation.

Examples

Neural network

A neural network is a machine learning algorithm that uses multiple layers of nonlinear functions to process input data and make predictions. Input data is fed through multiple layers of neurons, each applying nonlinear functions to extract features and enable pattern recognition — for example, classifying images of animals based on features such as size, shape, and color.

Nonlinear filters (median filter)

Nonlinear filters in image processing modify pixel values by applying nonlinear functions to neighborhoods. One common example is the median filter, which replaces the value of each pixel with the median value of the pixels in its neighborhood; it is often used to remove noise or unwanted artifacts from an image.

Use cases

Machine learning, where relationships between input and output data are often highly nonlinear and difficult to represent with linear models.
Image processing, for tasks such as noise removal and appearance enhancement.
Other areas where input–output relationships are complex and not easily represented by linear functions.

Notes or pitfalls

Nonlinear models can be more difficult to interpret than linear models due to complex input–output relationships.
Nonlinear functions can amplify small differences or variations in input data, making models more sensitive to noise and interference.
Training and evaluation of nonlinear models can require more computational power and time because of the increased complexity of the functions involved.

Neural network
Nonlinear filter
Median filter
Sigmoidal function
ReLU (Rectified Linear Unit)
Polynomial function