1. Introduction

Adding noise can be useful for simulating real-world scenarios where measurements are often accompanied with some level of random error.

2. Gaussian Noise

Gaussian noise, also known as normal noise, has several properties that make it a commonly used type of noise in various applications (including machine learning).

2.1 Statistical Properties

Gaussian noise is characterized by a normal distribution, which is well-studied and has known statistical properties.
This makes it easy to model and analyze mathematically.

2.2 Central Limit Theorem:

The Central Limit Theorem states that the sum (or average) of a large number of independent, identically distributed random variables, each with finite mean and variance, will be approximately normally distributed.
This property makes Gaussian noise a natural choice in many scenarios where the noise is a result of multiple independent factors.

2.3 Mathematical Simplicity:

The normal distribution has simple and well-defined mathematical properties, making it easy to work with in analytical and computational contexts.

2.4 Robustness in Estimation:

Many statistical estimation methods, including maximum likelihood estimation, assume that the underlying noise follows a Gaussian distribution.
This can lead to more robust parameter estimates when the actual noise is close to Gaussian.

2.5 Convenient in Machine Learning:

In machine learning, adding Gaussian noise can act as a form of regularization, preventing overfitting by introducing a controlled amount of randomness during training. It is also commonly used in generative models, such as Gaussian Mixture Models (GMMs).

3. The Function

Lets write a simple function named noise and add_noise that add Gaussian noise to an existing array.

import numpy as np
def noise(y, scale): 
    return np.random.normal(scale=scale, size=y.shape)

3.1 `noise` function

This function generates random noise using NumPy’s random.normal function.

y: numpy or pytorch array
scale: standard deviation of the normal distribution from which the noise is drawn.
output: array of random values with the same shape as the input array y.

def add_noise(x, mult, add): 
    return x * (1+noise(x, mult)) + noise(x, add)

3.2 `add_noise` function

This function uses the noise function to add noise to the input x:

* (1 + noise(x, mult)): The multiplicative noise is applied by multiplying x with.
+ noise(x, add): The additive noise is added directly to the result of the multiplicative part.

3.3 Differences:

The noise function generates:

random noise independently of any input array,
using a specified scale.
It’s a standalone function for generating random noise.

The add_noise function is specifically designed to:

apply noise to an input array x.
combines both multiplicative and additive noise components,
allowing for a more complex noise model.

4. Real World Applications

4.1 Regularization and Preventing Memorization:

Avoid and Discouraging Overfitting:

Adding random noise to the input data can act as a form of regularization, preventing the model from fitting the training data too closely.
This can improve the generalization of the model to new, unseen data.
Models that are too complex may memorize the training data instead of learning the underlying patterns.
Adding noise makes it more challenging for the model to memorize specific examples and encourages it to focus on general patterns.

4.2 Data Augmentation:

Increased Variability:

Introducing noise during training can artificially increase the variability in the dataset.
This can be particularly useful when dealing with limited training data, helping the model generalize better to different variations of the input.

4.3 Robustness Testing:

Model Robustness:

Adding noise during training can make the model more robust to variations and uncertainties in real-world data.
This is especially important when the model needs to perform well on data that may have different levels of noise or unexpected variations.

4.4 Stochasticity in Training:

Encouraging Exploration:

During the training process, introducing randomness can encourage the model to explore different parts of the parameter space.
This can be especially beneficial in reinforcement learning or optimization problems, helping to avoid getting stuck in local minima.