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What is Gaussian function in neural networks?

What is Gaussian function in neural networks?

Normalized Gaussian curves with expected value μ and variance σ2. The corresponding parameters are , b = μ and c = σ.

The Gaussian function is a widely used activation function in neural networks, particularly in machine learning and artificial intelligence. Moreover, the Gaussian function is a type of probability density function that is symmetric about the mean and has a bell shape, resembling the shape of a normal distribution. The Gaussian function models the distribution of random variables that are continuous and have a normal distribution.

A Gaussian function is a function of the base form:

In neural networks, the Gaussian function becomes typically used as a radial basis function. Radial basis functions used in artificial neural networks compute the distance between input data points and the center of the radial basis function. The Gaussian function is a popular radial basis function because it is smooth, differentiable, and has a unique peak, making it useful for clustering and similarity measures.

The Gaussian function becomes defined by the formula:

f(x) = exp(-(x – μ)^2 / 2σ^2)

where x is the input to the function, μ is the mean or center of the distribution, and σ is the standard deviation or spread of the distribution. The output of the function is a value between 0 and 1, with the maximum value of 1 occurring at x = μ.

In neural networks, the Gaussian function is often used as a probability distribution function to model the output of a neuron.

The output of the neuron becomes calculated as the weighted sum of the input data points. And the output then passes through the Gaussian function to obtain the final output value. The Gaussian function is useful for this purpose because it can model the probability distribution of the output values of the neuron, allowing the neural network to make probabilistic predictions.

The Gaussian function, also used in convolutional neural networks (CNNs) as an activation function. In this context, the Gaussian function becomes referred to as the softplus function, defined as:

f(x) = log(1 + exp(x))

The softplus function, similar to the rectified linear unit (ReLU) function. Which commonly becomes used as an activation function in CNNs. The advantage of the softplus function over the ReLU function is that it is smoother, which can help prevent overfitting in some cases.

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3d plot of a Gaussian function with a two-dimensional domain
Kopak999 – Own work
Created in Python with Numpy and Matplotlib.

In summary, the Gaussian function is a commonly used activation function in neural networks due to its smoothness, differentiability, and ability to model probability distributions.

Its use as a radial basis function makes it useful for clustering and similarity measures, while its use as an activation function in CNNs provides a smooth and non-linear function for processing image data.

In the review of the paper “Robust Rolling PCA: Managing Time Series and Multiple Dimensions,” the paper proposes a new method for performing Principal Component Analysis (PCA) on time series data. The authors argue that the traditional method of PCA is not suitable for time series data because it does not take into account the temporal correlation between data points. It uses a sliding window approach to perform PCA on time series data. The authors demonstrate that the RRPCA method is more effective at capturing the underlying structure of the time series data than the traditional method of PCA.

The paper is well-written and provides a detailed description of the RRPCA method. The authors provide a theoretical framework for the method and provide several examples to demonstrate its effectiveness. The authors also compare the RRPCA method to other methods for performing PCA on time series data and demonstrate that the RRPCA method outperforms these methods.

In conclusion, the Gaussian function is a powerful tool in neural networks. One that can become used to introduce nonlinearity, smoothness, and regularization into the output. The function has many applications in machine learning. Moreover, including as an activation function, a radial basis function, a kernel function, and as the basis for the output probability distribution in probabilistic neural networks. Lastly, the use of Gaussian functions in neural networks is likely to continue to grow in importance. Furthermore, as machine learning techniques become more widely used in a range of applications.

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4400158

What is Gaussian function in neural networks? Written by Jiatong Li