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Table of contents
1.
Introduction
2.
Prerequisites
3.
Representation Power
4.
Concept of Complex Functions
5.
Demonstration
6.
Frequently Asked Questions
6.1.
What do we understand by linear separable data?
6.2.
What is the most important advantage of using neural networks?
6.3.
Why is it required to use neural networks to express complex functions?
6.4.
What is the universal approximation theorem?
7.
Conclusion
Last Updated: Mar 27, 2024

Representation Power of Neural Networks

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Speaker
Prerita Agarwal
Data Specialist @
23 Jul, 2024 @ 01:30 PM

Introduction

We are familiar with neural networks and their innumerable accomplishments in fields ranging from data science to computer vision. They have a reputation for excelling at generalisation-based complex problem-solving. They are excellent in roughly simulating the behaviour of any complex function mathematically. We are expected to be familiar with basic forwarding and backward propagation fundamentals. Let us learn how to approximate a neural network's capabilities using a different visual explanation that uses fundamental math and associated analyses.

Prerequisites

The blog assumes that readers are familiar with perceptrons and MP neurons (McCulloch-Pitts Neuron), two types of simple neural networks. To quickly review the two, the reader might wish to look at our previous blogs on the topic.

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Representation Power

In order to estimate the presented function, we will mathematically examine the representation power of a particular neural network. The ability of a neural network to construct well-defined, precise decision boundaries for a class and to assign the appropriate labels to a given instance is referred to as representation power. This article will look at a visual method for understanding how to approximate a neural network's behaviour. They are intimately correlated with the neural network's representation of Power.

Concept of Complex Functions

Let us use the construction of a bridge as an example. It might be considered a highly complex output. We want to create something extremely complex from scratch.

Bride Image

We do not begin by constructing the entire bridge at once. What we do is begin with the fundamental building component that is the "Brick ".

Brick and Mortor Image

We lay the foundation of the bridge. The first layer follows. We add another layer to that and keep doing so until we get this incredibly complex result. 

JCB, Bridge Image

As a result, everything involved in making this output has become simple and essential. They make up the pillars. 

Bride Image
Pillars

We successfully combine them to get this complex result: a bridge. Additionally, we can now have a variety of bridges and different styles of pillars using the same building blocks.

Demonstration

Let us take a mathematical example of it. We are interested in complex functions rather than bricks and bridges. The sigmoid neuron functions like a brick here when looking at this through mathematics. It is the cornerstone of complex functions.

Formula
Formula
Formula

We can now arrange these building blocks to produce the results we want.

Neural Network

In addition, the inputs x1, x2,...., xn would also affect the final output, y_hat. Given that they undergo several transitions—many of them at each layer—these inputs will have a very complex function. The dotted edges and arrows linking two neurons in the figure above have weights spread across numerous layers. The output of those transitions is merged in many cases.
No matter what the actual function is, we might approximate it with various arrangements. In other words, a Deep Neural Network should be able to estimate any function between the input and the output if it has a given number of hidden layers. This is also referred to as the Universal Approximation Theorem, which illustrates the Deep Neural Network's capacity for representation.

Frequently Asked Questions

What do we understand by linear separable data?

Suppose a straight line can be used to distinguish between beneficial and detrimental objects in a two-dimensional dataset. In that case, we claim that the dataset is linearly separable. It makes no difference if there are multiple such lines. Assume that a group of data points may be linearly isolated from the other data points when dispersed in a 2D plane. The data points are referred to as linearly separable in that situation.

What is the most important advantage of using neural networks?

Neural networks are capable of independent learning and can generate output independent of their input. Instead of storing the input in a database, its networks do so. Therefore, the loss of data has no impact on how it functions.

Why is it required to use neural networks to express complex functions?

The real-world data can typically be separated non-linearly. A more complicated function is needed to represent the input and output relationship.

What is the universal approximation theorem?

According to the Universal Approximation Theorem, a deep neural network should be able to approximate a function that links the input to the output with a specific number of hidden layers.

Conclusion

The blog explains why neural networks that simulate complex processes are needed to connect the input to the output. The two main lessons are how to recognise non-linearly separable data and depict complex functions using a sigmoidal function, which is simpler to understand. Our challenge involves dealing with real-world inputs and a non-linear relationship between the input and the outcome. We use sigmoid neurons as a fundamental component of these operations. We combine a number of these neurons to create a network that can roughly represent the relationship between input and output.

Check out this article - Padding In Convolutional Neural Network

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