Mathematics of neural network

In this video, I will guide you through the entire process of deriving a mathematical representation of an artificial neural network. You can use the following timestamps to browse through the content. Timecodes 0:00 Introduction 2:20 What does a neuron do? 10:17 Labeling the weights and biases for the math. 29:40 How to represent weights and biases in matrix form? 01:03:17 Mathematical representation of the forward pass 01:10:29 Derive the math for Backward Pass. 01:11:04 Bringing cost function into the picture with an example 01:32:50 Cost function optimization. Gradient descent Start 01:39:15 Computation of gradients. Chain Rule starts. 04:24:40 Summarization of the Final Expressions 04:38:09 What’s next? Please like and subscribe. Link to the playlist Link to the e-book “Neural Networks and Deep Learning by Michael Nielson“ The mathematics behind neural networks, particularly feedforward networks, involves concepts from linear algebra, calculus, and optimization. Here is a step-by-step overview of the mathematics involved in the operation of a neural network: Representation of Neurons: In a feedforward neural network, neurons are represented as mathematical functions. Each neuron takes inputs, performs a weighted sum of those inputs, applies an activation function to the sum, and produces an output. Weighted Sum: The weighted sum of inputs for a neuron is calculated by multiplying each input by its corresponding weight and summing them up. Mathematically, for a neuron with n inputs, the weighted sum can be represented as: z = w₁x₁ w₂x₂ ... wₙxₙ Here, w₁, w₂, ..., wₙ represent the weights, and x₁, x₂, ..., xₙ represent the inputs. Activation Function: The weighted sum is then passed through an activation function to introduce non-linearity into the neural network. Common activation functions include sigmoid, ReLU, tanh, and softmax. The choice of activation function depends on the specific problem and network architecture. Forward Propagation: Forward propagation refers to the process of passing inputs through the neural network layer by layer, starting from the input layer and moving towards the output layer. Each layer performs the weighted sum and activation function operations described above. Loss Function: A loss function is used to measure the difference between the predicted output of the neural network and the actual output (ground truth) for a given input. The choice of loss function depends on the specific problem being solved. For example, mean squared error (MSE) is commonly used for regression tasks, while cross-entropy loss is often used for classification tasks. Backpropagation: Backpropagation is a key algorithm for training neural networks. It involves computing the gradients of the loss function with respect to the weights and biases of the network. This process utilizes the chain rule of calculus to efficiently propagate the gradients backward through the layers of the network. Gradient Descent: Once the gradients are computed, the weights and biases are updated using an optimization algorithm such as gradient descent. Gradient descent adjusts the weights and biases in the direction opposite to the gradient to minimize the loss function. Training Iterations: The process of forward propagation, backpropagation, and weight updates is repeated iteratively until the network’s performance reaches a satisfactory level. Each iteration is called an epoch, and the entire dataset is typically used for each epoch. By understanding these mathematical concepts and algorithms, you can implement a neural network from scratch by creating the necessary data structures and implementing the mathematical operations described above. However, it’s important to note that implementing a neural network from scratch requires a solid understanding of linear algebra, calculus, and programming, as well as knowledge of efficient algorithms and numerical optimization techniques. Mathematics behind neural networks Neural network workings Implementing neural networks from scratch Artificial neural network mathematics Understanding neural networks step by step Mathematics of artificial neural networks Backpropagation explained step by step Chain rule in neural networks Chain rule of partial differentiation in backpropagation Gradient computation in neural networks Deep learning mathematics Mathematics of feedforward networks Exploring the math behind neural networks Introduction to deep learning mathematics Neural network implementation math Mathematics behind neural networks. How does neural network work? Implement a neural network from scratch.