In neural networks, specific mathematical notation is used to represent different components. Key variables include:
- $X$: The input features.
- $W$: The weight parameter. Its dimensions vary; it can be a scalar ($W \in \mathbb{R}$) in simple models or a matrix in multi-neuron layers.
- $b$: The bias parameter. It can be a scalar ($b \in \mathbb{R}$) or a vector.
- $a$: The activation value. For example, $a^{[l]}_i$ denotes the activation of the $i$-th neuron in the $l$-th layer.
- $\hat{y}$: The predicted output value.

Mathematical Notation in Neural Networks

A neural network is typically structured with an input layer, one or more hidden layers, and an output layer. Its design is historically inspired by biological neurons: dendrites are modeled as inputs $$x_i$$, synapses as synaptic weights $$w_i$$, and the nucleus as an aggregator that computes a weighted sum $$y = \sum_i x_i w_i + b$$, potentially followed by a nonlinear activation function $$\sigma(y)$$. The resulting signal is then transmitted to the output, analogous to a biological axon. A positive weight reflects an excitatory connection, while a negative value signifies an inhibitory connection.

Neural Network Representation

http://datahacker.rs/neural-networks-representation/

University of Michigan - Ann Arbor

https://www.coursera.org/learn/neural-networks-deep-learning?specialization=deep-learning

Neural Networks and Deep Learning

(1) Standard neural networks: real estate, online advertising

(2) Convolutional neural networks: applied in image recognition including photo tagging

(3) Recurrent Neural Networks: applied in sequence data including speech recognition, machine translation

(4) custom/hybrid neural networks: complex applications such as autonomous driving

application of neural networks in supervised learning

deep-learning representation

Because we have access to large columns of data, more computational power, and advanced algorithms.

For big data, large neural networks perform better than traditional ML models (SVM, logistic regression...)

Why is deep learning taking off?

(1) initialize parameters
(2) Implement the forward propagation to get the prediction
(3) compute the loss
(4) Implement the backward propagation to compute the derivatives of the parameters
(5) update the parameters
(6) repeat steps (2) to (5) to reduce the cost.

The learning circle of the neural network

- Start off trying to have one neural network do multiple things (can actually lead to better performance)
- differences and similarities amongst tasks is utilized to output a better result

Multi-task Learning

Benchmarking is a way to analyze how well your neural network is performing on a given standard set.
-Ensembling
-Multicrop at test time
-using open source code


Benchmarking

• Single neuron == linear regression without applying activation(perceptron)

• Basically a single neuron will calculate weighted sum of input(W.T*X) and then we can set a threshold to predict output in a perceptron. If weighted sum of input cross the threshold, perceptron fires and if not then perceptron doesn’t predict.

What is a Neural Network(NN)?

The working memory is the system that enables the ability to explicitly hold and manipulate pieces of information that are used to achieve a goal. Unlike humans, neural networks do not have a working memory.

Working Memory

Beyond standard categorical and numeric variables, deep learning practitioners frequently work with myriad unstructured data formats. These complex data types include text strings for natural language processing, images and point clouds for computer vision, and audio data. Handling these diverse formats requires specialized approaches and advanced tools compared to traditional tabular data.

Learn Before

Related

Learn After