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Questions and Answers
What is the purpose of finding the right set of offsets (δ𝑖) in the context of the neural network?
Adjust the output of the hidden layer throughout the training phase
How is each row of 𝑌ℎ related to the output neuron i in the output layer?
Represents the output of all neurons inside the hidden layer (except h) to output neuron i
Why might the linear system described in the text not have a solution?
Linear system might not have a solution due to the lack of compatibility between equations
What is the objective of minimizing the expression ||𝑌ℎδ − 𝑏ℎ||2 δ?
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How is the residual concept introduced in the text relevant to the optimization problem?
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Study Notes
Vanishing Gradient Problem
- If the derived function is 0, the weights won't be updated, and the convergence to an optimal solution is slow or unreachable
- This issue is less prevalent in ReLU compared to other activation functions, due to its graph properties
- Small or zero values can occur in sigmoid and tanh functions when input values are too small (negative) or too large
Pruning
- Assumes that inactive neurons can be removed from the network to improve efficiency and reduce overfitting
- Inactive neurons are characterized by outputs close to zero
- Two approaches to find and remove inactive neurons:
- Split the graph into subgraphs until eigenvalues are small enough
- Compute all eigenpairs
Algorithm Clustering as Graph Partitioning
- Maps feature space to nodes and computes relationships between nodes as edge weights
- Splits the graph into subgraphs until the desired number of clusters is met
Min Cut
- Splits the graph into subclusters by computing the cut with the minimum value possible (sum of edge weights)
- Cons: may split isolated nodes with respect to clusters
Graph Theory Notions
- Degree of a node: 𝑑𝑖 = ∑ 𝑤𝑖𝑗 𝑗
- Volume of a set: 𝑣𝑜𝑙(𝐴) = ∑ 𝑑𝑖 𝑖∈𝐴
Normalized Cut
- Objective: split the graph while minimizing the value of Ncut
- Goal: achieve high cohesiveness within clusters and well-separated clusters
- Problem: normalized cut is NPC, so approximations are required
Graph Laplacian
- A matrix used in graph theory and image analysis to represent a graph's relations and connection properties
- Compact representation of a graph
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Description
Learn about the challenges of neural networks convergence when the derived function approaches zero, impacting the weight updates and optimization process. Explore the implications and reasons behind slow convergence rates in ReLu and other activation functions.
