Design a custom neural network layer that implements a fully connected layer with ReLU activation, and explain the time and space complexity of your implementation.

To design a custom fully connected layer with ReLU, first define a class inheriting from a framework's base layer (e.g., PyTorch's nn.Module). Initialize weights and biases using random initialization (e.g., Kaiming for ReLU). Implement the forward pass with matrix multiplication for linear transformation, followed by ReLU activation. For complexity analysis, time complexity during forward pass is O(n * m) where n is input size and m is output size. Space complexity includes O(m) for parameters and O(n) for activations. Backward pass has similar time complexity due to gradient computation, with additional space for gradients.

The custom layer uses matrix multiplication (Wx + b) for linear transformation, followed by ReLU (max(0, x)). Time complexity for forward pass is O(n * m) due to matrix operations, where n is input features and m is output features. Space complexity is O(m) for weights and biases, plus O(n) for storing activations. During backpropagation, gradients require O(n * m) time and O(n + m) space. ReLU introduces no additional parameters but has O(1) computation per neuron. Using PyTorch's optimized operations ensures efficiency. The layer's implementation adheres to standard practices for weight initialization and gradient flow, ensuring stability during training.