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About this paper
Saeed Damadi, Golnaz Moharrer, Mostafa Cham
Researchers at the University of Maryland, Baltimore County, present a mathematically rigorous exposition of the backpropagation algorithm, articulating gradient computation as matrix multiplications using the Jacobian operator. The work extends this systematic derivation to convolutional neural networks, offering a foundational understanding of gradient mechanics in complex deep learning architectures.
Key insights
- The gradient of a scalar-valued function can be directly related to the transpose of its Jacobian matrix, enabling consistent application of matrix calculus.
- A 2D convolution operation can be precisely modeled as a matrix multiplication by converting filters into Toeplitz matrices and vectorizing inputs, allowing the Jacobian framework to cover CNNs.
- The overall gradient calculation in deep networks can be effectively decomposed into two main components: the gradient of the loss with respect to the last layer's input and the Jacobian of that input with respect to all parameters.
Summary by alphaXiv.
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