Convolution as Matrix Operation

Linear Algebra for Machine Learning · Axiom Academy

Convolution as Matrix Multiplication Understanding CNNs through linear algebra Step 1: Conv Filters as Matrices Convolution is: sliding a filter over the input, computing dot products. Matrix Representation of Convolution Convolution can be represented as matrix multiplication using Im2col (Image to Column) transformation: Advantages of Matrix Representation Hardware optimization: GEMM (General Matrix Multiply) is highly optimized on GPUs Unified framework: Convolution becomes standard matrix multiplication Backpropagation: Uses the same chain rule as dense layers Step 3: Im2col (Image to Column) Trick Step 4: Multiple Filters and Channels Convolution = structured matrix multiplication: The structure enforces parameter sharing and locality Im2col transformation: Converts convolution to GEMM, enabling GPU acceleration Backpropagation: Same chain rule applies; gradients flow through the same matrix operations Efficiency: Modern frameworks use GEMM-based implementations for convolution Memory-efficient alternatives: Directly computing convolution can be faster for small kernels Unit 8: Linear Algebra in Neural Networks | Module 13: Convolution as Matrix Multiplication

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