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For a given matrices $A$ of size $4\times 4$ and $B$ of size $3\times 3$ then I construct a blocked Toeplitz matrix and perform the convolution. The resulting output is of size $6 \times 6$. I have no doubt in setting up the blocked Toeplitz matrix and perform the convolution.

But in Neural networks the convolution of a $4 \times 4$ matrix with the so called kernel of size $3 \times 3$ results in $2 \times 2$ matrix. Presumably without zero-padding and stride of one.

I have no problem in understanding the results but the difference in output size is something I do not understand.

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The former is calculating a full sized convolution resulting in an N + M - 1 size, while the later is probably calculating a "valid" sized convolution. This case only returns the values from the convolution where there is complete overlap between the matrices, resulting in an N - M + 1 size. See MATLAB's documentation on the conv function (specifically the shape parameter) for more.

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