A one-dimensional lattice of sites with periodic boundary conditions carries spins…

PHYSICS · Multi-part · Fermi University Physics · Computational Physics · High-performance and data-driven computational physics · High-performance and data-driven computational physics

Scientific Computing for Physicists, Ch Neural networks on physical data: backpropagation, convolution and symmetry

Problem

A one-dimensional lattice of

N=6

sites with periodic boundary conditions carries spins

x_0,\dots,x_5 = (1,-1,1,1,-1,-1)

(indices taken modulo 6). A dense layer

h(r) = \sum_{r'=0}^{5} W_{r,r'}x_{r'}

is restricted by weight sharing to

W_{r,r'} = K(r'-r \bmod 6)

, with

K(-1)=0.2

K(0)=0.5

K(1)=-0.1

and

K(\delta)=0

for every other offset (offsets read as the representative in

\{-2,-1,0,1,2,3\}

, i.e. the nearest of the two equivalent values modulo 6). ((a)) Show that the weight-shared sum reduces to

h(r) = \sum_{\delta=-1}^{1} K(\delta)\,x_{(r+\delta)\bmod 6}

, and identify this as the circular cross-correlation of

x

with

K

. ((b)) Show that this equals the circular convolution

\big(x * \tilde K\big)(r) = \sum_{m=0}^{5} x_m\,\tilde K\big((r-m)\bmod 6\big)

x

with the flipped kernel

\tilde K(\delta) = K(-\delta)

. ((c)) Compute

h(2)

directly from the sum in (a), then check it against the convolution formula in (b).

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