This slide shows the formula for forward propagation in a neural network in an easy-to-understand diagram. This slide will help you understand how the values of the formula are used in the network.
Detailed and Visual Understanding ofForward Propagation in Neural Networks
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Weight matrix representation
𝑥1𝑥2
𝑥1𝑥2𝑤1,11𝑊(1) =𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)𝑤2,11𝑤3,11𝑤1,21𝑤2,21𝑤3,21
𝑥1𝑥2𝑤1,11𝑊(1) =𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)𝑤2,11𝑤3,11𝑤1,21𝑤2,21𝑤3,21The weights from a 2-node layer to 3-node layercan be expressed as a (2×3) matrix.
𝑥1𝑥2𝑤1,11𝑊(1) =𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)𝑤2,11𝑤3,11𝑤1,21𝑤2,21𝑤3,21𝑊(2) =𝑤1,1(2) 𝑤2,1(2)𝑤1,2(2) 𝑤2,2(2)𝑤1,3(2) 𝑤2,3(2)𝑤1,12𝑤2,12𝑤1,22𝑤2,22𝑤1,32𝑤2,32
𝑥1𝑥2𝑤1,11𝑊(1) =𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)𝑤2,11𝑤3,11𝑤1,21𝑤2,21𝑤3,21𝑊(2) =𝑤1,1(2) 𝑤2,1(2)𝑤1,2(2) 𝑤2,2(2)𝑤1,3(2) 𝑤2,3(2)𝑤1,12𝑤2,12𝑤1,22𝑤2,22𝑤1,32𝑤2,32𝑊(3) =𝑤1,1(3) 𝑤2,1(3)𝑤1,2(3) 𝑤2,2(3)𝑤1,1(3)𝑤2,1(3)𝑤1,2(3)𝑤2,2(3)
Forward propagation
Forward propagationFirst Layer
𝑥1(1)𝑥2(1)𝑋(1) = 𝑥1(1) 𝑥2(1)𝑊(1) =𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)𝑤1,1(1)𝑤2,1(1)𝑤3,1(1)𝑤1,2(1)𝑤2,2(1)𝑤3,2(1)𝐵(1) = 𝑏1(1) 𝑏2(1) 𝑏3(1)𝑏3(1)𝑏2(1)𝑏1(1)
𝑥1(1)𝑥2(1)𝑋(1) = 𝑥1(1) 𝑥2(1)𝑤1,2(1)𝑤2,2(1)𝑤3,2(1)𝐵(1) = 𝑏1(1) 𝑏2(1) 𝑏3(1)𝑊(1) =𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)𝑤1,1(1)𝑤2,1(1)𝑤3,1(1)𝑏3(1)𝑏2(1)𝑏1(1)
𝑥1(1)𝑥2(1)𝑋(1) = 𝑥1(1) 𝑥2(1)𝑊(1) =𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)𝑤1,1(1)𝑤2,1(1)𝑤3,1(1)𝑤1,2(1)𝑤2,2(1)𝑤3,2(1)𝒙𝟏(𝟐)𝒙𝟐(𝟐)𝒙𝟑(𝟐)𝑋(2) = 𝑓(𝑋(1)𝑊(1) + 𝐵(1)) = 𝑥1(2) 𝑥2(2) 𝑥3(2)𝐵(1) = 𝑏1(1) 𝑏2(1) 𝑏3(1)𝑏3(1)𝑏2(1)𝑏1(1)
𝑥1(1)𝑥2(1)𝑋(1) = 𝑥1(1) 𝑥2(1)𝑊(1) =𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)𝑤1,1(1)𝑤2,1(1)𝑤3,1(1)𝑤1,2(1)𝑤2,2(1)𝑤3,2(1)𝒙𝟏(𝟐)𝒙𝟐(𝟐)𝒙𝟑(𝟐)𝑋(2) = 𝑓(𝑋(1)𝑊(1) + 𝐵(1)) = 𝑓 𝑥1(1) 𝑥2(1)𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)+ 𝑏1(1) 𝑏2(1) 𝑏3(1)𝐵(1) = 𝑏1(1) 𝑏2(1) 𝑏3(1)𝑏3(1)𝑏2(1)𝑏1(1)
𝑥1(1)𝑥2(1)𝑋(1) = 𝑥1(1) 𝑥2(1)𝑊(1) =𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)𝑤1,1(1)𝑤2,1(1)𝑤3,1(1)𝑤1,2(1)𝑤2,2(1)𝑤3,2(1)𝒙𝟏(𝟐)𝒙𝟐(𝟐)𝒙𝟑(𝟐)𝑋(2) = 𝑓(𝑋(1)𝑊(1) + 𝐵(1)) = 𝒇 𝑥1(1) 𝑥2(1)𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)+ 𝑏1(1) 𝑏2(1) 𝑏3(1)Activation Function𝐵(1) = 𝑏1(1) 𝑏2(1) 𝑏3(1)𝑏3(1)𝑏2(1)𝑏1(1)
𝑥1(1)𝑥2(1)𝑋(1) = 𝑥1(1) 𝑥2(1)𝑊(1) =𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)𝑤1,1(1)𝑤2,1(1)𝑤3,1(1)𝑤1,2(1)𝑤2,2(1)𝑤3,2(1)𝒙𝟏(𝟐)𝒙𝟐(𝟐)𝒙𝟑(𝟐)𝑋(2) = 𝑓(𝑋(1)𝑊(1) + 𝐵(1)) = 𝑓 𝑥1(1) 𝑥2(1)𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)+ 𝑏1(1) 𝑏2(1) 𝑏3(1)𝒙𝟏(𝟐) = 𝑓(𝑥11 𝑤1,11 + 𝑥21 𝑤1,21 + 𝑏1(1))𝐵(1) = 𝑏1(1) 𝑏2(1) 𝑏3(1)𝑏3(1)𝑏2(1)𝑏1(1)
𝒙𝟏(𝟏)𝒙𝟐(𝟏)𝑋(1) = 𝑥1(1) 𝑥2(1)𝑊(1) =𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)𝒘𝟏,𝟏(𝟏)𝑤2,1(1)𝑤3,1(1)𝒘𝟏,𝟐(𝟏)𝑤2,2(1)𝑤3,2(1)𝒙𝟏(𝟐)𝒙𝟐(𝟐)𝒙𝟑(𝟐)𝑋(2) = 𝑓(𝑋(1)𝑊(1) + 𝐵(1)) = 𝑓 𝑥1(1) 𝑥2(1)𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)+ 𝑏1(1) 𝑏2(1) 𝑏3(1)𝒙𝟏(𝟐) = 𝒇(𝑥11 𝑤1,11 + 𝑥21 𝑤1,21 + 𝑏1(1))𝐵(1) = 𝑏1(1) 𝑏2(1) 𝑏3(1)𝑏3(1)𝑏2(1)𝒃𝟏(𝟏)
𝒙𝟏(𝟏)𝒙𝟐(𝟏)𝑋(1) = 𝑥1(1) 𝑥2(1)𝑊(1) =𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)𝑤1,1(1)𝒘𝟐,𝟏(𝟏)𝑤3,1(1)𝑤1,2(1)𝒘𝟐,𝟐(𝟏)𝑤3,2(1)𝒙𝟏(𝟐)𝒙𝟐(𝟐)𝒙𝟑(𝟐)𝑋(2) = 𝑓(𝑋(1)𝑊(1) + 𝐵(1)) = 𝑓 𝑥1(1) 𝑥2(1)𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)+ 𝑏1(1) 𝑏2(1) 𝑏3(1)𝒙𝟏(𝟐) = 𝒇(𝑥11 𝑤1,11 + 𝑥21 𝑤1,21 + 𝑏1(1))𝒙𝟐(𝟐) = 𝒇(𝑥11 𝑤2,11 + 𝑥21 𝑤2,21 + 𝑏2(1))𝐵(1) = 𝑏1(1) 𝑏2(1) 𝑏3(1)𝑏3(1)𝒃𝟐(𝟏)𝑏1(1)
𝒙𝟏(𝟏)𝒙𝟐(𝟏)𝑋(1) = 𝑥1(1) 𝑥2(1)𝑊(1) =𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)𝑤1,1(1)𝑤2,1(1)𝒘𝟑,𝟏(𝟏)𝑤1,2(1)𝑤2,2(1)𝒘𝟑,𝟐(𝟏)𝒙𝟏(𝟐)𝒙𝟐(𝟐)𝒙𝟑(𝟐)𝑋(2) = 𝑓(𝑋(1)𝑊(1) + 𝐵(1)) = 𝑓 𝑥1(1) 𝑥2(1)𝑤1,1(1) 𝑤2,1(1) 𝑤3,1(1)𝑤1,2(1) 𝑤2,2(1) 𝑤3,2(1)+ 𝑏1(1) 𝑏2(1) 𝑏3(1)𝒙𝟏(𝟐) = 𝒇(𝑥11 𝑤1,11 + 𝑥21 𝑤1,21 + 𝑏1(1))𝒙𝟐(𝟐) = 𝒇(𝑥11 𝑤2,11 + 𝑥21 𝑤2,21 + 𝑏2(1))𝒙𝟑(𝟐) = 𝒇(𝑥11 𝑤3,11 + 𝑥21 𝑤3,21 + 𝑏3(1))𝐵(1) = 𝑏1(1) 𝑏2(1) 𝑏3(1)𝒃𝟑(𝟏)𝑏2(1)𝑏1(1)
Forward propagationSecond Layer
𝑊(2) =𝑤1,1(2) 𝑤2,1(2)𝑤1,2(2) 𝑤2,2(2)𝑤1,3(2) 𝑤2,3(2)𝑥1(1)𝑥2(1)𝑥1(2)𝑥2(2)𝑥3(2)𝑤1,12𝑤2,12𝑤1,22𝑤2,22𝑤1,32𝑤2,32𝒙𝟏(𝟑)𝒙𝟐(𝟑)𝑋(2) = 𝑥1(2) 𝑥2(2) 𝑥3(2)𝐵(2) = 𝑏1(2) 𝑏2(2)𝑏2(2)𝑏1(2)
𝑊(2) =𝑤1,1(2) 𝑤2,1(2)𝑤1,2(2) 𝑤2,2(2)𝑤1,3(2) 𝑤2,3(2)𝑥1(1)𝑥2(1)𝑥1(2)𝑥2(2)𝑥3(2)𝑤1,12𝑤2,12𝑤1,22𝑤2,22𝑤1,32𝑤2,32𝒙𝟏(𝟑)𝒙𝟐(𝟑)𝑋(2) = 𝑥1(2) 𝑥2(2) 𝑥3(2)𝐵(2) = 𝑏1(2) 𝑏2(2)𝑋(3) = 𝑓(𝑋(2)𝑊(2) + 𝐵(2)) = 𝑥1(3) 𝑥2(3)𝑏2(2)𝑏1(2)
𝑊(2) =𝑤1,1(2) 𝑤2,1(2)𝑤1,2(2) 𝑤2,2(2)𝑤1,3(2) 𝑤2,3(2)𝑥1(1)𝑥2(1)𝑥1(2)𝑥2(2)𝑥3(2)𝑥1(2) 𝑥2(2) 𝑥3(2)𝑤1,1(2) 𝑤2,1(2)𝑤1,2(2) 𝑤2,2(2)𝑤1,3(2) 𝑤2,3(2)+ 𝑏1(2) 𝑏2(2)𝑤1,12𝑤2,12𝑤1,22𝑤2,22𝑤1,32𝑤2,32𝒙𝟏(𝟑)𝒙𝟐(𝟑)𝑋(3) = 𝑓(𝑋(2)𝑊(2) + 𝐵(2)) = 𝑓𝑋(2) = 𝑥1(2) 𝑥2(2) 𝑥3(2)𝐵(2) = 𝑏1(2) 𝑏2(2)𝑏2(2)𝑏1(2)
𝑊(2) =𝑤1,1(2) 𝑤2,1(2)𝑤1,2(2) 𝑤2,2(2)𝑤1,3(2) 𝑤2,3(2)𝑥1(1)𝑥2(1)𝑥1(2)𝑥2(2)𝑥3(2)𝑥1(2) 𝑥2(2) 𝑥3(2)𝑤1,1(2) 𝑤2,1(2)𝑤1,2(2) 𝑤2,2(2)𝑤1,3(2) 𝑤2,3(2)+ 𝑏1(2) 𝑏2(2)𝒙𝟏(𝟑) = 𝒇(𝑥12 𝑤1,12 + 𝑥22 𝑤1,22 + 𝑥32 𝑤1,32 + 𝑏1(2))𝑤1,12𝑤2,12𝑤1,22𝑤2,22𝑤1,32𝑤2,32𝒙𝟏(𝟑)𝒙𝟐(𝟑)𝑋(3) = 𝑓(𝑋(2)𝑊(2) + 𝐵(2)) = 𝑓𝑋(2) = 𝑥1(2) 𝑥2(2) 𝑥3(2)𝐵(2) = 𝑏1(2) 𝑏2(2)𝑏2(2)𝑏1(2)
𝑊(2) =𝑤1,1(2) 𝑤2,1(2)𝑤1,2(2) 𝑤2,2(2)𝑤1,3(2) 𝑤2,3(2)𝑥1(1)𝑥2(1)𝑥1(2)𝑥2(2)𝑥3(2)𝑥1(2) 𝑥2(2) 𝑥3(2)𝑤1,1(2) 𝑤2,1(2)𝑤1,2(2) 𝑤2,2(2)𝑤1,3(2) 𝑤2,3(2)+ 𝑏1(2) 𝑏2(2)𝒙𝟏(𝟑) = 𝒇(𝑥12 𝑤1,12 + 𝑥22 𝑤1,22 + 𝑥32 𝑤1,32 + 𝑏1(2))𝒘𝟏,𝟏𝟐𝑤2,12𝒘𝟏,𝟐𝟐𝑤2,22𝒘𝟏,𝟑𝟐𝑤2,32𝒙𝟏(𝟑)𝒙𝟐(𝟑)𝑋(3) = 𝑓(𝑋(2)𝑊(2) + 𝐵(2)) = 𝑓𝑋(2) = 𝑥1(2) 𝑥2(2) 𝑥3(2)𝐵(2) = 𝑏1(2) 𝑏2(2)𝑏2(2)𝒃𝟏(𝟐)
𝑊(2) =𝑤1,1(2) 𝑤2,1(2)𝑤1,2(2) 𝑤2,2(2)𝑤1,3(2) 𝑤2,3(2)𝑥1(1)𝑥2(1)𝑥1(2)𝑥2(2)𝑥3(2)𝑥1(2) 𝑥2(2) 𝑥3(2)𝑤1,1(2) 𝑤2,1(2)𝑤1,2(2) 𝑤2,2(2)𝑤1,3(2) 𝑤2,3(2)+ 𝑏1(2) 𝑏2(2)𝒙𝟏(𝟑) = 𝒇(𝑥12 𝑤1,12 + 𝑥22 𝑤1,22 + 𝑥32 𝑤1,32 + 𝑏1(2))𝑤1,12𝒘𝟐,𝟏𝟐𝑤1,22𝒘𝟐,𝟐𝟐𝑤1,32𝒘𝟐,𝟑𝟐𝒙𝟏(𝟑)𝒙𝟐(𝟑)𝒙𝟐(𝟑) = 𝒇(𝑥12 𝑤2,12 + 𝑥22 𝑤2,22 + 𝑥32 𝑤2,32 + 𝑏2(2))𝑋(3) = 𝑓(𝑋(2)𝑊(2) + 𝐵(2)) = 𝑓𝑋(2) = 𝑥1(2) 𝑥2(2) 𝑥3(2)𝐵(2) = 𝑏1(2) 𝑏2(2)𝒃𝟐(𝟐)𝑏1(2)
Forward propagationThird Layer
𝑥1(1)𝑥2(1)𝑥1(2)𝑥2(2)𝑥3(2)𝑤1,13𝑤2,13𝑤1,23𝑤2,23𝒙𝟏(𝟑)𝒙𝟐(𝟑)𝐵(3) = 𝑏1(3) 𝑏2(3)𝒙𝟏(𝟒)𝒙𝟐(𝟒)𝑋(3) = 𝑥1(3) 𝑥2(3)𝑊(3) =𝑤1,1(3) 𝑤2,1(3)𝑤1,2(3) 𝑤2,2(3)𝑏2(3)𝑏1(3)
𝑥1(1)𝑥2(1)𝑥1(2)𝑥2(2)𝑥3(2)𝑋(4) = 𝑓(𝑋(3)𝑊(3) + 𝐵(3)) = 𝑥1(4) 𝑥2(4)𝑤1,13𝑤2,13𝑤1,23𝑤2,23𝒙𝟏(𝟑)𝒙𝟐(𝟑)𝐵(3) = 𝑏1(3) 𝑏2(3)𝒙𝟏(𝟒)𝒙𝟐(𝟒)𝑋(3) = 𝑥1(3) 𝑥2(3)𝑊(3) =𝑤1,1(3) 𝑤2,1(3)𝑤1,2(3) 𝑤2,2(3)𝑏2(3)𝑏1(3)
𝑥1(1)𝑥2(1)𝑥1(2)𝑥2(2)𝑥3(2)𝑋(4) = 𝑓(𝑋(3)𝑊(3) + 𝐵(3)) = 𝑓 𝑥1(3) 𝑥2(3)𝑤1,1(3) 𝑤2,1(3)𝑤1,2(3) 𝑤2,2(3)+ 𝑏1(3) 𝑏2(3)𝒙𝟏(𝟒) = 𝒇(𝑥13 𝑤1,13 + 𝑥23 𝑤1,23 + 𝑏1(3))𝑤1,13𝑤2,13𝑤1,23𝑤2,23𝒙𝟏(𝟑)𝒙𝟐(𝟑)𝐵(3) = 𝑏1(3) 𝑏2(3)𝒙𝟏(𝟒)𝒙𝟐(𝟒)𝑋(3) = 𝑥1(3) 𝑥2(3)𝑊(3) =𝑤1,1(3) 𝑤2,1(3)𝑤1,2(3) 𝑤2,2(3)𝑏2(3)𝑏1(3)
𝑥1(1)𝑥2(1)𝑥1(2)𝑥2(2)𝑥3(2)𝑋(4) = 𝑓(𝑋(3)𝑊(3) + 𝐵(3)) = 𝑓 𝑥1(3) 𝑥2(3)𝑤1,1(3) 𝑤2,1(3)𝑤1,2(3) 𝑤2,2(3)+ 𝑏1(3) 𝑏2(3)𝒙𝟏(𝟒) = 𝒇(𝑥13 𝑤1,13 + 𝑥23 𝑤1,23 + 𝑏1(3))𝒘𝟏,𝟏𝟑𝑤2,13𝒘𝟏,𝟐𝟑𝑤2,23𝒙𝟏(𝟑)𝒙𝟐(𝟑)𝐵(3) = 𝑏1(3) 𝑏2(3)𝒙𝟏(𝟒)𝒙𝟐(𝟒)𝑋(3) = 𝑥1(3) 𝑥2(3)𝑊(3) =𝑤1,1(3) 𝑤2,1(3)𝑤1,2(3) 𝑤2,2(3)𝑏2(3)𝒃𝟏(𝟑)
𝑥1(1)𝑥2(1)𝑥1(2)𝑥2(2)𝑥3(2)𝑋(4) = 𝑓(𝑋(3)𝑊(3) + 𝐵(3)) = 𝑓 𝑥1(3) 𝑥2(3)𝑤1,1(3) 𝑤2,1(3)𝑤1,2(3) 𝑤2,2(3)+ 𝑏1(3) 𝑏2(3)𝒙𝟏(𝟒) = 𝒇(𝑥13 𝑤1,13 + 𝑥23 𝑤1,23 + 𝑏1(3))𝑤1,13𝒘𝟐,𝟏𝟑𝑤1,23𝒘𝟐,𝟐𝟑𝒙𝟏(𝟑)𝒙𝟐(𝟑)𝒙𝟐(𝟒) = 𝒇(𝑥13 𝑤2,13 + 𝑥23 𝑤2,23 + 𝑏2(3))𝐵(3) = 𝑏1(3) 𝑏2(3)𝒙𝟏(𝟒)𝒙𝟐(𝟒)𝑋(3) = 𝑥1(3) 𝑥2(3)𝑊(3) =𝑤1,1(3) 𝑤2,1(3)𝑤1,2(3) 𝑤2,2(3)𝒃𝟐(𝟑)𝑏1(3)
Forward propagationSummary
𝑾(𝟏)𝑩(𝟏)𝑾(𝟐)𝑩(𝟐)𝑾(𝟑)𝑩(𝟑)
𝑥1(1)𝑥2(1)𝑿(𝟏)𝑾(𝟏)𝑩(𝟏)𝑾(𝟐)𝑩(𝟐)𝑾(𝟑)𝑩(𝟑)
𝑥1(1)𝑥2(1)𝑥1(2)𝑥2(2)𝑥3(2)𝑿(𝟏)𝑾(𝟏)𝑩(𝟏)𝑿(𝟐) = 𝒇(𝑿(𝟏)𝑾(𝟏) + 𝑩(𝟏))𝑾(𝟐)𝑩(𝟐)𝑾(𝟑)𝑩(𝟑)
𝑥1(1)𝑥2(1)𝑥1(2)𝑥2(2)𝑥3(2)𝒙𝟏(𝟑)𝒙𝟐(𝟑)𝑿(𝟏)𝑾(𝟏)𝑩(𝟏)𝑿(𝟐) = 𝒇(𝑿(𝟏)𝑾(𝟏) + 𝑩(𝟏))𝑾(𝟐)𝑩(𝟐)𝑿(𝟑) = 𝒇(𝑿(𝟐)𝑾(𝟐) + 𝑩(𝟐))𝑾(𝟑)𝑩(𝟑)
𝑥1(1)𝑥2(1)𝑥1(2)𝑥2(2)𝑥3(2)𝒙𝟏(𝟑)𝒙𝟐(𝟑)𝒙𝟏(𝟒)𝒙𝟐(𝟒)𝑿(𝟏)𝑾(𝟏)𝑩(𝟏)𝑿(𝟐) = 𝒇(𝑿(𝟏)𝑾(𝟏) + 𝑩(𝟏))𝑾(𝟐)𝑩(𝟐)𝑿(𝟑) = 𝒇(𝑿(𝟐)𝑾(𝟐) + 𝑩(𝟐))𝑾(𝟑)𝑩(𝟑)𝑿(𝟒) = 𝒇(𝑿(𝟑)𝑾(𝟑) + 𝑩(𝟑))
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