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Drill 44 Inception concatenation and parameter count

MT2 numerical practice from direct to mixed

Question

Question. Input 10 × 10 × 8. Parallel branches: A uses 1 × 1 conv 4 filters ; B uses 3 × 3 same Conv 6 filters; C max pool 3 same, stride 1 then 1 × 1 conv 2 filters. Find combined shape and trainable parameters.

Answer — step by step

Given

Input = 10 × 10 × 8.

Branch A: 1 × 1 convolution, 4 filters.

Branch B: 3 × 3 same convolution, 6 filters.

Branch C: 3 × 3 same max pool with stride 1, then 1 × 1 convolution, 2 filters.

To find

Find each branch shape and parameters, then the concatenated shape and total parameters.

Formula — which one and why

Inception channel concatenation. Parallel branches retain their separate features. Add their channel counts while keeping the common height and width. Add trainable parameters across branches.

C₁,C₂,… = branch output channels. Concatenation requires matching spatial dimensions.

$C_{\mathrm{out}}=\sum_k C_k$ LaTeX: C_{\mathrm{out}}=\sum_k C_k

$P_{\mathrm{total}}=\sum_k P_k$ LaTeX: P_{\mathrm{total}}=\sum_k P_k

Yaad rakhne ka cue: Channels jodo; height-width same rakho.

Convolution parameters. Every filter spans all input channels. Count its spatial coefficients across those channels, add one bias and multiply by the number of filters.

K_h,K_w = filter height/width; C_in = input channels; C_out = number of filters and output channels. Spatial output positions reuse these weights.

$N_w=K_hK_wC_{\mathrm{in}}C_{\mathrm{out}}$ LaTeX: N_w=K_hK_wC_{\mathrm{in}}C_{\mathrm{out}}

$N_b=C_{\mathrm{out}}$ LaTeX: N_b=C_{\mathrm{out}}

$P=N_w+N_b$ LaTeX: P=N_w+N_b

Yaad rakhne ka cue: Filter area × input channels × filters; phir ek bias per filter.

Solution — step by step

Meaning and formulas

Each parallel branch receives the same original 10 × 10 × 8 input. “Same” with stride 1 preserves height and width. Concatenation joins the channel lists without adding the feature values.

$C_{\mathrm{combined}}=C_A+C_B+C_C$ LaTeX: C_{\mathrm{combined}}=C_A+C_B+C_C

$P_{\mathrm{conv}}=(K^2C_{\mathrm{in}}+1)C_{\mathrm{out}}$ LaTeX: P_{\mathrm{conv}}=(K^2C_{\mathrm{in}}+1)C_{\mathrm{out}}

Assume one bias per filter. Pooling has no learned parameters.

Step 1: Branch A.

Its 4 filters produce shape 10 × 10 × 4.

Step 2: Branch B.

Its 6 filters produce shape 10 × 10 × 6.

Step 3: Branch C.

Pooling preserves 10 × 10 × 8 here because stride is 1 and padding is same. Its parameter count is zero. The following 2-filter projection creates 10 × 10 × 2.

Step 4: Join the shapes and add the parameters.

$C_{\mathrm{combined}}=4+6+2=12$ LaTeX: C_{\mathrm{combined}}=4+6+2=12

Final answer: combined shape 10 × 10 × 12; 492 parameters. Branch B does not take branch A's four channels as input: they are parallel branches.