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GATE DA 2024 Q43: Equivalent single-layer weights of a ReLU network

Question

Consider the two neural networks (NNs) shown in Figures 1 and 2, with ReLUReLU activation (ReLU(z)=max⁡{0,z},∀z∈RReLU(z) = \max\{0, z\}, \forall z \in \mathbb{R}). R\mathbb{R} denotes the set of real numbers. The connections and their corresponding weights are shown in the Figures. The biases at every neuron are set to 0. For what values of p,q,rp, q, r in Figure 2 are the two NNs equivalent, when x1,x2,x3x_1, x_2, x_3 are positive?

Figure 1: a network with inputs x1, x2, x3, a hidden layer of three neurons (weights 1), a layer of two neurons (weights 2) and one output (weights 3). Figure 2: inputs x1, x2, x3 connected directly to one output with weights p, q, r
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Answer: A (official GATE 2024 final answer key)

Concept used: With positive inputs and non-negative weights, ReLU acts as the identity, so the network is linear.

First step: The hidden neurons get x1+x2x_1 + x_2, x1+x3x_1 + x_3 and x1+x2+x3x_1 + x_2 + x_3; each next neuron gets twice their sum, and the output adds 3×3 \times both.

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Source: official GATE 2024 DA question paper and final answer key, organised by IISc Bangalore. Question paper · Answer key

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