Let $S_1$ and $S_2$ be sets of vectors from $\mathbb{R}^d$ that are distinct and let $\sigma(\cdot)$ be a non-linearity, e.g., a componentwise sigmoid function.

Does there exist a random matrix $R \in \mathbb{R}^{d \times k}$, e.g., a gaussian matrix, such that the probability of $ \sum_{s \in S_1} \sigma(s R) = \sum_{t \in S_2} \sigma(t R)$ tends to 0 as $k$ tends to infinity