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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

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  • $\begingroup$ I think you mean: Let $S_1$ and $S_2$ be distinct sets of vectors....Let $\sigma$ be a non-linear function. Does there exist a set of matrices $R_k$ such that the probability...tends to 0 as $k$ tends to infinity? If so, it would help to say it that way. $\endgroup$
    – user44143
    Commented Apr 29, 2018 at 14:15
  • $\begingroup$ I think this probability will usually be $0$ if e.g. $R$ is a Gaussian matrix and $\sigma(\cdot)$ is a smooth function with strictly positive partial derivatives. $\endgroup$
    – Iosif Pinelis
    Commented Apr 29, 2018 at 15:32
  • $\begingroup$ @MattF There is only one matrix, not a set. $\endgroup$
    – Christopher
    Commented Apr 30, 2018 at 10:31
  • $\begingroup$ @IosifPinelis Could you please elaborate on your comment. $\endgroup$
    – Christopher
    Commented Apr 30, 2018 at 10:32

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