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Random Fourier features provide approximations to kernel functions. They're used for various kernel methods, like SVMs and Gaussian processes.

Today, I tried using the TensorFlow implementation and I got negative values for half of my features. As I understand it, this shouldn't happen.

So I went back to the original paper, which---like I expected---says that the features should live in [0,1]. But its explanation (highlighted below) doesn't make sense to me: the cosine function can produce values anywhere in [-1,1] and most of the points displayed have negative cosine values.

I'm probably missing something obvious, but would appreciate it if someone can point out what it is.

Figure 1

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Apparently, the highlighted sentence is wrong (or at least confusing): $z(x)$ can be negative. This isn't a problem because we only care about the inner product of $z$, not $z$ itself.

The "inner product" of $z$ only seemed incorrect when I used this method because I mixed up $z'z$ and $zz'$. Not because $z$ was wrong.

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