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Comment by Sniffnoy | Hacker News Reader

While this phrases things in terms of Riemannian geometry, it seems to me like this is really about more fundamental metric properties, but I find its terminology a bit unclear. Would it be correct to summarize this as saying is that color space is not a length space / geodesic space? (See e.g. https://en.wikipedia.org/wiki/Intrinsic_metric )

(Yes "length space" (or "path space") and "geodesic space" are not exactly the same thing, but "length space but not geodesic space" doesn't exactly seem like a very likely possibility.)

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Yes, it's important to note the measure of distance (metric) used. Quoting from the paper:

    Humans are not good at judging questions of the type “How big is the difference?” that form the basis for many large difference experiments.

Instead, the authors use data from decisions of "which color step is larger". They find that as the steps get larger, you need a higher difference in color steps to reach the same certainty.

This is not that surprising, since people generally get worse at noticing small differences as the absolute quantities get larger. Unfortunately, the model of perception uncertainty used in the paper (Thurstone’s theory of Gaussian perceptual process) does not take this effect into account.

Quoting from the cartoon: ( perceptual jump) "Watch out for that tree." -George.

That's an interesting related thought that is not quite the same as the paper. They claim that what we perceive as "color similarity" is not a metric at all. That would be required in order to define arclength and geodesy.

I agree with others that it is not surprising and is a technicality. If you put that aside then the situation is much closer to your intuition.

Are you sure it's claiming that? Looking quickly I didn't see anything to indicate that, I didn't see anything claiming triangle inequality violations. (But this is why I wish it had been phrased explicitly in these terms, instead of talking about whether it's specifically Riemannian!)

At one point, the article says "importantly, [the principle of diminishing returns] holds even along geodesics, making it distinct from and stronger than the triangle inequality." Later, they say "it is not trivial to verify whether any given path through color space is a geodesic. We chose the neutral axis because it is the one path on which all available data agree that it is indeed a geodesic", and go on to argue (if I am following it correctly) that it is unlikely that their principle of diminishing returns is just an artifact from this choice.

I do not know if the authors are claiming that what we perceive as color similarity is not a metric at all, but personally, I would not be surprised if it were not. See my other post for how my subjective perception of color differences seems to me.

> stronger than the triangle inequality

That's not a violation -- "stronger than" is the opposite of a violation!

You're right, I misinterpreted. I don't know what they get out of calling the space non-Riemannian if not to say the inner product fails. Good idea, ignoring that term.

I mean, what they mean by "non-Riemannina" specifically is that it's a metric that can't be realized as the metric on a Riemannian manifold. But while that much is clear, getting beyond that is not. Which is why I'm asking, are they saying it's not a length space...

My thoughts exactly. It made me wonder if color space has any interesting topology. Topological semigroup might be a way of looking at it.

Rienmann geometry is smooth. Human perception is anything but. delta-p exists, and it varies in multiple way.

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