Assuming that the Manifold Hypothesis is valid, or that real-world high-dimensional data lie on low-dimensional manifolds embedded within the high-dimensional space, How can one describe the hypothetical manifold that, for example, contains all $W\times L$ pixel images of a dog?

Let $D$ be the set of all images of dogs, such that the intrinsic dimension of any image $dim_{in}(d)=n$ for $d\in D$. Then the manifold $(D,\mathcal{T})$ is the representative manifold of all dogs. Let us define the chart $p: D\mapsto \mathbb{R}^{WL}$. We can then see that the common practice of representing a $W\times L$ image as a vector $v\in \mathbb{R}^{WL}$ is calling this function $p$ on an image, or that we are mapping the "true" image (whatever that means) to a euclidean vector representation.

For example, a binary classification neural network is then a function $f(\cdot,\theta):\mathbb{R}^{WL}\to \mathbb{R}^2$

This characterization then illustrates the point that the pixel vector representation does not necessarily reflect the intrinsic characteristics of an image. Indeed, if one crops the image from $W\times L$ to $W'\times L$, then you are simply choosing another chart to map from the dog manifold.

However, there are some questions I have. For one, the intrinsic dimension of an image is claimed to be much less than the extrinsic pixel dimension, i.e. $n<< WL$. If so, can the chart $p$ as defined above possibly even be homeomorphic?

Secondly, how could one argue somewhat rigorously that the A topological space formed by dog images even is a manifold? i.e. that it has the following properties

  1. A Hausdorff Space
  2. second-countable
  3. has local Euclidean of dimension n
  • 1
    $\begingroup$ Please ask only one question per post. I'm not sure what kind of answer you're expecting or hoping for to "How can one describe...?" Are you able to identify a concrete technical question? Statements like "images of dogs form a manifold" are probably often intended as informal statements, not necessarily something that can be proven rigorously. $\endgroup$
    – D.W.
    Commented Jun 13, 2022 at 7:56
  • $\begingroup$ The question is under-specified. All pixel images of a dog obviously form a 0-dimensional manifold, as they are just a finite collection of points in some high-dimensional $\mathbb{R}^n$. That is not what you meant, is it? But then, what did you mean? $\endgroup$ Commented Jun 15, 2022 at 6:16


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