0
$\begingroup$

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
$\endgroup$
2
  • 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

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.