<p>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 <span class="math-container">$W\times L$</span> pixel images of a dog?</p>
<p>Let <span class="math-container">$D$</span> be the set of all images of dogs, such that the intrinsic dimension of any image <span class="math-container">$dim_{in}(d)=n$</span> for <span class="math-container">$d\in D$</span>. Then the manifold <span class="math-container">$(D,\mathcal{T})$</span> is the representative manifold of all dogs. Let us define the chart <span class="math-container">$p: D\mapsto \mathbb{R}^{WL}$</span>. We can then see that the common practice of representing a <span class="math-container">$W\times L$</span> image as a vector <span class="math-container">$v\in \mathbb{R}^{WL}$</span> is calling this function <span class="math-container">$p$</span> on an image, or that we are mapping the &quot;true&quot; image (whatever that means) to a euclidean vector representation.</p>
<p>For example, a binary classification neural network is then a function <span class="math-container">$f(\cdot,\theta):\mathbb{R}^{WL}\to \mathbb{R}^2$</span></p>
<p>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 <span class="math-container">$W\times L$</span> to <span class="math-container">$W'\times L$</span>, then you are simply choosing another chart to map from the dog manifold.</p>
<p>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. <span class="math-container">$n&lt;&lt; WL$</span>. If so, can the chart <span class="math-container">$p$</span> as defined above possibly even be homeomorphic?</p>
<p>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</p>
<ol>
<li>A Hausdorff Space</li>
<li>second-countable</li>
<li>has local Euclidean of dimension n</li>
</ol>

CC BY-SA 4.0
2022-06-13T07:56:29.237Z
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.
D.W.
CC BY-SA 4.0
2022-06-15T06:16:48.707Z
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?
Andrej Bauer
<machine-learning>
characterising the manifold representing images
jeb2
# characterising the manifold representing images

(Asked by: username_0 on 2022-06-13)

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

**Tags:** machine-learning

> **Comments:**
> **username_1**: 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.  
> **username_2**: 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?  

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## 0 answers

