Dimension reduction using space filling curve to avoid "Curse of dimensionality"? In machine learning, we want to train a model. While training, if the dimension of data is high, we have a problem (Curse of Dimensionality), so we want to reduce the dimension of our data.
Since we know $\mathbb{R}^n$ and $\mathbb{R}$ have the same cardinality. So we always have some space-filling curve that maps each point uniquely in both directions. i.e. we can always bijectively map any n-dimensional data to 1-dimension. So what is the problem we have in the first place?
I come up with two problems that can still be there:

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*With this space-filling curve map, we can't reduce the size of data. i.e., we have to increase the precision when we are writing it in 1-dimension.

*This is the place where I have doubts. I am thinking that while representing data in $\mathbb{R}^n$ we have more information than representing it in $\mathbb{R}$. There is some structure when we write data in $\mathbb{R}^n$ like which point is near to which point. That is lost in this map (space-filling curve), i.e, the map is not homomorphic.

My question:

*

*Is it right what I am thinking?

*I don't know what information I am talking about in point 2. could you help me make it more rigorous is.

*Is there any other problem?

Example:
Suppose we have a training data with $x^i \in \mathbb{R}^n $ with label $y^i \in \mathbb{R}$  where $i \in \{1,2,..,N\}$. When we are training a neural network to fit this data. If we change the order of the bases i.e.
$$x^i = (x_1^i,x_2^i,..,x_n^i)$$
if we take
$$\tilde{x}^i = (x_{\rho(1)}^i,x_{\rho(2)}^i,..,x_{\rho(n)}^i)$$
where $\rho$ is some permutation function(same for all $i$). then the training of neural network doesn't affect. So when order doesn't matter. What if I transform all the data from $\mathbb{R}^n$ to $\mathbb{R}$? It should also don't matter. But it did matter. Otherwise, there is nothing like the curse of dimensions.
I think When we transform the data from $\mathbb{R}^n$ to $\mathbb{R}$, we lose some information or do something wrong. What is that?

The problem of the curse of dimensionality is caused by the amount of data needed, in the worst case, to adequately represent the underlying distribution goes up exponentially in the number of features/attributes.  Unfortunately using a space-filling curve to reduce the number of dimensions will not eliminate the curse of dimensionality.  Consider a Gaussian distribution in a d-dimensional space, which could be described by a mean vector and a covariance matrix.  This is a nice smooth and "simple" distribution.  Now consider where the areas of high density would lie on the space-filling curve.  Instead of one contiguous smooth distributions, there would now be exponentially (in the dimension of the original space) many bumps of high data density along the "string".  We would still need exponentially large amounts of data to specify where along the "string" those bumps should be, and how high and how wide they would be.  So sadly this approach is highly unlikely to work.
