Can a function be used to approximate a matrix I am interested to know class of functions that can be used to approximate large matrices. The reason of this is dimensionality reduction, so if we will have to use Taylor series approximation methods then we fall into the same curse of dimensionality situation. So are there any class of function that are compact in a sense that they are quite versatile and can mimick any matrices just by changing few parameters, maybe the order of parameters can add versatility.
 A: You can transform to Fourier series and find the principal frequencies to help determine the major frequencies that contribute to most information in the matrix. But then you will loose some information. Ideally there are no class of functions that can achieve this with few parameters without loosing information.
A: One approach is to use Lagrange interpolation setting the $x_i$ of Lagrange interpolation as the first $n^2$ whole number. So
$$L(X)=\sum_{i=1}^{n^2}a_i\prod_{j\neq i}\left(\dfrac{X-i}{i-j}\right)$$
And
$$\forall k\leq n^2 f(k) \simeq a_k $$
If sufficient to describe your matrix

One of an approach in terms of approximation of your matrix is to use the of Piantadosi function. The advantage of that method is that, even if it is approximation (in fact very thin approximation), it provides a certain regularity between the approximated values where Lagrange does not

Let $A$ in $ M_n(\mathbb{R})$:
$$A=(a_{i,j})=({a_k})_{k\in[1,n^2]}$$

Let define the matrix $A'$ as the reduced matrix in sort of the coefficent of $A'$ are all inferior to 1
$$A'=\max(c,10^{-c}|a_k|<1, k\leq n^2)\times A$$

Now define the Piantadosi function associated to $A'$
$$f_{k, p} (x) = \sin^2(2^{kx}\arcsin(p))$$

Where $k$ is a natural number (the precision)
And $p$ is the number binary constructed by the concatenations of the $k$ first digits of each of the $n^2$ coefficients of your matrix.
So all you matrix is in $f$.
Example :
With the matrix $$A=\begin{pmatrix}0.4 & 0.2\\
0.5 & 0.25 \end{pmatrix}$$
$a_1=0.0110011..., a_2=0.0011001100..., a_3=0.1,a_4=0.01$ in binary.
So if we choose $k=5$ we take the first five digit of each $a_k$ and put one after other to construct $p$
Concretely $p=0.01100001101000001000$ in binary.
And we have the fundamental relation : $$\forall k\leq n^2 f(k) \simeq a_k $$
For further explaination on Piantadosi function see :
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&ved=2ahUKEwjlt634g5PrAhVH3IUKHZB-BI8QFjADegQIAxAB&url=https%3A%2F%2Fpdfs.semanticscholar.org%2F9f4a%2F4d01294fd1fcc3f80a3a7c876055971b7663.pdf&usg=AOvVaw2H8MGA2h9mIyse
