Let X be a m × n (m: number of records, and n: number of attributes) normalized dataset (between 0 and 1). Denote Y = X R, where R is an n × p matrix, and p &lt; n. I understand if R was drawn randomly from Gaussian distribution, e.g., N ( 0 , 1 ) then the transformation preserve the Euclidean distances between instances (all of the pairwise distances between the points in the feature space will be preserved). But what if R ∼ U ( 0 , 1 ), does the transformation still preserve the distance between instances?

Question

Let   X be a   m  ×  n (m: number of records, and n: number of attributes) normalized dataset (between 0 and 1). Denote   Y  =  X  R, where   R is an   n  ×  p matrix, and   p  &amp;lt;  n.  I understand if   R was drawn randomly from Gaussian distribution, e.g.,  N  (  0  ,  1  ) then the transformation preserve the Euclidean distances between instances (all of the pairwise distances between the points in the feature space will be preserved). But what if   R  ∼  U  (  0  ,  1  ), does the transformation still preserve the distance between instances?

Nola Rivera · Accepted Answer

Suppose we have   m records   (      X    i    :  1  ≤  i  ≤  m  ) of normalized n-dimensional data (that is, for each   i  ≤  m, we can write       X    i    =  (      x    1          (      i      )        ,  …  ,      x    n          (      i      )        )  ∈  [  0  ,  1      ]    n  ). Then, multiplying any of the       X    i   by a   n  ×  p matrix   R (where   p  &amp;lt;  n) can be seen as projecting the vector       X    i   onto         R    p  . Upon reading your question, I understand that you&#039;re interested in projections that can be seen as somewhat faithful, in that there will be little difference between the Euclidean distance between vectors   ‖      X    i    −      X    j        ‖    2   in         R    n   and their projection   ‖  (      X    i    −      X    j    )  R      ‖    2   on         R    p  .First, I believe your claim that if the entries of   R are i.i.d.   N  (  0  ,  1  ) random variables, then the transformation preserves the distance is false. To see this, consider the simple counterexample: Suppose we have two instances with two attributes       X    1    =  [  1  ,  1  ] and       X    2    =  [  0  ,  1  ]. Then, we have that  ‖      X    1    −      X    2        ‖    2    =            1      2        +    0    =  1.Let   R  =  [      R    1    ,      R    2        ]      ⊤   be our transformation. Then, if we want the norm to be preserved, it is necessary that  1  =  |      X    1    R  −      X    2    R  |  =  |  (      R    1    +      R    2    )  −      R    2    |  =  |      R    1    |  ,in which case       R    1   can only take the values -1 and 1, and hence is not   N  (  0  ,  1  ).

Let X be a m × n (m: number of records, and n: number of attribut

Answered question

Answer & Explanation

New Questions in Linear algebra