title: Does it make sense to randomly project high-dimensional data into low-dimensional ones in outliers detection?
body: <p>I have some high-dimensional data from which I want to detect outliers. I know that if I'm working with low-dimensional data, I can cluster and then check if a data point belongs to a cluster, or calculate the average distance from it to its k nearest neighbors, etc. But I can't do these on high-dimensional data because of the curse of dimensions.</p>
  
  <p>So I think maybe I can randomly project the high-dimensional data to lower dimensional ones, and check if the projections of a data point are outliers in most of the transformed dataset. My assumption is that an outlier in higher-dimension should also appears to be outliers in most projections to lower-dimension.</p>
  
  <p>For example, randomly generate some projections from <a href="https://i.stack.imgur.com/ECukN.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ECukN.gif" alt="R^4"></a> (suppose we have the curse of dimensions in <a href="https://i.stack.imgur.com/ECukN.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/ECukN.gif" alt="R^4"></a>) to <a href="https://i.stack.imgur.com/YllkD.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/YllkD.gif" alt="R^2"></a> (where we can cluster by traditional methods), denoted by <a href="https://i.stack.imgur.com/iipmu.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/iipmu.gif" alt="P1, P2, P3 .. Pn"></a> (all of them are <a href="https://i.stack.imgur.com/sNDmx.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/sNDmx.gif" alt="2x4"></a> matrices with random elements). Suppose we want to detect outliers in <a href="https://i.stack.imgur.com/5uupN.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5uupN.gif" alt="a1, a2 .. an"></a>. If for many <a href="https://i.stack.imgur.com/I34BA.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/I34BA.gif" alt="k"></a>, <a href="https://i.stack.imgur.com/T0vpy.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/T0vpy.gif" alt="Pk*am"></a> is an outlier in <a href="https://i.stack.imgur.com/FNIEg.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/FNIEg.gif" alt="Pk*a1, Pk*a2 .. Pk*an"></a>, than <a href="https://i.stack.imgur.com/5tcyx.gif" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/5tcyx.gif" alt="am"></a> is an outlier.</p>
  
  <p>Does it makes sense?</p>