[jasomill]

PowerShell's ~ can be counterintuitive, as it's relative to the current location's path and defined by a property of the current location's PSProvider:

  PS C:\> Get-PSDrive C,S,HKLM | select Name,{$_.Provider.Name},{$_.Provider.Home}
  
  Name $_.Provider.Name $_.Provider.Home
  ---- ---------------- ----------------
  C    FileSystem       C:\Users\jtm
  S    FileSystem       C:\Users\jtm
  HKLM Registry
  
  PS C:\Users\jtm> Get-PSDrive C,S,HKCU,HKLM | select Name,{$_.Provider.Name},{$_.Provider.Home}
  
  Name $_.Provider.Name $_.Provider.Home
  ---- ---------------- ----------------
  C    FileSystem       C:\Users\jtm
  S    FileSystem       C:\Users\jtm
  HKCU Registry
  HKLM Registry
  
  PS C:\Users\jtm> cd HKLM:
  PS HKLM:\> cd ~
  Set-Location: Home location for this provider is not set. To set the home location, call "(get-psprovider 'Registry').Home = 'path'".
  PS HKLM:\> (Get-Location).Provider.Home = 'C:\Program Files\Microsoft Office'
  PS HKLM:\> cd ~
  PS C:\Program Files\Microsoft Office> (Get-Location).Provider.Home = 'HKLM:\SYSTEM\CurrentControlSet'
  PS C:\Program Files\Microsoft Office> cd S:
  PS S:\> cd ~
  PS HKLM:\SYSTEM\CurrentControlSet> (Get-Location).Provider.Home = '..'
  PS HKLM:\SYSTEM\CurrentControlSet> cd ~
  PS HKLM:\SYSTEM> cd ~
  PS HKLM:\> Get-PSDrive C,S,HKCU,HKLM | select Name,{$_.Provider.Name},{$_.Provider.Home}
  
  Name $_.Provider.Name $_.Provider.Home
  ---- ---------------- ----------------
  C    FileSystem       HKLM:\SYSTEM\CurrentControlSet
  S    FileSystem       HKLM:\SYSTEM\CurrentControlSet
  HKCU Registry         ..
  HKLM Registry         ..
  
  PS HKLM:\>