Information Geometry

  • Home
  • About Us
  • Privacy Policy
  • Contact Us

Daily Archives: March 18, 2010

Log-euclidean matrix vector space

Frank March 18, 2010 News Comments are off

Tensors are square symmetric positive-definite matrix. They are surprisingly in 1-to-1 mapping with symmetric matrices through matrix exponentiation (exp.log computed on the diagonal elements of the spectral decomposition). I said surprisingly because tensors are symmetric and therefore a proper subset of the set they are in bijection with. Yes, nobody knows what is going on at infinity, and Cantor set theory and life was not so happy due to this… Anyway, obviously symmetric matrices are closed by addition and scalar multiplication, so that it is a vector space. Now using the bjiection tensor-symmetric matrix, we deduce a vector space for tensors. Euclidean distance (or other norm induced distance) hold, and we can transpose that to distances on tensors… These are called log-euclidean metrics, for distance metrics on matrix logarithm. And yes the metric with triangle inequality is derived from Cauchy-Schwarz inner product ineq.

tensor-smat

Spaces are beautiful lands to explore!
Frank.

Recent Posts

  • When Do You Need to Call a Locksmith?
  • Should I See a Podiatrist or Physiotherapist for Sports Injury?
  • 8 Benefits of Hiring a Digital Marketing Agency: Business Coaching
  • Reasons to Start a Cleaning Business
  • Digital Marketing for Tradesmen: Tips to Grow Your Business
  • Supplies for Art Painters: What You Need to Get Started
  • Things You Didn’t Know About Flinders Street Station
  • How to Find the Best Online Furniture Stores
  • 3D Printed Heat Exchangers: Titanium Additive Manufacturing
  • What Do Solar Developers Do?

Archives

  • August 2022
  • July 2022
  • June 2022
  • May 2022
  • April 2022
  • February 2022
  • January 2022
  • December 2021
  • November 2021
  • October 2021
  • September 2021
  • August 2021
  • June 2010
  • May 2010
  • April 2010
  • March 2010
March 2010
M T W T F S S
1234567
891011121314
15161718192021
22232425262728
293031  
    Apr »
Theme by ThemesPie | Proudly Powered by WordPress