LFD Book Forum  

Go Back   LFD Book Forum > Course Discussions > Online LFD course > Homework 4

 
 
Thread Tools Display Modes
Prev Previous Post   Next Post Next
  #1  
Old 04-24-2013, 04:21 AM
Elroch Elroch is offline
Invited Guest
 
Join Date: Mar 2013
Posts: 143
Default Perceptron VC dimension

First I'd like to say that Professor Abu-Mostafa's lectures are unsurpassed in clarity and effectiveness in communicating understanding of the key elements of this fascinating subject. So it is unusual (actually, unique so far) for me to think I can see a way in which clarity of a point can be improved. Maybe I'm wrong: please judge!

The proof I am referring to is the second side of the proof of the VC dimension of a perceptron, where it is necessary to show that no set of (n+2) points can be shattered. Here's my slightly different version.

Consider a set X of (n+2) points in (n+1)-dimensional space, all of which have first co-ordinate 1. There is a non-trivial linear relation on these points:

\sum\limits_{x\in X}a_x x = 0

Rearrange so all the coefficients are positive (changing labels for convenience)

\sum\limits_{x\in X_A}a_x x =\sum\limits_{x\in X_B}b_x x

X_A and X_B must be non-empty subsets of X because the relation is non-trivial, and the first co-ordinate of all the points is 1.

If some perceptron is positive on X_A and negative on X_B then the value of the perceptron on \sum\limits_{x\in X_A}a_x x is positive and its value on \sum\limits_{x\in X_B}b_x x is negative.

But from the above these are the same point. Hence such a perceptron does not exist, so X cannot be shattered, completing the proof.
Reply With Quote
 

Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off

Forum Jump


All times are GMT -7. The time now is 07:26 AM.


Powered by vBulletin® Version 3.8.3
Copyright ©2000 - 2019, Jelsoft Enterprises Ltd.
The contents of this forum are to be used ONLY by readers of the Learning From Data book by Yaser S. Abu-Mostafa, Malik Magdon-Ismail, and Hsuan-Tien Lin, and participants in the Learning From Data MOOC by Yaser S. Abu-Mostafa. No part of these contents is to be communicated or made accessible to ANY other person or entity.