Algorithm and associated equations
Sponsored Links
This presentation is the property of its rightful owner.
1 / 38

Path finding Framework using HRR PowerPoint PPT Presentation


  • 104 Views
  • Uploaded on
  • Presentation posted in: General

Surabhi Gupta ’11 Advisor: Prof. Audrey St. John. Algorithm and associated equations. Path finding Framework using HRR. Roadmap. Circular Convolution Associative Memory Path finding algorithm. Hierarchical environment. Locations are hierarchically clustered. X 1. X 4. j

Download Presentation

Path finding Framework using HRR

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Surabhi Gupta ’11

Advisor: Prof. Audrey St. John

Algorithm and associated equations

Path finding Framework using HRR


Roadmap

  • Circular Convolution

  • Associative Memory

  • Path finding algorithm


Hierarchical environment

  • Locations are hierarchically clustered

X1

X4

j

k l

a

b c

X2

X3

Z

Y2

Y1

m

n o

d

e f

g

h i

X6

p

q r

X5


Tree representation

  • The scale of a location corresponds to its height in the tree structure.

  • The node of a tree can be directly queried without pointer following

  • Maximum number of goal searches = height of the tree


Circular Convolution

Holographic Reduced Representations


Circular Convolution (HRR)

  • Developed by Tony Plate in 1991

  • Binding (encoding) operation – Convolution

  • Decoding operation – Involution followed by convolution


Basic Operations

  • Binding

  • Merge


Binding - encoding

C≁A

C≁B


Circular Convolution ( )

  • Elements are summed along the trans-diagonals (1991, Plate).


Involution

  • Involution is the approximate inverse.


Decoding


Basic Operations

  • Binding

  • Merge


Merge

  • Normalized Dot product


Properties

  • Commutativity:

  • Distributivity:(shown by sufficiently long vectors)

  • Associativity:


Associative Memory

Recall and retrieval of locations


Framework

X1

X4

j

k l

a

b c

X2

X3

Z

Y2

Y1

m

n o

d

e f

g

h i

X6

p

q r

X5


Assumptions

  • Perfect tree – each leaf has the same depth

  • Locations within a scale are fully connected e.g. a,b and c, X4, X5 and X6 etc.

  • Each constituent has the same contribution to the scale location (no bias).

X1

X4

a

X2

X5

X3

Z

Y2

X6

Y1

p


Associative Memory

  • Consists of a list of locations

  • Inputs a location and returns the most similar location from the list.

What do we store?


Scales

  • Locations a-r are each2048-bit vectors taken from a normal distribution (0,1/2048).

  • Higher scales - Recursive auto-convolution of constituents


X1 =

Constructing scales

X1

+

a

b c

+


Across Scale sequences

  • Between each location and corresponding locations at higher scales.

X1

a

b c

+


Path finding algorithm

Quite different from standard graph search algorithms…


Path finding algorithm

Start==Goal?

Start

Move towards the Goal

Go to a higher scale and

search for the goal

If goal not found at this scale

If goal found at this scale

Retrieve the scales corresponding to the goal


Retrieving the next scale

  • If at scale-0, query the AS memory to retrieve the AS sequence. Else use the sequence retrieved in a previous step.

  • Query the L memory with


Retrieving the next scale

  • Helllo

  • Query the L memory with


Path finding algorithm

Start==Goal?

Start

Move towards the Goal

Go to a higher scale and

search for the goal

If goal not found at this scale

If goal found at this scale

Retrieve the scales corresponding to the goal


Locating the goal

  • For example:location:

  • and goal: c


Locating the goal

  • Goal: p

  • Not contained in X1

X1

X4

a

X2

X5

X3

Z

Y2

X6

Y1

p


Path finding algorithm

Start==Goal?

Start

Move towards the Goal

Go to a higher scale and

search for the goal

If goal not found at this scale

If goal found at this scale

Retrieve the scales corresponding to the goal


Goal not found at Y1

X1

X4

a

X2

X5

X3

Z

Y1

Y2

p

X6


Goal found at Z!

X1

X4

a

X2

X5

X3

Z

Y1

Y2

p

X6


Path finding algorithm

Start==Goal?

Start

Move towards the Goal

Go to a higher scale and

search for the goal

If goal not found at this scale

If goal found at this scale

Retrieve the scales corresponding to the goal


Decoding scales

  • Same decoding operation


Decoding scales

  • Using the retrieved scales


Path finding algorithm

Start==Goal?

Start

Move towards the Goal

Go to a higher scale and

search for the goal

If goal not found at this scale

If goal found at this scale

Retrieve the scales corresponding to the goal


Moving to the Goal

X1

X4

j

k l

a

b c

X2

X3

Z

Y2

Y1

m

n o

d

e f

g

h i

X6

p

q r

X5


To work on

  • Relax the assumption of a perfect tree.

  • Relax the assumption of a fully connected graph within a scale location.


References

  • Kanerva, P., Distributed Representations, Encyclopedia of Cognitive Science 2002. 59.

  • Plate, T. A. (1991). Holographic reduced representations: Convolution algebra for compositional distributed representations. In J. Mylopoulos & R. Reiter (Eds.), Proceedings of the 12th International Joint Conference on Artificial Intelligence (pp. 30-35). San Mateo, CA: Morgan Kaufmann.


  • Login