Path finding Framework using HRR

1 / 38

# Path finding Framework using HRR - PowerPoint PPT Presentation

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

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Path finding Framework using HRR' - herman

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

Algorithm and associated equations

### Path finding Framework using HRR

• 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
Circular Convolution ( )
• Elements are summed along the trans-diagonals (1991, Plate).
Involution
• Involution is the approximate inverse.
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
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 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 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 found at this scale

Retrieve the scales corresponding to the goal

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 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 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.