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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

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Path finding Framework using HRR

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#### 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

• Binding

• Merge

C≁A

C≁B

### Circular Convolution ( )

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

### Involution

• Involution is the approximate inverse.

• Binding

• Merge

### Merge

• Normalized Dot product

### Properties

• Commutativity:

• Distributivity:(shown by sufficiently long vectors)

• Associativity:

### Associative Memory

Recall and retrieval of locations

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 =

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 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

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

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.