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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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algorithm and associated equations

Surabhi Gupta ’11

Advisor: Prof. Audrey St. John

Algorithm and associated equations

Path finding Framework using HRR

roadmap
Roadmap
  • Circular Convolution
  • Associative Memory
  • Path finding algorithm
hierarchical environment
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
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
Circular Convolution

Holographic Reduced Representations

circular convolution hrr
Circular Convolution (HRR)
  • Developed by Tony Plate in 1991
  • Binding (encoding) operation – Convolution
  • Decoding operation – Involution followed by convolution
basic operations
Basic Operations
  • Binding
  • Merge
circular convolution1
Circular Convolution ( )
  • Elements are summed along the trans-diagonals (1991, Plate).
involution
Involution
  • Involution is the approximate inverse.
basic operations1
Basic Operations
  • Binding
  • Merge
merge
Merge
  • Normalized Dot product
properties
Properties
  • Commutativity:
  • Distributivity:(shown by sufficiently long vectors)
  • Associativity:
associative memory
Associative Memory

Recall and retrieval of locations

framework
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
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 memory1
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
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
Across Scale sequences
  • Between each location and corresponding locations at higher scales.

X1

a

b c

+

path finding algorithm
Path finding algorithm

Quite different from standard graph search algorithms…

path finding algorithm1
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
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 scale1
Retrieving the next scale
  • Helllo
  • Query the L memory with
path finding algorithm2
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
Locating the goal
  • For example:location:
  • and goal: c
locating the goal1
Locating the goal
  • Goal: p
  • Not contained in X1

X1

X4

a

X2

X5

X3

Z

Y2

X6

Y1

p

path finding algorithm3
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
Goal not found at Y1

X1

X4

a

X2

X5

X3

Z

Y1

Y2

p

X6

goal found at z
Goal found at Z!

X1

X4

a

X2

X5

X3

Z

Y1

Y2

p

X6

path finding algorithm4
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
Decoding scales
  • Same decoding operation
decoding scales1
Decoding scales
  • Using the retrieved scales
path finding algorithm5
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
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
To work on
  • Relax the assumption of a perfect tree.
  • Relax the assumption of a fully connected graph within a scale location.
references
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.
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