path finding framework using hrr
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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Path finding Framework using HRRAlgorithm and associated equations
Surabhi Gupta ’11Advisor: Prof. Audrey St. John
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Roadmap
Circular Convolution Associative Memory Path finding algorithm
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Hierarchical environment
Locations are hierarchically clustered
d e f
a b c
j k l
m n o
Z
X1
X2 X3Y1
X5
X4
X6Y2
g h i
p q r
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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
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Circular ConvolutionHolographic Reduced Representations
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Circular Convolution (HRR) Developed by Tony Plate in 1991 Binding (encoding) operation –
Convolution Decoding operation – Involution
followed by convolution
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Basic Operations
1) Binding2) Merge
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Binding - encoding
C≁AC≁B
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Circular Convolution ( )
Elements are summed along the trans-diagonals (1991, Plate).
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Involution
Involution is the approximate inverse.
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Decoding
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Basic Operations
1) Binding2) Merge
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Merge
Normalized Dot product
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Properties
Commutativity: Distributivity:
(shown by sufficiently long vectors) Associativity:
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Associative MemoryRecall and retrieval of locations
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Framework
d e f
a b c
j k l
m n o
Z
X1
X2 X3Y1
X5
X4
X6Y2
g h i
p q r
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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).
a
Z
X1
X2 X3Y1X5
X4
X6Y2 p
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Associative Memory
Consists of a list of locations Inputs a location and returns the
most similar location from the list.Memory Input OutputWhat do we store?
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Scales
Locations a-r are each2048-bit vectors taken from a normal distribution (0,1/2048).
Higher scales - Recursive auto-convolution of constituents
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Constructing scales
a b c
X1
X1 =
a
b
c
++
a
b
c
a
b
c
X1
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Across Scale sequences
Between each location and corresponding locations at higher scales. a
b c
X1
+a
a X1
a
X1
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Path finding algorithmQuite different from standard graph search algorithms…
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Path finding algorithm
Start Move towards the Goal
Start==Goal?
Go to a higher scale andsearch for the goal
If goal found at this scale
Retrieve the scales corresponding to the goal
If goal not found at this scale
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Retrieving the next scale1) If at scale-0, query the AS memory
to retrieve the AS sequence. Else use the sequence retrieved in a previous step.
2) Query the L memory with
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Retrieving the next scale1) Helllo2) Query the L memory with
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Path finding algorithm
Start Move towards the Goal
Start==Goal?
Go to a higher scale andsearch for the goal
If goal found at this scale
Retrieve the scales corresponding to the goal
If goal not found at this scale
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Locating the goal
For example:location:
and goal: c
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Locating the goal
Goal: p Not contained in X1
a
Z
X1
X2 X3Y1X5
X4
X6Y2 p
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Path finding algorithm
Start Move towards the Goal
Start==Goal?
Go to a higher scale andsearch for the goal
If goal found at this scale
Retrieve the scales corresponding to the goal
If goal not found at this scale
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Goal not found at Y1
a
Z
X1
X2 X3Y1X5
X4
X6Y2 p
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Goal found at Z!
a
Z
X1
X2 X3Y1X5
X4
X6Y2 p
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Path finding algorithm
Start Move towards the Goal
Start==Goal?
Go to a higher scale andsearch for the goal
If goal found at this scale
Retrieve the scales corresponding to the goal
If goal not found at this scale
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Decoding scales
Same decoding operation
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Decoding scales
Using the retrieved scales
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Path finding algorithm
Start Move towards the Goal
Start==Goal?
Go to a higher scale andsearch for the goal
If goal found at this scale
Retrieve the scales corresponding to the goal
If goal not found at this scale
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Moving to the Goal
d e f
a b c
j k l
m n o
Z
X1
X2 X3Y1
X5
X4
X6Y2
g h i
p q r
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To work on
Relax the assumption of a perfect tree.
Relax the assumption of a fully connected graph within a scale location.
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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.