Download - Path finding Framework using HRR
Path finding Framework using HRRAlgorithm and associated equations
Surabhi Gupta ’11Advisor: Prof. Audrey St. John
Roadmap
Circular Convolution Associative Memory Path finding algorithm
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
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 ConvolutionHolographic Reduced Representations
Circular Convolution (HRR) Developed by Tony Plate in 1991 Binding (encoding) operation –
Convolution Decoding operation – Involution
followed by convolution
Basic Operations
1) Binding2) Merge
Binding - encoding
C≁AC≁B
Circular Convolution ( )
Elements are summed along the trans-diagonals (1991, Plate).
Involution
Involution is the approximate inverse.
Decoding
Basic Operations
1) Binding2) Merge
Merge
Normalized Dot product
Properties
Commutativity: Distributivity:
(shown by sufficiently long vectors) Associativity:
Associative MemoryRecall and retrieval of locations
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
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
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?
Scales
Locations a-r are each2048-bit vectors taken from a normal distribution (0,1/2048).
Higher scales - Recursive auto-convolution of constituents
Constructing scales
a b c
X1
X1 =
a
b
c
++
a
b
c
a
b
c
X1
Across Scale sequences
Between each location and corresponding locations at higher scales. a
b c
X1
+a
a X1
a
X1
Path finding algorithmQuite different from standard graph search algorithms…
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
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
Retrieving the next scale1) Helllo2) Query the L memory with
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
Locating the goal
For example:location:
and goal: c
Locating the goal
Goal: p Not contained in X1
a
Z
X1
X2 X3Y1X5
X4
X6Y2 p
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
Goal not found at Y1
a
Z
X1
X2 X3Y1X5
X4
X6Y2 p
Goal found at Z!
a
Z
X1
X2 X3Y1X5
X4
X6Y2 p
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
Decoding scales
Same decoding operation
Decoding scales
Using the retrieved scales
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
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
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.