Representations can be designed to achieve objectives
-
What features needed for standard regression?
-
Create algebraic terms that implement those features
+
What features needed for standard regression?
+
Create algebraic terms that implement those features
E.g. Epileptic Seizure Challenge needed interactions of time-series features with time of day (bindings)
-
-
Understand everything at the element level
+
+
+
Indices and permutation
-
-
-
Permutation and indices
+
+
Element indices as unique labels
+
+
Computer people tend to think of vector indices as consecutive integers: \(a_i\) where \(i = 1, 2, \ldots\)
+
+
This imposes more structure than necessary
+
+
Indices only need to be unique : \(i = sad, bee, hot, \ldots\)
+
Indices do not need to be ordered
+
+
Ordering convenient for 2D electronic implementation
+
Ordering is an imposition for 3D neural implementation
+
+
Hypervector is a set of key-value pairs where the values are from the VSA base field (sound familiar?)
+
+
+
+
Permutation and operators
+
+
It doesn’t make sense to talk of permuting an isolated hypervector (interpreted as set of key-value pairs) because it’s unordered
+
Makes sense to talk of permutation:
+
+
relative to another vector,
+
when they are being combined by an operator,
+
because it’s about tracking which elements are combined \[\{a_{a1}, a_{a2}, \ldots\} + \rho\{b_{b1}, b_{b2}, \ldots\} = \{x_{a1.b2}, x_{a2.b3}, \ldots\}\]
+
+
+
+
+
Possible implications
+
+
What makes a tensor product a tensor product is the pattern of combination of the elements of the arguments and the availability of that pattern to guide tensor operations (e.g. tensor contraction)
+
Key-value pairs (hypervector elements) can be represented and operated on as VSA hypervectors
+
Is it possible to self-embed?
+
+
Implement tensor product operations with VSA?
+
Have dynamic elements (add/remove elements)?
+
+
-
+
diff --git a/presentation_Gayler_MidnightVSA_2023-06-15.qmd b/presentation_Gayler_MidnightVSA_2023-06-15.qmd
index f8dbfd4..456af94 100644
--- a/presentation_Gayler_MidnightVSA_2023-06-15.qmd
+++ b/presentation_Gayler_MidnightVSA_2023-06-15.qmd
@@ -227,11 +227,33 @@ Interpret hypervector as specifying a set of indistinguishable realities rather
- E.g. Integer Echo State Network builds standard sequence representation (Interpretable as set of lagged inputs)
- Representations can be designed to achieve objectives
- What features needed for standard regression?
-
- Create algebraic terms that implement those features
-
- E.g. Epileptic Seizure Challenge needed interactions of time-series features with time of day (bindings)
-# Understand everything at the element level
+# Indices and permutation
+
+## Element indices as unique labels
+
+- Computer people tend to think of vector indices as consecutive integers: $a_i$ where $i = 1, 2, \ldots$
+ - This imposes more structure than necessary
+- Indices only need to be unique : $i = sad, bee, hot, \ldots$
+- Indices do *not* need to be ordered
+ - Ordering convenient for 2D electronic implementation
+ - Ordering is an imposition for 3D neural implementation
+- Hypervector is a set of key-value pairs where the values are from the VSA base field (sound familiar?)
+
+## Permutation and operators
+
+- It doesn't make sense to talk of permuting an isolated hypervector (interpreted as set of key-value pairs) because it's unordered
+- Makes sense to talk of permutation:
+ - relative to another vector,
+ - when they are being combined by an operator,
+ - because it's about tracking which elements are combined $$\{a_{a1}, a_{a2}, \ldots\} + \rho\{b_{b1}, b_{b2}, \ldots\} = \{x_{a1.b2}, x_{a2.b3}, \ldots\}$$
+
+## Possible implications
-# Permutation and indices
+- What makes a tensor product a tensor product is the pattern of combination of the elements of the arguments and the availability of that pattern to guide tensor operations (e.g. tensor contraction)
+- Key-value pairs (hypervector elements) can be represented and operated on as VSA hypervectors
+- Is it possible to self-embed?
+ - Implement tensor product operations with VSA?
+ - Have dynamic elements (add/remove elements)?