Suppose you have a symmetric tensor, but for the purpose of optimal memory management you only want to store and work with the the unique elements. For example, the symmetric matrix
A = [ 0 1 2 3 ]
[ 1 4 5 6 ]
[ 2 5 7 8 ]
[ 3 6 8 9 ]
can be fully represented by a one-dimensional vector containing the elements of the upper triangle:
A' = [ 0 1 2 3 4 5 6 7 8 9]
Mapping the index of elements in the full matrix to elements in the one-dimensional minimal representation is trivial in this case. For a given 0-based index pair
which indicate the address of an element in the matrix A, the corresponding 1d index in the flattened upper triangle A' can be found by
where n is the dimension size, in this case 4.
It is much more difficult to derive an analogous equation for the general case of an arbitrary-rank symmetric tensor. That is, for a symmetric tensor of rank r, given a multidimensional index, we seek an expression to find the 1d index of the corresponding element in the flattened upper hypertriangle of that same tensor, which we call z.
It is very likely such a relation has been derived before, but I could not find a reference after quite a lot of seaching around.
This repo serves as a reference implementation and a set of notes regarding the above problem. In the code folder is a recursive Python implementation of the index mapping. In the future, this repo may be extended to find the reverse mapping (1d index to multidimensional index, for a certain rank and dimension size) or perhaps explore implementations which take advantage of these concepts for the case of exploiting efficiencies in tensor contractions.
Without proof or derivation, this is the result, in its non-optimal but most descriptive form:
