Triplet Computation

Define a sparse n-tensor as two-tuple (indices, values) of parallel arrays where indices is an array of n-tuples specifying the location of an element and where values is an array of element values.

Define a graph as a four-tuple (nodes, edges, senders, receivers) where nodes is an array of size N containing node features, and edges, senders, receivers are parallel arrays of size E containing edge features, sending nodes indices, and recieving node indices respectively.

Truth Coefficient Matrix

The truth coefficient matrix TCM is an N x N sparse 2-tensor where each element a_(i,j) in TCM equals the truth value t for the edge connecting the ith and jth node (n_i and n_j). If an edge does not exist between n_i and nodes_j then the element is zero. Note that each column of the TCM represents the outgoing edges for n_i and that each row of TCM represents the incoming edges for n_j. This sparse 2-tensor can be constructed directly from our dataset. We define our indices as the transpose of our E x 2 senders and recievers arrays. We define our values as the truth feature for each edge.

Truth Coefficient Tensor

The truth coefficient tensor TCT is a sparse N x N x N 3-tensor where each element a_(i,j,k) in TCT contains the product of the truth value from the ith incoming edge and the kth outgoing edge of node j. This tensor can be constructed by repeating our TCM N-times, transposing it to align the nth column of the TCM with the nth row, and performing elementwise multiplication by the transposed tensor.

Triplet Parameter Tensor

The Triplet Parameter Tensor TPT is a N x N x N sparse 3-tensor where each element a_(i,j,k) in the TPT contains the helix parameter for the triple (n_i, n_j, n_k). This sparse 3-tensor can be constructed directly from our triplet parameter dataset since each element in the dataset contain a triplet of node indices and an associated radius.

Node Parameter Regression

We can compute our node parameters by calculating the non-zero mean of each matrix in the TPT. This operation will output a size N vector. We can the train our edge classifier by regression using the node parameters computed from truth values output from the edge classifier.

Edge Parameter Regression

We can compute our edge parameters by averaging the node parameter for the sender and receiver of each edge. This operation will output a size E vector. We can the train our edge classifier by regression using the edge parameters computed from truth values output from the edge classifier.