3-Tensor Graph Visualizer

0. Description

Summary: Generates a random 3-tensor represented as a 3-dimensional array, computes the associated graph and displays the result after applying a set of filters (see section 2.)

In particular, given the finite dimensional vector spaces $U,V,W$ of dimensions $n,m,k$ over $\mathbb{F}_q$, we represent a 3-tensor as the trilinear form $\mathcal{C} : U \times V \times W \rightarrow \mathbb{F}_q$.

The computed set of vertices $\mathcal{V}(\mathcal{C})$, corresponds to the disjoint union of projective spaces $\mathcal{V}(\mathcal{C}) := \mathbb{P}(U) \cup \mathbb{P}(V) \cup \mathbb{P}(W)$. On a high level, the set of edges $\mathcal{E}(\mathcal{C})$ is computed as the set of pairs of representatives $(u,v) \in \mathbb{P}(U) \times \mathbb{P}(V)$ (for instance), for which the missing coordinate vanishes, e.g. $(u,v) \in \mathcal{E}(\mathcal{C}) \iff \forall w \in W \mathcal{C}(u,v,w) = 0$ (for implementation details see section 4.)

1. Requirements

• SageMath

2. Command-line arguments

Option	Description	Default
-n N	Dimension n for the first vector space	4
-m M	Dimension m for the second vector space	4
-k K	Dimension k for the third vector space	4
-q Q	Prime field size	5
-c C	Highlights all cycles of length c in the final graph	None
--loose	Highlights all cycles of length 2 < c' <= c	false
--deg_lbound D	Filters out all nodes of degree less or equal than specified	0
--deg_ubound D	Filters out all nodes of degree greater or equal than specified	1000
--isolated_nodes	Displays nodes of degree zero on the final graph	false
--labeled	Show graph with vertex labels	false
--verbose	Show extra info on terminal	false
--isometry	Applies a random isometry to the original tensor and displays it	false
--no_visualization	Prevents the display of a new tab with the resulting tensor graph	False
--minimal	Only prints in terminal the random tensor, and, if enabled, all cycles of various lengths	false
--load_tensor	Loads tensor from file instead of generating a random one [TODO]	""
--load_graph	Loads tensor graph from file instead of calculating one given a random tensor [TODO]	""

3. Sample execution

sage main.py -n=5 -m=5 -k=5 -q=7 --deg_lbound=1 --deg_ubound=10 --verbose

4. Implementation and Specifications

4.1 Vanishing test

Given $(u,v) \in \mathbb{P}(U) \times \mathbb{P}(V)$, the vanishing test for this couple of points is the following:

If $\forall l \in [1,k], \sum_{i = 1}^{n} \sum_{j = 1}^{m} u_i \cdot v_j \cdot \mathcal{C}_{i,j,l} = 0$, then $(u,v) \in \mathcal{E}(\mathcal{C})$

4.2 Limitations

This implementation is only suited for small values $n,m,k,q$. No tests have been done with values higher than 10.

Authors

Developed by David Pulido Cornejo under the supervision of Laurane Marco as part of a combinatorial-algebraic study of the 3-Tensor Isomorphism Problem @ EPFL/LASEC.