Supplementary Files for
Kronecker Powers of Tensors and Strassen's Laser Method
by A. Conner, F. Gesmundo, J. M. Landsberg, E. Ventura
We collect the supplementary files required to complete the computer
calculations described in the paper. The softwares we use are Sage and Macaulay2.
The files are collected in different appendices:
Appendix A: A randomly chosen tensor in C3 x C3 x C3
This data file addresses the Claim of Theorem 2.8. We provide the
coordinates of a randomly chosen tensor in C3 \otimes C3 \otimes C3 and
an approximate decomposition of its Kronecker square.
We include one file, with enclosed documentation. It contains the following two pieces of data:
the coordinates of a tensor T in C3 \otimes C3 \otimes C3,
represented as three matrices of size 3x3, M_1,M_2,M_3: in coordinates
T_ijk = (M_i)_jk. The entries of the matrices were randomly chosen from
the uniform distribution on [-1,1];
the coordinates of 22
rank one tensors in C9 \otimes C9 \otimes C9, forming a rank 22
decomposition of T \boxtimes T. This is a list of 22 elements
Z_1,...,Z_22. Each element Z_j is a list of 3 elements a,b,c. The
elements a,b,c are matrices of size 3x3 representing the coordinates of
three vectors in C9 = C3 \otimes C3. This is the same representation as
the one of Section 5 of the paper.
Download the file here.
Appendix B: Rank 18 Decomposition of 3x3 determinant
This is a Macaulay2 file which verifies the Rank 18 decomposition for the determinant polynomial, Theorem 4.1. It is one file, with enclosed
documentation.
Download the file here.
Appendix C: Border Rank 17 Decomposition of 3x3 determinant
This is a set of files addressing Claims made in the proof of Theorem 4.2. The set consists of 7 files, which can be downloaded in
an archive tar.gz file.
We include seven files with enclosed documentation. The role of each file is briefly explained below:
checkingType1eqns.m2: A Macaulay2 file verifying the equations of type 1, as explained in the paper;
checkingType2eqns.m2: A Macaulay2 file verifying the equations of type 2, as explained in the paper;
eqnsType1.m2: The list of equations of type 1, for easy access: they can be computed directly as well;
eqnsType2.m2: The list of equations of type 2, for easy access: they can be computed directly as well;
monomialsType1.m2: The monomials appearing in equations of type 1, with the corresponding expression in terms of the root y*;
monomialsType2.m2:
The monomials appearing in equations of type 2, with the corresponding
expression in terms of the chosen root u in each equation;
yy_exps.m2: The expressions of the elements y_j in terms of the root y*.
Download the archive file here.
For the reader's convenience, we include the minimal polynomials of the
roots z_i with an approximation of their value in
det_br_17_entries.txt on the border rank 17 decomposition.
Appendix D: Matrices of Koszul flattenings of Coppersmith-Winograd powers
This
set of files addresses the Claims on the matrices representing the
Koszul flattenings of the Kronecker square and the Kronecker cube of
the small Coppersmith-Winograd tensor, in the proof of Theorem 3.3 and
Theorem 3.4. The set consists of three files, which can be downloaded in an archive tar.gz file.
We include three files with enclosed documentation. The role of each file is briefly explained below:
cwsquare.txt: A data file for easy access to the four
matrices Phi_1, ... , Phi_4 describing the map on the multiplicity
spaces in the proof of Theorem 2.1;
cwcube.txt: A data file
for easy access to the eight matrices Psi_1, ... , Psi_8 describing the
map on the multiplicity spaces in the proof of Theorem 2.2;
cwmatrices.sage: A sage file: it generates the relevant matrices using the method
described in Section 6 of the paper, and computes their ranks
Download the archive file here.
Appendix E: Tight tensors in C3 x C3 x C3
This is a Macaulay2
file addressing the Claims made in the proofs of Section 5.
Documentation is enclosed and a description of the various parts of the
file is given in the paper.
Download the file here.
Appendix F: Decompositions of Squares of Tight tensors
This set of files contains the numerical decomposition showing the upper bounds for the Kronecker square of the tight tensors listed in Section 5.2. The archive contains 6 files, corresponding to the six tensors with numerical upper bound.
To give a decomposition with r summands, a file provides r groups of 27 lines, recording the coefficients of the three factors of the corresponding rank one element.
Download the file here.