Dleto, which means chisel, is a package of tools to carve information out of tensor data. This is a simplified, early release implementation of a larger project of research methods developed in multiple languages. Visit TheTensor.Space for more information about the main project.
Using the algebra of operators on tensors, Dleto methods determine change of coordinates relative to which tensor data is supported on a smaller valence.
For a 3-tensor represented as a point cloud
this means a to cluster the data near a block array of 2-tensors.
Here are some examples of what this can look like:
| Strata | Channels |
|---|---|
 |
 |
|
| Diagonal Blocks | Step Blocks |
|---|---|
 |
 |
We have packaged the essentials into a single file Delto.jl having rudimentary but stand-alone implementations of some of the chiseling methods. You may download that file alone, but you may also benefit from using the examples in examples/... Start with examples/Demo.jl.
- A recent installation of the Julia Language is needed (ver. 1.7.0 or later seems to be compatible with the features required for
OpenDleto). If you do not have an installation of Julia, follow the installation instructions for the Julia system available here. - Clone or Download the
OpenDletorelease from github here. Make surejuliacan be run from whatever folder contains yourOpenDletodownload, typically by ensuring thatjuliais in the path of your operating system shell. - From the command line start julia and load the
OpenDletopackage by usinginclude("$path$/OpenDleto/Dleto.jl")
The functions mostly require standard Julia packages like Random and LinearAlgebra. The only other requirement is the package Arpack which will be installed automatically by the include command above. It is recommended that this package is manually installed; the following are commented out of Dleto.jl (lines 5 & 6):
julia> import Pkg
julia> Pkg.add("Arpack")The code can be used without Arpack but it will run significantly
more slowly. To do so, one must comment out the function ArpackEigen (lines 274-282) and change the default function from ArpackEigen to LinearAlgebraEigen in the function toSurfaceTensor, toFaceCurveTensor and ToCurveTensor (lines 358, 396 and 436)
Our algorithms are provided in a number of platforms.
- The bleeding edge algorithms are developed for, and tested in the [Magma] Computer Algebra System (http://magma.maths.usyd.edu.au/magma/). Core tensor algorithms are distributed with that system. Further information about extensions and experimental additions can be found at TheTensor.Space.
- Python access is available to core algorithms through SageTensorSpace for the Sage Math (in Python).
- Julia language port is being developed as OpenDelto.
The algorithms presented in this tutorial are for instructional purposes. For detailed treatments and improved performance follow the attached references.
We invite you explore the repository and join our team. We welcome and encourage any contributions to the repository. If you need help getting started, please feel free to @-mention any of the contributors below or you can read the repository's Projects tab.
| Name | Username | Affiliation | |
|---|---|---|---|
 |
Prof. Peter A. Brooksbank, Ph.D. | @galois60 |
Bucknell University |
 |
Prof. Martin Kassabov, Ph.D. | @kassabov |
Cornell University |
 |
Joshua Maglione, Ph.D. | @joshmaglione |
University of Galway |
 |
Amaury V. Miniño | @amaury-minino |
Colorado State University |
 |
Prof. James B. Wilson, Ph.D. | @algeboy |
Colorado State University |
The project has received partial support from the following granting organizations:
Portions of the project sponsored by:
- The National Science Foundation (USA) to Peter A. Brooksbank (DMS-1620454), to Martin Kassabov (DMS-1620454) to James B. Wilson (DMS-1620454).
- The Simons Foundation to Peter A. Brooksbank (281435) to Martin Kassabov, and to James B. Wilson (636189).
- The National Security Agency Math Sciences Program to Peter A. Brooksbank (Grant Number H98230-11-1-0146) to James B. Wilson (Grant Number H98230-19-1-00).
We also acknowledge the institutes that hosted research on related projects over the years:
- The Colorado State University
- Kent State University
- The University of Auckland
- Bucknell University
- University Bielefeld
- Hausdorff Institute For Mathematics
- Isaac Newton Institute (EPSRC Grant Number EP/R014604/1)
A tensor supported on a surface:
A random change of basis to the above tensor:
The functions randomSurfaceTensor, randomFaceCurveTensor and randomCurveTensor produce tensors supported near a surface/face-curve/curve. The input to the these functions consists of 3 arrays that define the
restriction; and a parameter cutoff that governs how thick the support is
The helper functions testSurfaceTensor, testFaceCurveTensor and testCurveTensor measure if a tensor is supported near a surface/facecurve/curve with the given equation.
The output is between 0 and 1, where 0 means that the tensor is exactly supported on restriction. These functions are not perfectly normalized: for many restrictions random tensors have values around 0.5.
The functions toSurfaceTensor, toFaceCurveTensor and toCurveTensor attempt to stratify a given tensor by making orthogonal transformations of the 3 coordinates spaces, producing a tensor with a restricted support. The input is a 3-dimensional array. There is a second (optional) argument, which is a function performing an SVD of a large system of linear equations.
The output is a named tuple with 7 components: .tensor is the transformed tensor; .Xchange, .Ychange and .Zchange are the 3 orthogonal matrices used to do the transformation; and .Xes, .Yes and .Zes are vectors that restrict the support.
We recommend that these functions are applied only to non-degenerate tensors (i.e. tensors that cannot be shrunk using HoSVD). If the input tensor is degenerate, these functions are likely to discover the degeneracy and not find any additional structure.
The functions have not been tested on abstract arrays. If the input is a sparse tensor represented as some AbstractArray, it might be necessary to
first convert it to a normal Array.
The files included provide examples of how these functions can be used. Here is a basic example showing that this code can be used to recover diagonal block decomposition of tensors.
First load the file
julia> include("stratification.jl") Define 3 arrays which describe the block structure:
julia> Xes = [1,1,2,2,2,3,3,3]
julia> Yes = [1,1,1,2,2,3,3,3]
julia> Zes = [1,1,1,2,2,2,3,3]8-element Vector{Int64}:
1
1
2
2
2
3
3
3
8-element Vector{Int64}:
1
1
1
2
2
3
3
3
8-element Vector{Int64}:
1
1
1
2
2
2
3
3
Generate a 8x8x8 tensor consisting of blocks of size 2x3x3 3x2x3 and 3x3x2:
julia> tensorDiagonalBlock = randomCurveTensor( Xes, Yes, Zes, 0.1)8×8×8 Array{Float64, 3}:
[:, :, 1] =
0.281444 -0.771449 -0.230731 0.0 0.0 0.0 0.0 0.0
0.68989 0.353554 -0.963161 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[:, :, 2] =
0.831795 -0.558895 1.73975 0.0 0.0 0.0 0.0 0.0
-0.769265 1.03611 0.600005 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[:, :, 3] =
0.470298 -0.678797 0.197142 0.0 0.0 0.0 0.0 0.0
1.41462 -1.02874 0.253569 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[:, :, 4] =
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 -1.67667 -2.17101 0.0 0.0 0.0
0.0 0.0 0.0 -1.16819 0.205952 0.0 0.0 0.0
0.0 0.0 0.0 1.75122 -0.767267 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[:, :, 5] =
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 -1.57631 1.23094 0.0 0.0 0.0
0.0 0.0 0.0 -0.590718 0.951459 0.0 0.0 0.0
0.0 0.0 0.0 -0.456224 -1.44161 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[:, :, 6] =
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 1.43575 -1.07418 0.0 0.0 0.0
0.0 0.0 0.0 -0.237536 -0.631201 0.0 0.0 0.0
0.0 0.0 0.0 0.320036 -0.783304 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
[:, :, 7] =
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 1.28909 0.205456 2.22163
0.0 0.0 0.0 0.0 0.0 0.658314 -0.182946 -0.892084
0.0 0.0 0.0 0.0 0.0 0.322976 1.39161 0.83425
[:, :, 8] =
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 -0.380632 -1.22248 1.88636
0.0 0.0 0.0 0.0 0.0 -1.04584 1.52227 0.376
0.0 0.0 0.0 0.0 0.0 -1.31282 -1.16861 0.00662803
Randomize the tensor by generating three 8x8 orthogonal matrices and performing the change of basis:
julia> rTensorDiagonalBlock = randomizeTensor(tensorDiagonalBlock)(tensor = [-0.7024746276104429 0.2963471007139938 … 0.1253521132232702 -0.23687256022017183; -0.45865089367528006 0.016337187635288155 … -0.22612587931949799 0.06041606592870537; … ; -0.2563707902130204 -0.24131170735091761 … -0.16487227524045395 0.008929327353449163; 0.03108960314383061 -0.007970200573260824 … 0.06600293704539476 -0.7767924232426673;;; 0.17215267649320465 0.5279826208177606 … -0.30737386702533814 0.06261041792500442; 0.025670359458514633 0.4363149673506537 … 0.6348412014860125 -0.16416012911710126; … ; 0.07071570841449397 0.000405164340436413 … 0.12002904521231536 0.1460881197735838; 0.8339145938983152 -0.8600486025999846 … -0.04488674234304724 -0.1659782122918398;;; 0.3077280505781425 -0.16669703274753525 … 0.24312774440072013 -0.13971623080004564; -0.0028081143958506097 0.7073393976346791 … 0.47207333789018396 -0.4623091303747916; … ; -0.1577127916675091 -0.03846313180963299 … -0.0733907706806753 0.32042203586418383; 0.5808429007280725 0.026924918165430457 … -0.4192224118414693 -0.3041357120853166;;; 0.27376822924953426 0.14670564491930838 … 0.16314011895435693 0.14549450615965745; -0.22609705908083771 0.18674534297040812 … 0.3505409252246221 0.754302045081575; … ; 0.3147576224123127 -0.21065952426400014 … -0.11449745239753754 -0.016886070065897235; 0.104196222273735 0.8904678672071237 … 0.3776315743197468 -0.2690904374015832;;; -0.2222043354717667 0.016611735719920903 … -0.12027468260084041 -0.32877368121925893; 0.22628597857093963 0.30569763810461786 … 0.09064176560334732 -0.38840299035583453; … ; -0.31669968800672976 -0.15143232286399796 … -0.15342087554252004 0.3790755632358135; 0.15536125477604426 0.1846989003199384 … 0.14269325649118003 -0.5399927811607663;;; -0.8450534611467111 -0.3444415369606425 … 0.0034731112877044636 -0.38777284007413; 0.052313798990970224 0.052135969241617904 … -0.5075489375730434 -0.014108904610416691; … ; -0.4083102782429647 -0.10130782689805153 … -0.20873158194616379 -0.4735001474552224; -0.15152593852746163 -0.09557737182663935 … -0.618814174948011 0.17332882686295284;;; -0.036320019205084465 -0.13377859028407804 … -0.04363011189969535 -0.2078098147895931; -0.5325619093768249 0.412043838916253 … 0.2827449957835387 0.30306238279288483; … ; 0.01719075969753439 0.08447212632098523 … -0.07089280073125985 -0.03290524905544018; 0.1596917359556732 -0.14035696691206537 … 0.0015949950726324243 -0.435208644446437;;; 0.0014902689258033555 0.06474145222948877 … -0.274174083096012 0.12567985751366126; 0.6011212069014038 -0.35012447518192336 … -0.09123144883709783 0.4388412135290433; … ; 0.0010940474822821904 0.17497787872327328 … -0.01612379164743524 -0.15352081296032266; 0.1533341090670312 -0.059494131630283845 … 0.29375990461560275 0.4061122899637345], Xchange = [0.4744167209451655 -0.1296375671813731 … -0.012087726879920574 -0.5886988008656266; 0.06612714846898976 0.021704090163151 … -0.19636524043757542 0.3067559502282672; … ; -0.6283746195807594 -0.26810125433120097 … -0.5858150602428649 -0.299802172385621; 0.5394919937599101 -0.3836748188990333 … -0.44085939330645313 -0.08906163078837666], Ychange = [0.08701473096783985 0.6940240478195556 … -0.25972423096579006 0.2712638074316892; 0.34529335636032543 -0.08639930381652763 … -0.43013624477157253 -0.0358525901371598; … ; 0.3997686156141235 0.10861763183087642 … -0.25454657840375794 0.13790871547913763; 0.5018035337509246 0.10679006071176456 … 0.17455422379379232 0.28553005126967135], Zchange = [0.49479909752742235 -0.024868077103067543 … -0.31626889487766274 0.32789131052857295; 0.3938416253385604 0.612693825853946 … 0.06046705920253505 0.294662706624177; … ; -0.0909145238816364 -0.012437687502620587 … -0.07060232475092618 -0.48496235569884527; 0.5124789641010667 -0.33249134518184065 … 0.071533630976055 -0.5326894477876711]
Show the randomized tensor:
julia> rTensorDiagonalBlock.tensor8×8×8 Array{Float64, 3}:
[:, :, 1] =
-0.702475 0.296347 0.00247473 -0.287948 0.241706 0.418138 0.125352 -0.236873
-0.458651 0.0163372 0.259663 0.372776 -0.0265987 -0.482275 -0.226126 0.0604161
0.20742 0.24801 0.0610537 0.222524 0.394471 0.0483035 0.115085 0.198717
-0.13426 0.283201 -0.165056 0.226389 0.2382 -0.323128 0.531379 0.107345
-0.118286 0.548111 0.0531631 0.0731164 -0.751058 -0.494927 -0.387676 -0.212955
-0.067795 0.89809 -0.175167 0.454094 0.088632 0.014222 0.7071 -0.58209
-0.256371 -0.241312 0.123928 0.308604 -0.000613099 0.614024 -0.164872 0.00892933
0.0310896 -0.0079702 0.402639 0.333913 0.441475 -0.518071 0.0660029 -0.776792
[:, :, 2] =
0.172153 0.527983 -0.332397 -0.0273464 -0.493023 0.0771282 -0.307374 0.0626104
0.0256704 0.436315 -0.0696738 0.352501 0.342413 -0.233131 0.634841 -0.16416
-0.191051 0.429594 -0.331955 -0.134498 0.0220322 -0.0794263 -0.384593 -0.229604
0.111402 -0.0608521 0.312597 0.167226 0.0310144 -0.138406 0.204569 -0.520867
0.0359766 0.320652 -0.417782 -0.354215 -0.113355 0.868436 -0.108717 0.0234906
0.33696 -0.192297 0.0454511 -0.588523 -0.409182 0.267245 -0.373037 -0.474059
0.0707157 0.000405164 -0.0333864 -0.288611 -0.0801911 -0.443863 0.120029 0.146088
0.833915 -0.860049 0.147022 -0.160017 0.134369 0.218815 -0.0448867 -0.165978
[:, :, 3] =
0.307728 -0.166697 0.321589 0.10504 -0.136093 -0.164758 0.243128 -0.139716
-0.00280811 0.707339 -0.11406 0.329135 0.197757 -0.199973 0.472073 -0.462309
0.143506 0.0638417 0.859034 -0.107018 0.036768 -0.206582 0.397439 -0.115088
-0.0644163 0.413575 -0.291898 0.151935 0.0200681 -0.0465648 -0.0307345 -0.551136
0.321374 -0.427418 0.241959 -0.0564599 0.0359606 0.282802 0.297945 -0.104407
0.687186 -0.157065 -0.026119 -0.442352 -0.283112 0.259708 -0.1488 -1.19558
-0.157713 -0.0384631 -0.239535 -0.128573 -0.00761089 -0.297398 -0.0733908 0.320422
0.580843 0.0269249 -0.105542 -0.440249 -0.219728 0.566895 -0.419222 -0.304136
[:, :, 4] =
0.273768 0.146706 0.154395 0.321822 -0.28801 -0.37823 0.16314 0.145495
-0.226097 0.186745 -0.243589 0.240824 0.273216 -0.0532978 0.350541 0.754302
-0.468163 0.559502 0.491102 -0.376334 -0.462962 -0.0549783 0.169507 0.205869
-0.44377 0.215609 -0.0902764 0.500775 0.307063 -0.413063 0.0472074 -0.051226
0.125337 0.0926563 0.0434762 0.220054 0.388473 0.3851 0.678224 -0.154299
-0.442523 0.558775 0.0333715 0.428467 0.352476 -0.923862 0.453365 -0.471618
0.314758 -0.21066 -0.218337 -0.526293 -0.0643646 -0.496211 -0.114497 -0.0168861
0.104196 0.890468 -0.192724 0.204366 -0.0506031 0.0771721 0.377632 -0.26909
[:, :, 5] =
-0.222204 0.0166117 -0.0298128 0.0679532 0.24579 0.0649964 -0.120275 -0.328774
0.226286 0.305698 0.0759898 0.401641 -0.192561 -0.0322167 0.0906418 -0.388403
-0.217139 -0.00729125 -0.601355 -0.117896 0.0715049 -0.254061 0.0152805 -0.274018
-0.128594 0.314261 0.186989 0.049009 0.153264 -0.115334 0.348303 -0.0805861
0.397388 0.149734 -0.113519 0.620894 -0.0923104 0.0380862 -0.147295 0.0531028
0.447229 0.552056 0.118877 -0.33665 0.101955 -0.163949 0.31943 -1.35773
-0.3167 -0.151432 0.136947 -0.382661 0.0139397 -0.413117 -0.153421 0.379076
0.155361 0.184699 0.0206701 0.24769 -0.303881 0.143143 0.142693 -0.539993
[:, :, 6] =
-0.845053 -0.344442 0.410077 -0.282004 0.463111 0.534818 0.00347311 -0.387773
0.0523138 0.052136 0.240219 0.697709 -0.496851 -0.130202 -0.507549 -0.0141089
-0.050787 0.307767 0.148974 -0.70008 -0.177233 0.0972879 0.326851 0.175916
0.202517 -0.170581 -0.289877 -0.312865 0.269599 0.157883 0.111042 -0.0614107
0.782345 -0.0718234 0.254037 0.998983 -0.903972 -0.224231 -0.468759 0.286396
-0.0617606 -0.562675 -0.175735 -0.643401 0.238106 -0.0281522 -0.0546479 0.430692
-0.40831 -0.101308 0.131137 -0.298939 0.116913 -0.357528 -0.208732 -0.4735
-0.151526 -0.0955774 0.186176 0.212185 -0.399937 0.326532 -0.618814 0.173329
[:, :, 7] =
-0.03632 -0.133779 -0.102505 0.132605 0.360818 -0.107398 -0.0436301 -0.20781
-0.532562 0.412044 0.0346942 0.478511 0.126058 -0.437275 0.282745 0.303062
0.0225267 -0.590152 -0.637609 0.51462 0.535709 -0.494483 0.0753524 -0.342361
-0.149266 0.64897 0.205562 0.0235813 -0.0865639 -0.0402893 0.350529 -0.0979107
0.336398 -0.123433 -0.0536402 0.11789 0.0916253 -0.102047 -0.218299 -0.307972
0.0420125 0.326338 0.117449 -0.0269389 0.0472393 0.00314654 0.119853 -0.156518
0.0171908 0.0844721 0.170145 -0.162205 -0.120738 0.179703 -0.0708928 -0.0329052
0.159692 -0.140357 -0.0471676 -0.103209 -0.279229 0.173 0.001595 -0.435209
[:, :, 8] =
0.00149027 0.0647415 -0.0312331 0.199073 -0.245487 -0.0904413 -0.274174 0.12568
0.601121 -0.350124 -0.131066 0.10503 -0.158538 0.472491 -0.0912314 0.438841
-0.0404055 -0.549661 -0.0472311 -0.373463 0.507555 -0.656963 -0.272479 -0.164481
-0.271 -0.207561 0.137217 -0.186691 -0.123771 0.209469 -0.177623 0.222741
0.258489 0.325605 -0.295374 0.727436 0.0557086 0.713551 0.324661 0.45089
-0.757087 0.122226 -0.0174641 -0.507927 -0.0364498 -0.624282 -0.0562659 0.475568
0.00109405 0.174978 -0.099142 -0.996827 -0.282081 -0.957327 -0.0161238 -0.153521
0.153334 -0.0594941 0.0159036 0.768976 0.328615 -0.199072 0.29376 0.406112
Run the algorithm on the resulting tensor:
julia> resRandomDiagonalBlock=toCurveTensor(rTensorDiagonalBlock.tensor)(tensor = [2.522167441185261 1.1205903752896191 … 4.663513226650628e-16 7.944375935783431e-16; 1.4706704329365592 0.005728378852994986 … 7.527732293872729e-16 6.596708505411562e-16; … ; 1.4457065595388649e-15 7.988439986282894e-16 … 8.519457477147969e-16 1.0956121334688278e-15; -7.503027232723816e-16 1.0995788660076533e-15 … 2.123396700476875e-16 -8.75114787461572e-16;;; 0.8846373128626783 -1.3100718398801987 … 7.724206699978667e-16 -8.369596809421778e-16; 0.04000825619020491 1.0998022676212074 … -6.880571515817601e-16 -9.199924325467899e-17; … ; 4.788099823024108e-16 -8.29150270367448e-16 … 1.3887055829946707e-16 -4.1165826027083053e-16; 5.689860403802004e-16 -2.6745682755962364e-15 … 1.0958035016443158e-16 -6.982681721920173e-16;;; -2.108355599173611e-15 4.580203197690917e-17 … -3.5040673131298044e-17 7.172464184101075e-18; -7.199364977463047e-16 -7.906906063181214e-16 … 1.5293236169512242e-17 -6.985708151737266e-17; … ; 5.193338698775587e-17 -5.575771830359343e-18 … -1.9351776528218928e-15 -1.790077349324323e-15; 7.11714654367747e-17 -3.2742788568521557e-17 … -2.1922179439439517e-16 1.0115074623858426e-15;;; -2.334783904209992e-15 -1.4443087233156743e-15 … 7.135064112274865e-18 2.6324892646381753e-17; -1.4983923409695667e-15 3.6087681890593516e-16 … 3.839308393829004e-17 4.348394409941564e-17; … ; -3.6373447097258985e-19 9.9204594633917e-18 … -1.418613113533548e-15 -1.4567273517816767e-15; 6.836224487020565e-18 -7.831477234102301e-17 … -5.226804812284285e-16 2.3090987739615883e-15;;; -4.877734661962187e-15 -1.903701952733988e-15 … 4.140046776652075e-18 -3.237622590251839e-17; -2.582225472245621e-15 -6.740261977186425e-17 … 3.967741697578755e-17 4.8910625512160073e-17; … ; -1.1883176586491551e-17 4.9380003305519774e-17 … -7.142897090334068e-16 1.374818477270037e-15; 9.863962881996122e-18 -2.946167637516214e-17 … -2.676534162613924e-16 1.9709309822032316e-16;;; -1.4753031931194904e-15 7.9740065836597125e-16 … 3.6417886250331893e-16 -3.3276033800280633e-16; -4.800993321153344e-16 -9.289479217926204e-16 … 2.890460837605447e-16 -2.7334763581668492e-15; … ; -9.10323143942879e-16 4.549051397765812e-16 … 0.6617432077842805 -0.4422350214249645; -1.597027607757424e-16 1.4664763474759782e-15 … 0.49234232819009843 -1.8043053392039574;;; -1.97132378543493e-16 2.3230026636912776e-16 … -1.06502593262581e-15 -1.7914043465672285e-16; 1.6929045586602933e-16 -1.5964272022012813e-16 … 7.55205964554474e-16 1.0414498198611812e-15; … ; 1.0506214296080965e-15 7.448303566507221e-16 … -1.7556782300013851 -0.3647117126455474; 8.656487117304513e-16 -4.921755959251218e-16 … -0.2787317830800355 0.45695140127367023;;; 1.1469071264434575e-15 3.818826538989276e-16 … -2.0486001311755876e-16 -9.495114689100094e-16; 5.690796070050924e-16 9.635657282887038e-17 … 1.9945890906140387e-16 1.990226423401654e-15; … ; 4.0304901547913027e-17 1.6197436152923054e-15 … -0.2666756833710847 -1.7373040425865105; 1.0009783710781039e-16 -4.86057790105166e-16 … 0.0024221870233757502 0.5070589280924723], Xchange = [-0.30391648613709155 0.49492217286520146 … -0.30741452783384354 -0.3673422558782188; -0.38459635349410204 0.22296100230806623 … 0.05026916838399825 0.1214494835909029; … ; 0.3063146820225401 0.7390939309540691 … 0.17294980433521212 -0.09377518847564459; -0.3778245990601943 0.2192718962913001 … 0.051698257426139885 0.6618102305949757], Ychange = [-0.09463120032903263 -0.4988957617347311 … 0.09308815850170468 -0.29630698692843754; 0.003536757550233565 -0.1321651779224803 … -0.18363147802189336 0.5568959994911196; … ; 0.010200790188643089 -0.5676875269027027 … 0.5235385860313793 0.029298967612758575; -0.0588626673825768 -0.3455651359353808 … 0.11256543269428784 0.14717137506051753], Zchange = [-0.36040773857175484 0.375508190691389 … 0.19230178000746806 -0.09541674541714074; 0.2759266036784542 -0.18592928783200177 … -0.31142307444176637 0.32188756471820107; … ; -0.016172453576257184 0.09919778407896743 … -0.3783949026145276 -0.563325964410051; 0.7161672244405628 0.07778843658545467 … 0.05498177119767794 -0.45966162524243376], Xes = [-0.3061464133293273, -0.30614641332932696, -0.3061464133293266, -0.1855828993807031, -0.18558289938070266, -0.18558289938070252, -0.07973683604986744, -0.07973683604986723], Yes = [-0.3061464133293272, -0.30614641332932707, -0.30614641332932657, -0.18558289938070288, -0.18558289938070271, -0.07973683604986775, -0.07973683604986755, -0.0797368360498672], Zes = [-0.30614641332932774, -0.3061464133293272, -0.18558289938070288, -0.18558289938070266, -0.1855828993807025, -0.07973683604986735, -0.07973683604986725, -0.07973683604986709])
Extract the tensor from the output:
julia> resRandomDiagonalBlock.tensor
8×8×8 Array{Float64, 3}:
[:, :, 1] =
2.52217 1.12059 -0.491479 4.0323e-15 -4.08632e-16 -4.04046e-16 4.66351e-16 7.94438e-16
1.47067 0.00572838 0.830886 -2.73681e-16 1.06175e-15 1.62469e-15 7.52773e-16 6.59671e-16
-0.135353 0.88032 0.118945 1.41011e-15 -3.00197e-16 -7.01585e-16 -4.76114e-16 4.09818e-16
-2.24043e-15 -2.16845e-16 6.57912e-16 -3.96042e-16 7.92829e-17 8.67136e-18 2.55006e-17 3.18406e-17
2.22845e-16 1.19605e-15 1.84197e-15 -1.38946e-15 3.48968e-15 -3.53214e-17 -1.30206e-17 -2.15168e-17
-4.15721e-15 -6.33216e-17 2.05346e-15 3.88009e-15 -8.51112e-16 4.74293e-17 1.96653e-17 1.66924e-17
1.44571e-15 7.98844e-16 -2.30331e-16 -1.93393e-17 -7.67856e-17 -4.09608e-16 8.51946e-16 1.09561e-15
-7.50303e-16 1.09958e-15 -1.40573e-15 -8.21914e-18 -1.24638e-16 1.89832e-16 2.1234e-16 -8.75115e-16
[:, :, 2] =
0.884637 -1.31007 -1.04278 -6.87793e-16 1.64758e-16 2.6239e-16 7.72421e-16 -8.3696e-16
0.0400083 1.0998 -1.31048 3.57497e-15 -1.61414e-15 -2.42903e-15 -6.88057e-16 -9.19992e-17
0.286426 -0.585098 -2.00498 1.44944e-15 -1.35172e-15 -1.85796e-15 1.01281e-16 -1.02827e-15
-5.95974e-16 8.48881e-16 -1.23115e-15 6.50744e-17 -6.8735e-16 -5.39626e-17 4.06383e-18 -3.23056e-19
1.21473e-16 1.27586e-15 -5.3657e-15 -1.05402e-15 -8.51879e-16 -1.52024e-17 1.03399e-17 4.11197e-18
-1.02187e-15 2.04807e-15 -3.98539e-15 6.87546e-16 5.35547e-16 -8.06305e-17 -9.43021e-18 1.50217e-17
4.7881e-16 -8.2915e-16 -6.44141e-16 5.93032e-18 -2.00895e-17 2.51905e-16 1.38871e-16 -4.11658e-16
5.68986e-16 -2.67457e-15 2.60366e-16 -1.34718e-17 2.54442e-17 -1.1347e-16 1.0958e-16 -6.98268e-16
[:, :, 3] =
-2.10836e-15 4.5802e-17 8.49539e-16 1.43192e-16 -6.7975e-16 -2.35891e-17 -3.50407e-17 7.17246e-18
-7.19936e-16 -7.90691e-16 4.28171e-16 2.22919e-15 -1.01366e-15 2.10893e-18 1.52932e-17 -6.98571e-17
-6.75437e-17 -1.27758e-16 1.04161e-15 9.79529e-16 -1.7383e-16 -7.80908e-17 -1.59726e-17 2.92271e-17
7.4671e-18 -7.60379e-16 1.04979e-15 0.312846 -0.589409 -3.08086e-16 -1.32131e-16 3.47127e-17
5.41306e-16 1.47269e-15 -2.73771e-16 -1.1175 -0.761157 -3.78684e-17 -4.51735e-16 4.80032e-16
-3.46077e-16 -1.63198e-15 1.3577e-15 0.950679 -0.26616 -2.73867e-16 6.26477e-17 -3.02027e-16
5.19334e-17 -5.57577e-18 5.84988e-17 -1.23883e-15 5.27237e-16 7.44866e-16 -1.93518e-15 -1.79008e-15
7.11715e-17 -3.27428e-17 -8.4649e-18 3.75267e-15 1.42157e-15 -8.41695e-16 -2.19222e-16 1.01151e-15
[:, :, 4] =
-2.33478e-15 -1.44431e-15 7.02981e-17 2.90215e-16 1.03974e-15 -1.98968e-17 7.13506e-18 2.63249e-17
-1.49839e-15 3.60877e-16 -1.24068e-15 1.124e-15 -3.70058e-15 -2.31977e-17 3.83931e-17 4.34839e-17
1.64393e-16 -1.04865e-15 -8.08823e-16 -1.73971e-16 -1.46424e-15 -3.14197e-17 -2.02231e-17 -4.44486e-17
-2.83302e-16 -1.2728e-16 -5.66828e-16 0.353077 0.802246 2.51121e-16 3.10493e-16 -2.16298e-16
-6.62355e-16 -1.7489e-15 -1.65719e-16 1.48504 1.38239 2.2366e-16 6.71003e-16 -6.76284e-16
2.42742e-16 -1.38377e-15 2.5804e-15 0.306609 -1.91255 -8.37346e-16 -5.09604e-16 2.62656e-16
-3.63734e-19 9.92046e-18 1.15547e-17 -6.13356e-16 1.86024e-15 1.07037e-17 -1.41861e-15 -1.45673e-15
6.83622e-18 -7.83148e-17 1.78188e-17 -3.14375e-15 -5.1101e-15 -2.18782e-16 -5.2268e-16 2.3091e-15
[:, :, 5] =
-4.87773e-15 -1.9037e-15 9.68498e-16 -1.01059e-15 1.28853e-16 4.64504e-17 4.14005e-18 -3.23762e-17
-2.58223e-15 -6.74026e-17 -1.4203e-15 4.38384e-15 2.22177e-15 -1.84385e-17 3.96774e-17 4.89106e-17
2.30029e-16 -1.51177e-15 1.85867e-17 1.68188e-15 -5.86924e-17 -6.40744e-17 5.21938e-19 4.37701e-17
3.0844e-16 1.1187e-15 -6.38881e-16 -0.703288 -0.0889086 9.76923e-17 -1.7535e-16 1.98935e-16
1.53974e-16 3.03407e-15 -3.90559e-15 -1.26312 2.22003 1.09573e-15 4.59876e-16 -7.48645e-17
-1.03638e-15 -3.09007e-15 1.29335e-15 2.20638 0.857058 -1.96355e-17 6.51616e-16 -7.58594e-16
-1.18832e-17 4.938e-17 -3.93119e-17 -2.29144e-15 -1.04717e-15 -1.88503e-16 -7.1429e-16 1.37482e-15
9.86396e-18 -2.94617e-17 -8.23456e-17 5.15378e-15 -4.5351e-15 2.0728e-17 -2.67653e-16 1.97093e-16
[:, :, 6] =
-1.4753e-15 7.97401e-16 1.15953e-15 -5.55897e-17 -8.33385e-17 2.88498e-16 3.64179e-16 -3.3276e-16
-4.80099e-16 -9.28948e-16 9.49965e-16 4.22232e-17 -2.5465e-17 -5.35015e-16 2.89046e-16 -2.73348e-15
-1.27275e-16 1.83079e-16 1.60557e-15 1.44136e-17 -9.28417e-17 2.2212e-16 8.37038e-16 -2.02637e-15
5.64899e-18 5.94163e-17 -5.8559e-17 6.17671e-16 5.90854e-16 9.02735e-17 -5.24019e-16 1.48076e-15
3.83886e-17 -3.85817e-17 -7.43163e-17 1.43884e-15 -1.32056e-16 -5.21071e-16 2.16527e-16 -2.14919e-15
6.34745e-17 -1.58692e-17 -1.72794e-17 -4.8812e-16 -1.77176e-15 1.03633e-16 -1.93586e-15 5.75154e-15
-9.10323e-16 4.54905e-16 -2.13709e-16 -2.06862e-16 -6.21183e-17 0.486715 0.661743 -0.442235
-1.59703e-16 1.46648e-15 1.7463e-15 -3.57737e-17 4.49338e-16 -0.121927 0.492342 -1.80431
[:, :, 7] =
-1.97132e-16 2.323e-16 6.74864e-17 4.94184e-17 7.2583e-17 1.82507e-16 -1.06503e-15 -1.7914e-16
1.6929e-16 -1.59643e-16 2.74046e-16 5.22125e-17 -9.57143e-18 -1.4657e-15 7.55206e-16 1.04145e-15
-6.13693e-17 9.62704e-17 1.99755e-16 -5.25127e-17 8.68069e-19 -3.24992e-16 -1.28287e-15 2.15638e-16
-4.00154e-18 4.14104e-17 -9.12827e-17 -1.21056e-16 3.90874e-17 5.23553e-16 3.48697e-16 -3.94672e-16
-2.65662e-17 -5.16069e-17 -2.80219e-17 7.08206e-16 -1.32845e-15 -1.33575e-15 9.60838e-16 9.05151e-16
5.3941e-17 5.4411e-17 -1.70619e-17 -2.42244e-15 1.20941e-15 1.80486e-15 1.86809e-15 -1.27527e-15
1.05062e-15 7.4483e-16 -8.5731e-18 7.18491e-16 2.63906e-16 0.529983 -1.75568 -0.364712
8.65649e-16 -4.92176e-16 3.86665e-16 5.53585e-17 6.72167e-17 -0.690685 -0.278732 0.456951
[:, :, 8] =
1.14691e-15 3.81883e-16 -4.52888e-16 -9.19638e-18 -2.32177e-17 1.083e-16 -2.0486e-16 -9.49511e-16
5.6908e-16 9.63566e-17 3.21682e-16 -2.73522e-17 -2.85056e-17 -5.2123e-16 1.99459e-16 1.99023e-15
-9.15864e-17 3.40839e-16 -2.02543e-16 -2.10722e-17 5.85528e-17 9.12366e-18 -1.72792e-16 -5.29594e-16
2.97407e-17 -1.70129e-17 1.96499e-17 -1.32255e-15 -8.58097e-17 5.22297e-17 4.93885e-17 -3.29781e-16
9.49004e-17 7.1126e-17 -1.71732e-18 -1.22493e-15 1.16244e-15 -5.32166e-16 1.53294e-16 1.93945e-15
3.30005e-17 6.5724e-17 -5.24975e-17 1.16948e-16 2.98927e-15 4.43232e-16 1.59224e-16 -4.68934e-16
4.03049e-17 1.61974e-15 9.73959e-16 3.83415e-16 4.40655e-16 0.356929 -0.266676 -1.7373
1.00098e-16 -4.86058e-16 -2.04602e-16 -7.61651e-17 -1.3487e-16 -0.200466 0.00242219 0.507059
Remove small entries:
julia> resRandomDiagonalBlock.tensor .|> x -> (abs(x) < 1e-10 ? 0 : x )8×8×8 Array{Real, 3}:
[:, :, 1] =
2.52217 1.12059 -0.491479 0 0 0 0 0
1.47067 0.00572838 0.830886 0 0 0 0 0
-0.135353 0.88032 0.118945 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
[:, :, 2] =
0.884637 -1.31007 -1.04278 0 0 0 0 0
0.0400083 1.0998 -1.31048 0 0 0 0 0
0.286426 -0.585098 -2.00498 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
[:, :, 3] =
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0.312846 -0.589409 0 0 0
0 0 0 -1.1175 -0.761157 0 0 0
0 0 0 0.950679 -0.26616 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
[:, :, 4] =
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0.353077 0.802246 0 0 0
0 0 0 1.48504 1.38239 0 0 0
0 0 0 0.306609 -1.91255 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
[:, :, 5] =
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 -0.703288 -0.0889086 0 0 0
0 0 0 -1.26312 2.22003 0 0 0
0 0 0 2.20638 0.857058 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
[:, :, 6] =
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0.486715 0.661743 -0.442235
0 0 0 0 0 -0.121927 0.492342 -1.80431
[:, :, 7] =
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0.529983 -1.75568 -0.364712
0 0 0 0 0 -0.690685 -0.278732 0.456951
[:, :, 8] =
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0.356929 -0.266676 -1.7373
0 0 0 0 0 -0.200466 0.00242219 0.507059
Show the matrices used to transform the tensor:
julia> resRandomDiagonalBlock.Xchange
julia> resRandomDiagonalBlock.Ychange
julia> resRandomDiagonalBlock.Zchange8×8 Matrix{Float64}:
-0.303916 0.494922 0.632854 -0.143851 -0.0373864 0.10319 -0.307415 -0.367342
-0.384596 0.222961 -0.318225 0.166888 0.394633 0.707279 0.0502692 0.121449
0.387186 -0.0124055 -0.0960277 -0.205225 0.0881188 0.176643 -0.817366 0.302556
0.13711 -0.230104 0.207308 -0.803393 0.170952 0.304994 0.341533 0.0307622
-0.439395 -0.230556 -0.518642 -0.358599 -0.214386 -0.0941928 -0.27336 -0.476061
0.401697 0.0688668 -0.0760821 0.128203 -0.652667 0.538857 0.104539 -0.290522
0.306315 0.739094 -0.41329 -0.266497 0.122192 -0.253862 0.17295 -0.0937752
-0.377825 0.219272 -0.021103 -0.218741 -0.56481 -0.0346282 0.0516983 0.66181
8×8 Matrix{Float64}:
-0.0946312 -0.498896 0.537428 -0.252344 0.467874 -0.272531 0.0930882 -0.296307
0.00353676 -0.132165 0.134446 -0.414726 -0.428801 -0.514512 -0.183631 0.556896
-0.0864046 0.410597 -0.0488672 -0.0286676 0.0144864 -0.44312 0.788472 -0.0498069
-0.739176 0.196425 0.328384 -0.0744205 -0.428715 0.163906 -0.0839553 -0.289745
-0.00283802 -0.228625 -0.533242 -0.00334265 -0.23736 -0.45526 -0.203729 -0.598546
-0.65849 -0.180302 -0.459143 0.0430081 0.426502 -0.0620232 -0.0246559 0.367234
0.0102008 -0.567688 -0.198525 -0.247837 -0.345849 0.426883 0.523539 0.029299
-0.0588627 -0.345565 0.219772 0.833475 -0.235292 -0.210822 0.112565 0.147171
8×8 Matrix{Float64}:
-0.360408 0.375508 -0.210573 0.485683 -0.211007 0.59855 0.192302 -0.0954167
0.275927 -0.185929 0.330606 -0.225611 -0.440694 0.578176 -0.311423 0.321888
0.167548 -0.077759 -0.0760857 -0.283268 -0.592515 -0.113154 0.663671 -0.2748
0.388095 -0.215582 0.0504954 0.703291 -0.119785 -0.226575 0.218699 0.438435
0.258629 0.291875 -0.681301 -0.0347941 -0.359565 -0.209335 -0.452529 0.0681743
0.187709 0.819036 0.370281 -0.15042 0.0434866 -0.138887 0.163584 0.29372
-0.0161725 0.0991978 0.479413 0.329772 -0.349298 -0.262268 -0.378395 -0.563326
0.716167 0.0777884 -0.0830973 0.0759328 0.377698 0.333788 0.0549818 -0.459662
Check that these matrices are almost the inverses of the matrices used to randomize the tensor:
julia> rTensorDiagonalBlock.Xchange * resRandomDiagonalBlock.Xchange .|> x -> (abs(x) < 1e-10 ? 0 : x )
julia> rTensorDiagonalBlock.Ychange * resRandomDiagonalBlock.Ychange .|> x -> (abs(x) < 1e-10 ? 0 : x )
julia> rTensorDiagonalBlock.Zchange * resRandomDiagonalBlock.Zchange .|> x -> (abs(x) < 1e-10 ? 0 : x )The product is not the identity, but it is a product of a block orthogonal matrix plus a block permutation:
8×8 Matrix{Real}:
0 0 0 0 0 0 -0.529764 -0.848145
0 0 0 0 0 0 -0.848145 0.529764
0 0 0 -0.0237067 0.926948 -0.374439 0 0
0 0 0 0.962862 -0.0795914 -0.257995 0 0
0 0 0 0.26895 0.36665 0.890637 0 0
0.962225 0.254428 -0.0969028 0 0 0 0 0
0.241948 -0.962313 -0.124156 0 0 0 0 0
0.12484 -0.0960203 0.98752 0 0 0 0 0
8×8 Matrix{Real}:
0 0 0 0 0 -0.836569 0.156648 0.52499
0 0 0 0 0 -0.545099 -0.334099 -0.768924
0 0 0 0 0 0.0549485 -0.92943 0.364886
0 0 0 0.953037 0.302855 0 0 0
0 0 0 0.302855 -0.953037 0 0 0
-0.467403 0.88245 0.053061 0 0 0 0 0
0.351696 0.130544 0.926967 0 0 0 0 0
-0.811076 -0.451929 0.371371 0 0 0 0 0
8×8 Matrix{Real}:
0 0 0 0 0 0.526927 0.817369 -0.23293
0 0 0 0 0 0.820246 -0.56084 -0.112496
0 0 0 0 0 0.222587 0.131782 0.965965
0 0 0.286454 0.0331824 0.957519 0 0 0
0 0 -0.198673 -0.97562 0.0932453 0 0 0
0 0 -0.937269 0.216944 0.272878 0 0 0
-0.589419 -0.807827 0 0 0 0 0 0
-0.807827 0.589419 0 0 0 0 0 0