Add SymCPD type

[alexmul1114]

Adding Symmetric CPD (SymCPD) type to support symmetric generalized CP decompositions in the future.

[codecov]

Codecov Report

Attention: Patch coverage is 65.90909% with 60 lines in your changes missing coverage. Please review.

Project coverage is 89.11%. Comparing base (4900abe) to head (4b1f410).

Files with missing lines	Patch %	Lines
src/gcp-algorithms/lbfgsb.jl	0.00%	27 Missing ⚠️
src/gcp-losses.jl	0.00%	19 Missing ⚠️
src/GCPDecompositions.jl	0.00%	14 Missing ⚠️

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HEAD has 1 upload less than BASE

Flag	BASE (4900abe)	HEAD (4b1f410)
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Additional details and impacted files

@@             Coverage Diff              @@
##            master      #70       +/-   ##
============================================
- Coverage   100.00%   89.11%   -10.89%     
============================================
  Files           13       13               
  Lines          378      551      +173     
============================================
+ Hits           378      491      +113     
- Misses           0       60       +60     

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[alexmul1114]

Added struct definition and constructor, as well as ncomps, ndims, size, and getindex functions for the SymCPD type. The way I have it set up currently is with $\lambda$ the same as in CPD, $U$ a k-tuple (instead of n-tuple as in CPD), where k is the number of groups of symmetric indices, and $S$ is a n-tuple of ints mapping indices to their group of symmetric indices.

[alexmul1114]

Current implementation works sometimes but gets a message from LBFGSB about the line search failing sometimes. Need to check the implementation of the gradients.

julia> using GCPDecompositions: symgcp

julia> using GCPDecompositions: GCPAlgorithms.LBFGSB

julia> U1 = randn(50, 5);

julia> U2 = randn(100,5);

julia> X = zeros(50,100,50);

julia> for i1 in axes(U1, 1), i2 in axes(U2, 1), i3 in axes(U1, 1)
       X[i1,i2,i3] = sum(U1[i1,:].*U2[i2,:].*U1[i3,:])
       end

julia> r=5;

julia> S=(1,2,1);

julia> symgcp(X,r,S,algorithm=LBFGSB(iprint=0))
RUNNING THE L-BFGS-B CODE

           * * *

Machine precision = 2.220D-16
 N =          750     M =           10
 This problem is unconstrained.

           * * *

Tit   = total number of iterations
Skip  = number of BFGS updates skipped
Nact  = number of active bounds at final generalized Cauchy point
Projg = norm of the final projected gradient
F     = final function value

           * * *

   N    Tit     Tnf  Tnint  Skip  Nact     Projg        F
  750    118    214      2     0     0   1.641D-03   5.044D-09

CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH

 Total User time 4.016E+00 seconds.

GCPDecompositions.SymCPD{Float64, 3, 2, Vector{Float64}, Matrix{Float64}}([1.0, 1.0, 1.0, 1.0, 1.0], ([-0.11918929749329864 -1.2724469007624526 … -0.870107400330269 0.04966934885230875; -1.2991765380062754 -0.37177018362825986 … 0.5150619404504531 0.15462793146561693; … ; -0.6843013982199081 0.5827294596219785 … 0.47958196246631474 1.9701944770269133; 1.3196244341593173 0.032043865029284635 … -0.7615196194105708 0.11409868794415903], [-0.8120270258170258 -2.4931121824474904 … -0.7798001841923826 2.930681591824272; -1.3169065074544186 -0.21828985220189268 … -3.9932367288487898 1.8689807048764457; … ; 1.0646765213905314 -1.882228373392413 … 2.9068016565247605 0.6349502467851968; -0.1921145188068539 1.6504620292246774 … -0.9600423953104262 2.2952375803193386]), (1, 2, 1))

julia> symgcp(X,r,S,algorithm=LBFGSB(iprint=0))
RUNNING THE L-BFGS-B CODE

           * * *

Machine precision = 2.220D-16
 N =          750     M =           10
 This problem is unconstrained.

           * * *

Tit   = total number of iterations
Tnf   = total number of function evaluations
Tnint = total number of segments explored during Cauchy searches
Skip  = number of BFGS updates skipped
Nact  = number of active bounds at final generalized Cauchy point
Projg = norm of the final projected gradient
F     = final function value

           * * *

   N    Tit     Tnf  Tnint  Skip  Nact     Projg        F
  750     10     37      1     0     0   6.997D+03   1.778D+06

CONVERGENCE: REL_REDUCTION_OF_F_<=_FACTR*EPSMCH

 Warning:  more than 10 function and gradient
   evaluations in the last line search.  Termination
   may possibly be caused by a bad search direction.

 Total User time 2.500E-01 seconds.