# Problem with the positivedefinitivenesse of a kernel matrix

**URL:** <https://discourse.julialang.org/t/problem-with-the-positivedefinitivenesse-of-a-kernel-matrix/28976>\
**Category:** Statistics\
**Created:** [September 20, 2019, 3:30am UTC](https://discourse.julialang.org/t/problem-with-the-positivedefinitivenesse-of-a-kernel-matrix/28976 "2019-09-20T03:30:47Z")\
**Posts on this page:** 9\
**Page:** 1

<div class="post-metadata">

**Author:** ![Ujku\_KU](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ujku_ku/32/13159_2.png) [@Ujku\_KU](https://discourse.julialang.org/u/Ujku_KU)\
**Post date:** [September 20, 2019, 3:30am UTC](https://discourse.julialang.org/t/problem-with-the-positivedefinitivenesse-of-a-kernel-matrix/28976/1 "2019-09-20T03:30:47Z")

</div>

Hi guys,

I am trying to generate a positive definite kernel matrix. Below is the minimalistic code:

```julia

julia> using MultivariateStats, ForwardDiff, StatsBase, MLKernels, LinearAlgebra, Optim, Plots, Distributions, ForwardDiff, StatsBase, MLKernels

julia> ν = 0.2
0.2

julia> n = 550
550

julia> Nf = 2
2

julia> lowerBound = 2
2

julia> upperBound = 7
7

julia> leftBound = 2
2

julia> rightBound = 7
7

julia> lowerLeft = [leftBound, lowerBound]
2-element Array{Int64,1}:
 2
 2

julia> upperRight = [rightBound, upperBound]
2-element Array{Int64,1}:
 7
 7

julia> polyConst = 1
1

julia> polyDegree = 2
2

julia> abstract type AbstractKernel <: Function end

julia> struct PolynomialKernel{T₁ <: Real, T₂ <: Real} <: AbstractKernel
           c::T₁
           degree::T₂
       end

julia> function (kernel::PolynomialKernel)(xᵢ::AbstractVecOrMat{<: Number}, xⱼ::AbstractVecOrMat{<: Number})
           c = kernel.c
           degree = kernel.degree
           (c + dot(xᵢ, xⱼ))^degree
       end

julia> function generate_frame_data(N, Nf, lowerLeft, upperRight, ν)
           N = 4 * floor(Int,N/4)
           d = Array{Float64,2}(undef, Nf, N)
           x = rand(Uniform(lowerLeft[1], upperRight[1]), floor(Int,N/2))
           y = rand(Uniform(lowerLeft[2], upperRight[2]), floor(Int,N/2))
           ξ = rand(Normal(0, ν), N)
           for n = 1:4:N
               d[:,n] = [x[trunc(Int, ceil(n/2))], lowerLeft[2] + ξ[n]]
               d[:,n+1] = [x[trunc(Int, ceil(n/2))+1], upperRight[2] + ξ[n+1]]
               d[:,n+2] = [lowerLeft[1] + ξ[n+2], y[trunc(Int, ceil(n/2))]]
               d[:,n+3] = [upperRight[1] + ξ[n+3], y[trunc(Int, ceil(n/2))+1]]
           end
           d
       end
generate_frame_data (generic function with 1 method)

julia> data = generate_frame_data(n, Nf, lowerLeft, upperRight, ν)
2×548 Array{Float64,2}:
 2.31205 3.14641 1.89181 6.91895 2.3989 3.19527 1.90201 6.82866 2.50081 4.44663 … 2.32862 7.038 4.6699 5.29233 2.22841 6.94015 5.80691 5.66697 2.07714 6.8466 
 1.82655 7.05153 2.64666 6.88816 2.18666 7.44543 3.99939 2.91274 1.91899 6.97779 5.48881 6.07231 1.99929 6.67752 5.86748 3.64929 1.57316 7.38312 6.76614 5.66168

julia> fig = plot(data[1,:], data[2,:], seriestype = :scatter, color = :green, legend = false)

julia> tempNormData = fit(ZScoreTransform, data)
ZScoreTransform{Float64}(2, [4.4492, 4.50386], [2.04428, 2.08999])

julia> normalizedData = StatsBase.transform(tempNormData, data)
2×548 Array{Float64,2}:
 -1.04543 -0.637285 -1.251 1.20813 -1.00294 -0.613382 -1.24601 1.16396 -0.953093 … 0.107961 0.412433 -1.08634 1.21849 0.66415 0.595694 -1.16034 1.17273 
 -1.28101 1.21898 -0.888618 1.14082 -1.10871 1.40745 -0.241375 -0.761307 -1.23679 -1.19836 1.04003 0.652453 -0.40889 -1.40226 1.37764 1.08243 0.553983

julia> fig = plot(normalizedData[1,:], normalizedData[2,:], seriestype = :scatter, color = :green, legend = false)

julia> function kernelMatrixConstruct(X::AbstractArray{T, 2};
                                       kernel::PolynomialKernel) where {T<:Real}
               dims = size(X) # the dimensions are: no. of features x number of datapoints
               kernelMatrix = Array{T,2}(undef, dims[2], dims[2]) # dims[2] is the number of datapoints
               for i in 1:dims[2]
                   for j in 1:dims[2]
                       kernelMatrix[i, j] = kernel(X[:, i], X[:, j])
                   end
               end
               kernelMatrix
       end
kernelMatrixConstruct (generic function with 1 method)

julia> function centreKernelMatrix(kernelMatrix::AbstractVecOrMat{T}) where {T<:Real}
               dims = size(kernelMatrix)
               kernelMatrix = (I - ones(dims[1], dims[2])) * kernelMatrix * (I - ones(dims[1], dims[2]))
       end
centreKernelMatrix (generic function with 1 method)

julia> kernelMatrix = kernelMatrixConstruct(normalizedData; kernel = PolynomialKernel(polyConst, polyDegree))
548×548 Array{Float64,2}:
 13.9421 0.0109622 11.876 2.97359 12.0324 0.0261532 6.82157 … 0.582871 1.68971 0.062473 4.41836 1.92524 0.683005 0.87547   
  0.0109622 8.36396 0.509841 2.62672 0.0827482 9.65072 2.24949 4.01979 6.18834 0.0756017 1.28274 5.28859 9.35707 0.861056  
 11.876 0.509841 11.2536 2.32598 10.497 0.26693 7.69088 0.193725 3.16566 0.0259172 2.00285 0.939752 2.21925 0.920383  
  2.97359 2.62672 2.32598 14.1454 2.18011 3.47674 0.609493 7.2079 0.18653 4.02255 0.0410717 10.8327 0.693917 9.2952    
 12.0324 0.0827482 10.497 2.18011 10.4661 0.00299528 6.33675 0.321193 1.86638 0.0534809 3.56678 1.26529 0.928624 0.624719  
  0.0261532 9.65072 0.26693 3.47674 0.00299528 11.2705 2.02936 … 4.88769 6.68035 0.104262 1.90712 6.62329 10.4665 1.12439   
  6.82157 2.24949 7.69088 0.609493 6.33675 2.02936 6.81626 0.0552572 4.82287 0.176029 0.261052 0.00559012 4.77213 0.353968  
  0.575185 0.448624 0.0485757 2.36452 0.4579 0.616941 0.0710453 0.473721 0.579385 7.45055 8.06896 0.415454 1.37981 3.77627   
 12.8216 0.00995384 10.833 2.4411 11.0698 0.0243699 6.18064 0.461559 1.50906 0.118591 4.41544 1.61696 0.588557 0.644619  
  0.265247 5.97173 0.00252855 5.51717 0.0967962 7.11168 0.512443 4.97542 3.14592 0.264675 0.436506 6.91671 5.21087 2.73663   
  4.25603 2.85923 5.2252 0.0555727 4.09009 2.78902 5.26498 … 0.350586 4.6104 0.0782942 0.0767945 0.166615 4.98898 0.044695  
  0.03509 0.0475523 0.336443 6.03209 0.028491 0.0530199 0.537037 1.78562 0.524245 8.14512 5.29925 2.29966 0.877546 6.50328   
 12.8107 0.155979 11.4577 2.59507 11.1789 0.0264606 7.15347 0.359547 2.31207 0.00416521 3.14762 1.38024 1.2911 0.868469  
  1.95156 4.24591 1.11493 11.168 1.3178 5.36424 0.0557114 6.8634 1.02096 1.94745 0.0737499 10.1286 2.24183 6.63282   
  9.57633 0.577663 9.10699 1.35258 8.51617 0.349781 6.35808 0.0537468 2.7959 1.48241e-6 1.82277 0.468882 2.03197 0.462273  
  ⋮ ⋮ ⋱ ⋮                                  
  7.19816 2.64586 8.32455 0.769231 6.7109 2.42223 7.55804 0.052584 5.53546 0.338487 0.139357 0.00779258 5.60708 0.506335  
  0.635776 0.373064 0.852286 8.74462 0.464906 0.501425 0.473535 … 3.60218 0.02789 5.87114 1.84696 5.07578 0.0121682 7.31326   
  6.83792 0.101767 4.79788 0.225785 5.81204 0.29082 2.08584 0.0639736 0.154787 1.71192 6.3833 0.443875 0.00845457 0.0422178 
  1.30888 4.22842 0.64688 9.46469 0.836924 5.25987 0.00132596 6.13887 1.24569 1.56598 0.0658918 8.92289 2.49518 5.57742   
  2.19259 4.99764 3.53029 0.080893 2.30393 5.288 4.74698 1.1285 5.92614 0.208546 0.122345 1.06361 7.36451 0.00198693
  1.65182 1.22713 1.56521 11.4654 1.21468 1.63703 0.576139 5.30282 0.0129812 5.0006 0.622088 7.77418 0.117583 8.4149    
  5.86732 0.280461 3.72424 0.0560184 4.93 0.566799 1.33341 … 0.0407258 0.0101693 2.62942 7.57413 0.344101 0.178436 0.214125  
  0.582871 4.01979 0.193725 7.2079 0.321193 4.88769 0.0552572 5.07043 1.5142 1.16055 0.0340292 7.1742 2.71326 4.24291   
  1.68971 6.18834 3.16566 0.18653 1.86638 6.68035 4.82287 1.5142 6.79035 0.348668 0.405004 1.5668 8.80166 0.00764925
  0.062473 0.0756017 0.0259172 4.02255 0.0534809 0.104262 0.176029 1.16055 0.348668 7.03268 5.67693 1.35152 0.73353 4.85079   
  4.41836 1.28274 2.00285 0.0410717 3.56678 1.90712 0.261052 0.0340292 0.405004 5.67693 11.6105 0.287482 1.6602 1.00409   
  1.92524 5.28859 0.939752 10.8327 1.26529 6.62329 0.00559012 … 7.1742 1.5668 1.35152 0.287482 10.5803 3.23998 6.06036   
  0.683005 9.35707 2.21925 0.693917 0.928624 10.4665 4.77213 2.71326 8.80166 0.73353 1.6602 3.23998 12.3767 0.0570631 
  0.87547 0.861056 0.920383 9.2952 0.624719 1.12439 0.353968 4.24291 0.00764925 4.85079 1.00409 6.06036 0.0570631 7.19419   

julia> kernelMatrix = centreKernelMatrix(kernelMatrix)
548×548 Array{Float64,2}:
      8.954e5 8.95935e5 895615.0 8.95373e5 8.95681e5 8.95684e5 … 8.96054e5 8.95664e5 8.95673e5 8.95613e5 895991.0      
      8.95935e5 8.96491e5 896152.0 8.95921e5 8.96217e5 8.96242e5 8.96603e5 8.96209e5 8.96225e5 8.9617e5 8.96539e5
 895615.0 896152.0 8.95831e5 895589.0 8.95896e5 8.95901e5 8.96271e5 895878.0 8.95889e5 8.95831e5 896208.0      
      8.95373e5 8.95921e5 895589.0 8.95367e5 8.95654e5 8.95671e5 8.96042e5 8.95643e5 8.95665e5 8.95596e5 895983.0      
      8.95681e5 8.96217e5 8.95896e5 8.95654e5 8.95961e5 895966.0 8.96337e5 8.95945e5 8.95954e5 8.95895e5 896273.0      
      8.95684e5 8.96242e5 8.95901e5 8.95671e5 895966.0 8.95993e5 … 896352.0 8.95959e5 8.95975e5 8.9592e5 8.96289e5
      8.9605e5 8.96594e5 8.96268e5 8.96028e5 8.96332e5 8.96343e5 896712.0 8.96317e5 8.96328e5 8.96274e5 896648.0      
      8.95917e5 8.96465e5 8.96133e5 8.95902e5 8.96199e5 8.96214e5 8.96592e5 8.96197e5 8.96201e5 8.96143e5 8.96524e5
      8.95568e5 8.96103e5 8.95782e5 8.95541e5 8.95848e5 8.95852e5 896223.0 895832.0 8.95841e5 8.95781e5 8.96159e5
      8.96171e5 8.96725e5 8.96388e5 896160.0 8.96453e5 8.96476e5 8.96839e5 8.96444e5 8.96463e5 896402.0 8.96778e5
 896347.0 896894.0 8.96565e5 8.96326e5 8.96629e5 8.96643e5 … 897011.0 8.96616e5 8.96628e5 8.96574e5 8.96947e5
      8.95787e5 896335.0 8.96004e5 8.95776e5 8.96069e5 8.96084e5 8.96463e5 8.96065e5 8.96074e5 8.96013e5 896397.0      
      8.95548e5 8.96084e5 8.95764e5 8.95522e5 8.95829e5 895833.0 8.96204e5 8.95811e5 8.95822e5 8.95763e5 8.9614e5 
      8.95725e5 8.96275e5 8.95941e5 8.95717e5 8.96006e5 8.96025e5 8.96393e5 8.95996e5 8.96017e5 8.95951e5 8.96333e5
      8.95967e5 8.96507e5 8.96184e5 8.95942e5 8.96248e5 8.96256e5 8.96626e5 8.96232e5 8.96243e5 8.96185e5 896562.0      
      ⋮ ⋮ ⋱ ⋮                                        
      8.95892e5 8.96436e5 896110.0 895869.0 8.96174e5 8.96185e5 8.96553e5 8.96158e5 8.96169e5 8.96116e5 8.96489e5
      8.95946e5 8.96494e5 8.96163e5 8.95937e5 8.96228e5 896243.0 … 896619.0 8.9622e5 8.96235e5 8.96171e5 8.96556e5
      8.96202e5 896744.0 8.96417e5 8.96179e5 8.96484e5 8.96493e5 8.96865e5 8.96475e5 8.96481e5 8.96421e5 8.96799e5
      8.95946e5 8.96498e5 8.96163e5 895938.0 8.96228e5 8.96248e5 8.96615e5 896218.0 8.96239e5 8.96173e5 8.96555e5
      8.9625e5 8.96801e5 8.96468e5 8.96231e5 8.96532e5 8.9655e5 8.96916e5 8.96521e5 8.96533e5 8.96481e5 8.96851e5
      8.95714e5 8.96262e5 895931.0 8.95708e5 895996.0 8.96012e5 8.96386e5 895986.0 896005.0 8.95939e5 8.96325e5
      8.96157e5 8.96699e5 8.96371e5 8.96134e5 8.96438e5 8.96449e5 … 8.96822e5 8.96431e5 8.96436e5 8.96377e5 8.96755e5
      8.96235e5 8.96787e5 8.96452e5 8.96225e5 8.96517e5 8.96537e5 8.96904e5 8.96508e5 8.96527e5 8.96463e5 8.96843e5
 896090.0 8.96643e5 8.96308e5 896072.0 8.96372e5 8.96393e5 8.96757e5 8.96362e5 8.96375e5 896323.0 8.96692e5
      8.96054e5 8.96603e5 8.96271e5 8.96042e5 8.96337e5 896352.0 8.9673e5 8.96333e5 8.9634e5 8.96281e5 8.96663e5
      8.95664e5 8.96209e5 895878.0 8.95643e5 8.95945e5 8.95959e5 8.96333e5 8.95944e5 8.95944e5 8.95887e5 896264.0      
      8.95673e5 8.96225e5 8.95889e5 8.95665e5 8.95954e5 8.95975e5 … 8.9634e5 8.95944e5 8.95966e5 895900.0 8.96281e5
      8.95613e5 8.9617e5 8.95831e5 8.95596e5 8.95895e5 8.9592e5 8.96281e5 8.95887e5 895900.0 8.9585e5 896216.0      
 895991.0 8.96539e5 896208.0 895983.0 896273.0 8.96289e5 8.96663e5 896264.0 8.96281e5 896216.0 8.96601e5

julia> eigenValVec = eigen(kernelMatrix)
Eigen{Complex{Float64},Complex{Float64},Array{Complex{Float64},2},Array{Complex{Float64},1}}
eigenvalues:
548-element Array{Complex{Float64},1}:
     4.911747351880314e8 + 0.0im                   
      1137.8202188752973 + 0.0im                   
      1054.0479761456127 + 0.0im                   
       838.9836296970793 + 0.0im                   
       312.5398081227351 + 0.0im                   
      21.633901822518677 + 0.0im                   
    1.664423035365322e-8 + 3.584221841941111e-8im  
    1.664423035365322e-8 - 3.584221841941111e-8im  
  -2.7162862774100785e-8 + 0.0im                   
    2.502946340220498e-8 + 0.0im                   
  -2.3080504883470032e-8 + 0.0im                   
  -1.3999997646844292e-8 + 1.70315313702118e-8im   
  -1.3999997646844292e-8 - 1.70315313702118e-8im   
   4.7988399655563274e-9 + 1.66361824146358e-8im   
   4.7988399655563274e-9 - 1.66361824146358e-8im   
                         ⋮                         
    5.46022460876261e-11 + 3.7263223949169517e-10im
    5.46022460876261e-11 - 3.7263223949169517e-10im
   -4.46532142270148e-10 + 4.704761128069045e-11im 
   -4.46532142270148e-10 - 4.704761128069045e-11im 
  1.1263945579014663e-10 + 0.0im                   
  -4.289819408540298e-10 + 2.041056526897707e-10im 
  -4.289819408540298e-10 - 2.041056526897707e-10im 
  2.6214349981218933e-10 + 6.798206427991077e-11im 
  2.6214349981218933e-10 - 6.798206427991077e-11im 
 -1.8915121536809485e-10 + 2.3206075081781673e-10im
 -1.8915121536809485e-10 - 2.3206075081781673e-10im
 -1.5150420311178568e-10 + 8.831852238755217e-11im 
 -1.5150420311178568e-10 - 8.831852238755217e-11im 
  -7.075526147730452e-11 + 0.0im                   
eigenvectors:
548×548 Array{Complex{Float64},2}:
 0.0426961+0.0im -0.0720917+0.0im 0.00940534+0.0im 0.0579544+0.0im -0.0207216+0.0im … 0.0524645+0.0154957im 0.0524645-0.0154957im -0.0703808+0.0im
 0.0427222+0.0im 0.0175642+0.0im -0.0593175+0.0im -0.0375602+0.0im -0.0378221+0.0im 0.000343896+0.00127586im 0.000343896-0.00127586im -0.00479335+0.0im
 0.0427064+0.0im -0.0651934+0.0im -0.00826281+0.0im 0.0492872+0.0im 0.033543+0.0im 0.0102564+0.00158704im 0.0102564-0.00158704im -0.0121957+0.0im
 0.0426953+0.0im 0.066625+0.0im 0.00328951+0.0im 0.0693362+0.0im -0.00310441+0.0im 0.00502525-0.000974774im 0.00502525+0.000974774im 0.00678188+0.0im
 0.0427095+0.0im -0.0650449+0.0im 0.00523479+0.0im 0.0479469+0.0im -0.00746049+0.0im 0.0349251-0.00511892im 0.0349251+0.00511892im -0.0364507+0.0im
 0.0427103+0.0im 0.0236062+0.0im -0.0649393+0.0im -0.0418898+0.0im -0.0584519+0.0im … -0.0216943+0.00290838im -0.0216943-0.00290838im 0.00877036+0.0im
 0.0427274+0.0im -0.043739+0.0im -0.0288086+0.0im 0.0127114+0.0im 0.0643621+0.0im -0.0066288+0.000305589im -0.0066288-0.000305589im 0.0308194+0.0im
 0.0427213+0.0im 0.0152994+0.0im 0.058535+0.0im -0.0455283+0.0im 0.0285042+0.0im 0.000655473-0.00384114im 0.000655473+0.00384114im 0.027396+0.0im
 0.0427041+0.0im -0.067759+0.0im 0.0105112+0.0im 0.0503989+0.0im -0.0236829+0.0im -0.0295279+0.00965549im -0.0295279-0.00965549im 0.0179927+0.0im
 0.0427334+0.0im 0.0334583+0.0im -0.0382319+0.0im 0.000454383+0.0im -0.0560928+0.0im 0.0325476+0.0151013im 0.0325476-0.0151013im -0.0135687+0.0im
 0.0427416+0.0im -0.0296302+0.0im -0.0316837+0.0im -0.00226301+0.0im 0.0479902+0.0im … -0.0278107-0.00399492im -0.0278107+0.00399492im 0.00704777+0.0im
 0.0427152+0.0im 0.0383635+0.0im 0.0526997+0.0im -0.016897+0.0im 0.0748643+0.0im -0.0224047-0.00566027im -0.0224047+0.00566027im 0.00813065+0.0im
 0.0427032+0.0im -0.0683428+0.0im 0.00113766+0.0im 0.0540347+0.0im 0.00568535+0.0im 0.012311-0.0126567im 0.012311+0.0126567im -0.0319987+0.0im
 0.0427121+0.0im 0.0563294+0.0im -0.0161644+0.0im 0.0472602+0.0im -0.0473515+0.0im -0.0116512+0.000494527im -0.0116512-0.000494527im -0.000966125+0.0im
 0.0427233+0.0im -0.0556427+0.0im -0.00760667+0.0im 0.0354221+0.0im 0.0255586+0.0im -0.0194237+0.00641922im -0.0194237-0.00641922im 0.00717196+0.0im
          ⋮ ⋱ ⋮                                                            
 0.0427198+0.0im -0.0457168+0.0im -0.0334265+0.0im 0.0115639+0.0im 0.0782429+0.0im 0.105972-0.00589204im 0.105972+0.00589204im -0.00543094+0.0im
 0.0427227+0.0im 0.0490814+0.0im 0.0299769+0.0im 0.0238459+0.0im 0.0538103+0.0im … -0.0628663+0.0162232im -0.0628663-0.0162232im -0.0326258+0.0im
 0.0427346+0.0im -0.0390823+0.0im 0.0306605+0.0im 0.00118507+0.0im -0.0521176+0.0im 0.0338543+0.0105197im 0.0338543-0.0105197im -0.011491+0.0im
 0.0427227+0.0im 0.0501676+0.0im -0.0184792+0.0im 0.035861+0.0im -0.0462094+0.0im 0.0064186+0.0201039im 0.0064186-0.0201039im -0.0632509+0.0im
 0.0427371+0.0im -0.0154009+0.0im -0.0462731+0.0im -0.0239529+0.0im 0.0401102+0.0im 0.0325889-0.00914366im 0.0325889+0.00914366im 0.0149876+0.0im
 0.0427116+0.0im 0.058597+0.0im 0.0173045+0.0im 0.0485008+0.0im 0.0329596+0.0im 0.0508233-0.00471094im 0.0508233+0.00471094im -0.0283691+0.0im
 0.0427325+0.0im -0.0326595+0.0im 0.0389971+0.0im -0.0126914+0.0im -0.0569178+0.0im … 0.046509-0.0202266im 0.046509+0.0202266im 0.00637999+0.0im
 0.0427365+0.0im 0.0410269+0.0im -0.0202196+0.0im 0.0218283+0.0im -0.039804+0.0im -0.0404817+0.0218601im -0.0404817-0.0218601im 0.0674474+0.0im
 0.0427295+0.0im -0.0111131+0.0im -0.0537179+0.0im -0.0343847+0.0im 0.0371366+0.0im -0.0570779+0.00649837im -0.0570779-0.00649837im 0.0526048+0.0im
 0.0427279+0.0im 0.0267622+0.0im 0.0501617+0.0im -0.0249457+0.0im 0.0485728+0.0im -0.0377565+0.0195414im -0.0377565-0.0195414im 0.0869055+0.0im
 0.0427091+0.0im -0.0201301+0.0im 0.0612899+0.0im -0.0523741+0.0im -0.0606421+0.0im 0.046732-0.0170236im 0.046732+0.0170236im -0.0248631+0.0im
 0.0427097+0.0im 0.0550659+0.0im -0.0250049+0.0im 0.0410069+0.0im -0.0666905+0.0im … -0.0331885+0.000852603im -0.0331885-0.000852603im 0.0587821+0.0im
 0.0427069+0.0im 0.000123438+0.0im -0.0703981+0.0im -0.0607409+0.0im 0.0163231+0.0im -0.0700966+0.00680126im -0.0700966-0.00680126im 0.0249807+0.0im
 0.0427248+0.0im 0.0507146+0.0im 0.0201118+0.0im 0.0334471+0.0im 0.0352572+0.0im -0.0175624-0.0153147im -0.0175624+0.0153147im -0.0247107+0.0im

julia> 

```

As you can see from the result of the values of eigenvalues and eigenvectors it fails to do so.

I suspect it has to do with the overflow.

Your feedback would be very appreciated.

Cheers.

Ergnoor

---

<div class="post-metadata">

**Author:** ![cgeoga](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/cgeoga/32/216186_2.png) [@cgeoga](https://discourse.julialang.org/u/cgeoga)\
**Post date:** [September 21, 2019, 6:15pm UTC](https://discourse.julialang.org/t/problem-with-the-positivedefinitivenesse-of-a-kernel-matrix/28976/2 "2019-09-21T18:15:47Z")

</div>

Hi Ergnoor—I have not run your code, but from looking at the matrix there are a few things that stand out. For one, you can avoid the complex-valued eigenvalues by wrapping your matrix in a `Symmetric` or `Hermitian` structure. Running `eigvals(Symmetric(kernelMatrix))` would return real numbers.

Secondly, and more importantly, your matrix seems incredibly poorly conditioned. I don’t know what’s going on in the dots of the printed output, but every entry we’re seeing looks to be almost exactly the same. If you look at the magnitude of the eigenvalues, your first is is on the order of 10^8 and the second one is on the order of 10^3. So in some sense your matrix is almost a rank one matrix and is probably better thought of as being positive semi-definite from a numerical perspective.

If this is the exact matrix you need to work with, then you’ll probably need to think about some relatively specialized methods for working with such a numerically difficult matrix. Otherwise, it might pay to pick a different kernel or parameterization.

---

<div class="post-metadata">

**Author:** ![Ujku\_KU](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ujku_ku/32/13159_2.png) [@Ujku\_KU](https://discourse.julialang.org/u/Ujku_KU)\
**Post date:** [September 24, 2019, 8:11am UTC](https://discourse.julialang.org/t/problem-with-the-positivedefinitivenesse-of-a-kernel-matrix/28976/3 "2019-09-24T08:11:14Z")

</div>

@cgeoga Hi Chris and thank you very much for your suggestions. I investigated a little bit further the problem of the matrix that you pointed out and in one paper I found this:

> If, in the feature space, the origin is far away from the convex hull of the data, then the elements of K (the Gram matrix) have about the same value and, as a result, the matrix is ill-conditioned.

Hence I believe I need to center the data in feature space in order to avoid problems with, as you portrayed them, numerically difficult matrices.

Once more thank you very much for your support.

Cheers.

Ergnoor

---

<div class="post-metadata">

**Author:** ![jkbest2](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jkbest2/32/7350_2.png) [@jkbest2](https://discourse.julialang.org/u/jkbest2)\
**Post date:** [September 24, 2019, 7:58pm UTC](https://discourse.julialang.org/t/problem-with-the-positivedefinitivenesse-of-a-kernel-matrix/28976/4 "2019-09-24T19:58:34Z")

</div>

It looks like you center your data, so I don’t think that this is the issue. Your `kernelMatrix` is also effectively singular and is not positive definite. Are you sure that the kernel is positive definite? I’m not familiar with this particular kernel. It looks similar to the [rational quadratic](http://www.gaussianprocess.org/gpml/chapters/RW4.pdf) but with a different parameterization?

---

<div class="post-metadata">

**Author:** ![Ujku\_KU](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ujku_ku/32/13159_2.png) [@Ujku\_KU](https://discourse.julialang.org/u/Ujku_KU)\
**Post date:** [September 26, 2019, 6:46am UTC](https://discourse.julialang.org/t/problem-with-the-positivedefinitivenesse-of-a-kernel-matrix/28976/5 "2019-09-26T06:46:33Z")

</div>

@jkbest2 Hi John, and thank you very much for your support. Yes the kernel is positive definite, at least theoretically.

I found the solution using the MLKernels package, it generates much better eigenvalues. It seems the problem is a numerical one.

Thank you again for your support.

Cheers.

Ergnoor

---

<div class="post-metadata">

**Author:** ![jkbest2](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jkbest2/32/7350_2.png) [@jkbest2](https://discourse.julialang.org/u/jkbest2)\
**Post date:** [September 26, 2019, 12:51pm UTC](https://discourse.julialang.org/t/problem-with-the-positivedefinitivenesse-of-a-kernel-matrix/28976/6 "2019-09-26T12:51:55Z")

</div>

Glad you figured it out. I’m interested to see how MLKernels implements this kernel.

---

<div class="post-metadata">

**Author:** ![Ujku\_KU](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/ujku_ku/32/13159_2.png) [@Ujku\_KU](https://discourse.julialang.org/u/Ujku_KU)\
**Post date:** [September 30, 2019, 3:11am UTC](https://discourse.julialang.org/t/problem-with-the-positivedefinitivenesse-of-a-kernel-matrix/28976/7 "2019-09-30T03:11:50Z")

</div>

@jkbest2 Hi John, I think I found out what was wrong with my code. At the centering function I forgot to use `1/n` in the formula that does matrix centering. When I did that correction I got the same results as MLKernels package.

```julia
function centreKernelMatrix(kernelMatrix::AbstractVecOrMat{T}) where {T<:Real}
        dims = size(kernelMatrix)
        kernelMatrix = (I - 1/dims[2]*ones(dims[1], dims[2])) * kernelMatrix * (I - 1/dims[2]*ones(dims[1], dims[2]))
end

```

as you can see I use now `1/dims[2]` that is to say `1/n`.

Again thank you for your support.

Cheers.

Ergnoor

---

<div class="post-metadata">

**Author:** ![jkbest2](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/jkbest2/32/7350_2.png) [@jkbest2](https://discourse.julialang.org/u/jkbest2)\
**Post date:** [September 30, 2019, 4:02am UTC](https://discourse.julialang.org/t/problem-with-the-positivedefinitivenesse-of-a-kernel-matrix/28976/8 "2019-09-30T04:02:22Z")

</div>

Great, glad you figured it out!

---

<div class="post-metadata">

**Author:** ![SEA](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/sea/32/8956_2.png) [@SEA](https://discourse.julialang.org/u/SEA)\
**Post date:** [October 29, 2019, 2:37pm UTC](https://discourse.julialang.org/t/problem-with-the-positivedefinitivenesse-of-a-kernel-matrix/28976/9 "2019-10-29T14:37:51Z")

</div>

Just to clarify properties of the polynomial kernel:

The polynomial kernel is actually only positive-semidefinite. For any kernel matrix generated by the polynomial kernel, the highest rank is `degree`. Once the number of data points exceeds `degree`, you end up with a rank-deficient and non-definite matrix. Therefore, your problem is not _just_ numerical.