A simple comparison of TensorLy NNCPD with a CVXPY based optimization

In [1]:
import matplotlib.pyplot as plt
import math
import tensorly as tl
import numpy as np
import cvxpy as cvx
import random
import time
from scipy.optimize import linear_sum_assignment
from scipy.optimize import nnls
from scipy.sparse.linalg import svds
from tensorly.decomposition import non_negative_parafac as tlmu
from cvxpy.atoms.affine.transpose import transpose as cvxtranspose
from cvxpy.atoms.quad_form import quad_form
Using numpy backend.

Let us fix a random seed, for reproducibility (nldr: this is not working for now, I am too bad at Python...)

In [2]:
random.seed(0)
random_state = 0
np.random.seed = 0

Defining some of the problems parameters

In [3]:
d = 3
r = 2
dim = [5, 5, 5]
noise_lvl = 0
iter_max_mu = 10000
iter_max_cp = 10000

We now generate random (Uniform distribution) nonnegative factors. To increase the difficulty of the problem, let us resort to a simple trick: write the second column as a combination of the first and itself. Then some noise is added.

In [4]:
Factors = []
for i in range(d):
    Factors.append(np.random.rand(dim[i],r))
    
corr = 0.90
Factors[0][:,1] = corr*Factors[0][:,0] + (1-corr)*Factors[0][:,1]

Data = tl.kruskal_to_tensor(Factors) 
Perturb = noise_lvl*tl.tensor(np.random.standard_normal(dim))
Data_noisy = Data + Perturb

1/ A first method: the native NNCPD of Tensorly

Tensorly comes shipped with a Nonnegative CPD decomposition, which is based on multiplicative update. MU is usually not used for NNCPD, since it is supposedly quite slow. Let us see how it performs here.

In [5]:
tic = time.time()
Festim_tl = tlmu(Data_noisy, r, iter_max_mu, init='svd', tol=1e-07, random_state = random_state, verbose=1)
toc_tl = time.time() - tic
reconstruction error=0.15495120230919698, variation=0.019015509393179836.
reconstruction error=0.13586899933228866, variation=0.019082202976908325.
reconstruction error=0.11877718158055756, variation=0.017091817751731103.
reconstruction error=0.10419061674483379, variation=0.014586564835723764.
reconstruction error=0.09159566177889286, variation=0.01259495496594093.
reconstruction error=0.08036021964129833, variation=0.011235442137594531.
reconstruction error=0.07013019833283446, variation=0.010230021308463871.
reconstruction error=0.06081300048526008, variation=0.009317197847574378.
reconstruction error=0.052446348134493205, variation=0.008366652350766876.
reconstruction error=0.04509070822700029, variation=0.007355639907492918.
reconstruction error=0.038771645261909515, variation=0.0063190629650907715.
reconstruction error=0.033461651730899586, variation=0.0053099935310099294.
reconstruction error=0.02908501954749158, variation=0.004376632183408006.
reconstruction error=0.02553262673556397, variation=0.00355239281192761.
reconstruction error=0.022678652266822887, variation=0.002853974468741083.
reconstruction error=0.02039526235709763, variation=0.002283389909725257.
reconstruction error=0.01856359475673098, variation=0.0018316676003666509.
reconstruction error=0.017080532540788597, variation=0.0014830622159423823.
reconstruction error=0.015861536475628317, variation=0.00121899606516028.
reconstruction error=0.014840406899342245, variation=0.0010211295762860718.
reconstruction error=0.013967138814165744, variation=0.0008732680851765012.
reconstruction error=0.01320495602159354, variation=0.0007621827925722042.
reconstruction error=0.012527297455716334, variation=0.0006776585658772056.
reconstruction error=0.011915176567660262, variation=0.0006121208880560721.
reconstruction error=0.011355064602109112, variation=0.0005601119655511497.
reconstruction error=0.010837289618568925, variation=0.0005177749835401872.
reconstruction error=0.010354872132633367, variation=0.000482417485935558.
reconstruction error=0.009902700535336165, variation=0.0004521715972972025.
reconstruction error=0.009476957445572844, variation=0.00042574308976332104.
reconstruction error=0.009074725159416598, variation=0.00040223228615624516.
reconstruction error=0.008693716051802252, variation=0.0003810091076143461.
reconstruction error=0.008332088847584157, variation=0.00036162720421809534.
reconstruction error=0.007988323329703638, variation=0.00034376551788051907.
reconstruction error=0.00766113458381261, variation=0.0003271887458910277.
reconstruction error=0.007349413916554303, variation=0.00031172066725830734.
reconstruction error=0.007052187763780944, variation=0.00029722615277335846.
reconstruction error=0.006768588755064146, variation=0.0002835990087167988.
reconstruction error=0.006497835025486848, variation=0.0002707537295772973.
reconstruction error=0.006239215157020525, variation=0.000258619868466323.
reconstruction error=0.005992076994717367, variation=0.00024713816230315794.
reconstruction error=0.005755819158360192, variation=0.0002362578363571755.
reconstruction error=0.005529884453645297, variation=0.00022593470471489463.
reconstruction error=0.005313754642630652, variation=0.000216129811014645.
reconstruction error=0.005106946203941262, variation=0.00020680843868939053.
reconstruction error=0.004909006827613837, variation=0.00019793937632742436.
reconstruction error=0.004719512466405123, variation=0.0001894943612087141.
reconstruction error=0.004538064817428402, variation=0.00018144764897672083.
reconstruction error=0.004364289143424568, variation=0.00017377567400383466.
reconstruction error=0.004197832367317902, variation=0.00016645677610666564.
reconstruction error=0.004038361390604109, variation=0.00015947097671379327.
reconstruction error=0.003885561597990457, variation=0.0001527997926136517.
reconstruction error=0.003739135519166652, variation=0.00014642607882380497.
reconstruction error=0.0035988016247107428, variation=0.00014033389445590938.
reconstruction error=0.0034642932376441637, variation=0.00013450838706657907.
reconstruction error=0.0033353575455341653, variation=0.00012893569210999838.
reconstruction error=0.003211754700626947, variation=0.0001236028449072184.
reconstruction error=0.0030932569975067334, variation=0.00011849770312021349.
reconstruction error=0.002979648119369415, variation=0.00011360887813731835.
reconstruction error=0.002870722445281932, variation=0.00010892567408748305.
reconstruction error=0.0027662844118462966, variation=0.00010443803343563544.
reconstruction error=0.0026661479235537823, variation=0.0001001364882925143.
reconstruction error=0.002570135806839886, variation=9.601211671389628e-05.
reconstruction error=0.002478079303464044, variation=9.205650337584196e-05.
reconstruction error=0.0023898175993594716, variation=8.826170410457246e-05.
reconstruction error=0.0023051973855472484, variation=8.462021381222323e-05.
reconstruction error=0.002224072448094568, variation=8.112493745268028e-05.
reconstruction error=0.002146303284434866, variation=7.776916365970221e-05.
reconstruction error=0.002071756743658179, variation=7.454654077668687e-05.
reconstruction error=0.002000305688640512, variation=7.145105501766679e-05.
reconstruction error=0.0019318286781058968, variation=6.847701053461545e-05.
reconstruction error=0.0018662096669171215, variation=6.561901118877522e-05.
reconstruction error=0.0018033377230700315, variation=6.287194384709005e-05.
reconstruction error=0.0017431067600284878, variation=6.023096304154374e-05.
reconstruction error=0.0016854152831806608, variation=5.769147684782697e-05.
reconstruction error=0.0016301661493308016, variation=5.52491338498592e-05.
reconstruction error=0.0015772663382598807, variation=5.2899811070920844e-05.
reconstruction error=0.0015266267355009526, variation=5.0639602758928166e-05.
reconstruction error=0.0014781619255759793, variation=4.846480992497324e-05.
reconstruction error=0.0014317899950377331, variation=4.6371930538246196e-05.
reconstruction error=0.0013874323447486884, variation=4.4357650289044695e-05.
reconstruction error=0.0013450135109125511, variation=4.24188338361373e-05.
reconstruction error=0.0013044609944514941, variation=4.0552516461056986e-05.
reconstruction error=0.0012657050983952668, variation=3.875589605622739e-05.
reconstruction error=0.0012286787730148756, variation=3.702632538039115e-05.
reconstruction error=0.0011933174684957643, variation=3.5361304519111346e-05.
reconstruction error=0.0011595589949993886, variation=3.3758473496375624e-05.
reconstruction error=0.0011273433900121834, variation=3.221560498720523e-05.
reconstruction error=0.0010966127929196389, variation=3.073059709254454e-05.
reconstruction error=0.0010673113267784956, variation=2.9301466141143286e-05.
reconstruction error=0.0010393849872809703, variation=2.792633949752531e-05.
reconstruction error=0.0010127815389210924, variation=2.6603448359877914e-05.
reconstruction error=0.0009874504183759627, variation=2.5331120545129615e-05.
reconstruction error=0.0009633426451083342, variation=2.4107773267628523e-05.
reconstruction error=0.0009404107391800767, variation=2.2931905928257556e-05.
reconstruction error=0.0009186086462386743, variation=2.1802092941402315e-05.
reconstruction error=0.0008978916696025078, variation=2.071697663616652e-05.
reconstruction error=0.0008782164093251875, variation=1.9675260277320344e-05.
reconstruction error=0.0008595407080679558, variation=1.8675701257231652e-05.
reconstruction error=0.0008418236035523027, variation=1.7717104515653178e-05.
reconstruction error=0.0008250252873041354, variation=1.6798316248167245e-05.
reconstruction error=0.0008091070693425363, variation=1.5918217961599147e-05.
reconstruction error=0.0007940313484054163, variation=1.5075720937119926e-05.
reconstruction error=0.0007797615872525321, variation=1.4269761152884233e-05.
reconstruction error=0.0007662622925375046, variation=1.3499294715027489e-05.
reconstruction error=0.0007534989987025209, variation=1.2763293834983742e-05.
reconstruction error=0.0007414382553214365, variation=1.206074338108438e-05.
reconstruction error=0.0007300476172995235, variation=1.1390638021912983e-05.
reconstruction error=0.0007192956373353518, variation=1.0751979964171692e-05.
reconstruction error=0.0007091518600567997, variation=1.014377727855207e-05.
reconstruction error=0.000699586817265226, variation=9.565042791573786e-06.
reconstruction error=0.0006905720237521959, variation=9.01479351303003e-06.
reconstruction error=0.0006820799731955684, variation=8.492050556627487e-06.
reconstruction error=0.0006740841336906238, variation=7.995839504944626e-06.
reconstruction error=0.0006665589425287365, variation=7.5251911618872975e-06.
reconstruction error=0.0006594797998961641, variation=7.079142632572424e-06.
reconstruction error=0.0006528230612278308, variation=6.656738668333272e-06.
reconstruction error=0.0006465660280145553, variation=6.257033213275518e-06.
reconstruction error=0.0006406869369228058, variation=5.879091091749462e-06.
reconstruction error=0.0006351649471448108, variation=5.521989777995001e-06.
reconstruction error=0.0006299801259503241, variation=5.184821194486743e-06.
reconstruction error=0.0006251134324600483, variation=4.866693490275802e-06.
reconstruction error=0.0006205466997025753, variation=4.5667327574729815e-06.
reconstruction error=0.0006162626150525811, variation=4.284084649994169e-06.
reconstruction error=0.0006122446991775265, variation=4.017915875054656e-06.
reconstruction error=0.000608477283641555, variation=3.7674155359715238e-06.
reconstruction error=0.0006049454873327941, variation=3.5317963087608782e-06.
reconstruction error=0.0006016351918897052, variation=3.3102954430889054e-06.
reconstruction error=0.0005985330163078812, variation=3.102175581823981e-06.
reconstruction error=0.0005956262909093068, variation=2.9067253985744354e-06.
reconstruction error=0.0005929030308522784, variation=2.723260057028369e-06.
reconstruction error=0.0005903519093546111, variation=2.551121497667338e-06.
reconstruction error=0.0005879622307925083, variation=2.389678562102768e-06.
reconstruction error=0.000585723903827321, variation=2.2383269651872397e-06.
reconstruction error=0.0005836274146994891, variation=2.0964891278319213e-06.
reconstruction error=0.0005816638008154645, variation=1.9636138840246767e-06.
reconstruction error=0.0005798246247401195, variation=1.8391760753449582e-06.
reconstruction error=0.0005781019486925173, variation=1.7226760476022392e-06.
reconstruction error=0.0005764883096300231, variation=1.6136390624941392e-06.
reconstruction error=0.0005749766949918732, variation=1.511614638149885e-06.
reconstruction error=0.0005735605191609867, variation=1.4161758308864868e-06.
reconstruction error=0.0005722336006912226, variation=1.3269184697641727e-06.
reconstruction error=0.0005709901403361024, variation=1.2434603551201824e-06.
reconstruction error=0.0005698246999053783, variation=1.165440430724138e-06.
reconstruction error=0.0005687321819665339, variation=1.092517938844329e-06.
reconstruction error=0.0005677078104005598, variation=1.0243715659741378e-06.
reconstruction error=0.0005667471118141799, variation=9.606985863798694e-07.
reconstruction error=0.0005658458978045648, variation=9.012140096151342e-07.
reconstruction error=0.0005650002480672437, variation=8.456497373210978e-07.
reconstruction error=0.0005642064943333085, variation=7.937537339352011e-07.
reconstruction error=0.0005634612051185445, variation=7.452892147640292e-07.
reconstruction error=0.0005627611712636282, variation=7.00033854916229e-07.
reconstruction error=0.0005621033922423286, variation=6.577790212996615e-07.
reconstruction error=0.0005614850632126745, variation=6.183290296540812e-07.
reconstruction error=0.0005609035627841827, variation=5.815004284917633e-07.
reconstruction error=0.0005603564414740703, variation=5.471213101123935e-07.
reconstruction error=0.0005598414108236718, variation=5.150306503985376e-07.
reconstruction error=0.0005593563331467716, variation=4.850776769002238e-07.
reconstruction error=0.0005588992118808939, variation=4.5712126587769616e-07.
reconstruction error=0.0005584681825133878, variation=4.31029367506104e-07.
reconstruction error=0.0005580615040537572, variation=4.066784596305946e-07.
reconstruction error=0.0005576775510248385, variation=3.839530289186533e-07.
reconstruction error=0.0005573148059457579, variation=3.6274507908063466e-07.
reconstruction error=0.0005569718522803988, variation=3.429536653591029e-07.
reconstruction error=0.0005566473678260677, variation=3.2448445433104055e-07.
reconstruction error=0.0005563401185177331, variation=3.0724930833467366e-07.
reconstruction error=0.0005560489526243846, variation=2.911658933484572e-07.
reconstruction error=0.0005557727953148761, variation=2.761573095085273e-07.
reconstruction error=0.000555510643571626, variation=2.62151743250132e-07.
reconstruction error=0.0005552615614315463, variation=2.4908214007969467e-07.
reconstruction error=0.0005550246755346319, variation=2.3688589691434146e-07.
reconstruction error=0.000554799170961498, variation=2.2550457313389068e-07.
reconstruction error=0.0005545842873421012, variation=2.148836193968445e-07.
reconstruction error=0.0005543793152189546, variation=2.0497212314658873e-07.
reconstruction error=0.000554183592648992, variation=1.9572256996255692e-07.
reconstruction error=0.0005539965020288812, variation=1.8709062011082467e-07.
reconstruction error=0.0005538174671299432, variation=1.790348989380141e-07.
reconstruction error=0.0005536459503290957, variation=1.715168008475411e-07.
reconstruction error=0.0005534814500232744, variation=1.6450030582127427e-07.
reconstruction error=0.0005533234982156073, variation=1.579518076670573e-07.
reconstruction error=0.0005531716582623137, variation=1.5183995329365145e-07.
reconstruction error=0.0005530255227695454, variation=1.4613549276828574e-07.
reconstruction error=0.0005528847116307488, variation=1.408111387965743e-07.
reconstruction error=0.0005527488701951005, variation=1.3584143564832844e-07.
reconstruction error=0.0005526176675585454, variation=1.3120263655510764e-07.
reconstruction error=0.0005524907949692482, variation=1.268725892971832e-07.
reconstruction error=0.0005523679643398967, variation=1.22830629351511e-07.
reconstruction error=0.0005522489068600243, variation=1.1905747987234639e-07.
reconstruction error=0.0005521333717012181, variation=1.155351588062346e-07.
reconstruction error=0.0005520211248094958, variation=1.1224689172226448e-07.
reconstruction error=0.0005519119477787535, variation=1.0917703074236895e-07.
reconstruction error=0.0005518056368001375, variation=1.0631097861594791e-07.
reconstruction error=0.000551702001681834, variation=1.036351183035392e-07.
reconstruction error=0.0005516008649350075, variation=1.0113674682649988e-07.
reconstruction error=0.0005515020609212393, variation=9.880401376813168e-08.
converged in 195 iterations.

To estimate the reconstructed factors error, we normalize all the columns with the $\ell_2$ norm, and find the best permutation.

In [6]:
for i in range(d):
    for p in range(r):
        Factors[i][:,p] = Factors[i][:,p]/np.linalg.norm(Factors[i][:,p])
        Festim_tl[i][:,p] = Festim_tl[i][:,p]/np.linalg.norm(Festim_tl[i][:,p])
        
# using only the second factor for permutation ambiguity as a quick dirty heuristic
cost = np.transpose(Factors[1]).dot(Festim_tl[1])
perm = linear_sum_assignment(-cost)
for i in range(d):
    Festim_tl[i] = Festim_tl[i][:,perm[1]]

Finally, we can compute the factor error (after normalization) for the multiplicative update rule. Since factors are normalized, their Frobenius norm is $\sqrt(r)$, therefore the sum of their norms amounts to $d\sqrt(r)$. I normalized the final error by this quantity to have some relative error meaning.

In [7]:
err_tl = 0
for i in range(d):
    err_tl = err_tl + np.linalg.norm(Festim_tl[i] - Factors[i], 'fro')
    
err_tl/math.sqrt(r)/d, toc_tl
Out[7]:
(0.011512860596714568, 0.07651400566101074)

format: (error, time)

2/ Using Nonnegative Least Squares solver from scipy

At first I wanted to advocate for CVXPY as a versatile tool for constrained CPD, but maybe it is actually not so easy to use in this context, see my implementation below.

However, still convinced that Multiplicative update is not so great (plus being restricted to nonnegativity constraints), I wanted to try using a nonnegative least squares solver. But because of how little time I can spend on this proof of concept, I could not implement in Python one of my favorite solver such as HALS. Therefore, I used the numpy nnls solver based on active set (which is a good method, but will be slow since it is not vectorized and the structure of the mixing matrix is not used, see the work of Rasmus Bro on this topic).

First, let us initialize the factor matrices. I tried to mimick Tensorly implemetation using SVD. Sometimes, this gives a very poor initialization (a bug?) thus I also included a random initialization.

In [8]:
tic = time.time()
Festim_cp = []
for i in range(d):
    U, _, _ = svds(tl.base.unfold(Data_noisy, i), r)
    #U = np.random.rand(dim[i],r)
    Festim_cp.append(abs(U))
    

Now let us use alternatively the scipy nnls solver

In [9]:
# Perform alternating minimization.
residual = np.zeros(iter_max_cp)
iter_num = 0
err_old = 0
var = 1
normA = tl.norm(tl.base.tensor_to_vec(Data_noisy)) ** 2
while iter_num < iter_max_cp+1 and abs(var)>(10 ** -7):
    for p in range(d):
        
        # Initial results
        X_e = np.zeros(Festim_cp[p].shape)
       
        # Constants: Y = AX, X>=0
        Y = np.transpose(tl.base.unfold(Data_noisy, p))
        A = tl.tenalg.khatri_rao(Festim_cp, p)
           
        # Optimization, not vectorized, no warm start and suboptimal
        for l in range(X_e.shape[0]):
            output = nnls(A,Y[:,l])
            X_e[l,:] = output[0]
            
        # Affectation
        Festim_cp[p] = X_e
        
    # Error computation
    err_new = np.linalg.norm(Y - A.dot(np.transpose(X_e)),'fro')
    
    # Stopping criterion and prints
    var = err_old - err_new 
    err_old = err_new       
    print('Iteration', iter_num, 'residual norm', err_old , 'variation', var)
    residual[iter_num-1] = err_old
    iter_num = iter_num + 1    
        
toc_cp = time.time() - tic
print('converged in', iter_num, 'iterations')
Iteration 0 residual norm 0.2705165837127751 variation -0.2705165837127751
Iteration 1 residual norm 0.10805776579554714 variation 0.16245881791722794
Iteration 2 residual norm 0.07816470840418444 variation 0.029893057391362707
Iteration 3 residual norm 0.06526716847660179 variation 0.012897539927582649
Iteration 4 residual norm 0.054207993508009186 variation 0.0110591749685926
Iteration 5 residual norm 0.04480687322963799 variation 0.009401120278371193
Iteration 6 residual norm 0.03688858542814586 variation 0.007918287801492133
Iteration 7 residual norm 0.030272660739804017 variation 0.006615924688341844
Iteration 8 residual norm 0.024782400728446334 variation 0.0054902600113576824
Iteration 9 residual norm 0.020252247923639343 variation 0.004530152804806991
Iteration 10 residual norm 0.016532486171456286 variation 0.0037197617521830574
Iteration 11 residual norm 0.013491390043974925 variation 0.0030410961274813606
Iteration 12 residual norm 0.011015455502207764 variation 0.0024759345417671615
Iteration 13 residual norm 0.00900841142628829 variation 0.0020070440759194733
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Iteration 2286 residual norm 8.392748405828449e-05 variation 9.995865806790034e-08
converged in 2287 iterations

Let us now again compute the error on the factors, and see if we did a good job wrt the native Tensorly implementation

In [10]:
for i in range(d):
    for p in range(r):
        Festim_cp[i][:,p] = Festim_cp[i][:,p]/np.linalg.norm(Festim_cp[i][:,p])
        
# using only the second factor for permutation ambiguity as a quick dirty heuristic
cost = np.transpose(Factors[1]).dot(Festim_cp[1])
perm = linear_sum_assignment(-cost)
err_cp = 0
for i in range(d):
    Festim_cp[i] = Festim_cp[i][:,perm[1]]
    err_cp = err_cp + np.linalg.norm(Festim_cp[i] - Factors[i], 'fro')
    
err_cp/math.sqrt(r)/d, toc_cp
Out[10]:
(0.0010253987112930666, 1.6560418605804443)

Partial conclusion:

One can appreciate that the reconstruction error on the factors is usually lower using the NNLS optimizer. Furthermore, the cost function decreases faster with the NNLS optimizer per each iteration. The time consumption of NNLS is not however a problem, especially for larger dimensions, but this is an implementation issue. (note: I would print the residuals to show this in an image, but I cannot retrieve errors with the TensorLy implementation?)

Another comment from the future

Nonnegative least squares solvers can of course only be used for nonnegative PARAFAC, but I hoped to use CVX as a general framework. It appears however that, either I am a terrible programmer, or using CVX for tensor factorization, especially the outer loop stuff, is not straightforward. Therefore, there could be interest in implementing both specific optimization methods for constrained problems (such as a good tensor version of NNLS) and a general optimization toolbox for tensor problems, for instance based on AOADMM.

3/ A third approach: conic programming using CVXPY (bugged, not yet functional)

Conic programming is a generalization of linear and quadratic programming introduced by Nesterov and Nemirovsky. Since, for each individual factor, our optimization problem is quadratic programming, we can make use of the CVXPY toolbox to compute estimates of the factors sequentially. We are basically going to implement ANLS (different from NALS which is simply projected ALS). Personally, I prefear HALS for solving nonnegative least squares, but conic programming is flexible whereas HALS is not. So for the sake of this discussion, I will stick to using CVXPY. As a side note, the solver used is SCS, which is basically running ADMM on a transformed problem.

Fortunately for me, CVXPY provides an example on how to compute NMF: http://www.cvxpy.org/examples/applications/nonneg_matrix_fact.html. We can adapt this easily to compute our NNCPD. First, let us initialize our estimates through SVD just like is implemented in TensorLy, without the absolute value.

In [11]:
#tic = time.time()
#Festim_cp = []
#for i in range(d):
#    #U, _, _ = svds(tl.base.unfold(Data_noisy, i), r)
#    U = np.random.rand(dim[i],r)
#    Festim_cp.append(U)
    

Now I have basically copy-pasted the NMF implementation in the CVXPY examples, but adapted to d-way arrays. As a side note, TensorLy uses an unfolding order that matches maintains the Kronecker and Khatri Rao products orders, as mentionned in one of my works: https://arxiv.org/pdf/1511.01306.pdf (a little self promotions never hurts).

In [12]:
## Perform alternating minimization.
#residual = np.zeros(iter_max_cp)
#iter_num = 0
#err_old = 0
#var = 1
#normA = tl.norm(tl.base.tensor_to_vec(Data_noisy)) ** 2
#while iter_num < iter_max_cp+1 and abs(var)>(10 ** -7):
#    # At the beginning of an iteration, Factors are NumPy
#    # array types, NOT CVXPY variables.
#    for p in range(d):
#        # Optimize over only p-th factor variable
#        X = Festim_cp[p]
#        X = cvx.Variable(shape=(dim[p], r))
#        constraint = [X >= 0]
#        
#        # Solve the nnls subproblem.
#        A = tl.base.unfold(Data_noisy, p)
#        Y = np.transpose(tl.tenalg.khatri_rao(Festim_cp, p))
#        
#           
#        # This is bad because Y is structured !! 
#        obj = cvx.Minimize(cvx.norm(A - X*Y, 'fro'))
#        prob = cvx.Problem(obj, constraint)
#        prob.solve(solver=cvx.SCS) ##


        #if prob.status != cvx.OPTIMAL:
         #   raise Exception("Solver did not converge!")
        # Convert variable to NumPy array constant for next iteration.
#        Festim_cp[p] = X.value
    
#    var = err_old ** 2 - prob.value ** 2 
#    err_old = prob.value        
#    print('Iteration', iter_num, 'residual norm', err_old , 'variation', var)
#    residual[iter_num-1] = err_old
#    iter_num = iter_num + 1    
        
#toc_cp = time.time() - tic
#print('converged in', iter_num, 'iterations')

Let us now again post-process the results to obtain factor relative error.

In [13]:
#for i in range(d):
#    for p in range(r):
#        Festim_cp[i][:,p] = Festim_cp[i][:,p]/np.linalg.norm(Festim_cp[i][:,p])
#        
## using only the second factor for permutation ambiguity as a quick dirty heuristic
#cost = np.transpose(Factors[1]).dot(Festim_cp[1])
#perm = linear_sum_assignment(-cost)
#err_cp = 0
#for i in range(d):
#    Festim_cp[i] = Festim_cp[i][:,perm[1]]
#    err_cp = err_cp + np.linalg.norm(Festim_cp[i] - Factors[i], 'fro')
    
#err_cp/math.sqrt(r)/d, toc_cp