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spdesic.py
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238
spdesic.py
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import torch as th
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import geoopt
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import pymanopt
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from tqdm import tqdm
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n = 3
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spd_alt = geoopt.SymmetricPositiveDefinite()
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spd = pymanopt.manifolds.positive_definite.SymmetricPositiveDefinite(n, k=1)
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so = pymanopt.manifolds.special_orthogonal_group.SpecialOrthogonalGroup(n, k=1)
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dist = spd.dist
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shape = (n,n)
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eta = 1
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pos = (eta)/(1+eta)
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s = 3.14159
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d = 0.01
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eps = 0.001
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num = 1024
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blacklist = []
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blacklist = ['linear_riemann_eigen', 'linear_riemann_eigen_sqrt', 'riemann']
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#blacklist += ['linear_eigen', 'sqrt_eigen']
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#blacklist += ['scaled_eigen']
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#blacklist += ['euclidean_sqrt']
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def genRandSPDs(local=True):
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if local:
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a = spd.random_point()*s
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#a = spd.random(shape)
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#return a, a + spd.random(shape)*d
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return th.Tensor(a) + th.eye(n)*eps, th.Tensor(a + spd.random_point()*s*d) + th.eye(3)*eta
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return spd.random(shape), spd.random(shape)
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akku = 0
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def calcErrors(a, b):
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global akku
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### eigen decomp
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ewa, eva = th.linalg.eigh(a)
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ewb, evb = th.linalg.eigh(b)
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ewa, eva = ewa.real, eva.real
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ewb, evb = ewb.real, evb.real
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if th.norm(eva-evb) > d*2:
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# EVs flipped; try again...
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return False
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### euclidean approx (also depends on eigendecomp for fair comparison)
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ar = eva @ th.diag(ewa) @ eva.T
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br = evb @ th.diag(ewb) @ evb.T
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emb = (ar + eta*br)/(1+eta)
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### euclidean_sqrt
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asqrt = eva @ th.diag(th.sqrt(ewa)) @ eva.T
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bsqrt = evb @ th.diag(th.sqrt(ewb)) @ evb.T
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mbsqrt = (asqrt + eta*bsqrt)/(1+eta)
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sqrtmb = mbsqrt@mbsqrt
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### chol approx (also depends on eigendecomp for fair comparison)
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la = th.linalg.cholesky(ar)
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lb = th.linalg.cholesky(br)
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cholmb_chol = (la + eta*lb)/(1+eta)
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cholmb = cholmb_chol@cholmb_chol.T
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### riemann
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if 'riemann' in blacklist:
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riemannmb = ar
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else:
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riemannmb = th.Tensor(spd.exp(ar.numpy(), pos*(spd.log(ar.numpy(), br.numpy())))).real.float()
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### eigen approx
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# com
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ewmb, ecvmb = (ewa + eta*ewb)/(1+eta), eva
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# lin
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elvmb = (eva + eta*evb)/(1+eta)
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# not closed form, but stable gradients
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elvmb_retr = th.Tensor(so.retraction(eva, elvmb-eva))
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if so.norm(elvmb_retr, elvmb-elvmb_retr) > 1.0:
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elvmb = elvmb_retr
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# lin_riemann
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if 'linear_riemann_eigen' in blacklist and 'linear_riemann_eigen_sqrt' in blacklist:
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elrvmb = eva
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else:
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elrvmb = th.Tensor(so.exp(eva, pos*(so.log(eva, evb)))).real.float()
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# ew_sqrt
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ewsqrtmb_sqrt = (th.sqrt(ewa) + eta*th.sqrt(ewb))/(1+eta)
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ewsqrtmb = ewsqrtmb_sqrt**2
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# ew scaling
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esvmb = ((eva@th.diag(ewa) + eta*evb@th.diag(ewb))/(1+eta))/ewmb
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cmb = ecvmb @ th.diag(ewmb) @ ecvmb.T
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cmb = spd_alt.projx(cmb)
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lmb = elvmb @ th.diag(ewmb) @ elvmb.T
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#lmb = spd_alt.projx(lmb)
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# THIS
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lsqrtmb = elvmb @ th.diag(ewsqrtmb) @ elvmb.T
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lrmb = elrvmb @ th.diag(ewmb) @ elrvmb.T
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#lrmb = spd_alt.projx(lrmb)
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lrsqrtmb = elrvmb @ th.diag(ewsqrtmb) @ elrvmb.T
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smb = esvmb @ th.diag(ewmb) @ esvmb.T
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#smb = spd_alt.projx(smb)
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lssqrtmb = esvmb @ th.diag(ewsqrtmb) @ esvmb.T
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### checking
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if True:
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a = a.numpy() # Sigma_old
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b = b.numpy() # Sigma
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emb = emb.numpy() # euclidean line
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cholmb = cholmb.numpy() # line in chol space
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sqrtmb = sqrtmb.numpy() # line in spq-matrix space
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riemannmb = riemannmb.numpy() # spd geodesic (theoretical best case)
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cmb = cmb.numpy() # eigen under commutative assumption
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lmb = lmb.numpy() # eigen with linear basis interpolation
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lrmb = lrmb.numpy() # eigen with eigenbasis interpolation along so(n) geodesic
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lsqrtmb = lsqrtmb.numpy() # eigen with linear eigenbasis interpolation and sqrt interpol for EW (=std interpol)
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lrsqrtmb = lrsqrtmb.numpy() # eigen with eigenbasis interpolation along so(n) geodesic and sqrt interpol for EW
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lssqrtmb = lssqrtmb.numpy() # eigen with scaled eigenbasis interpolation and sqrt interpol for EW
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smb = smb.numpy() # eigen with scaled interpolation
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# ground truth
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tru_damb = dist(a, b)
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# euclid
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euc_damb = dist(a, emb) + dist(emb, b)
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# euclid
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riemann_damb = dist(a, riemannmb) + dist(riemannmb, b)
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# euclid_sqrt
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sqrt_damb = dist(a, sqrtmb) + dist(sqrtmb, b)
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# chol
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chol_damb = dist(a, cholmb) + dist(cholmb, b)
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# ew com
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ewc_damb = dist(a, cmb) + dist(cmb, b)
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# ew lin
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if 'linear_eigen' in blacklist:
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ewl_damb = 0
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else:
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ewl_damb = dist(a, lmb) + dist(lmb, b)
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# ew sqrt
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if 'sqrt_eigen' in blacklist:
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ewlsqrt_damb = 0
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else:
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ewlsqrt_damb = dist(a, lsqrtmb) + dist(lsqrtmb, b)
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# ew sca sqrt
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ewlssqrt_damb = dist(a, lssqrtmb) + dist(lssqrtmb, b)
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# ew lin riemann
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ewlr_damb = dist(a, lrmb) + dist(lrmb, b)
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# ew riemann sqrt
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ewlrsqrt_damb = dist(a, lrsqrtmb) + dist(lrsqrtmb, b)
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# ew sca
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ews_damb = dist(a, smb) + dist(smb, b)
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akku += dist(sqrtmb, lsqrtmb)/tru_damb
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return abs(euc_damb-tru_damb), abs(ewc_damb-tru_damb), abs(ewl_damb-tru_damb), abs(ews_damb-tru_damb), abs(chol_damb-tru_damb), abs(sqrt_damb-tru_damb), abs(ewlr_damb-tru_damb), abs(ewlrsqrt_damb-tru_damb), abs(ewlsqrt_damb-tru_damb), abs(riemann_damb-tru_damb), abs(ewlssqrt_damb-tru_damb)
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#except:
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# print('num issue')
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# return 0, 0, 0, 0
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def testSingle(local=True):
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a, b = genRandSPDs(local=local)
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return calcErrors(a, b)
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def test(num=1024, local=True):
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euc_errs, ewc_errs, ewl_errs, ews_errs, chol_errs, sqrt_errs, ewlr_errs, ewlrsqrt_errs, ewlsqrt_errs, rie_errs, ewlssqrt_errs = 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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for i in tqdm(range(num)):
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res = False
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while res == False:
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res = testSingle(local=local)
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euc_err, ewc_err, ewl_err, ews_err, chol_err, sqrt_err, ewlr_err, ewlrsqrt_err, ewlsqrt_err, rie_err, ewlssqrt_err = res
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euc_errs += euc_err
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ewc_errs += ewc_err
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ewl_errs += ewl_err
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ews_errs += ews_err
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chol_errs += chol_err
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sqrt_errs += sqrt_err
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ewlr_errs += ewlr_err
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ewlrsqrt_errs += ewlrsqrt_err
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ewlsqrt_errs += ewlsqrt_err
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rie_errs += rie_err
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ewlssqrt_errs += ewlssqrt_err
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return euc_errs/num, ewc_errs/num, ewl_errs/num, ews_errs/num, chol_errs/num, sqrt_errs/num, ewlr_errs/num, ewlrsqrt_errs/num, ewlsqrt_errs/num, rie_errs/num, ewlsqrt_errs/num
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names = ['euclidean', 'commutative_eigen', 'linear_eigen', 'scaled_eigen', 'euclidean_chol', 'euclidean_sqrt', 'linear_riemann_eigen', 'linear_riemann_eigen_sqrt', 'sqrt_eigen', 'riemann', 'scaled_sqrt_eigen']
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res = th.Tensor(test(num=num, local=True))/d*100
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for n,r in sorted(zip(names, res), key=lambda x: float(x[1].item()), reverse=False):
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if not n in blacklist:
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print(n+': '+'%.6f' % r+'%')
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print('---')
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print(str(akku/num*100) + '%')
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#---
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#s = 3.14159
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#d = 0.01
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#eps = 0.001
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#100%|██████████████████████████████████████████████████████████████████████████████████████████| 131072/131072 [08:24<00:00, 259.59it/s]
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#sqrt_eigen: 0.030458%
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#scaled_sqrt_eigen: 0.030458%
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#euclidean_chol: 0.033712%
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#euclidean: 0.119651%
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#scaled_eigen: 0.119896%
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#linear_eigen: 0.119899%
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#commutative_eigen: 0.154270%
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#---
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#0.012237632813561243%
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