/
jadoc.py
266 lines (246 loc) · 8.79 KB
/
jadoc.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
import scipy.linalg
from scipy.sparse import linalg
import numpy as np
import time
from numba import njit,prange
def PerformJADOC(mC,mB0=None,iT=100,iTmin=10,dTol=1E-4,dTauH=1E-2,dAlpha=0.9,\
iS=None):
"""Joint Approximate Diagonalization under Orthogonality Constraints
(JADOC)
Authors: Ronald de Vlaming and Eric Slob
Repository: https://www.github.com/devlaming/jadoc/
Input
------
mC : np.ndarray with shape (iK, iN, iN)
iK Hermitian iN-by-iN matrices to jointly diagonalize
mB0 : np.ndarray with shape (iN, iN), optional
starting value for unitary transformation matrix such
that mB@mC[i]@(mB.conj().T) is approximately diagonal for all i
iT : int, optional
maximum number of iterations; default=100
iTmin : int, optional
minimal number of iterations before convergence is tested; default=10
dTol : float, optional
stop if average magnitude elements gradient<dTol; default=1E-4
dTauH : float, optional
minimum value of second-order derivatives; default=1E-2
dAlpha : float, optional
regularization strength between zero and one; default=0.9
iS : int, optional
replace mC[i] by rank-iS approximation; default=None
(set to ceil(iN/iK) under the default value)
Output
------
mB : np.ndarray with shape (iN, iN)
unitary matrix such that mB@mC[i]@(mB.conj().T) is
approximately diagonal for all i
"""
print("Starting JADOC")
(iK,iN,_)=mC.shape
if iS is None:
iS=(iN/iK)
if (iS-int(iS))>0: iS=int(iS)+1
else: iS=int(iS)
if iS==iN: print("Computing decomposition of input matrices")
elif iS>iN:
raise ValueError("Desired rank (iS) exceeds dimensionality" \
+" of input matrices (iN)")
else: print("Computing low-dimensional approximation of input matrices")
if mB0 is None: mB=np.eye(iN)
elif mB0.shape!=(iN,iN):
raise ValueError("Starting value transformation matrix" \
+" has wrong shape")
else: mB=mB0
bComplex=np.iscomplexobj(mC)
if bComplex: mA=np.empty((iK,iN,iS),dtype="complex128")
else: mA=np.empty((iK,iN,iS))
print("Regularization strength = "+str(dAlpha))
vAlphaLambda=np.empty(iK)
for i in range(iK):
mD=mC[i]-ConjT(mC[i])
if bComplex: dMSD=(np.real(mD)**2).mean()+(np.imag(mD)**2).mean()
else: dMSD=(mD**2).mean()
if dMSD>np.finfo(float).eps:
if bComplex:
raise ValueError("Input matrices are not Hermitian")
else:
raise ValueError("Input matrices are not real symmetric")
if iS<iN:
(vD,mP)=linalg.eigsh(mC[i],k=iS)
else:
(vD,mP)=np.linalg.eigh(mC[i])
vD=abs(vD)
vAlphaLambda[i]=dAlpha*((vD.sum())/iN)
mA[i]=((1-dAlpha)**0.5)*mP*(np.sqrt(vD)[None,:])
if mB0 is not None: mA[i]=np.dot(mB,mA[i])
(mP,vD,mC)=(None,None,None)
print("Starting quasi-Newton algorithm with line search (golden section)")
bConverged=False
for t in range(iT):
(dLoss,mDiags,dRMSG,mU)=ComputeLoss(mA,vAlphaLambda,bComplex,dTauH)
if dRMSG<dTol and t>=iTmin:
bConverged=True
break
dStepSize=PerformGoldenSection(mA,mU,mB,vAlphaLambda,bComplex)
print("ITER "+str(t)+": L="+str(round(dLoss,3))+", RMSD(g)=" \
+str(round(dRMSG,6))+", step="+str(round(dStepSize,3)))
(mB,mA)=UpdateEstimates(mA,mU,mB,dStepSize)
if not(bConverged):
print("WARNING: JADOC did not converge. Reconsider data or thresholds")
print("Returning transformation matrix B")
return mB
def ComputeLoss(mA,vAlphaLambda,bComplex,dTauH=None,bLossOnly=False):
if bComplex:
mDiags=((np.real(mA)**2).sum(axis=2))+((np.imag(mA)**2).sum(axis=2))\
+vAlphaLambda[:,None]
else:
mDiags=((mA**2).sum(axis=2))+vAlphaLambda[:,None]
(iK,iN,iS)=mA.shape
dLoss=0.5*(np.log(mDiags).sum())/iK
if bLossOnly:
return dLoss
else:
if bComplex:
mF=np.zeros((iN,iN),dtype="complex128")
mF=ComputeFComplex(mF,mA,mDiags,iK,iN)
else:
mF=np.zeros((iN,iN))
mF=ComputeFReal(mF,mA,mDiags,iK,iN)
mG=(mF-ConjT(mF))
if bComplex:
dRMSG=np.sqrt((((np.real(mG)**2).sum())+((np.imag(mG)**2).sum()))\
/(iN*(iN-1)))
else:
dRMSG=np.sqrt(((mG**2).sum())/(iN*(iN-1)))
mH=(mDiags[:,:,None]/mDiags[:,None,:]).mean(axis=0)
mH=mH+mH.T-2.0
mH[mH<dTauH]=dTauH
mU=-mG/mH
return dLoss,mDiags,dRMSG,mU
@njit
def ComputeFComplex(mF,mA,mDiags,iK,iN):
for i in prange(iK):
vDiags=(mDiags[i]).reshape((iN,1))
mF+=np.dot(mA[i]/vDiags,mA[i].conj().T)
mF=mF/iK
return mF
@njit
def ComputeFReal(mF,mA,mDiags,iK,iN):
for i in prange(iK):
vDiags=(mDiags[i]).reshape((iN,1))
mF+=np.dot(mA[i]/vDiags,mA[i].T)
mF=mF/iK
return mF
def PerformGoldenSection(mA,mU,mB,vAlphaLambda,bComplex):
dTheta=2/(1+(5**0.5))
iIter=0
iMaxIter=15
iGuesses=4
(dStepLB,dStepUB)=(0,1)
bLossOnlyGold=True
(iK,iN,iS)=mA.shape
mR=scipy.linalg.expm(mU)
if bComplex: mAS=np.empty((iGuesses,iK,iN,iS),dtype="complex128")
else: mAS=np.empty((iGuesses,iK,iN,iS))
mAS[0]=mA.copy()
mAS[1]=RotateData(mR,mA.copy())
mAS[2]=(1-dTheta)*mAS[1]+dTheta*mAS[0]
mAS[3]=(1-dTheta)*mAS[0]+dTheta*mAS[1]
(mA,mR)=(None,None)
dLoss2=ComputeLoss(mAS[2],vAlphaLambda,bComplex,bLossOnly=bLossOnlyGold)
dLoss3=ComputeLoss(mAS[3],vAlphaLambda,bComplex,bLossOnly=bLossOnlyGold)
while iIter<iMaxIter:
if (dLoss2<dLoss3):
mAS[1]=mAS[3]
mAS[3]=mAS[2]
dLoss3=dLoss2
dStepUB=dStepLB+dTheta*(dStepUB-dStepLB)
mAS[2]=mAS[1]-dTheta*(mAS[1]-mAS[0])
dLoss2=ComputeLoss(mAS[2],vAlphaLambda,bComplex,\
bLossOnly=bLossOnlyGold)
else:
mAS[0]=mAS[2]
mAS[2]=mAS[3]
dLoss2=dLoss3
dStepLB=dStepUB-dTheta*(dStepUB-dStepLB)
mAS[3]=mAS[0]+dTheta*(mAS[1]-mAS[0])
dLoss3=ComputeLoss(mAS[3],vAlphaLambda,bComplex,\
bLossOnly=bLossOnlyGold)
iIter+=1
return np.log(1+(dStepLB*(np.exp(1)-1)))
def UpdateEstimates(mA,mU,mB,dStepSize):
mR=scipy.linalg.expm(dStepSize*mU)
mB=np.dot(mR,mB)
mA=RotateData(mR,mA)
return mB,mA
@njit
def RotateData(mR,mData):
iK=mData.shape[0]
for i in prange(iK):
mData[i]=np.dot(mR,mData[i])
return mData
def ConjT(mA):
if np.iscomplexobj(mA):
return mA.conj().T
else:
return mA.T
def SimulateData(iK,iN,iR,dAlpha,bComplex=False,bPSD=True):
if bComplex: sType1="Hermitian "
else: sType1="real symmetric "
if bPSD: sType2="positive (semi)-definite "
else: sType2=""
print("Simulating "+str(iK)+" distinct "+str(iN)+"-by-"+str(iN)+" " \
+sType1+sType2+"matrices with alpha="+str(dAlpha) \
+", for run "+str(iR))
iMainSeed=15348091
iRmax=10000
if iR>=iRmax:
return
rngMain=np.random.default_rng(iMainSeed)
vSeed=rngMain.integers(0,iMainSeed,iRmax)
iSeed=vSeed[iR]
rng=np.random.default_rng(iSeed)
if bComplex:
mX=rng.normal(size=(iN,iN))+1j*rng.normal(size=(iN,iN))
mC=np.empty((iK,iN,iN),dtype="complex128")
else:
mX=rng.normal(size=(iN,iN))
mC=np.empty((iK,iN,iN))
for i in range(0,iK):
if bComplex:
mXk=rng.normal(size=(iN,iN))+1j*rng.normal(size=(iN,iN))
else:
mXk=rng.normal(size=(iN,iN))
mXk=dAlpha*mX+(1-dAlpha)*mXk
mR=scipy.linalg.expm(mXk-ConjT(mXk))
vD=rng.normal(size=iN)
if bPSD:
vD=vD**2
mC[i]=np.dot(mR*(vD[None,:]),ConjT(mR))
return mC
def Test():
iK=5
iN=500
iR=1
dAlpha=0.9
mC=SimulateData(iK,iN,iR,dAlpha)
dTimeStart=time.time()
mB=PerformJADOC(mC,dAlpha=.95,dTol=1E-5,iT=1000)
dTime=time.time()-dTimeStart
print("Runtime: "+str(round(dTime,3))+" seconds")
mD=np.empty((iK,iN,iN))
for i in range(iK):
mD[i]=np.dot(np.dot(mB,mC[i]),mB.T)
dSS_C=0
dSS_D=0
for i in range(iK):
mOffPre=mC[i]-np.diag(np.diag(mC[i]))
mOffPost=mD[i]-np.diag(np.diag(mD[i]))
dSS_C+=(mOffPre**2).sum()
dSS_D+=(mOffPost**2).sum()
dRMS_C=np.sqrt(dSS_C/(iN*(iN-1)*iK))
dRMS_D=np.sqrt(dSS_D/(iN*(iN-1)*iK))
print("Root-mean-square deviation off-diagonals before transformation: " \
+str(round(dRMS_C,6)))
print("Root-mean-square deviation off-diagonals after transformation: " \
+str(round(dRMS_D,6)))