/
linalg.py
304 lines (254 loc) · 11.1 KB
/
linalg.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
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
from __future__ import absolute_import
from functools import partial
import numpy.linalg as npla
from .numpy_wrapper import wrap_namespace
from . import numpy_wrapper as anp
from autograd.extend import defvjp, defjvp
wrap_namespace(npla.__dict__, globals())
# Some formulas are from
# "An extended collection of matrix derivative results
# for forward and reverse mode algorithmic differentiation"
# by Mike Giles
# https://people.maths.ox.ac.uk/gilesm/files/NA-08-01.pdf
# transpose by swapping last two dimensions
def T(x): return anp.swapaxes(x, -1, -2)
_dot = partial(anp.einsum, '...ij,...jk->...ik')
# batched diag
_diag = lambda a: anp.eye(a.shape[-1])*a
# batched diagonal, similar to matrix_diag in tensorflow
def _matrix_diag(a):
reps = anp.array(a.shape)
reps[:-1] = 1
reps[-1] = a.shape[-1]
newshape = list(a.shape) + [a.shape[-1]]
return _diag(anp.tile(a, reps).reshape(newshape))
# add two dimensions to the end of x
def add2d(x): return anp.reshape(x, anp.shape(x) + (1, 1))
defvjp(det, lambda ans, x: lambda g: add2d(g) * add2d(ans) * T(inv(x)))
defvjp(slogdet, lambda ans, x: lambda g: add2d(g[1]) * T(inv(x)))
def grad_inv(ans, x):
return lambda g: -_dot(_dot(T(ans), g), T(ans))
defvjp(inv, grad_inv)
def grad_pinv(ans, x):
# https://mathoverflow.net/questions/25778/analytical-formula-for-numerical-derivative-of-the-matrix-pseudo-inverse
return lambda g: T(
-_dot(_dot(ans, T(g)), ans)
+ _dot(_dot(_dot(ans, T(ans)), g), anp.eye(x.shape[-2]) - _dot(x,ans))
+ _dot(_dot(_dot(anp.eye(ans.shape[-2]) - _dot(ans,x), g), T(ans)), ans)
)
defvjp(pinv, grad_pinv)
def grad_solve(argnum, ans, a, b):
updim = lambda x: x if x.ndim == a.ndim else x[...,None]
if argnum == 0:
return lambda g: -_dot(updim(solve(T(a), g)), T(updim(ans)))
else:
return lambda g: solve(T(a), g)
defvjp(solve, partial(grad_solve, 0), partial(grad_solve, 1))
def norm_vjp(ans, x, ord=None, axis=None):
def check_implemented():
matrix_norm = (x.ndim == 2 and axis is None) or isinstance(axis, tuple)
if matrix_norm:
if not (ord is None or ord == 'fro' or ord == 'nuc'):
raise NotImplementedError('Gradient of matrix norm not '
'implemented for ord={}'.format(ord))
elif not (ord is None or ord > 1):
raise NotImplementedError('Gradient of norm not '
'implemented for ord={}'.format(ord))
if axis is None:
expand = lambda a: a
elif isinstance(axis, tuple):
row_axis, col_axis = axis
if row_axis > col_axis:
row_axis = row_axis - 1
expand = lambda a: anp.expand_dims(anp.expand_dims(a,
row_axis), col_axis)
else:
expand = lambda a: anp.expand_dims(a, axis=axis)
if ord == 'nuc':
if axis is None:
roll = lambda a: a
unroll = lambda a: a
else:
row_axis, col_axis = axis
if row_axis > col_axis:
row_axis = row_axis - 1
# Roll matrix axes to the back
roll = lambda a: anp.rollaxis(anp.rollaxis(a, col_axis, a.ndim),
row_axis, a.ndim-1)
# Roll matrix axes to their original position
unroll = lambda a: anp.rollaxis(anp.rollaxis(a, a.ndim-2, row_axis),
a.ndim-1, col_axis)
check_implemented()
def vjp(g):
if ord in (None, 2, 'fro'):
return expand(g / ans) * x
elif ord == 'nuc':
x_rolled = roll(x)
u, s, vt = svd(x_rolled, full_matrices=False)
uvt_rolled = _dot(u, vt)
# Roll the matrix axes back to their correct positions
uvt = unroll(uvt_rolled)
g = expand(g)
return g * uvt
else:
# see https://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm
return expand(g / ans**(ord-1)) * x * anp.abs(x)**(ord-2)
return vjp
defvjp(norm, norm_vjp)
def norm_jvp(g, ans, x, ord=None, axis=None):
def check_implemented():
matrix_norm = (x.ndim == 2 and axis is None) or isinstance(axis, tuple)
if matrix_norm:
if not (ord is None or ord == 'fro' or ord == 'nuc'):
raise NotImplementedError('Gradient of matrix norm not '
'implemented for ord={}'.format(ord))
elif not (ord is None or ord > 1):
raise NotImplementedError('Gradient of norm not '
'implemented for ord={}'.format(ord))
if axis is None:
contract = lambda a: anp.sum(a)
else:
contract = partial(anp.sum, axis=axis)
if ord == 'nuc':
if axis is None:
roll = lambda a: a
unroll = lambda a: a
else:
row_axis, col_axis = axis
if row_axis > col_axis:
row_axis = row_axis - 1
# Roll matrix axes to the back
roll = lambda a: anp.rollaxis(anp.rollaxis(a, col_axis, a.ndim),
row_axis, a.ndim-1)
# Roll matrix axes to their original position
unroll = lambda a: anp.rollaxis(anp.rollaxis(a, a.ndim-2, row_axis),
a.ndim-1, col_axis)
check_implemented()
if ord in (None, 2, 'fro'):
return contract(g * x) / ans
elif ord == 'nuc':
x_rolled = roll(x)
u, s, vt = svd(x_rolled, full_matrices=False)
uvt_rolled = _dot(u, vt)
# Roll the matrix axes back to their correct positions
uvt = unroll(uvt_rolled)
return contract(g * uvt)
else:
# see https://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm
return contract(g * x * anp.abs(x)**(ord-2)) / ans**(ord-1)
defjvp(norm, norm_jvp)
def grad_eigh(ans, x, UPLO='L'):
"""Gradient for eigenvalues and vectors of a symmetric matrix."""
N = x.shape[-1]
w, v = ans # Eigenvalues, eigenvectors.
vc = anp.conj(v)
def vjp(g):
wg, vg = g # Gradient w.r.t. eigenvalues, eigenvectors.
w_repeated = anp.repeat(w[..., anp.newaxis], N, axis=-1)
# Eigenvalue part
vjp_temp = _dot(vc * wg[..., anp.newaxis, :], T(v))
# Add eigenvector part only if non-zero backward signal is present.
# This can avoid NaN results for degenerate cases if the function depends
# on the eigenvalues only.
if anp.any(vg):
off_diag = anp.ones((N, N)) - anp.eye(N)
F = off_diag / (T(w_repeated) - w_repeated + anp.eye(N))
vjp_temp += _dot(_dot(vc, F * _dot(T(v), vg)), T(v))
# eigh always uses only the lower or the upper part of the matrix
# we also have to make sure broadcasting works
reps = anp.array(x.shape)
reps[-2:] = 1
if UPLO == 'L':
tri = anp.tile(anp.tril(anp.ones(N), -1), reps)
elif UPLO == 'U':
tri = anp.tile(anp.triu(anp.ones(N), 1), reps)
return anp.real(vjp_temp)*anp.eye(vjp_temp.shape[-1]) + \
(vjp_temp + anp.conj(T(vjp_temp))) * tri
return vjp
defvjp(eigh, grad_eigh)
# https://arxiv.org/pdf/1701.00392.pdf Eq(4.77)
# Note the formula from Sec3.1 in https://people.maths.ox.ac.uk/gilesm/files/NA-08-01.pdf is incomplete
def grad_eig(ans, x):
"""Gradient of a general square (complex valued) matrix"""
e, u = ans # eigenvalues as 1d array, eigenvectors in columns
n = e.shape[-1]
def vjp(g):
ge, gu = g
ge = _matrix_diag(ge)
f = 1/(e[..., anp.newaxis, :] - e[..., :, anp.newaxis] + 1.e-20)
f -= _diag(f)
ut = anp.swapaxes(u, -1, -2)
r1 = f * _dot(ut, gu)
r2 = -f * (_dot(_dot(ut, anp.conj(u)), anp.real(_dot(ut,gu)) * anp.eye(n)))
r = _dot(_dot(inv(ut), ge + r1 + r2), ut)
if not anp.iscomplexobj(x):
r = anp.real(r)
# the derivative is still complex for real input (imaginary delta is allowed), real output
# but the derivative should be real in real input case when imaginary delta is forbidden
return r
return vjp
defvjp(eig, grad_eig)
def grad_cholesky(L, A):
# Based on Iain Murray's note http://arxiv.org/abs/1602.07527
# scipy's dtrtrs wrapper, solve_triangular, doesn't broadcast along leading
# dimensions, so we just call a generic LU solve instead of directly using
# backsubstitution (also, we factor twice...)
solve_trans = lambda a, b: solve(T(a), b)
phi = lambda X: anp.tril(X) / (1. + anp.eye(X.shape[-1]))
def conjugate_solve(L, X):
# X -> L^{-T} X L^{-1}
return solve_trans(L, T(solve_trans(L, T(X))))
def vjp(g):
S = conjugate_solve(L, phi(anp.einsum('...ki,...kj->...ij', L, g)))
return (S + T(S)) / 2.
return vjp
defvjp(cholesky, grad_cholesky)
# https://j-towns.github.io/papers/svd-derivative.pdf
# https://arxiv.org/abs/1909.02659
def grad_svd(usv_, a, full_matrices=True, compute_uv=True):
def vjp(g):
usv = usv_
if not compute_uv:
s = usv
# Need U and V so do the whole svd anyway...
usv = svd(a, full_matrices=False)
u = usv[0]
v = anp.conj(T(usv[2]))
return _dot(anp.conj(u) * g[..., anp.newaxis, :], T(v))
elif full_matrices:
raise NotImplementedError(
"Gradient of svd not implemented for full_matrices=True")
else:
u = usv[0]
s = usv[1]
v = anp.conj(T(usv[2]))
m, n = a.shape[-2:]
k = anp.min((m, n))
# broadcastable identity array with shape (1, 1, ..., 1, k, k)
i = anp.reshape(anp.eye(k), anp.concatenate((anp.ones(a.ndim - 2, dtype=int), (k, k))))
f = 1 / (s[..., anp.newaxis, :]**2 - s[..., :, anp.newaxis]**2 + i)
gu = g[0]
gs = g[1]
gv = anp.conj(T(g[2]))
utgu = _dot(T(u), gu)
vtgv = _dot(T(v), gv)
t1 = (f * (utgu - anp.conj(T(utgu)))) * s[..., anp.newaxis, :]
t1 = t1 + i * gs[..., :, anp.newaxis]
t1 = t1 + s[..., :, anp.newaxis] * (f * (vtgv - anp.conj(T(vtgv))))
if anp.iscomplexobj(u):
t1 = t1 + 1j*anp.imag(_diag(utgu)) / s[..., anp.newaxis, :]
t1 = _dot(_dot(anp.conj(u), t1), T(v))
if m < n:
i_minus_vvt = (anp.reshape(anp.eye(n), anp.concatenate((anp.ones(a.ndim - 2, dtype=int), (n, n)))) -
_dot(v, anp.conj(T(v))))
t1 = t1 + anp.conj(_dot(_dot(u / s[..., anp.newaxis, :], T(gv)), i_minus_vvt))
return t1
elif m == n:
return t1
elif m > n:
i_minus_uut = (anp.reshape(anp.eye(m), anp.concatenate((anp.ones(a.ndim - 2, dtype=int), (m, m)))) -
_dot(u, anp.conj(T(u))))
t1 = t1 + T(_dot(_dot(v/s[..., anp.newaxis, :], T(gu)), i_minus_uut) )
return t1
return vjp
defvjp(svd, grad_svd)