def kl_divergence(A, B):
"""
@param A: doubly centered symmetric nxn matrix of rank n-1
@param B: doubly centered symmetric nxn matrix of rank n-1
"""
assert_symmetric(A)
assert_symmetric(B)
n = A.shape[0]
B_pinv = restored(inv(augmented(B)))
stein_loss = (
trace(dot(B_pinv, A)) -
(log_pdet(B_pinv) + log_pdet(A)) - (n-1))
return 0.5 * stein_loss
def clever_cross_entropy_trees(B, nleaves, va, vb):
"""
Try being a little more clever.
@param B: augmented incidence matrix
@param nleaves: number of leaves
@param va: augmented reference point edge variances
@param vb: augmented test point edge variances
"""
# deduce some quantities assuming an unrooted bifurcating tree
ninternal = nleaves - 2
nvertices = nleaves + ninternal
nedges = nvertices - 1
# define an index for taking schur complements
n = nvertices
k = nleaves + 1
# Construct the full Laplacian matrix plus J/n.
# Take a block of the diagonal, corresponding to the inverse
# of a schur complement.
Wa = diag(reciprocal(va))
La_plus = dot(B.T, dot(Wa, B))
print(La_plus)
print(scipy.linalg.eigh(La_plus))
Laa = La_plus[:k, :k]
Lab = La_plus[:k, k:]
Lba = La_plus[k:, :k]
Lbb = La_plus[k:, k:]
L_schur_plus = Laa - dot(Lab, dot(inv(Lbb), Lba))
assert_allclose(inv(L_schur_plus), inv(La_plus)[:k, :k])
A = inv(La_plus)[:k, :k]
print(scipy.linalg.eigh(A))
# Construct the Schur complement of the test point matrix.
Wb = diag(reciprocal(vb))
L_plus = dot(B.T, dot(Wb, B))
Laa = L_plus[:k, :k]
Lab = L_plus[:k, k:]
Lba = L_plus[k:, :k]
Lbb = L_plus[k:, k:]
L_schur_plus = Laa - dot(Lab, dot(inv(Lbb), Lba))
B_inv = L_schur_plus
#return 0.5 * ((n-1) * LOG2PI + trace(dot(B_inv, A)) - log(det(B_inv)))
return 0.5 * (n * LOG2PI + trace(dot(B_inv, A) - 1) - log(det(B_inv)))
def test_pullback_gradient(self):
(D,M,N) = 3,3,2
P = M*N
A = UTPM(numpy.zeros((D,P,M,M)))
A0 = numpy.random.rand(M,N)
for m in range(M):
for n in range(N):
p = m*N + n
A.data[0,p,:M,:N] = A0
A.data[1,p,m,n] = 1.
cg = CGraph()
A = Function(A)
Q,R = qr(A)
B = dot(Q,R)
y = trace(B)
cg.independentFunctionList = [A]
cg.dependentFunctionList = [y]
# print cg
# print y.x.data
g1 = y.x.data[1]
g11 = y.x.data[2]
# print g1
ybar = y.x.zeros_like()
ybar.data[0,:] = 1.
cg.pullback([ybar])
for m in range(M):
for n in range(N):
p = m*N + n
#check gradient
assert_array_almost_equal(y.x.data[1,p], A.xbar.data[0,p,m,n])
#check hessian
assert_array_almost_equal(0, A.xbar.data[1,p,m,n])
def test_outer(self):
x = numpy.arange(4)
cg = algopy.CGraph()
x = algopy.Function(x)
x1 = x[:x.size//2]
x2 = x[x.size//2:]
y = algopy.trace(algopy.outer(x1,x2))
cg.trace_off()
cg.independentFunctionList = [x]
cg.dependentFunctionList = [y]
cg2 = algopy.CGraph()
x = algopy.Function(x)
x1 = x[:x.size//2]
x2 = x[x.size//2:]
z = algopy.dot(x1,x2)
cg2.trace_off()
cg2.independentFunctionList = [x]
cg2.dependentFunctionList = [z]
assert_array_almost_equal(cg.jacobian(numpy.arange(4)), cg2.jacobian(numpy.arange(4)))
def test_pullback_gradient2(self):
(D,P,M,N) = 3,9,3,3
A = UTPM(numpy.zeros((D,P,M,M)))
A0 = numpy.random.rand(M,N)
for m in range(M):
for n in range(N):
p = m*N + n
A.data[0,p,:M,:N] = A0
A.data[1,p,m,n] = 1.
cg = CGraph()
A = Function(A)
B = inv(A)
y = trace(B)
cg.independentFunctionList = [A]
cg.dependentFunctionList = [y]
ybar = y.x.zeros_like()
ybar.data[0,:] = 1.
cg.pullback([ybar])
g1 = y.x.data[1]
g2 = A.xbar.data[0,0].ravel()
assert_array_almost_equal(g1, g2)
tmp = []
for m in range(M):
for n in range(N):
p = m*N + n
tmp.append( A.xbar.data[1,p,m,n])
h1 = y.x.data[2]
h2 = numpy.array(tmp)
assert_array_almost_equal(2*h1, h2)
xdata[0] = [1, 2, 3]
xdata[1] = dirs
# STEP 3: compute function F in UTP arithmetic
x = algopy.UTPM(xdata)
y = eval_F(x)
# STEP 4: use polarization identity to build univariate Taylor polynomial
# of the Jacobian J
Jdata = numpy.zeros((D - 1, N, N, N))
count, count2 = 0, 0
# build J_0
for n in range(N):
Jdata[0, :, :, n] = y.data[1, 2 * n * (N + 1), :] / 2.
# build J_1
count = 0
for n1 in range(N):
for n2 in range(N):
Jdata[1, n2, :, n1] = 0.5 * (y.data[2, count] - y.data[2, count + 1])
count += 2
# initialize UTPM instance
J = algopy.UTPM(Jdata)
# STEP 5: evaluate Phi in UTP arithmetic
Phi = algopy.trace(algopy.dot(J.T, J))
print('Phi=', Phi)
print('gradient of Phi =', Phi.data[1, :])
P = M*N
# generate badly conditioned matrix A
A = UTPM(numpy.zeros((D,P,M,N)))
A.data[0,:] = numpy.random.rand(*(M,N))
for m in range(M):
for n in range(N):
p = m*N + n
A.data[1,p,m,n] = 1.
cg = CGraph()
A = Function(A)
y = trace(inv(A))
cg.trace_off()
cg.independentFunctionList = [A]
cg.dependentFunctionList = [y]
ybar = y.x.zeros_like()
ybar.data[0,:] = 1.
cg.pullback([ybar])
# check gradient
g_forward = numpy.zeros(N*N)
g_reverse = numpy.zeros(N*N)
for m in range(M):
for n in range(N):
# STEP 3: compute function F in UTP arithmetic
x = algopy.UTPM(xdata)
y = eval_F(x)
# STEP 4: use polarization identity to build univariate Taylor polynomial
# of the Jacobian J
Jdata = numpy.zeros((D-1, N, N, N))
count, count2 = 0,0
# build J_0
for n in range(N):
Jdata[0,:,:,n] = y.data[1, 2*n*(N+1), :]/2.
# build J_1
count = 0
for n1 in range(N):
for n2 in range(N):
Jdata[1,n2,:,n1] = 0.5*( y.data[2, count] - y.data[2, count+1])
count += 2
# initialize UTPM instance
J = algopy.UTPM(Jdata)
# STEP 5: evaluate Phi in UTP arithmetic
Phi = algopy.trace(algopy.dot(J.T, J))
print 'Phi=',Phi
print 'gradient of Phi =', Phi.data[1,:]
D, M, N = 2, 3, 3
P = M * N
# generate badly conditioned matrix A
A = UTPM(numpy.zeros((D, P, M, N)))
A.data[0, :] = numpy.random.rand(*(M, N))
for m in range(M):
for n in range(N):
p = m * N + n
A.data[1, p, m, n] = 1.
cg = CGraph()
A = Function(A)
y = trace(inv(A))
cg.trace_off()
cg.independentFunctionList = [A]
cg.dependentFunctionList = [y]
ybar = y.x.zeros_like()
ybar.data[0, :] = 1.
cg.pullback([ybar])
# check gradient
g_forward = numpy.zeros(N * N)
g_reverse = numpy.zeros(N * N)
for m in range(M):
for n in range(N):
