def backward(ctx, dval, dvec):
eigval = ctx.eigval
eigvec = ctx.eigvec
n = len(eigval)
eigval_dist = eigval[:, None] - eigval[None, :]
idx = _arange(n).long().tolist()
eigval_dist[idx, idx] = 1.0
dval_out = eigvec[:, None, :] * eigvec[None, :, :]
dvec_out = _zeros(n, n, n, n).type(eigval.type())
omega = _zeros(n, n).type(eigval.type())
for i in range(n):
for j in range(n):
omega[:, :] = eigvec[i, :, None] * eigvec[j, None, :]
omega[idx, idx] = 0.0
omega.div_(eigval_dist)
dvec_out[i, j, :, :] = -_mm(eigvec, omega)
dval = _sum(dval[None, None, :] * dval_out, -1)
dvec = _sum(_sum(dvec[None, None, :, :] * dvec_out, -1), -1)
return dval + dvec
def get_koopman_matrix(x, y):
ixx = sqrtinv(covar(x, x))
iyy = sqrtinv(covar(y, y))
cxy = covar(x, y)
kmm = _mm(ixx, _mm(cxy, iyy))
return kmm.t()
def sqrtinv(matrix):
eigval, eigvec = DecomposeRSPDMatrix.apply(matrix)
diag = _diag(1.0 / _sqrt(eigval))
return _mm(eigvec, _mm(diag, eigvec.t()))
def covar(x, y):
return _mm(x.t(), y).div_(len(x))