def _solve(self, HH, HY):
"""Compute output weights from HH and HY using Dask functionality.
"""
# make HH/HY divisible by chunk size
n_features, _ = HH.shape
padding = 0
if n_features > self.bsize_ and n_features % self.bsize_ > 0:
print("Adjusting batch size {} to n_features {}".format(
self.bsize_, n_features))
padding = self.bsize_ - (n_features % self.bsize_)
P01 = da.zeros((n_features, padding))
P10 = da.zeros((padding, n_features))
P11 = da.zeros((padding, padding))
HH = da.block([[HH, P01], [P10, P11]])
P1 = da.zeros((padding, HY.shape[1]))
HY = da.block([[HY], [P1]])
# rechunk, add bias, and solve
HH = HH.rechunk(
self.bsize_) + self.alpha * da.eye(HH.shape[1], chunks=self.bsize_)
HY = HY.rechunk(self.bsize_)
B = da.linalg.solve(HH, HY, sym_pos=True)
if padding > 0:
B = B[:n_features]
return B
def mvn_random_DASK(mean, cov, N, dim):
da.random.seed(10)
epsilon = 0.0001
A = da.linalg.cholesky(cov + epsilon * da.eye(dim), lower=True)
z = da.random.standard_normal(size=(N, dim))
x = da.outer(da.ones((N, )), mean).transpose() + da.dot(A, z.transpose())
return x
def eig_dask(A, nofIt=None):
"""
A dask eigenvalue solver: assumes A is symmetric and used the QR method
to find eigenvalues and eigenvectors.
nofIt: number of iterations (default is the size of A)
"""
A_new = A
if nofIt is None:
nofIt = A.shape[0]
V = da.eye(A.shape[0], 100)
for i in range(nofIt):
Q, R = da.linalg.qr(A_new)
A_new = da.dot(R, Q)
V = da.dot(V, Q)
return (da.diag(A_new), V)
def nearestPD(A, threads=1):
"""
Find the nearest positive-definite matrix to input
A Python/Numpy port of John D'Errico's `nearestSPD` MATLAB code [1], which
credits [2] from Ahmed Fasih
[1] https://www.mathworks.com/matlabcentral/fileexchange/42885-nearestspd
[2] N.J. Higham, "Computing a nearest symmetric positive semidefinite
matrix" (1988): https://doi.org/10.1016/0024-3795(88)90223-6
"""
isPD = lambda x: da.all(np.linalg.eigvals(x) > 0).compute()
B = (A + A.T) / 2
_, s, V = da.linalg.svd(B)
H = da.dot(V.T, da.dot(da.diag(s), V))
A2 = (B + H) / 2
A3 = (A2 + A2.T) / 2
if isPD(A3):
return A3
spacing = da.spacing(da.linalg.norm(A))
# The above is different from [1]. It appears that MATLAB's `chol` Cholesky
# decomposition will accept matrixes with exactly 0-eigenvalue, whereas
# Numpy's will not. So where [1] uses `eps(mineig)` (where `eps` is Matlab
# for `np.spacing`), we use the above definition. CAVEAT: our `spacing`
# will be much larger than [1]'s `eps(mineig)`, since `mineig` is usually on
# the order of 1e-16, and `eps(1e-16)` is on the order of 1e-34, whereas
# `spacing` will, for Gaussian random matrixes of small dimension, be on
# othe order of 1e-16. In practice, both ways converge, as the unit test
# below suggests.
eye_chunk = estimate_chunks((A.shape[0], A.shape[0]), threads=threads)[0]
I = da.eye(A.shape[0], chunks=eye_chunk)
k = 1
while not isPD(A3):
mineig = da.min(da.real(np.linalg.eigvals(A3)))
A3 += I * (-mineig * k**2 + spacing)
k += 1
return A3
def test_subspace_to_SVD_case2():
"""
A = USV
U = [e1 e2, ..., ek] \
[0, 0, ..., 0] | N by K
[1, 1, ..., 1] /
S = np.range(N, 0, -1)
V = I
A = np.diag(np.range(N, 0, -1))
subspace_to_SVD should recover I and S from sections of A
"""
for N in range(2, 10):
for K in range(2, N + 1):
U = np.zeros((N, K))
U[N - 1, :] = np.ones(K)
U[:K, :K] = np.eye(K)
V = da.eye(K)
U = da.array(U)
U_q, _ = da.linalg.qr(U)
S = da.arange(K, 0, -1)
A = U.dot(da.diag(S))
for j in range(K, N + 1):
subspace = A[:, 0:j]
U_s, S_s, V_s = subspace_to_SVD(subspace, A, full_v=True)
np.testing.assert_almost_equal(subspace_dist(V_s, V, S), 0, decimal=decimals)
_, l, _ = da.linalg.svd(U_q.dot(U_s.T))
np.testing.assert_almost_equal(l[:K].compute(), np.ones(K))
np.testing.assert_almost_equal(l[K:].compute(), np.zeros(N - K))
def test_eye():
assert_eq(da.eye(9, chunks=3), np.eye(9))
assert_eq(da.eye(9), np.eye(9))
assert_eq(da.eye(10, chunks=3), np.eye(10))
assert_eq(da.eye(9, chunks=3, M=11), np.eye(9, M=11))
assert_eq(da.eye(11, chunks=3, M=9), np.eye(11, M=9))
assert_eq(da.eye(7, chunks=3, M=11), np.eye(7, M=11))
assert_eq(da.eye(11, chunks=3, M=7), np.eye(11, M=7))
assert_eq(da.eye(9, chunks=3, k=2), np.eye(9, k=2))
assert_eq(da.eye(9, chunks=3, k=-2), np.eye(9, k=-2))
assert_eq(da.eye(7, chunks=3, M=11, k=5), np.eye(7, M=11, k=5))
assert_eq(da.eye(11, chunks=3, M=7, k=-6), np.eye(11, M=7, k=-6))
assert_eq(da.eye(6, chunks=3, M=9, k=7), np.eye(6, M=9, k=7))
assert_eq(da.eye(12, chunks=3, M=6, k=-3), np.eye(12, M=6, k=-3))
assert_eq(da.eye(9, chunks=3, dtype=int), np.eye(9, dtype=int))
assert_eq(da.eye(10, chunks=3, dtype=int), np.eye(10, dtype=int))
with dask.config.set({"array.chunk-size": "50 MiB"}):
x = da.eye(10000, "auto")
assert 4 < x.npartitions < 32
def test_map_blocks():
x = da.eye(10, chunks=5)
y = x.map_blocks(sparse.COO.from_numpy,
meta=sparse.COO.from_numpy(np.eye(1)))
assert isinstance(y._meta, sparse.COO)
assert_eq(y, y)
def test_eye():
assert_eq(da.eye(9, chunks=3), np.eye(9))
assert_eq(da.eye(10, chunks=3), np.eye(10))
assert_eq(da.eye(9, chunks=3, M=11), np.eye(9, M=11))
assert_eq(da.eye(11, chunks=3, M=9), np.eye(11, M=9))
assert_eq(da.eye(7, chunks=3, M=11), np.eye(7, M=11))
assert_eq(da.eye(11, chunks=3, M=7), np.eye(11, M=7))
assert_eq(da.eye(9, chunks=3, k=2), np.eye(9, k=2))
assert_eq(da.eye(9, chunks=3, k=-2), np.eye(9, k=-2))
assert_eq(da.eye(7, chunks=3, M=11, k=5), np.eye(7, M=11, k=5))
assert_eq(da.eye(11, chunks=3, M=7, k=-6), np.eye(11, M=7, k=-6))
assert_eq(da.eye(6, chunks=3, M=9, k=7), np.eye(6, M=9, k=7))
assert_eq(da.eye(12, chunks=3, M=6, k=-3), np.eye(12, M=6, k=-3))
assert_eq(da.eye(9, chunks=3, dtype=int), np.eye(9, dtype=int))
assert_eq(da.eye(10, chunks=3, dtype=int), np.eye(10, dtype=int))
def _setup_cluster_matrix(n_clusters, cluster_idx):
""" Set cluster occupation matrix """
return da.eye(n_clusters, dtype=np.bool, chunks=n_clusters)[cluster_idx]
def test_eye():
assert_eq(da.eye(9, chunks=3), np.eye(9))
assert_eq(da.eye(10, chunks=3), np.eye(10))
assert_eq(da.eye(9, chunks=3, M=11), np.eye(9, M=11))
assert_eq(da.eye(11, chunks=3, M=9), np.eye(11, M=9))
assert_eq(da.eye(7, chunks=3, M=11), np.eye(7, M=11))
assert_eq(da.eye(11, chunks=3, M=7), np.eye(11, M=7))
assert_eq(da.eye(9, chunks=3, k=2), np.eye(9, k=2))
assert_eq(da.eye(9, chunks=3, k=-2), np.eye(9, k=-2))
assert_eq(da.eye(7, chunks=3, M=11, k=5), np.eye(7, M=11, k=5))
assert_eq(da.eye(11, chunks=3, M=7, k=-6), np.eye(11, M=7, k=-6))
assert_eq(da.eye(6, chunks=3, M=9, k=7), np.eye(6, M=9, k=7))
assert_eq(da.eye(12, chunks=3, M=6, k=-3), np.eye(12, M=6, k=-3))
assert_eq(da.eye(9, chunks=3, dtype=int), np.eye(9, dtype=int))
assert_eq(da.eye(10, chunks=3, dtype=int), np.eye(10, dtype=int))