def compute_svd(mat: Matrix) -> (Matrix, Matrix, Matrix):
if mat.num_cols() > mat.num_rows():
u, s, v = compute_svd(mat.transpose())
return v, s.transpose(), u
elif mat.num_rows() > mat.num_cols():
# mat is m x n, m > n
# q should be m x m, r should be m x n
q, r = compute_qr_factorization(mat)
# truncate r to be n x n, truncate q to be m x n
r_truncated = MatrixView.with_size(
r, (0, 0), (mat.num_cols(), mat.num_cols())).to_matrix()
u, s, v = compute_svd(r_truncated)
u_padded = Matrix.identity(mat.num_rows())
MatrixView.with_size(u_padded, (0, 0),
(mat.num_cols(), mat.num_cols())).set(
MatrixView.whole(u))
u = q.multiply(u_padded)
s_padded = Matrix.zeroes(mat.num_rows(), mat.num_cols())
MatrixView.with_size(s_padded, (0, 0),
(mat.num_cols(), mat.num_cols())).set(
MatrixView.whole(s))
s = s_padded
return u, s, v
else:
# matrix is square
b, left, right = reduce_to_bidiagonal(mat)
u, s, v = compute_svd_bidiagonal(b)
for index, hh in list(enumerate(left))[::-1]:
u = hh.multiply_left(u, index)
v_transpose = v.transpose()
for index, hh in list(enumerate(right))[::-1]:
v_transpose = hh.multiply_right(v_transpose, index + 1)
return u, s, v_transpose.transpose()
def compute_qr_factorization(mat: Matrix) -> (Matrix, Matrix):
# Do not overwrite original matrix
mat = mat.copy()
householders = [] # store householder transformations
iterations = min(mat.num_rows(), mat.num_cols())
for iteration in range(iterations):
col = mat.get_col(iteration)
# Zero out the entries below the diagonal
hh = Householder(col[iteration:])
householders.append((iteration, hh))
mat = hh.multiply_left(mat, pad_top=iteration)
# Accumulate the householder transformations
q_mat = Matrix.identity(mat.num_rows())
for iteration, hh in householders[::-1]:
q_mat = hh.multiply_left(q_mat, pad_top=iteration)
return (q_mat, mat)
def test_identity_triangular(self):
identity = Matrix.identity(3)
utils.assert_upper_triangular(identity)
utils.assert_lower_triangular(identity)
def test_householder_inverse(self):
vec = [5, 4, 3, 2, 1]
householder = Householder(vec)
householder_mat = householder.to_matrix()
product = householder_mat.multiply(householder_mat)
utils.assert_matrix_equal(product, Matrix.identity(5))
def test_identity(self):
identity = Matrix.identity(3)
assert identity.all_cols() == [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
def test_givens_inverse(self):
givens = Givens(1, 1).to_matrix()
product = givens.transpose().multiply(givens)
utils.assert_matrix_equal(product, Matrix.identity(2))
def compute_svd_bidiagonal(mat: Matrix) -> (Matrix, Matrix, Matrix):
dims = mat.num_cols()
u = Matrix.identity(dims)
v = Matrix.identity(dims)
while True:
# while not converged
# find max off-diagonal
max_off_diag = 0
diag_sum = 0
for i in range(dims - 1):
max_off_diag = max(max_off_diag, abs(mat.get(i, i + 1)))
for i in range(dims):
diag_sum += abs(mat.get(i, i))
diag_sum /= dims
if max_off_diag < diag_sum * 1e-6 and max_off_diag < 1e-8:
break
# introduce the bulge
givens = Givens(
mat.get(0, 0)**2 - mat.get(dims - 1, dims - 1)**2 -
mat.get(dims - 2, dims - 1)**2,
mat.get(0, 1) * mat.get(0, 0)).transpose()
mat = givens.multiply_right(mat)
v = givens.transpose().multiply_left(v)
# chase the bulge
for iteration in range(0, dims - 1):
# zero subdiagonal
givens_lower = Givens(mat.get(iteration, iteration),
mat.get(iteration + 1, iteration))
mat = givens_lower.multiply_left(mat, pad_top=iteration)
u = givens_lower.transpose().multiply_right(u, pad_top=iteration)
# zero above superdiagonal
if iteration != dims - 2:
givens_upper = Givens(mat.get(iteration, iteration + 1),
mat.get(iteration,
iteration + 2)).transpose()
mat = givens_upper.multiply_right(mat, pad_top=iteration + 1)
v = givens_upper.transpose().multiply_left(v,
pad_top=iteration +
1)
v = v.transpose()
# Ensure singular values are non-negative
for i in range(dims):
if mat.get(i, i) < 0:
MatrixView.with_size(mat, (i, i), (1, 1)).scale(-1)
MatrixView(u, (0, i), (u.num_rows() - 1, i)).scale(-1)
# reorder columns
sv = [(mat.get(i, i), i) for i in range(dims)]
sv.sort()
sv = sv[::-1]
sorted_v_cols = [v.get_col(index) for value, index in sv]
sorted_u_cols = [u.get_col(index) for value, index in sv]
v = Matrix.from_cols(sorted_v_cols)
u = Matrix.from_cols(sorted_u_cols)
mat = Matrix.zeroes(dims, dims)
for index, (value, _) in enumerate(sv):
MatrixView.with_size(mat, (index, index),
(1, 1)).set_element(0, 0, value)
return u, mat, v