def projx(self, x):
warnings.warn(("{}.projx performs a projection onto the open set of" +
" PSD matrices").format(self.__class__.__name__))
x_sym = (utils.transposem(x) + x) / 2.0
s, u, v = tf.linalg.svd(x_sym)
sigma = tf.linalg.diag(tf.maximum(s, 0.0))
return v @ sigma @ utils.transposem(v)
def retr(self, x, u):
xut = x @ utils.transposem(u)
xut_sym = (xut - utils.transposem(xut)) / 2.0
eye = tf.eye(tf.shape(xut)[-1],
batch_shape=tf.shape(xut)[:-2],
dtype=x.dtype)
return tf.linalg.solve(xut_sym + eye, x - xut_sym @ x)
def diff_from_cholesky(self, x, u):
"""Inverse of the differential of diffeomorphism to the Cholesky space"""
assert_x = tf.debugging.Assert(
self._cholesky.check_point_on_manifold(x), [x])
assert_u = tf.debugging.Assert(
self._cholesky.check_vector_on_tangent(x, u), [u])
with tf.control_dependencies([assert_x, assert_u]):
return x @ utils.transposem(u) + u @ utils.transposem(x)
def retr(self, x, u):
xtu = utils.transposem(x) @ u
w_ = (u - x @ xtu) @ utils.transposem(x)
w = w_ - utils.transposem(w_)
y = x + u
for _ in range(self.num_iter):
y = x + w @ ((x + y) / 2.0)
return y
def inner(self, x, u, v, keepdims=False):
xtu = utils.transposem(x) @ u
xtv = utils.transposem(x) @ v
u_v_inner = tf.reduce_sum(u * v, axis=[-2, -1], keepdims=keepdims)
xtu_xtv_inner = tf.reduce_sum(xtu * xtv,
axis=[-2, -1],
keepdims=keepdims)
return u_v_inner - 0.5 * xtu_xtv_inner
def geodesic(self, x, u, t):
xtu = utils.transposem(x) @ u
w_ = (u - x @ xtu) @ utils.transposem(x)
w = w_ - utils.transposem(w_)
eye = tf.linalg.eye(tf.shape(w)[-1],
batch_shape=tf.shape(w)[:-2],
dtype=x.dtype)
cayley_t = tf.linalg.inv(eye - t * w / 2.0) @ (eye + t * w / 2.0)
return cayley_t @ x
def ptransp(self, x, y, v):
log_xy = self.log(x, y)
s, u, vt = tf.linalg.svd(log_xy, full_matrices=False)
cos_s = tf.linalg.diag(tf.math.cos(s))
sin_s = tf.linalg.diag(tf.math.sin(s))
geod = ((-x @ utils.transposem(vt) @ sin_s + u @ cos_s)
@ utils.transposem(u) @ v)
proj = v - u @ utils.transposem(u) @ v
return geod + proj
def geodesic(self, x, u, t):
xtu = utils.transposem(x) @ u
utu = utils.transposem(u) @ u
eye = tf.eye(tf.shape(utu)[-1],
batch_shape=tf.shape(utu)[:-2],
dtype=x.dtype)
logw = blockm(xtu, -utu, eye, xtu)
w = tf.linalg.expm(t * logw)
z = tf.concat([tf.linalg.expm(-xtu * t), tf.zeros_like(utu)], axis=-2)
y = tf.concat([x, u], axis=-1) @ w @ z
return y
def _check_point_on_manifold(self, x, atol, rtol):
x_t = utils.transposem(x)
eigvals, _ = tf.linalg.eigh(x)
is_symmetric = utils.allclose(x, x_t, atol, rtol)
is_pos_vals = utils.allclose(eigvals, tf.abs(eigvals), atol, rtol)
is_zero_vals = utils.allclose(eigvals, tf.zeros_like(eigvals), atol,
rtol)
return is_symmetric & is_pos_vals & tf.logical_not(is_zero_vals)
def _check_point_on_manifold(self, x, atol, rtol):
xtx = utils.transposem(x) @ x
eye = tf.eye(
tf.shape(xtx)[-1], batch_shape=tf.shape(xtx)[:-2], dtype=x.dtype
)
is_orth = utils.allclose(xtx, eye, atol, rtol)
det = tf.linalg.det(x)
is_unit_det = utils.allclose(det, tf.ones_like(det), atol, rtol)
return is_orth & is_unit_det
def diff_to_cholesky(self, x, u):
"""Differential of the diffeomorphism to the Cholesky space"""
assert_x = tf.debugging.Assert(self.check_point_on_manifold(x), [x])
assert_u = tf.debugging.Assert(self.check_vector_on_tangent(x, u), [u])
with tf.control_dependencies([assert_x, assert_u]):
y = self.to_cholesky(x)
y_inv = tf.linalg.inv(y)
p = y_inv @ u @ utils.transposem(y_inv)
p_diag, p_lower = self._cholesky._diag_and_strictly_lower(p)
return y @ (p_lower + 0.5 * tf.linalg.diag(p_diag))
def _check_point_on_manifold(self, x, atol, rtol):
xtx = utils.transposem(x) @ x
shape = xtx.shape.as_list()
eye = tf.eye(shape[-1], batch_shape=shape[:-2])
is_idempotent = utils.allclose(xtx, tf.cast(eye, x.dtype), atol, rtol)
s = tf.linalg.svd(x, compute_uv=False)
rank = tf.math.count_nonzero(s, axis=-1, dtype=tf.float32)
k = tf.ones_like(rank) * int(x.shape[-1])
is_col_rank = utils.allclose(rank, k, atol, rtol)
return is_idempotent & is_col_rank
def _diff_power(self, x, d, power):
e, v = tf.linalg.eigh(x)
v_t = utils.transposem(v)
e = tf.expand_dims(e, -2)
if power == "log":
pow_e = tf.math.log(e)
elif power == "exp":
pow_e = tf.math.exp(e)
s = utils.transposem(tf.ones_like(e)) @ e
pow_s = utils.transposem(tf.ones_like(pow_e)) @ pow_e
denom = utils.transposem(s) - s
numer = utils.transposem(pow_s) - pow_s
abs_denom = tf.math.abs(denom)
eps = utils.get_eps(x)
if power == "log":
numer = tf.where(abs_denom < eps, tf.ones_like(numer), numer)
denom = tf.where(abs_denom < eps, utils.transposem(s), denom)
elif power == "exp":
numer = tf.where(abs_denom < eps, utils.transposem(pow_s), numer)
denom = tf.where(abs_denom < eps, tf.ones_like(denom), denom)
t = v_t @ d @ v * numer / denom
return v @ t @ v_t
def proju(self, x, u):
xtu = utils.transposem(x) @ u
return u - x @ xtu
def _check_vector_on_tangent(self, x, u, atol, rtol):
xtu = utils.transposem(x) @ u
return utils.allclose(xtu, tf.zeros_like(xtu), atol, rtol)
def ptransp(self, x, y, v):
e = tf.linalg.sqrtm(y @ tf.linalg.inv(x))
return e @ v @ utils.transposem(e)
def proju(self, x, u):
return 0.5 * (utils.transposem(u) + u)
def log(self, x, y):
xty = utils.transposem(x) @ y
u = utils.logm(xty)
return (u - utils.transposem(u)) / 2.0
def _check_vector_on_tangent(self, x, u, atol, rtol):
u_t = utils.transposem(u)
return utils.allclose(u, u_t, atol, rtol)
def proju(self, x, u):
xtu = utils.transposem(x) @ u
return (xtu - utils.transposem(xtu)) / 2.0
def geodesic(self, x, u, t):
s, u, vt = tf.linalg.svd(u, full_matrices=False)
cos_ts = tf.linalg.diag(tf.math.cos(t * s))
sin_ts = tf.linalg.diag(tf.math.sin(t * s))
return (x @ utils.transposem(vt) @ cos_ts + u @ sin_ts) @ vt
def dist(self, x, y, keepdims=False):
s = tf.linalg.svd(utils.transposem(x) @ y, compute_uv=False)
theta = tf.math.acos(tf.clip_by_value(s, -1.0, 1.0))
norm = tf.linalg.norm(theta, axis=-1, keepdims=False)
return norm[..., tf.newaxis, tf.newaxis] if keepdims else norm
def call(self, inputs):
if self._expand:
inputs = tf.expand_dims(inputs, -3)
return utils.transposem(self.w) @ inputs
def proju(self, x, u):
u_sym = (utils.transposem(u) + u) / 2.0
u_diag, u_lower = self._diag_and_strictly_lower(u_sym)
x_diag = tf.linalg.diag_part(x)
return u_lower + tf.linalg.diag(u_diag * x_diag**2)
def projx(self, x):
x_sym = (utils.transposem(x) + x) / 2.0
s, _u, v = tf.linalg.svd(x_sym)
sigma = tf.linalg.diag(tf.maximum(s, 0.0))
spd = v @ sigma @ utils.transposem(v)
return tf.linalg.cholesky(spd)
def call(self, inputs):
return inputs @ utils.transposem(inputs)
def exp(self, x, u):
s, u, vt = tf.linalg.svd(u, full_matrices=False)
cos_s = tf.linalg.diag(tf.math.cos(s))
sin_s = tf.linalg.diag(tf.math.sin(s))
return (x @ utils.transposem(vt) @ cos_s + u @ sin_s) @ vt
def call(self, inputs):
s, u, v = tf.linalg.svd(inputs)
log_s = tf.math.log(s)
return u @ tf.linalg.diag(log_s) @ utils.transposem(v)
def from_cholesky(self, x):
"""Inverse of the diffeomorphism to the Cholesky space"""
assert_x = tf.debugging.Assert(
self._cholesky.check_point_on_manifold(x), [x])
with tf.control_dependencies([assert_x]):
return x @ utils.transposem(x)
def _check_vector_on_tangent(self, x, u, atol, rtol):
diff = utils.transposem(u) + u
return utils.allclose(diff, tf.zeros_like(diff), atol, rtol)
