def _use_cholesky(u, params): """Uses Cholesky decomposition.""" a, b, c = params _, n = u.shape x = c * u.T.conj() @ u + jnp.eye(n) # `y` is lower triangular. y = lax_linalg.cholesky(x, symmetrize_input=False) z = lax_linalg.triangular_solve(y, u.T, left_side=True, lower=True, conjugate_a=True).conj() z = lax_linalg.triangular_solve(y, z, left_side=True, lower=True, transpose_a=True, conjugate_a=True).T.conj() e = b / c u = e * u + (a - e) * z return u
def _use_cholesky(u, m, n, params): """QDWH iteration using Cholesky decomposition. Args: u: a matrix, with static (padded) shape M x N m, n: the dynamic shape of the matrix, where m <= M and n <= N. params: the QDWH parameters. """ a, b, c = params _, N = u.shape x = c * (u.T.conj() @ u) + jnp.eye(N, dtype=jnp.dtype(u)) # Pads the lower-right corner with the identity matrix to prevent the Cholesky # decomposition from failing due to the matrix not being PSD if padded with # zeros. x = _mask(x, (n, n), jnp.eye(N, dtype=x.dtype)) # `y` is lower triangular. y = lax_linalg.cholesky(x, symmetrize_input=False) z = lax_linalg.triangular_solve(y, u.T, left_side=True, lower=True, conjugate_a=True).conj() z = lax_linalg.triangular_solve(y, z, left_side=True, lower=True, transpose_a=True, conjugate_a=True).T.conj() e = b / c u = e * u + (a - e) * z return u
def _cho_solve(c, b, lower): c, b = np_linalg._promote_arg_dtypes(jnp.asarray(c), jnp.asarray(b)) lax_linalg._check_solve_shapes(c, b) b = lax_linalg.triangular_solve(c, b, left_side=True, lower=lower, transpose_a=not lower, conjugate_a=not lower) b = lax_linalg.triangular_solve(c, b, left_side=True, lower=lower, transpose_a=lower, conjugate_a=lower) return b
def _solve_triangular(a, b, trans, lower, unit_diagonal): if trans == 0 or trans == "N": transpose_a, conjugate_a = False, False elif trans == 1 or trans == "T": transpose_a, conjugate_a = True, False elif trans == 2 or trans == "C": transpose_a, conjugate_a = True, True else: raise ValueError(f"Invalid 'trans' value {trans}") a, b = _promote_dtypes_inexact(jnp.asarray(a), jnp.asarray(b)) # lax_linalg.triangular_solve only supports matrix 'b's at the moment. b_is_vector = jnp.ndim(a) == jnp.ndim(b) + 1 if b_is_vector: b = b[..., None] out = lax_linalg.triangular_solve(a, b, left_side=True, lower=lower, transpose_a=transpose_a, conjugate_a=conjugate_a, unit_diagonal=unit_diagonal) if b_is_vector: return out[..., 0] else: return out
def _cofactor_solve(a, b): """Equivalent to det(a)*solve(a, b) for nonsingular mat. Intermediate function used for jvp and vjp of det. This function borrows heavily from jax.numpy.linalg.solve and jax.numpy.linalg.slogdet to compute the gradient of the determinant in a way that is well defined even for low rank matrices. This function handles two different cases: * rank(a) == n or n-1 * rank(a) < n-1 For rank n-1 matrices, the gradient of the determinant is a rank 1 matrix. Rather than computing det(a)*solve(a, b), which would return NaN, we work directly with the LU decomposition. If a = p @ l @ u, then det(a)*solve(a, b) = prod(diag(u)) * u^-1 @ l^-1 @ p^-1 b = prod(diag(u)) * triangular_solve(u, solve(p @ l, b)) If a is rank n-1, then the lower right corner of u will be zero and the triangular_solve will fail. Let x = solve(p @ l, b) and y = det(a)*solve(a, b). Then y_{n} x_{n} / u_{nn} * prod_{i=1...n}(u_{ii}) = x_{n} * prod_{i=1...n-1}(u_{ii}) So by replacing the lower-right corner of u with prod_{i=1...n-1}(u_{ii})^-1 we can avoid the triangular_solve failing. To correctly compute the rest of y_{i} for i != n, we simply multiply x_{i} by det(a) for all i != n, which will be zero if rank(a) = n-1. For the second case, a check is done on the matrix to see if `solve` returns NaN or Inf, and gives a matrix of zeros as a result, as the gradient of the determinant of a matrix with rank less than n-1 is 0. This will still return the correct value for rank n-1 matrices, as the check is applied *after* the lower right corner of u has been updated. Args: a: A square matrix or batch of matrices, possibly singular. b: A matrix, or batch of matrices of the same dimension as a. Returns: det(a) and cofactor(a)^T*b, aka adjugate(a)*b """ a = _promote_arg_dtypes(jnp.asarray(a)) b = _promote_arg_dtypes(jnp.asarray(b)) a_shape = jnp.shape(a) b_shape = jnp.shape(b) a_ndims = len(a_shape) if not (a_ndims >= 2 and a_shape[-1] == a_shape[-2] and b_shape[-2:] == a_shape[-2:]): msg = ("The arguments to _cofactor_solve must have shapes " "a=[..., m, m] and b=[..., m, m]; got a={} and b={}") raise ValueError(msg.format(a_shape, b_shape)) if a_shape[-1] == 1: return a[..., 0, 0], b # lu contains u in the upper triangular matrix and l in the strict lower # triangular matrix. # The diagonal of l is set to ones without loss of generality. lu, pivots, permutation = lax_linalg.lu(a) dtype = lax.dtype(a) batch_dims = lax.broadcast_shapes(lu.shape[:-2], b.shape[:-2]) x = jnp.broadcast_to(b, batch_dims + b.shape[-2:]) lu = jnp.broadcast_to(lu, batch_dims + lu.shape[-2:]) # Compute (partial) determinant, ignoring last diagonal of LU diag = jnp.diagonal(lu, axis1=-2, axis2=-1) parity = jnp.count_nonzero(pivots != jnp.arange(a_shape[-1]), axis=-1) sign = jnp.asarray(-2 * (parity % 2) + 1, dtype=dtype) # partial_det[:, -1] contains the full determinant and # partial_det[:, -2] contains det(u) / u_{nn}. partial_det = jnp.cumprod(diag, axis=-1) * sign[..., None] lu = lu.at[..., -1, -1].set(1.0 / partial_det[..., -2]) permutation = jnp.broadcast_to(permutation, batch_dims + (a_shape[-1], )) iotas = jnp.ix_(*(lax.iota(jnp.int32, b) for b in batch_dims + (1, ))) # filter out any matrices that are not full rank d = jnp.ones(x.shape[:-1], x.dtype) d = lax_linalg.triangular_solve(lu, d, left_side=True, lower=False) d = jnp.any(jnp.logical_or(jnp.isnan(d), jnp.isinf(d)), axis=-1) d = jnp.tile(d[..., None, None], d.ndim * (1, ) + x.shape[-2:]) x = jnp.where(d, jnp.zeros_like(x), x) # first filter x = x[iotas[:-1] + (permutation, slice(None))] x = lax_linalg.triangular_solve(lu, x, left_side=True, lower=True, unit_diagonal=True) x = jnp.concatenate( (x[..., :-1, :] * partial_det[..., -1, None, None], x[..., -1:, :]), axis=-2) x = lax_linalg.triangular_solve(lu, x, left_side=True, lower=False) x = jnp.where(d, jnp.zeros_like(x), x) # second filter return partial_det[..., -1], x