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Cholupdate (#673)

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Cholupdate
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Sébastien Crozet 2019-11-17 15:17:25 +01:00 committed by GitHub
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3 changed files with 217 additions and 21 deletions
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173
src/linalg/cholesky.rs
View File

@ -1,33 +1,31 @@
#[cfg(feature = "serde-serialize")]
use serde::{Deserialize, Serialize};
use num::One;
use alga::general::ComplexField;
use crate::allocator::Allocator;
use crate::base::{DefaultAllocator, Matrix, MatrixMN, MatrixN, SquareMatrix};
use crate::base::{DefaultAllocator, Matrix, MatrixMN, MatrixN, SquareMatrix, Vector};
use crate::constraint::{SameNumberOfRows, ShapeConstraint};
use crate::dimension::{Dim, DimSub, Dynamic};
use crate::dimension::{Dim, DimAdd, DimSum, DimDiff, DimSub, Dynamic, U1};
use crate::storage::{Storage, StorageMut};
/// The Cholesky decomposition of a symmetric-definite-positive matrix.
#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
#[cfg_attr(
feature = "serde-serialize",
serde(bound(
serialize = "DefaultAllocator: Allocator<N, D>,
MatrixN<N, D>: Serialize"
))
serde(bound(serialize = "DefaultAllocator: Allocator<N, D>,
MatrixN<N, D>: Serialize"))
)]
#[cfg_attr(
feature = "serde-serialize",
serde(bound(
deserialize = "DefaultAllocator: Allocator<N, D>,
MatrixN<N, D>: Deserialize<'de>"
))
serde(bound(deserialize = "DefaultAllocator: Allocator<N, D>,
MatrixN<N, D>: Deserialize<'de>"))
)]
#[derive(Clone, Debug)]
pub struct Cholesky<N: ComplexField, D: Dim>
where DefaultAllocator: Allocator<N, D, D>
where
DefaultAllocator: Allocator<N, D, D>,
{
chol: MatrixN<N, D>,
}
@ -36,10 +34,12 @@ impl<N: ComplexField, D: Dim> Copy for Cholesky<N, D>
where
DefaultAllocator: Allocator<N, D, D>,
MatrixN<N, D>: Copy,
{}
{
}
impl<N: ComplexField, D: DimSub<Dynamic>> Cholesky<N, D>
where DefaultAllocator: Allocator<N, D, D>
where
DefaultAllocator: Allocator<N, D, D>,
{
/// Attempts to compute the Cholesky decomposition of `matrix`.
///
@ -146,10 +146,155 @@ where DefaultAllocator: Allocator<N, D, D>
self.solve_mut(&mut res);
res
}
/// Given the Cholesky decomposition of a matrix `M`, a scalar `sigma` and a vector `v`,
/// performs a rank one update such that we end up with the decomposition of `M + sigma * (v * v.adjoint())`.
#[inline]
pub fn rank_one_update<R2: Dim, S2>(&mut self, x: &Vector<N, R2, S2>, sigma: N::RealField)
where
S2: Storage<N, R2, U1>,
DefaultAllocator: Allocator<N, R2, U1>,
ShapeConstraint: SameNumberOfRows<R2, D>,
{
Self::xx_rank_one_update(&mut self.chol, &mut x.clone_owned(), sigma)
}
/// Updates the decomposition such that we get the decomposition of a matrix with the given column `col` in the `j`th position.
/// Since the matrix is square, an identical row will be added in the `j`th row.
pub fn insert_column<R2, S2>(
&self,
j: usize,
col: Vector<N, R2, S2>,
) -> Cholesky<N, DimSum<D, U1>>
where
D: DimAdd<U1>,
R2: Dim,
S2: Storage<N, R2, U1>,
DefaultAllocator: Allocator<N, DimSum<D, U1>, DimSum<D, U1>> + Allocator<N, R2>,
ShapeConstraint: SameNumberOfRows<R2, DimSum<D, U1>>,
{
let mut col = col.into_owned();
// for an explanation of the formulas, see https://en.wikipedia.org/wiki/Cholesky_decomposition#Updating_the_decomposition
let n = col.nrows();
assert_eq!(n, self.chol.nrows() + 1, "The new column must have the size of the factored matrix plus one.");
assert!(j < n, "j needs to be within the bound of the new matrix.");
// loads the data into a new matrix with an additional jth row/column
let mut chol = unsafe { Matrix::new_uninitialized_generic(self.chol.data.shape().0.add(U1), self.chol.data.shape().1.add(U1)) };
chol.slice_range_mut(..j, ..j).copy_from(&self.chol.slice_range(..j, ..j));
chol.slice_range_mut(..j, j + 1..).copy_from(&self.chol.slice_range(..j, j..));
chol.slice_range_mut(j + 1.., ..j).copy_from(&self.chol.slice_range(j.., ..j));
chol.slice_range_mut(j + 1.., j + 1..).copy_from(&self.chol.slice_range(j.., j..));
// update the jth row
let top_left_corner = self.chol.slice_range(..j, ..j);
let col_j = col[j];
let (mut new_rowj_adjoint, mut new_colj) = col.rows_range_pair_mut(..j, j + 1..);
assert!(top_left_corner.solve_lower_triangular_mut(&mut new_rowj_adjoint), "Cholesky::insert_column : Unable to solve lower triangular system!");
new_rowj_adjoint.adjoint_to(&mut chol.slice_range_mut(j, ..j));
// update the center element
let center_element = N::sqrt(col_j - N::from_real(new_rowj_adjoint.norm_squared()));
chol[(j, j)] = center_element;
// update the jth column
let bottom_left_corner = self.chol.slice_range(j.., ..j);
// new_colj = (col_jplus - bottom_left_corner * new_rowj.adjoint()) / center_element;
new_colj.gemm(-N::one() / center_element, &bottom_left_corner, &new_rowj_adjoint, N::one() / center_element);
chol.slice_range_mut(j + 1.., j).copy_from(&new_colj);
// update the bottom right corner
let mut bottom_right_corner = chol.slice_range_mut(j + 1.., j + 1..);
Self::xx_rank_one_update(&mut bottom_right_corner, &mut new_colj, -N::RealField::one());
Cholesky { chol }
}
/// Updates the decomposition such that we get the decomposition of the factored matrix with its `j`th column removed.
/// Since the matrix is square, the `j`th row will also be removed.
pub fn remove_column(
&self,
j: usize,
) -> Cholesky<N, DimDiff<D, U1>>
where
D: DimSub<U1>,
DefaultAllocator: Allocator<N, DimDiff<D, U1>, DimDiff<D, U1>> + Allocator<N, D>
{
let n = self.chol.nrows();
assert!(n > 0, "The matrix needs at least one column.");
assert!(j < n, "j needs to be within the bound of the matrix.");
// loads the data into a new matrix except for the jth row/column
let mut chol = unsafe { Matrix::new_uninitialized_generic(self.chol.data.shape().0.sub(U1), self.chol.data.shape().1.sub(U1)) };
chol.slice_range_mut(..j, ..j).copy_from(&self.chol.slice_range(..j, ..j));
chol.slice_range_mut(..j, j..).copy_from(&self.chol.slice_range(..j, j + 1..));
chol.slice_range_mut(j.., ..j).copy_from(&self.chol.slice_range(j + 1.., ..j));
chol.slice_range_mut(j.., j..).copy_from(&self.chol.slice_range(j + 1.., j + 1..));
// updates the bottom right corner
let mut bottom_right_corner = chol.slice_range_mut(j.., j..);
let mut workspace = self.chol.column(j).clone_owned();
let mut old_colj = workspace.rows_range_mut(j + 1..);
Self::xx_rank_one_update(&mut bottom_right_corner, &mut old_colj, N::RealField::one());
Cholesky { chol }
}
/// Given the Cholesky decomposition of a matrix `M`, a scalar `sigma` and a vector `x`,
/// performs a rank one update such that we end up with the decomposition of `M + sigma * (x * x.adjoint())`.
///
/// This helper method is called by `rank_one_update` but also `insert_column` and `remove_column`
/// where it is used on a square slice of the decomposition
fn xx_rank_one_update<Dm, Sm, Rx, Sx>(chol : &mut Matrix<N, Dm, Dm, Sm>, x: &mut Vector<N, Rx, Sx>, sigma: N::RealField)
where
//N: ComplexField,
Dm: Dim,
Rx: Dim,
Sm: StorageMut<N, Dm, Dm>,
Sx: StorageMut<N, Rx, U1>,
{
// heavily inspired by Eigen's `llt_rank_update_lower` implementation https://eigen.tuxfamily.org/dox/LLT_8h_source.html
let n = x.nrows();
assert_eq!(
n,
chol.nrows(),
"The input vector must be of the same size as the factorized matrix."
);
let mut beta = crate::one::<N::RealField>();
for j in 0..n {
// updates the diagonal
let diag = N::real(unsafe { *chol.get_unchecked((j, j)) });
let diag2 = diag * diag;
let xj = unsafe { *x.get_unchecked(j) };
let sigma_xj2 = sigma * N::modulus_squared(xj);
let gamma = diag2 * beta + sigma_xj2;
let new_diag = (diag2 + sigma_xj2 / beta).sqrt();
unsafe { *chol.get_unchecked_mut((j, j)) = N::from_real(new_diag) };
beta += sigma_xj2 / diag2;
// updates the terms of L
let mut xjplus = x.rows_range_mut(j + 1..);
let mut col_j = chol.slice_range_mut(j + 1.., j);
// temp_jplus -= (wj / N::from_real(diag)) * col_j;
xjplus.axpy(-xj / N::from_real(diag), &col_j, N::one());
if gamma != crate::zero::<N::RealField>() {
// col_j = N::from_real(nljj / diag) * col_j + (N::from_real(nljj * sigma / gamma) * N::conjugate(wj)) * temp_jplus;
col_j.axpy(
N::from_real(new_diag * sigma / gamma) * N::conjugate(xj),
&xjplus,
N::from_real(new_diag / diag),
);
}
}
}
}
impl<N: ComplexField, D: DimSub<Dynamic>, S: Storage<N, D, D>> SquareMatrix<N, D, S>
where DefaultAllocator: Allocator<N, D, D>
where
DefaultAllocator: Allocator<N, D, D>,
{
/// Attempts to compute the Cholesky decomposition of this matrix.
///

12
src/linalg/solve.rs
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@ -15,7 +15,7 @@ impl<N: ComplexField, D: Dim, S: Storage<N, D, D>> SquareMatrix<N, D, S> {
b: &Matrix<N, R2, C2, S2>,
) -> Option<MatrixMN<N, R2, C2>>
where
S2: StorageMut<N, R2, C2>,
S2: Storage<N, R2, C2>,
DefaultAllocator: Allocator<N, R2, C2>,
ShapeConstraint: SameNumberOfRows<R2, D>,
{
@ -35,7 +35,7 @@ impl<N: ComplexField, D: Dim, S: Storage<N, D, D>> SquareMatrix<N, D, S> {
b: &Matrix<N, R2, C2, S2>,
) -> Option<MatrixMN<N, R2, C2>>
where
S2: StorageMut<N, R2, C2>,
S2: Storage<N, R2, C2>,
DefaultAllocator: Allocator<N, R2, C2>,
ShapeConstraint: SameNumberOfRows<R2, D>,
{
@ -191,7 +191,7 @@ impl<N: ComplexField, D: Dim, S: Storage<N, D, D>> SquareMatrix<N, D, S> {
b: &Matrix<N, R2, C2, S2>,
) -> Option<MatrixMN<N, R2, C2>>
where
S2: StorageMut<N, R2, C2>,
S2: Storage<N, R2, C2>,
DefaultAllocator: Allocator<N, R2, C2>,
ShapeConstraint: SameNumberOfRows<R2, D>,
{
@ -211,7 +211,7 @@ impl<N: ComplexField, D: Dim, S: Storage<N, D, D>> SquareMatrix<N, D, S> {
b: &Matrix<N, R2, C2, S2>,
) -> Option<MatrixMN<N, R2, C2>>
where
S2: StorageMut<N, R2, C2>,
S2: Storage<N, R2, C2>,
DefaultAllocator: Allocator<N, R2, C2>,
ShapeConstraint: SameNumberOfRows<R2, D>,
{
@ -273,7 +273,7 @@ impl<N: ComplexField, D: Dim, S: Storage<N, D, D>> SquareMatrix<N, D, S> {
b: &Matrix<N, R2, C2, S2>,
) -> Option<MatrixMN<N, R2, C2>>
where
S2: StorageMut<N, R2, C2>,
S2: Storage<N, R2, C2>,
DefaultAllocator: Allocator<N, R2, C2>,
ShapeConstraint: SameNumberOfRows<R2, D>,
{
@ -293,7 +293,7 @@ impl<N: ComplexField, D: Dim, S: Storage<N, D, D>> SquareMatrix<N, D, S> {
b: &Matrix<N, R2, C2, S2>,
) -> Option<MatrixMN<N, R2, C2>>
where
S2: StorageMut<N, R2, C2>,
S2: Storage<N, R2, C2>,
DefaultAllocator: Allocator<N, R2, C2>,
ShapeConstraint: SameNumberOfRows<R2, D>,
{

53
tests/linalg/cholesky.rs
View File

@ -1,6 +1,5 @@
#![cfg(all(feature = "arbitrary", feature = "debug"))]
macro_rules! gen_tests(
($module: ident, $scalar: ty) => {
mod $module {
@ -78,6 +77,58 @@ macro_rules! gen_tests(
id1.is_identity(1.0e-7) && id2.is_identity(1.0e-7)
}
fn cholesky_rank_one_update(_n: usize) -> bool {
let mut m = RandomSDP::new(U4, || random::<$scalar>().0).unwrap();
let x = Vector4::<$scalar>::new_random().map(|e| e.0);
// this is dirty but $scalar is not a scalar type (its a Rand) in this file
let zero = random::<$scalar>().0 * 0.;
let one = zero + 1.;
let sigma = random::<f64>(); // needs to be a real
let sigma_scalar = zero + sigma;
// updates cholesky decomposition and reconstructs m updated
let mut chol = m.clone().cholesky().unwrap();
chol.rank_one_update(&x, sigma);
let m_chol_updated = chol.l() * chol.l().adjoint();
// updates m manually
m.gerc(sigma_scalar, &x, &x, one); // m += sigma * x * x.adjoint()
relative_eq!(m, m_chol_updated, epsilon = 1.0e-7)
}
fn cholesky_insert_column(n: usize) -> bool {
let n = n.max(1).min(10);
let j = random::<usize>() % n;
let m_updated = RandomSDP::new(Dynamic::new(n), || random::<$scalar>().0).unwrap();
// build m and col from m_updated
let col = m_updated.column(j);
let m = m_updated.clone().remove_column(j).remove_row(j);
// remove column from cholesky decomposition and rebuild m
let chol = m.clone().cholesky().unwrap().insert_column(j, col);
let m_chol_updated = chol.l() * chol.l().adjoint();
relative_eq!(m_updated, m_chol_updated, epsilon = 1.0e-7)
}
fn cholesky_remove_column(n: usize) -> bool {
let n = n.max(1).min(10);
let j = random::<usize>() % n;
let m = RandomSDP::new(Dynamic::new(n), || random::<$scalar>().0).unwrap();
// remove column from cholesky decomposition and rebuild m
let chol = m.clone().cholesky().unwrap().remove_column(j);
let m_chol_updated = chol.l() * chol.l().adjoint();
// remove column from m
let m_updated = m.remove_column(j).remove_row(j);
relative_eq!(m_updated, m_chol_updated, epsilon = 1.0e-7)
}
}
}
}