The code in this post is indebted (and in some cases wholly ripped off from) work by the glorious Finn Lindgren, who emailed me some code to do this probably a decade ago. Yes I am behind on my emails.

Finn’s code can be found here as part of the glorious INLAbru project.

The code from this post can be found in my github repo.

The time has come once more to make a blog post truly untethered from context. This time, I’m going to show you how to compute entries of the inverse of a sparse symmetric positive definite matrix that correspond to the non-zero elements of the original matrix. And I am going to once again pull out that rusty spoon that is my C++ skill to do it.

## A little bit of motivation

Computing certain elements of the inverse of a matrix isn’t necessarily the most useless thing possible. It actually comes up quite a lot in statistical applications. For instance, if you are computing the score function while doing maximum likelihood estimation for a multivariate Gaussian you’re gonna need those values. Or, less specifically, if you happen to have a multivariate Gaussian \(N(0, Q^{-1})\) parameterized by its precision (inverse covariance) matrix \(Q\), if you are interested in the variance of each coordinate, you need the diagonal of \(Q^{-1}\).

A very real problem with computing \(Q^{-1}\) is that it is, infamously, quite expensive. The only really practical way to do it is to solve \(n\) linear systems, where \(n\) is the number of rows/columns in \(Q\). When \(n\) is big, this is going to be a bit of a computational disaster!

Thankfully, there is a convenient set of recursions due to Takahashi, Fagan, and Chen^{1} that allow us to compute these elements directly and cheaply from the Cholesky factorization of \(Q\).

In fact, I have blogged about this before.

Essentially, we need to implement the following pseudocode.

```
for i = n-1, ..., 0
for j = n-1, ..., i
if (L[j,i] not known to be 0)
Sigma[j,i] = Sigma[i,j] = (I(i==j)/L[i,i]
- sum_{k=i+1}^{n-1} L[k,i] Sigma[k,j] ) / L[i,i]
```

This is not going to be terribly complicated, but it does require a bit of C++ plumbing and dealing with the internal Eigen representation of the Choleksy factor. It’s always so fun to read documentation!

## Making this work in C++

One of the things about working with a library like Eigen is that we really want to use the official API for its functions as much as possible. Even when we itch to use the undocumented internal structure, we should desist: the API is, usually, pretty stable and it is considerably less likely that an Eigen update will materially break our code if we hold them to the promises they actually make rather than the ones we wish they made.

It might look like you need three iterators to build our algorithm, but we actually need four. Because the matrix is stored in column-major order, we are going to need a new iterator for every distinct column index. In this case, that is 1. A reverse iterator going up column `i`

of `Sigma`

2. A reverse iterator going up column `i`

of `L`

3. A reverse iterator going up column `j`

of `Sigma`

4. A reverse iterator going up column `i`

in sync with iterator 3.

The C++ code is pretty straightforward after that, you just need to keep your iterators in sync.

One wrinkle that I forgot about the first time I coded this is that there are a few things that I need to be true: firstly, I need the output to be the lower-triangle of a symmetric matrix, and secondly I need that matrix to have the same sparsity pattern as \(Q\). To do this, I wrote a RAII helper class, mainly because if I’m going to manipulate raw pointers I’m gonna want some safety.

This helper class is a *functor*, meaning that its objects are callable with similar syntax to functions. Is this strictly necessary? Of course not. But mummy I love him.

```
#include <Eigen/SparseCore>
#include <Eigen/SparseCholesky>
typedef Eigen::SparseMatrix<double>::StorageIndex StorageIndex;
template<typename SpMat> class MatchPattern {
using T = typename base_type<SpMat>::type;
StorageIndex* m_outer;
StorageIndex* m_inner;
T* m_val;
StorageIndex m_cols;
StorageIndex m_nnz;
public:
MatchPattern(const SpMat& A, const SpMat& pattern) {
/**
* MatchPattern(const SpMat& A, const SpMat& pattern)
* Constructs functor class designed to construct a sparse matrix with
* the same non-zero pattern as `pattern` and the same non-zero values
* as `A`.
*
* This function assumes that the sparsity pattern of `pattern` is a SUBSET
* of the sparsity pattern of `A`. Weird things will happen if this does not
* hold.
*
* Usage:
* ```
* typedef Eigen::SparseMatrix<double> SpMatrixd;
* SpMatrixd A_pattern = MatchPattern<SpMatrixd>(A, pattern)();
* ```
* */
m_cols = pattern.cols();
m_nnz = pattern.nonZeros();
m_outer = new StorageIndex[m_cols + 1];
std::copy(pattern.outerIndexPtr(), pattern.outerIndexPtr() + m_cols + 1, m_outer);
m_inner = new StorageIndex[m_nnz];
std::copy(pattern.innerIndexPtr(), pattern.innerIndexPtr() + m_nnz, m_inner);
m_val = new T[m_nnz];
T* valptr = m_val;
for (int j = 0; j < m_cols; ++j) {
typename SpMat::InnerIterator Acol(A, j);
for (typename SpMat::InnerIterator pattern_col(pattern, j);
pattern_col; ++pattern_col) {
while (Acol & (Acol.row() < pattern_col.row())){
++Acol;
}
valptr++ = Acol.value();
++Acol;
}
}
}
// Specialization for rank-1 matrices A = bc^T
MatchPattern(
const typename Eigen::Matrix<T,1,Eigen::Dynamic>& b,
const typename Eigen::Matrix<T,1,Eigen::Dynamic> c,
const SpMat& pattern
) {
/**
* MatchPattern(typename Eigen::Vector<T>& b, typename Eigen::Vector<T>& c, const SpMat& pattern)
* A specialization of the MatchPattern class where the matrix to be matched
* is a rank one matrix of the form $A = bc^T$.
*
* Usage:
* ```
* typedef Eigen::SparseMatrix<double> SpMatrixd;
* SpMatrixd A_pattern = MatchPattern<SpMatrixd>(b, c, pattern)();
* ```
* */
m_cols = pattern.cols();
m_nnz = pattern.nonZeros();
m_outer = new StorageIndex[m_cols + 1];
std::copy(pattern.outerIndexPtr(), pattern.outerIndexPtr() + m_cols + 1, m_outer);
m_inner = new StorageIndex[m_nnz];
std::copy(pattern.innerIndexPtr(), pattern.innerIndexPtr() + m_nnz, m_inner);
m_val = new T[m_nnz];
T* valptr = m_val;
for (int j = 0; j < m_cols; ++j) {
for (typename SpMat::InnerIterator pattern_col(pattern, j);
pattern_col; ++pattern_col) {
valptr++ = b.coeff(pattern_col.row()) * c.coeff(j);
}
}
}
~MatchPattern() {
delete[] m_inner;
delete[] m_outer;
delete[] m_val;
}
SpMat operator () () {
return typename SpMat::Map(
m_cols,
m_cols,
m_nnz,
m_outer,
m_inner,
m_val
);
}
};
```

The first thing to note here is that I am, fundamentally, quite lazy. As such I have made the convenient assumption that the target sparsity pattern is always a subset of the sparsity pattern of interest. This is true for the application that I have in mind, but you should probably be careful if you’re adapting this code to anything else.

The second thing you may have noticed is that there a second constructor that is not needed here at all. This is really a gift to future me and avoids me having to re-write this code at some point in the future. Nothing to see here.

With all of this in hand, we can jump over to the code indebted to^{2} Finn.

```
template<typename SpChol, typename SpMat>
typename SpChol::MatrixType partial_inverse(
const SpChol& llt,
const SpMat& pattern
) {
/**
* Input:
* - `llt`: a Sparse Cholesky factorization of a matrix `Q`.
* - `pattern`: a sparse matrix with the target sparsity
* Assumptions:
* - `pattern` has the same sparsity pattern as `Q` or is a subset of that pattern
* Output:
* - A sparse matrix with the same sparsity pattern as `pattern` who's non-zero
* elements correspond to the non-zero elements of $Q^{-1}$.
**/
typedef typename SpMat::ReverseInnerIterator reverse_it;
StorageIndex ncols = llt.cols();
const SpMat& L = llt.matrixL();
SpMat Qinv = L.template selfadjointView<Eigen::Lower>();
for (int i = ncols - 1; i >= 0; --i) {
reverse_it QinvcolI(Qinv, i);
for (reverse_it LcolI_slow(L, i); LcolI_slow; --LcolI_slow) {
// inner sum iterators
reverse_it LcolI(L, i);
reverse_it QinvcolJ(Qinv, LcolI_slow.row());
// Initialize Qinv[j,i]
QinvcolI.valueRef() = 0.0;
// Inner-most sum
while (LcolI.row() > i) {
// First up, sync the iterators
while ( QinvcolJ && (LcolI.row() < QinvcolJ.row())){
--QinvcolJ;
}
if (QinvcolJ && (QinvcolJ.row() == LcolI.row())) {
QinvcolI.valueRef() -= LcolI.value() * QinvcolJ.value();
--QinvcolJ;
}
--LcolI;
}
// At this point LcolI is the diagonal value
if (i == LcolI_slow.row()) {
QinvcolI.valueRef() += 1/ LcolI.value();
QinvcolI.valueRef() /= LcolI.value();
} else{
QinvcolI.valueRef() /= LcolI.value();
// Set Qinv[i,j] = Qinv[j,i]
while (QinvcolJ.row() > i) {
--QinvcolJ;
}
QinvcolJ.valueRef() = QinvcolI.value();
}
--QinvcolI;
}
}
// Undo the permutation
Qinv = Qinv.twistedBy(llt.permutationP().inverse());
// Return the non-zero elements of Qinv corresponding to the non-zero
// elements of Q
return MatchPattern(Qinv, Q)();
}
```

You’ll probably notice that there are far fewer template shenanigans here than in the block matrix code from yesterday. That is because this only needs to work with scalar types and doesn’t need to be part of the `math`

API. If needed, I guess we could always work out what the derivative of the partial inverse is and implement its reverse-mode specialization in Stan, but frankly why^{3} bother.

The other thing you may notice is the line

This exists because the Cholesky factor is not actually performed on \(Q\) but rather on a permuted matrix \(PQP^T\) for some permutation matrix \(P\). This line basically undoes the permutation and puts everything back into its right place.

## A quick test

Finally, we need to make sure this works. The easiest way to do this is with a simple example, which is a 25x25 sparse matrix. Everything is hard coded in, because why not.

```
#include <iostream>
#include "partial_inverse.hpp"
#include "Eigen/SparseCore"
#include "Eigen/Dense"
int main() {
using SparseMatrix = Eigen::SparseMatrix<double, Eigen::ColMajor>;
int Q_inner[] = {0,1,5,0,1,2,6,1,2,3,7,2,3,4,8,3,4,9,0,5,6,10,1,5,6,7,11,2,
6,7,8,12,3,7,8,9,13,4,8,9,14,5,10,11,15,6,10,11,12,16,7,11,
12,13,17,8,12,13,14,18,9,13,14,19,10,15,16,20,11,15,16,17,
21,12,16,17,18,22,13,17,18,19,23,14,18,19,24,15,20,21,16,20,
21,22,17,21,22,23,18,22,23,24,19,23,24};
int Q_outer[] = {0,3,7,11,15,18,22,27,32,37,41,45,50,55,60,64,68,73,78,83,
87,90,94,98,102,105};
double Q_val[] = {5.0,-1.0,-1.0,-1.0,5.0,-1.0,-1.0,-1.0,5.0,-1.0,-1.0,-1.0,
5.0,-1.0,-1.0,-1.0,5.0,-1.0,-1.0,5.0,-1.0,-1.0,-1.0,-1.0,5.0,
-1.0,-1.0,-1.0,-1.0,5.0,-1.0,-1.0,-1.0,-1.0,5.0,-1.0,-1.0,
-1.0,-1.0,5.0,-1.0,-1.0,5.0,-1.0,-1.0,-1.0,-1.0,5.0,-1.0,
-1.0,-1.0,-1.0,5.0,-1.0,-1.0,-1.0,-1.0,5.0,-1.0,-1.0,-1.0,
-1.0,5.0,-1.0,-1.0,5.0,-1.0,-1.0,-1.0,-1.0,5.0,-1.0,-1.0,
-1.0,-1.0,5.0,-1.0,-1.0,-1.0,-1.0,5.0,-1.0,-1.0,-1.0,-1.0,
5.0,-1.0,-1.0,5.0,-1.0,-1.0,-1.0,5.0,-1.0,-1.0,-1.0,5.0,-1.0,
-1.0,-1.0,5.0,-1.0,-1.0,-1.0,5.0};
Eigen::VectorXd Qinv_true(105);
Qinv_true << 0.220593295593296,0.051483238983239,0.051483238983239,0.051483238983239,0.233306970806971,0.0547785547785548,0.0602730602730603,0.0547785547785548,0.234139471639472,0.0547785547785548,0.0611402486402486,0.0547785547785548,0.233306970806971,0.051483238983239,0.0602730602730603,0.051483238983239,0.220593295593296,0.051483238983239,0.051483238983239,0.233306970806971,0.0602730602730603,0.0547785547785548,0.0602730602730603,0.0602730602730603,0.25021645021645,0.0652680652680653,0.0652680652680653,0.0611402486402486,0.0652680652680653,0.251621989121989,0.0652680652680653,0.0664335664335664,0.0602730602730603,0.0652680652680653,0.25021645021645,0.0602730602730603,0.0652680652680653,0.051483238983239,0.0602730602730603,0.233306970806971,0.0547785547785548,0.0547785547785548,0.234139471639472,0.0611402486402486,0.0547785547785548,0.0652680652680653,0.0611402486402486,0.251621989121989,0.0664335664335664,0.0652680652680653,0.0664335664335664,0.0664335664335664,0.253146853146853,0.0664335664335664,0.0664335664335664,0.0652680652680653,0.0664335664335664,0.251621989121989,0.0611402486402486,0.0652680652680653,0.0547785547785548,0.0611402486402486,0.234139471639472,0.0547785547785548,0.0547785547785548,0.233306970806971,0.0602730602730603,0.051483238983239,0.0652680652680653,0.0602730602730603,0.25021645021645,0.0652680652680653,0.0602730602730603,0.0664335664335664,0.0652680652680653,0.251621989121989,0.0652680652680653,0.0611402486402486,0.0652680652680653,0.0652680652680653,0.25021645021645,0.0602730602730603,0.0602730602730603,0.0547785547785548,0.0602730602730603,0.233306970806971,0.051483238983239,0.051483238983239,0.220593295593296,0.051483238983239,0.0602730602730603,0.051483238983239,0.233306970806971,0.0547785547785548,0.0611402486402486,0.0547785547785548,0.234139471639472,0.0547785547785548,0.0602730602730603,0.0547785547785548,0.233306970806971,0.051483238983239,0.051483238983239,0.051483238983239,0.220593295593296;
int Q_ncol = 25;
int Q_nnz = 105;
SparseMatrix Q = Eigen::Map<SparseMatrix>(Q_ncol, Q_ncol, Q_nnz, Q_outer, Q_inner, Q_val);
auto llt = Eigen::SimplicialLLT<SparseMatrix>(Q);
SparseMatrix Qinv = partial_inverse(llt, Q);
Eigen::VectorXd Qinv_val = Eigen::Map<Eigen::VectorXd>(Qinv.valuePtr(), Q_nnz);
std::cout << "The error in the partial inverse is " << (Qinv_val - Qinv_true).norm() << "!" << std::endl;
}
```

The output is

`The error in the partial inverse is 1.25852e-15!`

All good here.

That’s the end for this blog post. Hopefully I’ll be back soon-ish with a more interesting post that actually uses all of this stuff.

## Footnotes

Takahashi, K., Fagan, J., Chen, M.S., 1973. Formation of a sparse bus impedance matrix and its application to short circuit study. In: Eighth PICA Conference Proceedings.IEEE Power Engineering Society, pp. 63–69 (Papers Presented at the 1973 Power Industry Computer Application Conference in Minneapolis, MN)↩︎

stolen from↩︎

One reason would be to use gradient descent on the score function for a Gaussian MLE. Another is that this might be useful inside the

`generated quantities`

block to compute things like the marginal variances of the model, but, as the great lady said, not today Satan.↩︎

## Reuse

## Citation

```
@online{simpson2024,
author = {Simpson, Dan},
title = {Random {C++} {Part} 2: {Sparse} Partial Inverses in {Eigen}},
date = {2024-09-05},
url = {https://dansblog.netlify.app/posts/2024-05-08-laplace/laplace.html},
langid = {en}
}
```