Nearly-low-rank Hermitian matrices

a.k.a. perturbations of the identity, low-rank-plus-diagonal matrices

August 5, 2014 — January 25, 2024

Assumed audience:

People with undergrad linear algebra

Figure 1

Here is a decomposition that some matrix might possess that ends up being useful: \[\mathrm{K}= \sigma^2\mathrm{I} + a\mathrm{Z} \mathrm{Z}^{\dagger}\] where \(\mathrm{K}\in\mathbb{R}^{N\times N}\), \(\mathrm{Z}\in\mathbb{R}^{N\times D}\) with \(D\ll N\) and \(a\in\{-1, 1\}\). For compactness we call matrices of this form nearly-low-rank, as sibling of the actually low rank matrices. I write \(\mathrm{Z}^{\dagger}\) for the conjugate transpose of \(\mathrm{Z}\) and use that here rahter than the transpose, because sometimes I want to think of \(\mathrm{Z}\) as a complex matrix, and last I checked, most stuff here generalised to that case easily. Such lowish-rank matrices are clearly Hermitian, and thus arise in, e.g. covariance estimation.

This matrix structure is a workhorse. We might get such a low rank decompositions from matrix factorisation, or from some prior structuring of a problem; To pick one example, Ensemble filters have coavariance matrices that look a lot like this. In many applications \(\sigma^2\) is chosen so that the entire matrix is positive definite; whether we need this or not depends upon the application, but usually we do want that.

Sometimes we admit a more general version, where the diagonal is allowed to be other-than-constant, so that \(\mathrm{K}=\mathrm{V} + a\mathrm{Z} \mathrm{Z}^{\dagger}\) where \(\mathrm{V} = \operatorname{diag}(\boldsymbol{s})\) for \(s\in\mathbb{R}^D\).

1 Factorisations

1.1 Eigendecomposition

Nakatsukasa (2019) observes that, for general \(\mathrm{Y} \mathrm{Z}\),

The nonzero eigenvalues of \(\mathrm{Z} \mathrm{Y}\) are equal to those of \(\mathrm{Y} \mathrm{Z}\) : an identity that holds as long as the products are square, even when \(\mathrm{Y}, \mathrm{Z}\) are rectangular. This fact naturally suggests an efficient algorithm for computing eigenvalues and eigenvectors of a low-rank matrix \(\mathrm{K}=\mathrm{Z} \mathrm{Y}\) with \(\mathrm{Y}, \mathrm{Z}^{\dagger} \in \mathbb{C}^{D \times N}, D \gg N\) : form the small \(N \times N\) matrix \(\mathrm{Z} \mathrm{Y}\) and find its eigenvalues and eigenvectors. For nonzero eigenvalues, the eigenvectors are related by \(\mathrm{Z} \mathrm{Y} v=\lambda v \Leftrightarrow \mathrm{Y} \mathrm{Z} w=\lambda w\) with \(w=\mathrm{Y} v\)[…]

Cool. Our low-rank matrix \(\mathrm{K}\) has additional special structure \(\mathrm{Y}=\mathrm{Z}^\dagger\), i.e. symmetry. We use that the nonzero eigenvalues of \(\mathrm{Z} \mathrm{Z}^{\dagger}\) are equal to those of \(\mathrm{Z}^{\dagger} \mathrm{Z}\). So, to compute eigenvalues and eigenvectors of a low-rank matrix \(\mathrm{X}=\mathrm{Z} \mathrm{Z}^{\dagger}\): form the small \(N \times N\) matrix \(\mathrm{Z}^{\dagger} \mathrm{Z}\) and find its eigenvalues and eigenvectors. For nonzero eigenvalues, the eigenvectors are related by \(\mathrm{Z} \mathrm{Z}^{\dagger} v=\lambda v \Leftrightarrow \mathrm{Z}^{\dagger} \mathrm{Z} w=\lambda w\) with \(w=\mathrm{Z}^{\dagger} v\).

A classic piece of lore is cheap eigendecomposition of \(\mathrm{K}=\mathrm{Z}\mathrm{Z}^{\dagger}+\sigma^2\mathrm{I}\) by exploiting the low rank structure and SVD. First we calculate the SVD of \(\mathrm{Z}\) to obtain \(\mathrm{Z}=\mathrm{U}\mathrm{S}\mathrm{V}^{\dagger}\), where \(\mathrm{U}\in\mathbb{R}^{D\times N}\) and \(\mathrm{V}\in\mathbb{R}^{N\times N}\) are orthogonal and \(\mathrm{S}\in\mathbb{R}^{N\times N}\) is diagonal. Then we may write \[ \begin{aligned} \mathrm{K} &= \mathrm{Z} \mathrm{Z}^{\dagger} + \sigma^2 \mathrm{I} \\ &= \mathrm{U} \mathrm{S} \mathrm{V}^{\dagger} \mathrm{V} \mathrm{S} \mathrm{U}^{\dagger} + \sigma^2 \mathrm{I} \\ &= \mathrm{U} \mathrm{S}^2 \mathrm{U}^{\dagger} + \sigma^2 \mathrm{I} \end{aligned} \] Thus the top \(N\) eigenvalues of \(\mathrm{K}\) are \(\sigma^2+s_n^2\), and the corresponding eigenvectors are \(\boldsymbol{u}_n\). The remaining eigenvalues are \(\sigma^2\), and the corresponding eigenvectors are an arbitrary subset in the complement of the \(\mathrm{U}\) eigenvectors.

1.2 Square roots

No useful tricks that I know rn, See matrix square roots.

1.3 Cholesky

Louis Tiao, in Efficient Cholesky decomposition of low-rank updates summarises Seeger (2004).

1.4 Cholesky decomposition of a harmonic mean

This one pops up from time to time too. Suppose \(\mathrm{C},\mathrm{Z}_1,\mathrm{Z}_2\in\mathbb{R}^{D\times N }\) and \(N\ll D\). We wish to find \[ \begin{aligned} \mathrm{C}\mathrm{C}^{\dagger} &=\left((\mathrm{Z}_1\mathrm{Z}_1^{\dagger})^{-1}+(\mathrm{Z}_2\mathrm{Z}_2^{\dagger})^{-1}\right)^{-1}\\ &={\mathrm{Z}_1\mathrm{Z}_1^{\dagger}\left(\mathrm{Z}_1\mathrm{Z}_1^{\dagger}+\mathrm{Z}_2\mathrm{Z}_2^{\dagger}\right)^{-1}\mathrm{Z}_2\mathrm{Z}_2^{\dagger}}. \end{aligned} \] where that last line is the Woodbury identity. Can we do that in a low rank way?

2 Inverting

Specifically, solving \(\mathrm{X}=\mathrm{K}\mathrm{Y}\) for \(\mathrm{X}\) with \(\mathrm{Y}\in\mathbb{R}^{D\times M}\) and, in particular, solving it efficiently, in the sense that we

  1. exploit the computational efficiency of the low rank structure of \(\mathrm{K}\) so that it costs less than \(\mathcal{O}(D^3M)\) to compute \(\mathrm{K}^{-1}\mathrm{Y}\).
  2. avoid forming the explicit inverse matrix \(\mathrm{K}^{-1}\) which requires storage \(\mathcal{O}(D^2)\).

The workhorse here is the Woodbury identity, which has many variants, but often the basic version does what we want: Assuming both \(\mathrm{A}\) and \(\mathrm{B}\) are invertible, \[ \left(\mathrm{A}+\mathrm{C B C}^{\dagger}\right)^{-1}=\mathrm{A}^{-1}-\mathrm{A}^{-1} \mathrm{C}\left(\mathrm{B}^{-1}+\mathrm{C}^{\dagger} \mathrm{A}^{-1} \mathrm{C}\right)^{-1} \mathrm{C}^{\dagger} \mathrm{A}^{-1}. \] See Ken Tay’s intro to Woodbury identities for more about those. For now we note that they give us an efficient way of calculating matrix inverses.

There is a connection to the eigendecomposition clearly; we may also think about inversion by operating on the eigenvalues.

2.1 Classic

I have no idea who invented this; it seems to be part of the folklore now, but also it was not in my undergraduate degreee.

Assume \(a=1\). Applying the Woodbury identity, \[\begin{align*} \mathrm{K}^{-1}=\sigma^{-2}\mathrm{I}-\sigma^{-4} \mathrm{Z}\left(\mathrm{I}+\sigma^{-2}\mathrm{Z}^{\dagger} \mathrm{Z}\right)^{-1} \mathrm{Z}^{\dagger}. \end{align*}\] Computing the lower Cholesky decomposition \(\mathrm{L} \mathrm{L}^{\dagger}=\left(\mathrm{I}+\sigma^{-2}\mathrm{Z}^{\dagger} \mathrm{Z}\right)^{-1}\) at a cost of \(\mathcal{O}(N^3)\), and defining \(\mathrm{R}=\sigma^{-2}\mathrm{Z} \mathrm{L}\) we can write the inverse compactly as \[ \mathrm{K}^{-1}=\sigma^{-2}\mathrm{I}-\mathrm{R} \mathrm{R}^{\dagger}. \] Cool, so we have an explicit form for the inverse, which is still a nearly-low-rank operator. We may solve for \(\mathrm{X}\) by matrix multiplication, \[\begin{aligned} \mathrm{K}^{-1}\mathrm{Y} &=\left(\sigma^2 \mathrm{I}+\mathrm{Z} \mathrm{Z}^{\dagger}\right)^{-1}\mathrm{Y}\\ &=\left(\sigma^{-2}\mathrm{I}-\mathrm{R} \mathrm{R}^{\dagger}\right)\mathrm{Y}\\ &=\underbrace{\sigma^{-2}\mathrm{Y}}_{D \times M} - \underbrace{\mathrm{R}}_{D\times N} \underbrace{\mathrm{R}^{\dagger}\mathrm{Y}}_{N\times M} \end{aligned}\] The solution of the linear system is available at cost which looks something like \(\mathcal{O}\left(N^2 D + NDM +N^3\right)\) (hmm, should check that).

The price of this efficient inversion is that pre-multiplying by the nearly-low-rank inverse is not as numerically stable as classic matrix solution methods; but for many purposes this price is acceptable.

What else can we say now? Well, if \(\mathrm{K}\) is positive definite, then so is \(\mathrm{K}^{-1}\), i.e. both have positive eigenvalues. Suppose the eigenvalues of \(\mathrm{K}\) in descending order are \(\lambda^{\mathrm{K}}_{j}, j\in\{1, \ldots, D\}\). Note that for \(j\in \{N+1,\dots,D\}\) we have \(\lambda^{\mathrm{K}}_j=\sigma^2\). That is, all \(\lambda^{\mathrm{K}}_j\geq\sigma^2\). We can deduce that any non-zero eigenvalues \(\lambda^{\mathrm{Z}}_j\) of \(\mathrm{Z}\) have the form \(\lambda_j^{\mathrm{Z}}=\sqrt{\lambda^{\mathrm{K}}_{j}-\sigma^{2}}\).

The eigenvalues \(\lambda^{\mathrm{K}^{-1}}_j\) of \(\mathrm{K}^{-1}\) in descending order are \(\lambda^{\mathrm{K}^{-1}}_j=(\lambda^{\mathrm{K}}_{D+1-j})^{-1}\). Further, \(\lambda^{\mathrm{K}^{-1}}_j=\sigma^{-2}\) for \(j\in \{1, \dots, D-N\}\). The remaining eigenvalues are sandwiched in between \(\sigma^{-2}\) and \(0\), as we’d expect with a matrix inverse, from which it follows that the eigenvalues of \(-\mathrm{R}\mathrm{R}^{\dagger}\) are all negative, specifically \(\forall j,\,\lambda^{\mathrm{R}}_j\in[-\sigma^{-2},0]\), so the eigenvalues of \(\mathrm{R}\) are in the range \(\lambda^{\mathrm{R}}_j\in[0,\sigma^{-1}]\).

Note that the inverse is also a nearly-low-rank matrix, but with \(a=-1\). We can invert that guy also. Applying the Woodbury identity again, \[\begin{aligned} (\mathrm{K}^{-1})^{-1} &=\left(\sigma^{-2}\mathrm{I}-\mathrm{R} \mathrm{R}^{\dagger}\right)^{-1}\\ &=\sigma^{2}\mathrm{I}+\sigma^{4} \mathrm{R}\left(-\mathrm{I}+\sigma^{2}\mathrm{R}^{\dagger} \mathrm{R}\right)^{-1} \mathrm{R}^{\dagger} \end{aligned}\] Once again we compute the Cholesky decomposition \(\mathrm{M} \mathrm{M}^{\dagger}=\left(-\mathrm{I}+\sigma^{2}\mathrm{R}^{\dagger} \mathrm{R}\right)^{-1}\) at a cost of \(\mathcal{O}(N^3)\), and define \(\mathrm{Z}'=\sigma^{-2}\mathrm{R} \mathrm{M}\). We write the recovered inverse as \[ \mathrm{K}^{-1}=\sigma^{-2}\mathrm{I}+\mathrm{Z}' \mathrm{Z}'{}^{\dagger}. \]

In principle, \(\mathrm{Z}=\mathrm{Z}'\) if the Cholesky decomposition is unique, which is true if \(\left(-\mathrm{I}+\sigma^{2}\mathrm{R}^{\dagger} \mathrm{R}\right)\) resp. \(\left(\mathrm{I}+\sigma^{-2}\mathrm{Z}^{\dagger} \mathrm{Z}\right)\) is positive definite. In practice, none of this is terribly numerically stable, so I wouldn’t depend upon that in computational practice.

Terminology: For some reason, these \(\left(-\mathrm{I}+\sigma^{2}\mathrm{R}^{\dagger} \mathrm{R}\right)\) and \(\left(\mathrm{I}+\sigma^{-2}\mathrm{Z}^{\dagger} \mathrm{Z}\right)\) are referred to as capacitance matrices.

If such a capacitance is merely positive semidefinite then we need to do some more work to make it unique. And if it is indefinite (presumably because of numerical stability problems) then we are potentially in trouble, because the square root is not real. We can still have complex-valued solutions, if we want to go there.

2.2 General diagonals

OK, how about if we admit general diagonal matrices \(\mathrm{V}\)? Then \[\begin{align*} \mathrm{K}_{\mathrm{V}}^{-1}=\mathrm{V}^{-1}-\mathrm{V}^{-1} \mathrm{Z}\left(\mathrm{I}+\mathrm{Z}^{\dagger}\mathrm{V}^{-1} \mathrm{Z}\right)^{-1} \mathrm{Z}^{\dagger}\mathrm{V}^{-1}. \end{align*}\] Now we need the Cholesky decomposition of \(\mathrm{L} \mathrm{L}^{\dagger}=\left(\mathrm{I}+\mathrm{Z}^{\dagger}\mathrm{V}^{-1}\mathrm{Z}\right)^{-1}\), and define \(\mathrm{R}\) as before; the new low-rank inverse is \[ \mathrm{K}_{\mathrm{V}}^{-1}=\mathrm{V}^{-1}-\mathrm{R} \mathrm{R}^{\dagger}. \]

2.3 Centred

Suppose \(\mathrm{K}_{\mathrm{Z}\mathrm{C}}=\mathrm{Z}\mathrm{C}\mathrm{Z}^{\dagger}+\sigma^2\mathrm{I}\) where \(\mathrm{C}\) is a centering matrix. Then we can no longer use the Woodbury identity directly because \(\mathrm{C}\) is rank deficient, but a variant Woodbury identity (Harville 1977; Henderson and Searle 1981) applies, to wit: \[\begin{aligned} \left(\mathrm{V}+\mathrm{Z} \mathrm{C} \mathrm{Z}^{\dagger}\right)^{-1}=\mathrm{V}^{-1}-\mathrm{V}^{-1} \mathrm{Z} \mathrm{C}\left(\mathrm{I}+\mathrm{Z}^{\dagger} \mathrm{V}^{-1} \mathrm{Z} \mathrm{C}\right)^{-1} \mathrm{Z}^{\dagger} \mathrm{V}^{-1} \end{aligned}\] from which \[\begin{aligned} \mathrm{K}_{\mathrm{Z}\mathrm{C}}^{-1} &= \left(\mathrm{Z}\mathrm{C}\mathrm{Z}^{\dagger}+\sigma^2\mathrm{I}\right)^{-1}\\ &=\sigma^{-2}\mathrm{I}-\sigma^{-4}\mathrm{Z} \mathrm{C}\left(\mathrm{I}+\sigma^{-2}\mathrm{Z}^{\dagger} \mathrm{Z} \mathrm{C}\right)^{-1} \mathrm{Z}^{\dagger}\\ &=\sigma^{-2}\mathrm{I}-\sigma^{-4}\mathrm{Z} \left(\mathrm{C}+\sigma^{-2}\mathrm{C}\mathrm{Z}^{\dagger} \mathrm{Z} \mathrm{C}\right)^{-1} \mathrm{Z}^{\dagger} \end{aligned}\] We recover a different form for the \(\mathrm{R}\) factor, namely \[\begin{aligned} \mathrm{R}_{\mathrm{C}} &:=\sigma^{2}\mathrm{Z}\operatorname{chol}((\mathrm{C}+\sigma^{-2}\mathrm{C}\mathrm{Z}^{\dagger} \mathrm{Z} \mathrm{C})^{-1}). &=\sigma^{2}\mathrm{Z}\operatorname{chol}((\mathrm{C}+\sigma^{-2}\mathrm{Z}^{\dagger} \mathrm{Z} )^{-1}). \end{aligned}\] If \(\mathrm{Z}\) is already centred I think we get \[\begin{aligned} \mathrm{R}_{\mathrm{C}} &=\sigma^{2}\mathrm{Z}\operatorname{chol}((\mathrm{C}+\sigma^{-2}\mathrm{Z}^{\dagger} \mathrm{Z} )^{-1}). \end{aligned}\] There are a lot of alternate Woodbury identities for alternate setups (Ameli and Shadden 2023; Harville 1976, 1977; Henderson and Searle 1981).

3 Determinant

The matrix determinant lemma tells us:

Suppose \(\mathrm{A}\) is an invertible \(n\times n\) matrix and \(\mathrm{U}, \mathrm{V}\) are \(n\times m\) matrices. Then \[ \operatorname{det}\left(\mathrm{A}+\mathrm{U V}^{\boldsymbol{\top}}\right)=\operatorname{det}\left(\mathrm{I}_{\mathrm{m}}+\mathrm{V}^{\boldsymbol{\top}} \mathrm{A}^{-1} \mathrm{U}\right) \operatorname{det}(\mathrm{A}) . \]

In the special case \(\mathrm{A}=\mathrm{I}_{\mathrm{n}}\) this is the Weinstein-Aronszajn identity. Given additionally an invertible \(m\)-by- \(m\) matrix \(\mathrm{W}\), the relationship can also be expressed as \[ \operatorname{det}\left(\mathrm{A}+\mathrm{U} \mathrm{W} \mathrm{V}^{\boldsymbol{\top}}\right)=\operatorname{det}\left(\mathrm{W}^{-1}+\mathrm{V}^{\top} \mathrm{A}^{-1} \mathrm{U}\right) \operatorname{det}(\mathrm{W}) \operatorname{det}(\mathrm{A}). \]

Clearly, \[ \operatorname{det}\left(\mathrm{V}+\mathrm{Z}\mathrm{Z}^\top\right)=\operatorname{det}\left(\mathrm{I}_{\mathrm{D}}+\mathrm{Z}^\top \mathrm{V}^{-1} \mathrm{Z}\right) \operatorname{det}(\mathrm{V}) \]

4 Products

4.1 Primal

Specifically, \((\mathrm{Y} \mathrm{Y}^{\dagger}+\sigma^2\mathrm{I})(\mathrm{Z} \mathrm{Z}^{\dagger}+\sigma^2\mathrm{I})\). Are low rank products cheap?

\[ \begin{aligned} (\mathrm{Y} \mathrm{Y}^{\dagger}+\sigma^2\mathrm{I})(\mathrm{Z} \mathrm{Z}^{\dagger}+\sigma^2\mathrm{I}) &=\mathrm{Y} \mathrm{Y}^{\dagger} \mathrm{Z} \mathrm{Z}^{\dagger}+\sigma^2\mathrm{Y} \mathrm{Y}^{\dagger}+\sigma^2\mathrm{Z} \mathrm{Z}^{\dagger}+\sigma^4\mathrm{I}\\ &=\mathrm{Y} (\mathrm{Y}^{\dagger} \mathrm{Z} )\mathrm{Z}^{\dagger}+\sigma^2\mathrm{Y} \mathrm{Y}^{\dagger}+\sigma^2\mathrm{Z} \mathrm{Z}^{\dagger}+\sigma^4\mathrm{I} \end{aligned} \] which is still a sum of low-rank approximations. At this point it might be natural to consider a tensor decomposition.

4.2 Inverse

Suppose the low-rank inverse factors of \(\mathrm{Y}\) and \(\mathrm{X}\) are, respectively, \(\mathrm{R}\) and \(\mathrm{C}\). Then we have

\[ \begin{aligned} &(\mathrm{Y} \mathrm{Y}^{\dagger}+\sigma^2\mathrm{I})^{-1}(\mathrm{Z} \mathrm{Z}^{\dagger}+\sigma^2\mathrm{I})^{-1}\\ &=(\sigma^{-2}\mathrm{I}-\mathrm{R} \mathrm{R}^{\dagger})(\sigma^{-2}\mathrm{I}-\mathrm{C} \mathrm{C}^{\dagger})\\ &=\sigma^{-4}\mathrm{I}-\sigma^{-4}\mathrm{R} \mathrm{R}^{\dagger}-\sigma^{-4}\mathrm{C} \mathrm{C}^{\dagger}+\sigma^{-4}\mathrm{R} (\mathrm{R}^{\dagger}\mathrm{C}) \mathrm{C}^{\dagger}\\ \end{aligned} \]

Once again, cheap to evaluate, but not so obviously nice.

5 Distances

5.1 Frobenius

Suppose we want to measure the Frobenius distance between \(\mathrm{K}_{\mathrm{U},\sigma^2}\) and \(\mathrm{K}_{\mathrm{R},\gamma^2}\). We recall that we might expect things to be nice if they are exactly low rank because, e.g. \[ \begin{aligned} \|\mathrm{U}\mathrm{U}^{\dagger}\|_F^2 =\operatorname{tr}\left(\mathrm{U}\mathrm{U}^{\dagger}\mathrm{U}\mathrm{U}^{\dagger}\right) =\|\mathrm{U}^{\dagger}\mathrm{U}\|_F^2 \end{aligned} \] How does it come out as a distance between two nearly-low-rank matrices? The answer may be found without forming the full matrices. For compactness, we define \(\delta^2=\sigma^2-\gamma^2\). \[ \begin{aligned} &\|\mathrm{U}\mathrm{U}^{\dagger}+\sigma^{2}\mathrm{I}-\mathrm{R}\mathrm{R}^{\dagger}+\gamma^{2}\mathrm{I}\|_F^2\\ &=\left\|\mathrm{U}\mathrm{U}^{\dagger} -\mathrm{R}\mathrm{R}^{\dagger} + \delta^2\mathrm{I}\right\|_{F}^2\\ &=\left\|\mathrm{U}\mathrm{U}^{\dagger}+i\mathrm{R}i\mathrm{R}^{\dagger} + \delta^2\mathrm{I} \right\|_{F}^2\\ &=\left\|\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger} + \delta^2\mathrm{I} \right\|_{F}^2\\ &=\left\langle\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger} + \delta^2\mathrm{I},\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger} + \delta^2\mathrm{I} \right\rangle_{F}\\ &=\left\langle\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger},\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger}\right\rangle_{F} +\left\langle\delta^2\mathrm{I}, \delta^2\mathrm{I} \right\rangle_{F}\\ &\quad+2\operatorname{Re}\left(\left\langle\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger}, \delta^2\mathrm{I} \right\rangle_{F}\right)\\ &=\operatorname{Tr}\left(\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger}\right) +\delta^4D\\ &\quad+2\delta^2\operatorname{Re}\operatorname{Tr}\left(\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}\begin{bmatrix} \mathrm{U} &i\mathrm{R}\end{bmatrix}^{\dagger}\right)\\ &=\operatorname{Tr}\left(\left(\mathrm{U}\mathrm{U}^{\dagger} -\mathrm{R}\mathrm{R}^{\dagger}\right)\left(\mathrm{U}\mathrm{U}^{\dagger} -\mathrm{R}\mathrm{R}^{\dagger}\right)\right) +\delta^4D\\ &\quad+2\delta^2\operatorname{Tr}\left(\mathrm{U}\mathrm{U}^{\dagger} -\mathrm{R}\mathrm{R}^{\dagger}\right)\\ &=\operatorname{Tr}\left(\mathrm{U}\mathrm{U}^{\dagger}\mathrm{U}\mathrm{U}^{\dagger}\right) -2\operatorname{Tr}\left(\mathrm{U}\mathrm{U}^{\dagger}\mathrm{R}\mathrm{R}^{\dagger}\right) + \operatorname{Tr}\left(\mathrm{R}\mathrm{R}^{\dagger}\mathrm{R}\mathrm{R}^{\dagger}\right) +\delta^4D \\ &\quad+2\delta^2\left(\operatorname{Tr}\left(\mathrm{U}\mathrm{U}^{\dagger}\right) -\operatorname{Tr}\left(\mathrm{R}\mathrm{R}^{\dagger}\right)\right)\\ &=\operatorname{Tr}\left(\mathrm{U}^{\dagger}\mathrm{U}\mathrm{U}^{\dagger}\mathrm{U}\right) -2\operatorname{Tr}\left(\mathrm{U}^{\dagger}\mathrm{R}\mathrm{R}^{\dagger}\mathrm{U}\right) + \operatorname{Tr}\left(\mathrm{R}^{\dagger}\mathrm{R}\mathrm{R}^{\dagger}\mathrm{R}\right) +\delta^4D \\ &\quad+2\delta^2\left(\operatorname{Tr}\left(\mathrm{U}^{\dagger}\mathrm{U}\right) -\operatorname{Tr}\left(\mathrm{R}^{\dagger}\mathrm{R}\right)\right)\\ &=\left\|\mathrm{U}^{\dagger}\mathrm{U}\right\|^2_F -2\left\|\mathrm{U}^{\dagger}\mathrm{R}\right\|^2_F + \left\|\mathrm{R}^{\dagger}\mathrm{R}\right\|^2_F +\delta^4D +2\delta^2\left(\left\|\mathrm{U}\right\|^2_F -\left\|\mathrm{R}\right\|^2_F\right) \end{aligned} \]

6 Stochastic approximation

TBC

7 Tools

There is support for some of the simplifications mentioned for pytorch’s linear algebra in cornellius-gp/linear_operator.

8 Incoming

Discuss in terms of perturbation theory? (Rellich 1954; Bamieh 2022)

Bamieh (2022) in particular is compact and clear.

9 References

Akimoto. 2017. Fast Eigen Decomposition for Low-Rank Matrix Approximation.”
Ameli, and Shadden. 2023. A Singular Woodbury and Pseudo-Determinant Matrix Identities and Application to Gaussian Process Regression.” Applied Mathematics and Computation.
Babacan, Luessi, Molina, et al. 2012. Sparse Bayesian Methods for Low-Rank Matrix Estimation.” IEEE Transactions on Signal Processing.
Bach, Francis R. 2013. Sharp Analysis of Low-Rank Kernel Matrix Approximations. In COLT.
Bach, C, Ceglia, Song, et al. 2019. Randomized Low-Rank Approximation Methods for Projection-Based Model Order Reduction of Large Nonlinear Dynamical Problems.” International Journal for Numerical Methods in Engineering.
Bamieh. 2022. A Tutorial on Matrix Perturbation Theory (Using Compact Matrix Notation).”
Barbier, Macris, and Miolane. 2017. The Layered Structure of Tensor Estimation and Its Mutual Information.” arXiv:1709.10368 [Cond-Mat, Physics:math-Ph].
Bauckhage. 2015. K-Means Clustering Is Matrix Factorization.” arXiv:1512.07548 [Stat].
Berry, Browne, Langville, et al. 2007. Algorithms and Applications for Approximate Nonnegative Matrix Factorization.” Computational Statistics & Data Analysis.
Brand. 2002. Incremental Singular Value Decomposition of Uncertain Data with Missing Values.” In Computer Vision — ECCV 2002.
———. 2006. Fast Low-Rank Modifications of the Thin Singular Value Decomposition.” Linear Algebra and Its Applications, Special Issue on Large Scale Linear and Nonlinear Eigenvalue Problems,.
Chen, and Chi. 2018. Harnessing Structures in Big Data via Guaranteed Low-Rank Matrix Estimation: Recent Theory and Fast Algorithms via Convex and Nonconvex Optimization.” IEEE Signal Processing Magazine.
Chi, Lu, and Chen. 2019. Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview.” IEEE Transactions on Signal Processing.
Cichocki, Lee, Oseledets, et al. 2016. Low-Rank Tensor Networks for Dimensionality Reduction and Large-Scale Optimization Problems: Perspectives and Challenges PART 1.” arXiv:1609.00893 [Cs].
Dahleh, Dahleh, and Verghese. 1990. Matrix Perturbations.” In Lectures on Dynamic Systems and Control.
Drineas, and Mahoney. 2005. On the Nyström Method for Approximating a Gram Matrix for Improved Kernel-Based Learning.” Journal of Machine Learning Research.
Fasi, Higham, and Liu. 2023. Computing the Square Root of a Low-Rank Perturbation of the Scaled Identity Matrix.” SIAM Journal on Matrix Analysis and Applications.
Flammia, Gross, Liu, et al. 2012. Quantum Tomography via Compressed Sensing: Error Bounds, Sample Complexity, and Efficient Estimators.” New Journal of Physics.
Ghashami, Liberty, Phillips, et al. 2015. Frequent Directions : Simple and Deterministic Matrix Sketching.” arXiv:1501.01711 [Cs].
Gordon. 2002. Generalized² Linear² Models.” In Proceedings of the 15th International Conference on Neural Information Processing Systems. NIPS’02.
Gross, D. 2011. Recovering Low-Rank Matrices From Few Coefficients in Any Basis.” IEEE Transactions on Information Theory.
Gross, David, Liu, Flammia, et al. 2010. Quantum State Tomography via Compressed Sensing.” Physical Review Letters.
Hager. 1989. Updating the Inverse of a Matrix.” SIAM Review.
Halko, Martinsson, and Tropp. 2010. Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions.”
Harbrecht, Peters, and Schneider. 2012. On the Low-Rank Approximation by the Pivoted Cholesky Decomposition.” Applied Numerical Mathematics, Third Chilean Workshop on Numerical Analysis of Partial Differential Equations (WONAPDE 2010),.
Harville. 1976. Extension of the Gauss-Markov Theorem to Include the Estimation of Random Effects.” The Annals of Statistics.
———. 1977. Maximum Likelihood Approaches to Variance Component Estimation and to Related Problems.” Journal of the American Statistical Association.
Hastie, Mazumder, Lee, et al. 2015. Matrix Completion and Low-Rank SVD via Fast Alternating Least Squares.” In Journal of Machine Learning Research.
Henderson, and Searle. 1981. On Deriving the Inverse of a Sum of Matrices.” SIAM Review.
Hoaglin, and Welsch. 1978. The Hat Matrix in Regression and ANOVA.” The American Statistician.
Kala, and Klaczyński. 1994. Generalized Inverses of a Sum of Matrices.” Sankhyā: The Indian Journal of Statistics, Series A (1961-2002).
Kannan. 2016. Scalable and Distributed Constrained Low Rank Approximations.”
Kannan, Ballard, and Park. 2016. A High-Performance Parallel Algorithm for Nonnegative Matrix Factorization.” In Proceedings of the 21st ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming. PPoPP ’16.
Kim, and Park. 2008. Nonnegative Matrix Factorization Based on Alternating Nonnegativity Constrained Least Squares and Active Set Method.” SIAM Journal on Matrix Analysis and Applications.
Kumar, and Shneider. 2016. Literature Survey on Low Rank Approximation of Matrices.” arXiv:1606.06511 [Cs, Math].
Liberty, Woolfe, Martinsson, et al. 2007. Randomized Algorithms for the Low-Rank Approximation of Matrices.” Proceedings of the National Academy of Sciences.
Lim, and Teh. 2007. “Variational Bayesian Approach to Movie Rating Prediction.” In Proceedings of KDD Cup and Workshop.
Lin. 2016. A Review on Low-Rank Models in Signal and Data Analysis.”
Liu, and Tao. 2015. On the Performance of Manhattan Nonnegative Matrix Factorization.” IEEE Transactions on Neural Networks and Learning Systems.
Lu. 2022. A Rigorous Introduction to Linear Models.”
Mahoney. 2010. Randomized Algorithms for Matrices and Data.
Martinsson. 2016. Randomized Methods for Matrix Computations and Analysis of High Dimensional Data.” arXiv:1607.01649 [Math].
Miller. 1981. On the Inverse of the Sum of Matrices.” Mathematics Magazine.
Minka. 2000. Old and new matrix algebra useful for statistics.
Nakajima, and Sugiyama. 2012. “Theoretical Analysis of Bayesian Matrix Factorization.” Journal of Machine Learning Research.
Nakatsukasa. 2019. The Low-Rank Eigenvalue Problem.”
Nowak, and Litvinenko. 2013. Kriging and Spatial Design Accelerated by Orders of Magnitude: Combining Low-Rank Covariance Approximations with FFT-Techniques.” Mathematical Geosciences.
Petersen, and Pedersen. 2012. The Matrix Cookbook.”
Rellich. 1954. Perturbation theory of eigenvalue problems.
Rokhlin, Szlam, and Tygert. 2009. A Randomized Algorithm for Principal Component Analysis.” SIAM J. Matrix Anal. Appl.
Saad. 2003. Iterative Methods for Sparse Linear Systems: Second Edition.
Salakhutdinov, and Mnih. 2008. Bayesian Probabilistic Matrix Factorization Using Markov Chain Monte Carlo.” In Proceedings of the 25th International Conference on Machine Learning. ICML ’08.
Saul. 2023. A Geometrical Connection Between Sparse and Low-Rank Matrices and Its Application to Manifold Learning.” Transactions on Machine Learning Research.
Seeger, ed. 2004. Low Rank Updates for the Cholesky Decomposition.
Seshadhri, Sharma, Stolman, et al. 2020. The Impossibility of Low-Rank Representations for Triangle-Rich Complex Networks.” Proceedings of the National Academy of Sciences.
Shi, Zheng, and Yang. 2017. Survey on Probabilistic Models of Low-Rank Matrix Factorizations.” Entropy.
Srebro, Rennie, and Jaakkola. 2004. Maximum-Margin Matrix Factorization.” In Advances in Neural Information Processing Systems. NIPS’04.
Sundin. 2016. “Bayesian Methods for Sparse and Low-Rank Matrix Problems.”
Tropp, Yurtsever, Udell, et al. 2016. Randomized Single-View Algorithms for Low-Rank Matrix Approximation.” arXiv:1609.00048 [Cs, Math, Stat].
———, et al. 2017. Practical Sketching Algorithms for Low-Rank Matrix Approximation.” SIAM Journal on Matrix Analysis and Applications.
Tufts, and Kumaresan. 1982. Estimation of Frequencies of Multiple Sinusoids: Making Linear Prediction Perform Like Maximum Likelihood.” Proceedings of the IEEE.
Türkmen. 2015. A Review of Nonnegative Matrix Factorization Methods for Clustering.” arXiv:1507.03194 [Cs, Stat].
Udell, and Townsend. 2019. Why Are Big Data Matrices Approximately Low Rank? SIAM Journal on Mathematics of Data Science.
Wilkinson, Andersen, Reiss, et al. 2019. End-to-End Probabilistic Inference for Nonstationary Audio Analysis.” arXiv:1901.11436 [Cs, Eess, Stat].
Woodruff. 2014. Sketching as a Tool for Numerical Linear Algebra. Foundations and Trends in Theoretical Computer Science 1.0.
Woolfe, Liberty, Rokhlin, et al. 2008. A Fast Randomized Algorithm for the Approximation of Matrices.” Applied and Computational Harmonic Analysis.
Xinghao Ding, Lihan He, and Carin. 2011. Bayesian Robust Principal Component Analysis.” IEEE Transactions on Image Processing.
Yang, Fang, Duan, et al. 2018. Fast Low-Rank Bayesian Matrix Completion with Hierarchical Gaussian Prior Models.” IEEE Transactions on Signal Processing.
Yin, Gao, and Lin. 2016. Laplacian Regularized Low-Rank Representation and Its Applications.” IEEE Transactions on Pattern Analysis and Machine Intelligence.
Yu, Hsiang-Fu, Hsieh, Si, et al. 2012. Scalable Coordinate Descent Approaches to Parallel Matrix Factorization for Recommender Systems.” In IEEE International Conference of Data Mining.
———, et al. 2014. Parallel Matrix Factorization for Recommender Systems.” Knowledge and Information Systems.
Yu, Chenhan D., March, and Biros. 2017. An \(N \log N\) Parallel Fast Direct Solver for Kernel Matrices.” In arXiv:1701.02324 [Cs].
Zhang, Kai, Liu, Zhang, et al. 2017. Randomization or Condensation?: Linear-Cost Matrix Sketching Via Cascaded Compression Sampling.” In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. KDD ’17.
Zhang, Xiao, Wang, and Gu. 2017. Stochastic Variance-Reduced Gradient Descent for Low-Rank Matrix Recovery from Linear Measurements.” arXiv:1701.00481 [Stat].
Zhou, and Tao. 2011. GoDec: Randomized Low-Rank & Sparse Matrix Decomposition in Noisy Case.” In Proceedings of the 28th International Conference on International Conference on Machine Learning. ICML’11.
———. 2012. Multi-Label Subspace Ensemble.” Journal of Machine Learning Research.