Accuracy of approximate projection to the semidefinite cone

TL;DR

This paper derives gap-independent error bounds for approximate projections onto the semidefinite cone, showing that small spectral gaps do not necessarily impair accuracy, unlike traditional eigenvalue perturbation bounds.

Contribution

It introduces new error bounds for semidefinite cone projections that do not depend on spectral gaps, improving understanding of approximation accuracy.

Findings

Error bounds are gap-independent

Small eigenvalues do not significantly affect projection accuracy

Traditional bounds can be overly pessimistic in this context

Abstract

When a projection of a symmetric or Hermitian matrix to the positive semidefinite cone is computed approximately (or to working precision on a computer), a natural question is to quantify its accuracy. A straightforward bound invoking standard eigenvalue perturbation theory (e.g. Davis-Kahan and Weyl bounds) suggests that the accuracy would be inversely proportional to the spectral gap, implying it can be poor in the presence of small eigenvalues. This work shows that a small gap is not a concern for projection onto the semidefinite cone, by deriving error bounds that are gap-independent.

Figures

Equations

$$A=\left[\begin{array}{cc}{V}_{+}& V_{-}\end{array}\right]\left[\begin{array}{cc}{\Lambda }_{+}& \\ & {\Lambda }_{-}\end{array}\right]{\left[\begin{array}{cc}{V}_{+}& V_{-}\end{array}\right]}^T,$$

$${\mathbf{\Pi }}_{+}(A):=V_{+}{\Lambda }_{+}{V}_{+}^T.$$

$$\Vert \hat{V}\hat{\Lambda }{\hat{V}}^T-V_{+}{\Lambda }_{+}{V}_{+}^T\Vert \le 2\Vert \Delta V{\Lambda }_{+}{V}_{+}^T\Vert +\Vert V_{+}\Delta \Lambda V_{+}^T\Vert +O({\Vert \Delta \Vert }^2)$$

$$\Vert \sin \angle (V_{+},\hat{V})\Vert \le \frac{\Vert R\Vert }{\text{mbox}gap}$$

$$\text{mbox}gap:=\underset{i,j}{\min }\left|{\hat{\Lambda }}_{(i,i)}-{{\Lambda }_{-}}_{(j,j)}\right|.$$

$$\Vert \hat{V}\hat{\Lambda }{\hat{V}}^T-V_{+}{\Lambda }_{+}{V}_{+}^T\Vert \lesssim \frac{2\Vert R\Vert \Vert {\Lambda }_{+}{V}_{+}^T\Vert }{\text{mbox}gap}\le \frac{2\Vert R\Vert \Vert A\Vert }{\text{mbox}gap}.$$

$$\Vert \hat{V}\hat{\Lambda }{\hat{V}}^T-V_{+}{\Lambda }_{+}{V}_{+}^T{\Vert }_F\le \sqrt{2}\Vert R{\Vert }_F,$$

$$\Vert A(V_{+}{V}_{+}^T-\hat{V}{\hat{V}}^T)\Vert ,$$

$${\left[\begin{array}{cc}\hat{V}& {\hat{V}}_{\perp }\end{array}\right]}^{T}A\left[\begin{array}{cc}\hat{V}& {\hat{V}}_{\perp }\end{array}\right]=\left[\begin{array}{cc}\hat{\Lambda }& {\hat{R}}^T\\ \hat{R}& D\end{array}\right]$$

$$\Vert \hat{V}\hat{\Lambda }{\hat{V}}^T-V_{+}{\Lambda }_{+}{V}_{+}^T\Vert =\Vert {\mathbf{\Pi }}_{+}(\hat{A})-{\mathbf{\Pi }}_{+}(A)\Vert .$$

$${\Vert \hat{V}\hat{\Lambda }{\hat{V}}^T-{\mathbf{\Pi }}_{+}(A)\Vert }_F^2\le {\Vert R\Vert }_F^2+{\Vert {\hat{V}}_{\perp }^{T}A\hat{V}\Vert }_F^2+{\Vert D_{+}\Vert }_F^2.$$

$$B:=\hat{V}\hat{\Lambda }{\hat{V}}^T+{\hat{V}}_{\perp }{\mathbf{\Pi }}_{-}({\hat{V}}_{\perp }^{T}A{\hat{V}}_{\perp }){\hat{V}}_{\perp }^T.$$

$${\Vert \hat{V}\hat{\Lambda }{\hat{V}}^T-{\mathbf{\Pi }}_{+}(A)\Vert }_F^2$$

$$\le {\Vert B-A\Vert }_F^2,$$

$${\Vert B-A\Vert }_F^2$$

$$={\Vert \left[\begin{array}{cc}\hat{\Lambda }-{\hat{V}}^{T}A\hat{V}& {\hat{V}}^{T}A{\hat{V}}_{\perp }\\ {\hat{V}}_{\perp }^{T}A\hat{V}& -{\mathbf{\Pi }}_{+}({\hat{V}}_{\perp }^{T}A{\hat{V}}_{\perp })\end{array}\right]\Vert }_F^2,$$

$${\Vert R\Vert }_F^2={\Vert A\hat{V}-\hat{V}\hat{\Lambda }\Vert }_F^2={\Vert \left[\begin{array}{c}{\hat{V}}^T\\ {\hat{V}}_{\perp }^T\end{array}\right](A\hat{V}-\hat{V}\hat{\Lambda })\Vert }_F^2={\Vert \left[\begin{array}{c}{\hat{V}}^{T}A\hat{V}-\hat{\Lambda }\\ {\hat{V}}_{\perp }^{T}A\hat{V}\end{array}\right]\Vert }_F^2.$$

$${\Vert \hat{V}\hat{\Lambda }{\hat{V}}^T-{\mathbf{\Pi }}_{+}(A)\Vert }_F^2\le 2{\Vert R\Vert }_F^2+{\Vert D_{+}\Vert }_F^2.$$

$$\left\|{\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}-{\bm{\Pi}}_{+}(A)}\right\|_{F}^{2}\leq 2\left\|{R}\right\|_{F}^{2}.\$$

$$A=\left[\begin{array}{cc}1& -1\\ -1& 0\end{array}\right],\hat{V}=\left[\begin{array}{c}1\\ 0\end{array}\right],\text{and}\hat{\Lambda }={\hat{V}}^{T}A\hat{V}=[1],$$

$${\Vert \hat{V}\hat{\Lambda }{\hat{V}}^T-{\mathbf{\Pi }}_{+}(A)\Vert }_F^2/{\Vert R\Vert }_F^2\approx 1.2764>1.$$

$${\Vert (A-{\hat{V}}_{-}{\hat{\Lambda }}_{-}{\hat{V}}_{-}^T)-{\mathbf{\Pi }}_{+}(A)\Vert }_F^2={\Vert (A-{\hat{V}}_{-}{\hat{\Lambda }}_{-}{\hat{V}}_{-}^T)-(A-V_{-}{\Lambda }_{-}{V}_{-}^T)\Vert }_F^2$$

$$=$$

$${\Vert (A-{\hat{V}}_{-}{\hat{\Lambda }}_{-}{\hat{V}}_{-}^T)-{\mathbf{\Pi }}_{+}(A)\Vert }_F^2\le {\Vert R\Vert }_F^2+{\Vert {\hat{V}}_{-,\perp }^{T}A{\hat{V}}_{-}\Vert }_F^2+{\Vert D_{-}\Vert }_F^2.$$

$${\Vert (A-{\hat{V}}_{-}{\hat{\Lambda }}_{-}{\hat{V}}_{-}^T)-{\mathbf{\Pi }}_{+}(A)\Vert }_F^2\le 2{\Vert R\Vert }_F^2+{\Vert D_{-}\Vert }_F^2,$$

$$\left\|{(A-\smash{\widehat{V}}_{-}\smash{\widehat{\Lambda}}_{-}\smash{\widehat{V}}_{-}^{T})-{\bm{\Pi}}_{+}(A)}\right\|_{F}^{2}\leq 2\left\|{R}\right\|_{F}^{2}.\$$

$$\Vert A-X\Vert$$

$$\ge \Vert \mathrm{diag}(0,\dots ,0,{\alpha }_{k+1}-{\chi }_{k+1},\dots ,{\alpha }_n-{\chi }_n)\Vert$$

$$\ge \Vert \mathrm{diag}(0,\dots ,0,{\alpha }_{k+1},\dots ,{\alpha }_n)\Vert .$$

$$A=\left[\begin{array}{cc}M& 1\\ 1& 1/M\end{array}\right],B=\left[\begin{array}{cc}M+1& 0\\ 0& -1+1/M\end{array}\right],$$

$$B=\left[\begin{array}{c}{V}_{B_{+}}{V}_{B_{-}}\end{array}\right]\left[\begin{array}{cc}{\Lambda }_{B_{+}}& \\ & {\Lambda }_{B_{-}}\end{array}\right]{\left[\begin{array}{c}{V}_{B_{+}}{V}_{B_{-}}\end{array}\right]}^T,$$

$${\Vert {\mathbf{\Pi }}_{+}(A)-{\mathbf{\Pi }}_{+}(B)\Vert }_2={\Vert \left[\begin{array}{cc}{\tilde{A}}_{11}-{\Lambda }_{B_{+}}& {\tilde{A}}_{12}\\ {\tilde{A}}_{21}& {\tilde{A}}_{22}\end{array}\right]\Vert }_2,$$

Full text

††journal: Linear Algebra and its Applications

1 Introduction

Projection of symmetric or Hermitian matrices to the positive semidefinite cone is a standard operation that arises frequently in scientific computing. A common, practical, example is restoring positive definiteness of partially unknown or corrupted correlation matrices [17] arising in e.g., economics [12], integrated circuit design [20] and wireless communications  [11]. Further, more generic, examples include quasi-newton optimization methods [7, §4.2.2], incomplete matrix factorizations of sparse matrices [5, §15.11] and, finally, first order methods for solving semidefinite problems (SDPs) [2, 19] which, as we proceed to explain, was the motivating example for this work.

The projection operator ${\bm{\Pi}}_{+}$ maps a symmetric matrix to a nearest symmetric positive semidefinite matrix. As such, it belongs to the class of matrix nearness problems, a survey of which can be found in [10]. When a unitarily invariant norm is chosen as a distance metric, the projection operator can be expressed in “closed form”. Indeed, we show in §2.2 that if $A\in\mathbb{S}^{n}$ with eigenvalue decomposition

[TABLE]

where $[V_{+}\;V_{-}]$ is an $n\times n$ orthogonal matrix with $V_{+}$ having $k$ columns and $\Lambda_{+},\Lambda_{-}$ diagonal matrices containing the positive and non-positive eigenvalues respectively, then we can set:

[TABLE]

When ${\bm{\Pi}}_{+}(A)$ is computed in practice, either via the full eigenvalue decomposition or an iterative method such as the (block) Lanczos method, one obtains approximations $\smash{\widehat{\Lambda}}$ ($\tilde{k}\times\tilde{k}$ diagonal), $\smash{\widehat{V}}$ ($n\times\tilde{k}$ orthonormal) to $\Lambda_{+},V_{+}$ such that $\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}\approx V_{+}\Lambda_{+}V_{+}^{T}$. We usually (but not necessarily) take $\smash{\widehat{\Lambda}}=\smash{\widehat{V}}^{T}A\smash{\widehat{V}}$, which holds by the standard Rayleigh-Ritz process [16, Ch. 11]. Note that the sizes $k$ and $\tilde{k}$ are not assumed to agree since the number of positive eigenvalues computed may not be exact, especially in the presence of eigenvalues close to 0.

With a backward stable algorithm, we have $\|A\smash{\widehat{V}}-\smash{\widehat{V}}\smash{\widehat{\Lambda}}\|=O(u)\|A\|$, where $u$ is the unit roundoff. The question we address is: what does this imply about the size of the projection error $\|\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}-V_{+}\Lambda_{+}V_{+}^{T}\|$?

The first observation can arise if we assume that $k=\tilde{k}$ and define $\smash{\widehat{V}}=V_{+}+\Delta V$, $\smash{\widehat{\Lambda}}=\Lambda_{+}+\Delta\Lambda$, resulting in

[TABLE]

where $\left\|{\Delta}\right\|:=\max(\left\|{\Delta V}\right\|,\left\|{\Delta\Lambda}\right\|)$. This suggests bounding the term $\|\Delta V\|$, which usually dominates the $\|\Delta\Lambda\|$ term since eigenvalues of symmetric matrices are always well-conditioned and hence $\|\Delta\Lambda\|=O(u)\|A\|$ with a backward stable method. The term $\Delta V$ is the error in the computed eigenvectors, for which the classical Davis-Kahan $\sin\theta$ theorem [3] shows that

[TABLE]

in any unitarily invariant norm, where $\angle(V_{+},\smash{\widehat{V}})$ is a $k\times k$ diagonal matrix containing the principal angles between $V_{+}$ and $\smash{\widehat{V}}$. Here $R:=A\smash{\widehat{V}}-\smash{\widehat{V}}\smash{\widehat{\Lambda}}$ is the residual and ‘gap’ is the distance between the computed positive eigenvalues $\smash{\widehat{\Lambda}}$ and the exact nonpositive eigenvalues, i.e.

[TABLE]

The bound (4) is essentially sharp (although sometimes improvable [15]), and implies111 $\left\|{\Delta V}\right\|$ also depends on the “small gap” $:=\min_{i\neq j}\left|\smash{\widehat{\Lambda}}_{(i,i)}-{\Lambda}_{(j,j)}\right|$, where \Lambda:=\big{[}\begin{smallmatrix}\Lambda_{+}\\ &\Lambda_{-}\end{smallmatrix}\big{]}, but this is not a practical concern when $\left\|{R}\right\|\leq$ “small gap”. See [15, Theorem 3.1 and Remark 3.1] for a rigorous discussion on the effect of spectral gaps on $\left\|{\Delta V}\right\|$. $\|\Delta V\|\lesssim\frac{\|R\|}{\mbox{gap}}$. Together with (3), we obtain

[TABLE]

The problem here is that (3) suggests that the projection error would be large if gap is small, and we have no control over how small gap can be! Indeed, in many interesting cases $A$ has eigenvalues of small magnitude (both positive and negative). This causes problems in multiple ways—the gap indeed gets small, and since approximations to the small eigenvalues may have the wrong signs, $\smash{\widehat{\Lambda}}$ may not contain the correct number of positive eigenvalues.

The situation is further exacerbated when $\smash{\widehat{\Lambda}}$, $\smash{\widehat{V}}$ are computed not to full working precision, but only to a looser tolerance $\epsilon\gg u$. Then, $\|R\|=O(\epsilon\|A\|)$ instead of $O(u\|A\|)$. Indeed, this work was motivated in the context of devising efficient algorithms for semidefinite optimization with the Alternating Direction Method of Multipliers (ADMM) [8]. ADMM is an iterative method in which every iteration entails a projection to the semidefinite cone. These projections typically dominate the (total) computation time of ADMM, thus an inexact method for their computation is desirable to reduce ADMM’s execution time and scale ADMM to large SDPs. Naturally, the projection error must be quantified and controlled so that ADMM maintains its convergence properties. This can be achieved, for example, when the projection errors are summable, i.e. when the sum of the projection errors over all the ADMM iterations is bounded [6, Theorem 8]. However, since the gap is unknown and cannot be controlled, bounds like (6) are not very useful in such situations.

The purpose of this paper is to show that, fortunately, these problems suggested by (6) are not a concern, i.e., small gaps do not affect the projection accuracy. Specifically, our main result is

[TABLE]

which holds when $\smash{\widehat{\Lambda}}$ has the same size as $\Lambda_{+}$ (otherwise the bound worsens, but only slightly). Unlike (6), the bound (7) is sharp up to a constant smaller than $\sqrt{2}$. Since $\|R\|_{F}=\|A\smash{\widehat{V}}-\smash{\widehat{V}}\smash{\widehat{\Lambda}}\|_{F}$ is easy to compute, (7) provides a practical means to estimate the projection accuracy.

Noting that ${\bm{\Pi}}_{+}(A)=V_{+}\Lambda_{+}V_{+}^{T}=A(V_{+}V_{+}^{T})$, we also treat an alternative measure of the projection accuracy

[TABLE]

and prove similar bounds for this quantity. The two quantities $\|\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}-V_{+}\Lambda_{+}V_{+}^{T}\|$ and (8) are closely related, and we show in Section 3 that they must lie within $\|R\|$ of each other.

Here is an intuitive explanation for the gap-independence (which is easier to see with (8)): while $\|\Delta V\|$ does depend on $1/\mbox{gap}$, large errors in $\Delta V$ lie only in directions of eigenvectors $v_{i}$ of $A$ with small corresponding eigenvalues $\lambda_{i}$. Essentially, $\Delta V$ has $O(\|R\|/|\lambda_{i}|)$ magnitude in the direction of $v_{i}$. Crucially, such errors are suppressed when multiplied by $A$ as in (8), precisely by $\lambda_{i}$. Thus they cancel out to yield the quantity $\|R\|$ in our bounds for both $\|\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}-V_{+}\Lambda_{+}V_{+}^{T}\|_{F}$ and $\|\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}-V_{+}\Lambda_{+}V_{+}^{T}\|$ in (7) and (8), respectively. In what follows we make this intuition precise.

Notation. $\|\cdot\|_{F}$ denotes the Frobenius norm of a matrix, and $\|\cdot\|_{2}$ is the spectral norm (largest singular value). $\|\cdot\|$ denotes a generic norm. We sometimes state (in)equalities that hold for any unitarily invariant norm; these will be stated explicitly. $\mathbb{S}^{n}$ is the set of $n\times n$ Hermitian matrices, and $\mathbb{S}_{+}^{n}$ ($\mathbb{S}_{-}^{n}$) is the set of positive (negative) semidefinite matrices in $\mathbb{S}^{n}$. The projection operator onto the positive (negative) semidefinite cone is denoted ${\bm{\Pi}}_{+}$ (${\bm{\Pi}}_{-})$; the domain of these operator should always be clear from the context. Given an orthonormal matrix $V$, we denote with $V_{\perp}$ some orthonormal matrix that spans the nullspace of $V^{T}$. $\lambda_{\max}(A)$ denotes the largest eigenvalue of a Hermitian matrix $A$ and $A\succ 0$ $(A\succeq 0)$ the positive (semi)definiteness of $A$. For any matrix $B$, $\sigma_{i}(B)$ denotes the $i$th largest singular value. Finally, given a set of vector $\{a_{i}\}$, $i\in\mathcal{I}\subseteq\mathbb{N}$, $[a_{i}]_{i\in\mathcal{I}}$, denotes horizontal concatenation of $a_{\mathcal{I}_{[1]}},a_{\mathcal{I}_{[2]}},\dots$ where $\mathcal{I}_{[i]}$ is the $i-$th smallest element in $\mathcal{I}$.

2 Main result for the projection error

In this section we derive the main result for the projection error $\|\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}-V_{+}\Lambda_{+}V_{+}^{T}\|_{F}$. We begin with a presentation of the main idea used in this Section’s proofs. Assume, for the purposes of this introductory presentation, that we have $\smash{\widehat{\Lambda}}=\smash{\widehat{V}}^{T}A\smash{\widehat{V}}$, as it is common in practice, and that $\smash{\widehat{\Lambda}}\succeq 0$. Then, note that

[TABLE]

where $D:=\smash{\widehat{V}}_{\perp}^{T}A\smash{\widehat{V}}_{\perp}$ and $\left\|{{\smash{\widehat{R}}}}\right\|=\left\|{R}\right\|$ for any unitarily invariant norm (derived in detail in Theorem 2.1:(14)). Thus $\smash{\widehat{A}}:=\begin{bmatrix}\smash{\widehat{V}}&\smash{\widehat{V}}_{\perp}\end{bmatrix}\begin{bmatrix}\smash{\widehat{\Lambda}}&0\\ 0&D\end{bmatrix}\begin{bmatrix}\smash{\widehat{V}}&\smash{\widehat{V}}_{\perp}\end{bmatrix}^{T}$ approximates $A$ with $\left\|{A-\smash{\widehat{A}}}\right\|\leq\sqrt{2}\left\|{R}\right\|$. Now, if we assume that $\smash{\widehat{V}}_{\perp}$ contains ‘most’ of the negative eigenspace of $A$, in that $D\preceq 0$, then we have ${\bm{\Pi}}_{+}(\smash{\widehat{A}})=\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}$. We can then write

[TABLE]

The result (7) then follows immediately from the non-expansiveness of ${\bm{\Pi}}_{+}$ in the Frobenius norm [1, Proposition 4.16], i.e., $\left\|{{\bm{\Pi}}_{+}(\smash{\widehat{A}})-{\bm{\Pi}}_{+}(A)}\right\|_{F}\leq\left\|{\smash{\widehat{A}}-A}\right\|_{F}=\sqrt{2}\left\|{R}\right\|_{F}$.

The following proofs generalize this result for the cases where $\smash{\widehat{\Lambda}}\neq\smash{\widehat{V}}^{T}A\smash{\widehat{V}}$ and/or $D\not\preceq 0$.

Theorem 2.1

Suppose that the matrix $A\in\mathbb{S}^{n}$, $\smash{\widehat{V}}$ is an $n\times k$ orthonormal matrix with $k\leq n$ and $\smash{\widehat{\Lambda}}$ is a $k\times k$ positive semidefinite matrix. Then writing $R=A\smash{\widehat{V}}-\smash{\widehat{V}}\smash{\widehat{\Lambda}}$ and $D_{+}={\bm{\Pi}}_{+}(\smash{\widehat{V}}_{\perp}^{T}A\smash{\widehat{V}}_{\perp})$,

[TABLE]

Proof:

Define

[TABLE]

Then

[TABLE]

where the inequality in the second line follows from the nonexpansiveness of the projection operator in the Frobenius norm [1, Proposition 4.16]. Since this norm is invariant with respect to unitary transformation, we have

[TABLE]

where the term in the lower right hand corner is formed using the identity $(I-{\bm{\Pi}}_{-})={\bm{\Pi}}_{+}$. Considering next the norm of the residual $(A\smash{\widehat{V}}-\smash{\widehat{V}}\smash{\widehat{\Lambda}})$ and applying a unitary transformation again, we have

[TABLE]

The result then follows from combination of (12), (13) and (14). $\square$

Theorem 2.1 makes no assumption about the relationship of the matrix $A$ to the matrix $\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}$. If we further assume that the latter matrix has been constructed from an approximation of ${\bm{\Pi}}_{+}(A)$ based on the Rayleigh-Ritz procedure, then we can go a bit further:

Corollary 2.1

Suppose that $(\smash{\widehat{V}},\smash{\widehat{\Lambda}})$ in Theorem 2.1 satisfy the further relation $\smash{\widehat{\Lambda}}=\smash{\widehat{V}}^{T}A\smash{\widehat{V}}.$ Then

[TABLE]

Proof:

The top block in the rightmost expression in (14) becomes zero by assumption, leaving the relation $\left\|{A\smash{\widehat{V}}-\smash{\widehat{V}}\smash{\widehat{\Lambda}}}\right\|=\left\|{\smash{\widehat{V}}_{\perp}^{T}A\smash{\widehat{V}}}\right\|$ to be applied in (11). $\square$

The term $\left\|{D_{+}}\right\|_{F}$ can be bounded by approximate computation of the largest eigenvalues of $(\smash{\widehat{V}}_{\perp})^{T}A\smash{\widehat{V}}_{\perp}$ by e.g. Lanczos (in which instead of applying the unknown $\smash{\widehat{V}}_{\perp}$, we can apply $I-\smash{\widehat{V}}_{+}\smash{\widehat{V}}_{+}^{T}$). With a stable computation, we expect $\left\|{D_{+}}\right\|$ to be very small. It is identically zero if $\smash{\widehat{V}}_{\perp}$ contains ‘most’ of the negative eigenspace of $A$, in that $\smash{\widehat{V}}_{\perp}^{T}A\smash{\widehat{V}}_{\perp}\preceq 0$:

Corollary 2.2

Suppose that the assumptions of Theorem 2.1 and Corollary 2.1 hold, and in addition $\smash{\widehat{V}}_{\perp}^{T}A\smash{\widehat{V}}_{\perp}\preceq 0$. Then

[TABLE]

Proof:

Obvious since the assumption about $\smash{\widehat{V}}_{\perp}$ zeros the term being projected in (15). $\square$

It is noteworthy how concise the proofs are—perhaps verging on appearing trivial. The key fact enabling this is the nonexpansiveness of the operator ${\bm{\Pi}}_{+}$ (which itself is not trivial to establish [1, Proposition 4.16]), along with the introduction of the auxiliary matrix $B$. We also note that the above proof provides little insight into why a small gap does not harm the bound. In Section 3 we present a first-principle derivation for bounding (8), which demonstrates clearly why the bounds are independent of the gap.

Before concluding this subsection we present an example that establishes sharpness of Corollary 2.2 up to a constant smaller than 2:

Example 2.1

Consider

[TABLE]

in which $R=\begin{bmatrix}0\\ -1\end{bmatrix}$, $\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}=\begin{bmatrix}1&0\\ 0&0\end{bmatrix}$, ${\bm{\Pi}}_{+}(A)=\begin{bmatrix}\sqrt{5}+3&-\sqrt{5}-1\\ -\sqrt{5}-1&2\end{bmatrix}/(2\sqrt{5}),$ and

[TABLE]

2.1 When $A$ is nearly positive definite

In practice, projection onto the semidefinite cone can be done in two ways:

compute the positive eigenpairs $\smash{\widehat{V}},\smash{\widehat{\Lambda}}$ such that $\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}\approx V_{+}\Lambda_{+}V_{+}^{T}$ (which we implicitly assumed in the above arguments), or 2. 2.

compute the negative eigenpairs $\smash{\widehat{V}}_{-},\smash{\widehat{\Lambda}}_{-}$ such that $\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}\approx V_{-}\Lambda_{-}V_{-}^{T}$, and obtain the approximate projection as $A-\smash{\widehat{V}}_{-}\smash{\widehat{\Lambda}}_{-}\smash{\widehat{V}}_{-}^{T}$.

The former approach is conceptually more straightforward and is efficient when $A$ has a small number of positive eigenvalues. By contrast, the latter approach is much more efficient when $A$ is nearly positive definite, with the number of negative eigenvalues being small relative to the matrix size $n$.

We expect the second situation to be equally common if not more, and it is therefore important to derive analogous bounds applicable in case (ii). Fortunately, this is a trivial extension. We note that

[TABLE]

which is the accuracy of $\smash{\widehat{V}}_{-}\smash{\widehat{\Lambda}}_{-}\smash{\widehat{V}}_{-}^{T}$ as an approximate projection of the matrix $-A$ onto the semidefinite cone. We can therefore invoke the above results with $A\leftarrow-A$ to obtain the following.

Corollary 2.3

Suppose that the matrix $A\in\mathbb{S}^{n}$, $\smash{\widehat{V}}_{-}$ is a $n\times k$ orthonormal matrix with $k\leq n$ and $\smash{\widehat{\Lambda}}$ is a $k\times k$ negative semidefinite matrix. Then writing $R=A\smash{\widehat{V}}_{-}-\smash{\widehat{V}}_{-}\smash{\widehat{\Lambda}}_{-}$ and $D_{-}={\bm{\Pi}}_{-}(\smash{\widehat{V}}_{-,\perp}^{T}A\smash{\widehat{V}}_{-,\perp})$,

[TABLE]

If in addition we have $\smash{\widehat{\Lambda}}_{-}=\smash{\widehat{V}}_{-}^{T}A\smash{\widehat{V}}_{-}$, then

[TABLE]

and if furthermore $\smash{\widehat{V}}_{-,\perp}^{T}A\smash{\widehat{V}}_{-,\perp}\succeq 0$, then

[TABLE]

2.2 Extension to other norms

As we have already mentioned in the Introduction, a closest positive definite matrix to $A\in\mathbb{S}^{n}$ can be obtained by (2) for any unitarily invariant norm as a distance metric. This is shown in the following Lemma:

Lemma 2.1

Given $A\in\mathbb{S}^{n}$ with eigendecomposition defined in (1), $V_{+}\Lambda_{+}V_{+}^{T}$ is a solution to $\min_{X\succeq 0}\left\|{A-X}\right\|$ for any unitarily invariant norm.

Proof:

Consider any $X\in\mathbb{S}^{n}_{+}$ and denote with $\alpha_{1}\geq\dots\alpha_{k}\geq 0\geq\alpha_{k+1}\geq\alpha_{n}$ the eigenvalues of $A$ and $\chi_{1}\geq\dots\chi_{n}\geq 0$ those of $X$. Then according to [13, Corollary 7.4.9.3] and following [13, (7.4.9.2)] we have:

[TABLE]

Thus $\left\|{A-X}\right\|\geq\left\|{{\rm diag}(0,\dots,0,\alpha_{k},\dots,\alpha_{n})}\right\|$, with equality obtained in the final inequality by selecting $X={V_{+}\Lambda_{+}V_{+}^{T}}$. $\square$

Lemma 2.1 has already been proven for the Frobenius and the spectral norm in the literature [10], but we are unaware of a proof that considers any unitarily invariant norm.

The projector to the semidefinite cone, i.e. ${\bm{\Pi}}_{+}(A):=\text{argmin}_{X\succeq 0}\left\|{A-X}\right\|$, is not unique in general (though it is for the Frobenius norm). Thus the projection error could be defined as $\min_{X\in{\bm{\Pi}}_{+}(A)}\left\|{\smash{\widetilde{V}}\smash{\widetilde{\Lambda}}\smash{\widetilde{V}}^{T}-X}\right\|_{2}$. Nevertheless, for the rest of this section we consider the upper bound $\left\|{\smash{\widetilde{V}}\smash{\widetilde{\Lambda}}\smash{\widetilde{V}}^{T}-V_{+}\Lambda_{+}V_{+}^{T}}\right\|$ for reasons of simplicity and define ${\bm{\Pi}}_{+}(A)$ according to (2).

Perhaps surprisingly, the proof of Theorem 2.1 does not carry over to other norms, including the spectral norm. Specifically, the nonexpansiveness (12) of ${\bm{\Pi}}_{+}$ does not extend to every unitarily invariant norm; simple computations reveal counterexamples for e.g. the spectral norm $\|A\|_{2}=\sigma_{1}(A)$ and trace norm $\|A\|_{*}=\sum_{i}\sigma_{i}(A)$.

Let us further investigate the spectral norm in particular. The issue is that for $n\geq 2$ there exist $A,B\in\mathbb{S}^{n}$ such that $\left\|{{\bm{\Pi}}_{+}(A)-{\bm{\Pi}}_{+}(B)}\right\|_{2}>\left\|{A-B}\right\|_{2}$. An example is

[TABLE]

which as $M\rightarrow\infty$ gives $\left\|{{\bm{\Pi}}_{+}(A)-{\bm{\Pi}}_{+}(B)}\right\|_{2}/\left\|{A-B}\right\|_{2}\rightarrow\frac{\sqrt{5}+1}{2\sqrt{2}}\approx 1.1441$.

The potential expansiveness of ${\bm{\Pi}}_{+}$ might be explained as follows. Let $[V_{B_{+}}\;V_{B_{-}}]$ be the orthogonal matrix of $B$’s eigenvectors such that

[TABLE]

where $\Lambda_{B_{+}}\!\succ 0,\Lambda_{B_{-}}\!\preceq 0$ are diagonal. Then {\bm{\Pi}}_{+}(B)=[V_{B_{+}}\;V_{B_{-}}]\big{[}\begin{smallmatrix}\Lambda_{B_{+}}\\ &0\end{smallmatrix}\big{]}[V_{B_{+}}\;V_{B_{-}}]^{T}. Supposing $A\succeq 0$, write A=[V_{B_{+}}\;V_{B_{-}}]\biggl{[}{\begin{smallmatrix}\smash{\widetilde{A}}_{11}&\smash{\widetilde{A}}_{12}\\[4.30554pt] \smash{\widetilde{A}}_{21}&\smash{\widetilde{A}}_{22}\end{smallmatrix}}\biggr{]}[V_{B_{+}}\;V_{B_{-}}]^{T}. Thus

[TABLE]

and potential expansiveness of ${\bm{\Pi}}_{+}$ means that this might be larger than

[TABLE]

which is counterintuitive as we clearly have $\|\smash{\widetilde{A}}_{22}\|_{2}\leq\|\smash{\widetilde{A}}_{22}-\Lambda_{B_{-}}\|_{2}$, as $\smash{\widetilde{A}}_{22}$ and $-\Lambda_{B_{-}}$ are both positive semidefinite. This fact is related to a classical result by Davis, Kahan and Weinberger [4] on norm-preserving dilation, which implies that it is possible for the strict inequality

[TABLE]

to hold, even when the matrices are symmetric. For example, consider the case where $\smash{\widetilde{A}}_{22}$ is negligible relative to $-\Lambda_{B_{-}}$. Then the above comparison essentially reduces to that of \left\|{\biggl{[}\begin{smallmatrix}\smash{\widetilde{A}}_{11}-\Lambda_{B_{+}}&\smash{\widetilde{A}}_{12}\\ \smash{\widetilde{A}}_{21}&0\end{smallmatrix}\biggr{]}}\right\|_{2} and \left\|{\biggl{[}\begin{smallmatrix}\smash{\widetilde{A}}_{11}-\Lambda_{B_{+}}&\smash{\widetilde{A}}_{12}\\ \smash{\widetilde{A}}_{21}&-\Lambda_{B_{-}}\end{smallmatrix}\biggr{]}}\right\|_{2}, which is in the form treated by Davis-Kahan-Weinberger.

Note that while it is possible for (21) to hold, we still have

[TABLE]

Similarly, for any $A\succeq 0$ we have

[TABLE]

We conclude that $\left\|{{\bm{\Pi}}_{+}(A)-{\bm{\Pi}}_{+}(B)}\right\|_{2}\leq\sqrt{2}\left\|{{\bm{\Pi}}_{+}(A)-B}\right\|_{2}$ for any $A,B\in\mathbb{S}^{n\times n}$, which shows quasi-nonexpansiveness of $\frac{1}{\sqrt{2}}{\bm{\Pi}}_{+}$ in the spectral norm [1]. Unfortunately, this result is not sufficient to derive a bound like (16) using the techniques of Section 2.

Nevertheless, based on experimental evidence we conjecture that, when $\smash{\widehat{V}}^{T}A\smash{\widehat{V}}\succeq 0$, we have $\left\|{\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}-\Pi_{+}(A)}\right\|_{2}^{2}\leq\delta\left\|{R}\right\|_{2}^{2}$ for a “small” constant $\delta$, perhaps $\delta=2$. Note that the counterexample 2.1 gives

[TABLE]

thus the conjectured constant $\delta$ has to be larger than $1$.

3 Bounding $\|A(V_{+}V_{+}^{T}-\smash{\widehat{V}}\smash{\widehat{V}}^{T})\|_{F}$

We now turn to the alternative measure (8) for the projection accuracy. First note that from $AV_{+}=V_{+}\Lambda_{+}$ and $A\smash{\widehat{V}}=\smash{\widehat{V}}\smash{\widehat{\Lambda}}+R$ we have

[TABLE]

and hence

[TABLE]

Therefore we have

[TABLE]

The two accuracy measures $\|V\Lambda_{+}V^{T}-\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}\|$ and $\|A(V_{+}V_{+}^{T}-\smash{\widehat{V}}\smash{\widehat{V}}^{T})\|$ are therefore at most $\|R\|$ apart; this immediately gives the bound $\|A(V_{+}V_{+}^{T}-\smash{\widehat{V}}\smash{\widehat{V}}^{T})\|_{F}\leq\sqrt{2\left\|{R}\right\|_{F}^{2}+\left\|{D_{+}}\right\|_{F}^{2}}+\left\|{R}\right\|_{F}$ as a corollary of Theorem 2.1.

Here we follow a different argument to directly bound $\|A(V_{+}V_{+}^{T}-\smash{\widehat{V}}\smash{\widehat{V}}^{T})\|$, producing a tighter result. Moreover, while the proof is longer than in Theorem 2.1, it clearly reveals how the gap-independence comes about.

In what follows we assume $\smash{\widehat{\Lambda}}$ is obtained by the Rayleigh-Ritz process, i.e., $\smash{\widehat{\Lambda}}=\smash{\widehat{V}}^{T}A\smash{\widehat{V}}$. Furthermore, we define $[\lambda_{1}\;\dots\;\lambda_{n}]^{T}:=[{\rm diag}(\Lambda_{+})^{T}\;{\rm diag}(\Lambda_{-})^{T}]^{T}$ and $[v_{1}\;\dots v_{n}]:=[V_{+}\;V_{-}]$.

Theorem 3.1

Under the notation and assumptions in Theorem 2.1,

[TABLE]

Proof:

Following the arguments in [9, §2.5.3] we have for any unitarily invariant norm

[TABLE]

Let us examine the size of the $i$th row of the $(1,2)$ block of the rightmost matrix term, i.e. $\|v_{i}^{T}\smash{\widehat{V}}_{\perp}\|_{2}$. This is precisely the sine of the angle between $\smash{\widehat{V}}$ and $v_{i}$. Assume, without loss of generality, that $\smash{\widehat{V}}_{\perp}$ was chosen such that $A\smash{\widehat{V}}_{\perp}=\smash{\widehat{V}}_{\perp}D+\smash{\widehat{R}}$ where D=\big{[}\begin{smallmatrix}D_{+}&\\ &D_{-}\end{smallmatrix}\big{]} and \smash{\widetilde{D}}=\big{[}\begin{smallmatrix}0&\\ &D_{-}\end{smallmatrix}\big{]}, is diagonal and $\|\smash{\widehat{R}}\|=\|R\|$ in any unitarily invariant norm. Then,

[TABLE]

Hence we have

[TABLE]

thus v_{i}^{T}\smash{\widehat{V}}_{\perp}=v_{i}^{T}(\smash{\widehat{V}}_{\perp}\big{[}\begin{smallmatrix}D_{+}&\\ &0\end{smallmatrix}\big{]}+\smash{\widehat{R}})(\lambda_{i}I-\smash{\widetilde{D}})^{-1} and the inverse is guaranteed to exist since $\lambda_{i}$ is necessarily positive. Therefore, using the fact $\smash{\widetilde{D}}\preceq 0$ we obtain

[TABLE]

Since this holds for $i=1,\ldots,k$, it follows that the $i$th row of $\Lambda_{+}V_{+}^{T}\smash{\widehat{V}}_{\perp}$ is bounded by \|v_{i}^{T}(\smash{\widehat{V}}_{\perp}\big{[}\begin{smallmatrix}D_{+}&\\ &0\end{smallmatrix}\big{]}+\smash{\widehat{R}})\|_{2}. Note how the $\Lambda_{+,i,i}=\lambda_{i}$ and $1/\lambda_{i}$ terms neatly cancel out; this is why the result is gap-independent. Together with the bounds for $i=1,\ldots,k$, we obtain

[TABLE]

where we used the fact that $V_{+}$ has orthonormal columns, hence $\|V_{+}^{T}X\|_{F}\leq\|X\|_{F}$ for any matrix $X$.

Similarly, we examine the $i$th row of $V_{-}^{T}\smash{\widehat{V}}$, which appears in the $(2,1)$ block of (24). From $A\smash{\widehat{V}}=\smash{\widehat{V}}\smash{\widehat{\Lambda}}+R$ we have

[TABLE]

Hence we have

[TABLE]

so $v_{k+i}^{T}\smash{\widehat{V}}=v_{k+i}^{T}R(\lambda_{k+i}I-\smash{\widehat{\Lambda}})^{-1}$, giving

[TABLE]

Here we have used the facts that $\smash{\widehat{\Lambda}}\succ 0$ and $\lambda_{k+i}\leq 0$; we assumed $\lambda_{k+i}<0$, which can be done because the terms with $\lambda_{k+i}=0$ do not contribute to the terms in (24). Since this holds for $i=k+1,\ldots,n$, the $(k+i)$th row of (24) is bounded in norm by $\|v_{k+i}^{T}R\|_{2}$. Thus

[TABLE]

Putting  (24), (26) and (27) together, we obtain

[TABLE]

as required. It is easy to see the final expression is upper bounded by $\sqrt{2}(\|R\|_{F}+\|D_{+}\|_{F})$. $\square$

4 Experiments

In this section we demonstrate the results of this paper with a set of experiments. We begin with an experiment that demonstrates the independence of the projection accuracy on the spectral gap. We consider a set of parametric matrices $A\in\mathbb{S}^{500}$ with $20$ eigenvalues geometrically distributed on $[10^{-10},1]$, one at $\epsilon$, where $\epsilon$ is a parameter which is used to control the spectral gap, one at $-\epsilon$ and the rest uniformly distributed on $[-1,0]$. We compute approximate projections by running ARPACK [14] (accessed via MATLAB’s eigs with its default parameters) on $A$ requesting the $25$ largest eigenpairs and returning $\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}$, where $(\smash{\widehat{V}},\smash{\widehat{\Lambda}})$ are the positive Ritz pairs obtained by ARPACK. We let ARPACK iterate until convergence (set to its default value, $10^{-14}$) and we do this for varying values of $\epsilon$. In Figure 1 we plot the resulting projection errors as a function of the spectral gap. We observe that the projection accuracy is not impaired by small spectral gaps, and that the bound of Corollary 2.1 successfully estimates the high accuracy of the projection.

Next, we consider the accuracy of the projection for the iterates produced by ARPACK. We consider a matrix $A\in\mathbb{S}^{500}$ with eigenvalues distributed according to the previous experiment with $\epsilon=10^{-10}$. We compute approximate projections by running ARPACK as before. Figure 2 (left) shows the projection accuracy as a function of ARPACK’s iterations. The upper bound of Corollary 2.1 is compared with the “exact” projection error $\left\|{V_{+}\Lambda_{+}V_{+}^{T}-\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}}\right\|_{F}$ where $V_{+},\Lambda_{+}$ are obtained via a full eigenvalue decomposition. Although our bound eventually approximates the exact error, it exhibits oscillatory behaviour before convergence.

We will show that this oscillatory behaviour is caused by positive Ritz pairs with large residuals. The following lemma provides a criterion for excluding some positive Ritz pairs resulting in improved error bounds, which as we will see, exhibit significantly reduced oscillatory behaviour.

Lemma 4.1

Suppose that $\smash{\widehat{\Lambda}}=diag(\hat{\lambda}_{1},\dots,\hat{\lambda}_{k})\succeq 0$ and $\smash{\widehat{V}}=[\hat{v}_{1}\;\dots\;\hat{v}_{k}]$ are a set of Ritz pairs for some $A\in\mathbb{S}^{n}$. Furthermore, consider any $\smash{\widetilde{V}}=[\hat{v}_{i}]_{i\not\in\mathcal{I}}$ $\smash{\widetilde{\Lambda}}={\rm diag}([\hat{\lambda}_{i}]_{i\not\in\mathcal{I}})$ where $\mathcal{I}\subseteq\{1,\dots,k\}$ with every $i\in\mathcal{I}$ satisfying

[TABLE]

with $r_{i}:=A\hat{v}_{i}-\hat{\lambda}_{i}\hat{v}_{i}$. Then, the bound of Corollary 2.1 for the projection error

[TABLE]

is smaller (i.e. better) than the respective bound of $\left\|{\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}-V_{+}\Lambda_{+}V_{+}^{T}}\right\|_{F}$.

Proof:

We will first prove the case where $\mathcal{I}$ contains a single index and then generalize for the case where $\mathcal{I}$ has multiples elements.

Without loss of generality, assume that $\mathcal{I}=\{1\}$. Define $[\hat{v}_{1}\;\smash{\widehat{V}}_{2}]:=\smash{\widehat{V}}$, $\begin{bmatrix}\hat{\lambda}_{1}&0\\ 0&\smash{\widehat{\Lambda}}_{2}\end{bmatrix}:=\smash{\widehat{\Lambda}}$, and $[r_{1}\;R_{2}]:=R$. Corollary 2.1 gives the following bound for the approximate projection $\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}=\hat{\lambda}_{1}\hat{v}_{1}\hat{v}_{1}^{T}+\smash{\widehat{V}}_{2}\smash{\widehat{\Lambda}}_{2}\smash{\widehat{V}}_{2}^{T}$:

[TABLE]

where $D:=\smash{\widehat{V}}_{\perp}^{T}A\smash{\widehat{V}}_{\perp}$. If we do not include $\hat{\lambda}_{1}\hat{v}_{1}\hat{v}_{1}^{T}$ in our approximate projection, then we obtain the following bound:

[TABLE]

where $\tilde{D}:=[\hat{v}_{1}\;\smash{\widehat{V}}_{\perp}]^{T}A[\hat{v}_{1}\;\smash{\widehat{V}}_{\perp}]$. Note that

[TABLE]

and denote with $\mu_{1}\leq\dots\leq\mu_{k+1}$ the eigenvalues of $D$ and $\tilde{\mu}_{1}\leq\dots\leq\tilde{\mu}_{k}$ those of $\tilde{D}$. Then, using [16, Theorem 10.1.1] on $\tilde{D}=\begin{bmatrix}\hat{\lambda}_{1}&\hat{v}_{1}^{T}A\smash{\widehat{V}}_{\perp}\\ \smash{\widehat{V}}_{\perp}^{T}A\hat{v}_{1}&\smash{\widehat{V}}_{\perp}^{T}A\smash{\widehat{V}}_{\perp}\end{bmatrix}$ we get $\tilde{\mu}_{i}\leq\mu_{i}$ where $i=1,\dots k$. Thus,

[TABLE]

Furthermore, using [16, Theorem 10.3.1] on $\tilde{D}=\begin{bmatrix}\hat{\lambda}_{1}&0\\ 0&D\end{bmatrix}+\begin{bmatrix}0&\hat{v}_{1}^{T}A\smash{\widehat{V}}_{\perp}\\ \smash{\widehat{V}}_{\perp}^{T}A\hat{v}_{1}&0\end{bmatrix}$ we get

[TABLE]

where the last equality holds because $\smash{\widehat{V}}$ and $\smash{\widehat{\Lambda}}$ were obtained by performing the Rayleigh-Ritz process on $A$ (see (14)).

Combining (30), (32) and (33) gives

[TABLE]

This completes the proof for the case where $\mathcal{I}$ is a singleton.

Consider now the case where $\mathcal{I}=\{i_{1},\dots,i_{l}\}$ with $l>1$. According to the proceeding part of this proof, the Ritz pairs $([\hat{\lambda}_{i}]_{i\neq i_{1}},[\hat{v}_{i}]_{i\neq i_{1}})$ will produce an approximate projection with a lower approximation bound (obtained via Corollary 2.1) than those of $\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}$. Likewise, $([\hat{\lambda}_{i}]_{i\neq i_{1},i_{2}},[\hat{v}_{i}]_{i\neq i_{1},i_{2}})$ will produce an approximate projection with a further improved bound if

[TABLE]

where $W$ is an orthonormal matrix spanning the nullspace of $[\hat{v}_{i}]_{i\neq i_{1}}^{T}$. However, (35) is implied by (28) because $\text{span}(W)\subset\text{span}(\smash{\widehat{V}}_{\perp})$ and thus $\lambda_{\max}(W^{T}AW)\leq\lambda_{\max}(\smash{\widehat{V}}_{\perp}^{T}A\smash{\widehat{V}}_{\perp})$ [18, Corollary 4.1]. Since the same argument holds for the rest of the indices contained in $\mathcal{I}$, this concludes the proof. $\square$

Figure 2 (right) aims to demonstrate the usefulness of Lemma 4.1 in reducing the oscillations of Figure 2 (left). It considers the same experiment as Figure 2 (left), but compares the convergence of the “exact” accuracy of $\left\|{V_{+}\Lambda_{+}V_{+}^{T}-\smash{\widetilde{V}}\smash{\widetilde{\Lambda}}\smash{\widetilde{V}}^{T}}\right\|_{F}$ where $(\smash{\widetilde{\Lambda}},\smash{\widetilde{V}})$ consists of the positive Ritz pairs contained in $(\smash{\widehat{\Lambda}},\smash{\widehat{V}})$ for which $(\sqrt{2}-1)\left\|{r_{i}}\right\|_{2}\leq\max\left(\hat{\lambda}_{i},\lambda_{\max}(\smash{\widehat{V}}_{\perp}^{T}A\smash{\widehat{V}}_{\perp})\right)$, with the respective bound obtained by applying Corollary 2.1. We observe a significant reduction in the oscillatory behaviour of our bound as compared to Figure 2 (left). We further include in Figure 2 (right) the convergence of $\left\|{V_{+}\Lambda_{+}V_{+}^{T}-\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}}\right\|_{F}$ for comparison. It is worth noting that, in this experiment, $\left\|{V_{+}\Lambda_{+}V_{+}^{T}-\smash{\widehat{V}}\smash{\widehat{\Lambda}}\smash{\widehat{V}}^{T}}\right\|_{F}$ converges almost monotonically and it is never greater than either of $\left\|{V_{+}\Lambda_{+}V_{+}^{T}-\smash{\widetilde{V}}\smash{\widetilde{\Lambda}}\smash{\widetilde{V}}^{T}}\right\|_{F}$ or our bound.