The Optimal Error Bound for the Method of Simultaneous Projections

TL;DR

This paper derives the best possible linear convergence error bound for the simultaneous projection method on linear subspaces in Hilbert spaces, enhancing understanding of its efficiency and relation to other projection methods.

Contribution

It provides the first optimal error bound for the simultaneous projection method, expressed via the Friedrichs number, and compares it with existing bounds for the alternating projection method.

Findings

Established the smallest possible convergence rate estimate.

Expressed the error operator norm in terms of the Friedrichs number.

Connected the results to the alternating projection method and recent dichotomy theorems.

Abstract

In this paper we find the optimal error bound (smallest possible estimate, independent of the starting point) for the linear convergence rate of the simultaneous projection method applied to closed linear subspaces in a real Hilbert space. We achieve this by computing the norm of an error operator which we also express in terms of the Friedrichs number. We compare our estimate with the optimal one provided for the alternating projection method by Kayalar and Weinert (1988). Moreover, we relate our result to the alternating projection formalization of Pierra (1984) in a product space. Finally, we adjust our results to closed affine subspaces and put them in context with recent dichotomy theorems.

Equations

$$\underset{k\to \infty }{\lim }\Vert {\left(P_{M_r}\dots P_{M_1}\right)}^k(x)-P_M(x)\Vert =0.$$

$$\underset{k\to \infty }{\lim }\Vert {\left(\frac{1}{r}\sum _{i=1}^r{P}_{M_i}\right)}^k(x)-P_M(x)\Vert =0.$$

$$\Vert (P_{M_2}{P}_{M_1}{)}^k-P_M\Vert =\cos (M_1,M_2{)}^{2k-1},$$

$$\cos (M_1,M_2):=\sup \{|⟨x_1,x_2⟩|\mid x_i\in M_i\cap (M_1\cap M_2{)}^{\perp },\Vert x_i\Vert \le 1\}\in [0,1]$$

$$\mathbf{C}:=M_1\times \dots \times M_r\text{and}\mathbf{D}:=\{\mathbf{x}=(x,\dots ,x)\mid x\in \mathcal{H}\}$$

$$\Vert \mathbf{x}\Vert =\Vert x\Vert ,(P_{\mathbf{D}}{P}_{\mathbf{C}}{)}^k(\mathbf{x})=\left(T^k(x),\dots ,T^k(x)\right)\text{and}{P}_{\mathbf{C}\cap \mathbf{D}}(\mathbf{x})=(P_M(x),\dots ,P_M(x)),$$

$$\Vert T^k(x)-P_M(x)\Vert$$

$$=\Vert (P_{\mathbf{D}}{P}_{\mathbf{C}}{)}^k(\mathbf{x})-P_{\mathbf{C}\cap \mathbf{D}}(\mathbf{x})\Vert$$

$$\le \cos (\mathbf{C},\mathbf{D}{)}^{2k-1}\Vert \mathbf{x}\Vert$$

$$=\cos (\mathbf{C},\mathbf{D}{)}^{2k-1}\Vert x\Vert .$$

$$\Vert (P_{\mathbf{D}}{P}_{\mathbf{C}}{)}^k(\mathbf{x})-P_{\mathbf{C}\cap \mathbf{D}}(\mathbf{x})\Vert =\Vert (P_{\mathbf{D}}{P}_{\mathbf{C}}{)}^{k-1}(\mathbf{y})-P_{\mathbf{C}\cap \mathbf{D}}(\mathbf{y})\Vert =\Vert T^{k-1}(y)-P_M(y)\Vert .$$

$$T^k-P_M=(T-P_M{)}^k$$

$$\Vert T^k-P_M\Vert \le \Vert T-P_M{\Vert }^k.$$

$$\Vert T^k-P_M\Vert =\Vert T-P_M{\Vert }^k.$$

$$(T-P_M{)}^k=\sum _{l=0}^k(\genfrac{}{}{0ex}{}{k}{l})(-1{)}^l{T}^{k-l}{P}_M^l=T^k+\sum _{l=1}^k\left(\genfrac{}{}{0ex}{}{k}{l}\right)(-1{)}^l{P}_M=T^k-P_M.$$

$$⟨x-P_{F}T(x),z⟩=⟨x-P_{F}T(x),T(z)⟩=⟨T(I-P_F)T(x),z⟩=⟨(I-P_F)T(x),z⟩=0,$$

$$P_F(x)=\underset{k}{\lim }\frac{x+T(x)+\dots +T^{k-1}(x)}{k}=\underset{k}{\lim }\frac{T(x)+T(T(x))\dots +T^{k-1}(T(x))}{k}=P_{F}T(x),$$

$$\cos (M_1,\dots ,M_r):=\sup \left\{\frac{1}{r-1}\frac{{\sum }_{i\ne j}⟨x_i,x_j⟩}{{\sum }_{i=1}^r\Vert x_i{\Vert }^2}\mid x_i\in M_i\cap M^{\perp }\text{ and }\sum _{i=1}^r\Vert x_i{\Vert }^2\ne 0\right\}.$$

$$\Vert {\left(\frac{1}{r}\sum _{i=1}^r{P}_{M_i}\right)}^k-P_M\Vert$$

$$={\left(\frac{r-1}{r}\cos (M_1,\dots ,M_r)+\frac{1}{r}\right)}^k$$

$$=\cos (\mathbf{C},\mathbf{D}{)}^{2k}$$

$$=\Vert P_{\mathbf{D}}{P}_{\mathbf{C}}{P}_{\mathbf{D}}-P_{\mathbf{C}\cap \mathbf{D}}{\Vert }^k$$

$$=\Vert (P_{\mathbf{D}}{P}_{\mathbf{C}}{P}_{\mathbf{D}}{)}^k-P_{\mathbf{C}\cap \mathbf{D}}\Vert .$$

$$\Vert P_{\mathbf{D}}{P}_{\mathbf{C}}{P}_{\mathbf{D}}-P_{\mathbf{C}\cap \mathbf{D}}\Vert$$

$$=\Vert (P_{\mathbf{D}}{P}_{\mathbf{C}}-P_{\mathbf{C}\cap \mathbf{D}})(P_{\mathbf{C}}{P}_{\mathbf{D}}-P_{\mathbf{C}\cap \mathbf{D}})\Vert$$

$$=\Vert (P_{\mathbf{D}}{P}_{\mathbf{C}}-P_{\mathbf{C}\cap \mathbf{D}})(P_{\mathbf{D}}{P}_{\mathbf{C}}-P_{\mathbf{C}\cap \mathbf{D}}{)}^{\ast }\Vert$$

$$=\Vert P_{\mathbf{D}}{P}_{\mathbf{C}}-P_{\mathbf{C}\cap \mathbf{D}}{\Vert }^2$$

$$=\cos (\mathbf{C},\mathbf{D}{)}^2,$$

$$\Vert P_{\mathbf{D}}{P}_{\mathbf{C}}{P}_{\mathbf{D}}-P_{\mathbf{C}\cap \mathbf{D}}\Vert$$

$$=\sup \left\{\Vert P_{\mathbf{D}}{P}_{\mathbf{C}}{P}_{\mathbf{D}}(\mathbf{x})-P_{\mathbf{C}\cap \mathbf{D}}{P}_{\mathbf{D}}(\mathbf{x})\Vert \mid \mathbf{x}\in \mathbf{B}\right\}$$

$$=\sup \left\{\Vert P_{\mathbf{D}}{P}_{\mathbf{C}}(\mathbf{y})-P_{\mathbf{C}\cap \mathbf{D}}(\mathbf{y})\Vert \mid \mathbf{y}\in P_{\mathbf{D}}(\mathbf{B})\right\}$$

$$=\sup \left\{\Vert P_{\mathbf{D}}{P}_{\mathbf{C}}(\mathbf{y})-P_{\mathbf{C}\cap \mathbf{D}}(\mathbf{y})\Vert \mid \mathbf{y}\in \mathbf{D}\cap \mathbf{B}\right\}$$

$$=\sup \left\{\Vert T(y)-P_M(y)\Vert \mid y\in \mathcal{H}\text{ and }\Vert y\Vert \le 1\right\}$$

$$=\Vert T-P_M\Vert .$$

$$\cos (\mathbf{C},\mathbf{D}{)}^2$$

$$=\sup \left\{\frac{|⟨\frac{1}{r}{\sum }_{i=1}^r{x}_i,y⟩{|}^2}{\frac{1}{r}{\sum }_{i=1}^r\Vert x_i{\Vert }^2\Vert y{\Vert }^2}\mid x_i\in M_i\cap M^{\perp },\sum _{i=1}^r\Vert x_i{\Vert }^2\ne 0\text{ and }y\in M^{\perp },y\ne 0\right\}$$

$$=\sup \left\{\frac{|⟨\frac{1}{r}{\sum }_{i=1}^r{x}_i,y⟩{|}^2}{\frac{1}{r}{\sum }_{i=1}^r\Vert x_i{\Vert }^2\Vert y{\Vert }^2}\mid x_i\in M_i\cap M^{\perp },\sum _{i=1}^r\Vert x_i{\Vert }^2\ne 0\text{ and }y\in \mathcal{H},y\ne 0\right\}$$

$$=\sup \left\{\frac{\Vert {\sum }_{i=1}^r{x}_i{\Vert }^2}{r{\sum }_{i=1}^r\Vert x_i{\Vert }^2}\mid x_i\in M_i\cap M^{\perp },\sum _{i=1}^r\Vert x_i{\Vert }^2\ne 0\right\}$$

$$=\frac{r-1}{r}\cos (M_1,\dots ,M_r)+\frac{1}{r},$$

$$\frac{\Vert {\sum }_{i=1}^r{x}_i{\Vert }^2}{r{\sum }_{i=1}^r\Vert x_i{\Vert }^2}$$

Full text

Key words and phrases: Linear rate of convergence, optimal error bound, simultaneous projection method.

2010 Mathematics Subject Classification: 41A25, 41A28, 41A44, 41A65.

1 Introduction

Let $M_{1},\ldots,M_{r}$ be closed and linear subspaces of a real Hilbert space $\mathcal{H}$ and let $M:=\bigcap_{i=1}^{r}M_{i}$. By $P_{C}$ we denote the metric projection onto a nonempty, closed and convex set $C\subseteq\mathcal{H}$. In this paper we consider simultaneous and cyclic projection methods. The following two theorems are well known:

Theorem 1** (von Neumann [24] and Halperin [17]).**

For each $x\in\mathcal{H}$,

[TABLE]

Theorem 2** (Lapidus [19] and Reich [23]).**

For each $x\in\mathcal{H}$,

[TABLE]

Let $T,T^{\infty}\colon\mathcal{H}\rightarrow\mathcal{H}$ be such that $T^{k}(x)$ converges to $T^{\infty}(x)$ for every $x\in\mathcal{H}$. Following [8, 13], we say that $T^{k}$ converges arbitrarily slowly to $T^{\infty}$ if for every sequence $\{a_{k}\}_{k=0}^{\infty}\subseteq(0,\infty)$ satisfying $a_{k}\rightarrow 0$ as $k\rightarrow\infty$, there is $x\in\mathcal{H}$ such that $\|T^{k}(x)-T^{\infty}(x)\|\geq a_{k}$ for every $k=0,1,2,\ldots$. We also recall that $T^{k}$ converges linearly to $T^{\infty}$ if for some $c,f(x)>0$ and $q\in(0,1)$, we have $\|T^{k}(x)-T^{\infty}(x)\|\leq cq^{k}f(x)$ for all $k=0,1,2,\ldots$. In this paper $T$ and $T^{\infty}$ are related to projections onto linear or affine subspaces in which case we use $f(x)=\|x\|$ or $f(x)=\|x-T^{\infty}(0)\|$, respectively. Note that even in the case of closed linear subspaces the convergence in Theorems 1 and 2 does not have to be linear and moreover, it may indeed be arbitrarily slow. To see this, we now quote a relevant dichotomy result.

Theorem 3** (Bauschke, Deutsch and Hundal [8, 13]).**

Let $T:=P_{M_{r}}\ldots P_{M_{1}}$ or $T:=\frac{1}{r}\sum_{i=1}^{r}P_{M_{i}}$. Then exactly one of the following two statements holds:

1. (i)

$\sum_{i=1}^{r}M_{i}^{\perp}$* is closed. Then $T^{k}$ converges linearly to $P_{M}$.* 2. (ii)

$\sum_{i=1}^{r}M_{i}^{\perp}$* is not closed. Then $T^{k}$ converges arbitrarily slowly to $P_{M}$.*

Alternative (ii) of the above theorem has recently been extended in the case of the cyclic projection method. Following [3], we say that $T^{k}$ converges super-polynomially fast to $T^{\infty}$ on a nonempty set $X\subseteq\mathcal{H}$ if $\|T^{k}(x)-T^{\infty}(x)\|/k^{-\alpha}\rightarrow 0$ as $k\rightarrow\infty$ for all $\alpha>0$ and $x\in X$.

Theorem 4** (Badea and Seifert [3]).**

Let $T:=P_{M_{r}}\ldots P_{M_{1}}$. If $\sum_{i=1}^{r}M_{i}^{\perp}$ is not closed, then $T^{k}$ converges super-polynomially fast to $P_{M}$ on some dense linear subspace $X\subseteq\mathcal{H}$.

We remark here that Theorem 4 follows from [3, Theorem 4.3] which was only proved for a complex Hilbert space. In order to see this, one can apply a complexification argument which has kindly been provided to us by Catalin Badea and David Seifert; see the Appendix for more details. For more dichotomy and trichotomy results concerning arbitrarily slow convergence, we refer the interested reader to [2, 14, 15]. Note that using the above theorems, one can easily see that arbitrarily slow as well as super-polynomially fast convergence may only happen in the infinite dimensional case.

A very natural question related to Theorem 3 (i) is the following one: What is the optimal error bound (smallest possible estimate, independent of $x$) such that $\|T^{k}(x)-P_{M}(x)\|\leq cq^{k}\|x\|$? This question can be answered by finding the norm of the error operator $T^{k}-P_{M}$. In the case of the alternating projection method ($r=2$), we have, by Aronszajn [1] (inequality), and Kayalar and Weinert [18] (equality),

[TABLE]

where by

[TABLE]

we denote the cosine of the Friedrichs angle between the subspaces $M_{1}$ and $M_{2}$. In addition, one can show that $\cos(M_{1},M_{2})<1$ if and only if $M_{1}^{\perp}+M_{2}^{\perp}$ is closed; see, for example, [11, Theorem 9.35] and [11, p. 235] for a complete proof and detailed historical notes going back to [6, 10] and Simonič. As far as we are aware, for $r>2$ the exact computation of the error operator norm for both algorithmic operators is still unknown; see, for example, [2, 3, 14, 15, 21, 22]. Even for $r=2$, the norm $\|((P_{M_{1}}+P_{M_{2}})/2)^{k}-P_{M}\|$ seems to be unknown. Note that one could try to find the optimal estimate for the simultaneous projection method by using (3) and the corresponding alternating projection formalization of Pierra [20] in the product space $\mathcal{H}^{r}$. This approach, although very natural, is somewhat misleading and, when applied directly, provides a weaker result than the optimal one; compare Example 5, Theorem 8 and Example 10 below.

The main contribution of this paper is to extend Theorem 3 (i) in the case of the simultaneous projection method by finding the optimal error bound, that is, by computing the exact value of $\|(\frac{1}{r}\sum_{i=1}^{r}P_{M_{i}})^{k}-P_{M}\|$ for any $r\geq 2$; see Theorems 8 and 14 below. For $r=2$ we show that this norm is greater than the norm in (3), which somewhat explains why, in general, the alternating projection method is indeed faster than its simultaneous variant whenever we have linear convergence; see Remark 11.

In addition, we formally extend Theorem 4 for the simultaneous projection method with $T=\frac{1}{r}\sum_{i=1}^{r}P_{M_{i}}$ by using the alternating projection formalization in a product space.

Finally, by using a translation argument, we obtain analogous results in the case of affine subspaces; see Corollary 15.

2 Main result

We begin this section with a simple example showing that a direct application of Pierra’s alternating projection formalization in a product space indeed leads to linear convergence, but the obtained estimate, as we show in Theorem 8 below, is not optimal.

Example 5** (Alternating projection formalization of Pierra ).**

Let $M_{1},\ldots,M_{r}\subseteq\mathcal{H}$ be closed and linear subspaces and $M:=\bigcap_{i=1}^{r}M_{i}$. Moreover, following Pierra [20], we consider the subsets

[TABLE]

of the product space $\mathcal{H}^{r}$ equipped with the scalar product $\langle\mathbf{x},\mathbf{y}\rangle:=\frac{1}{r}\sum_{i=1}^{r}\langle x_{i},y_{i}\rangle$, where $\mathbf{x}=(x_{1},\ldots,x_{r})$, $\mathbf{y}=(y_{1},\ldots,y_{r})$, and $x_{i},y_{i}\in\mathcal{H}$, $i=1,\ldots,r$. We recall [12, Fact 3.2, Lemma 3.3] that for any $\mathbf{x}=(x,\ldots,x)\in\mathbf{D}$, we have

[TABLE]

where $T:=\frac{1}{r}\sum_{i=1}^{r}P_{M_{i}}$. Moreover, by (3), $\|(P_{\mathbf{D}}P_{\mathbf{C}})^{k}-P_{\mathbf{C}\cap\mathbf{D}}\|=\cos(\mathbf{C},\mathbf{D})^{2k-1}.$ This leads to the following estimate:

[TABLE]

Consequently, if $(P_{\mathbf{D}}P_{\mathbf{C}})^{k}$ converges linearly to $P_{\mathbf{C}\cap\mathbf{D}}$, that is, if $\cos(\mathbf{C},\mathbf{D})<1$, then $T^{k}$ converges linearly to $P_{M}$. On the other hand, $\mathbf{y}=P_{\mathbf{D}}P_{\mathbf{C}}\mathbf{x}\in\mathbf{D}$ for every $\mathbf{x}\in\mathcal{H}^{r}$ and we have

[TABLE]

Thus $(P_{\mathbf{D}}P_{\mathbf{C}})^{k}$ converges linearly to $P_{\mathbf{C}\cap\mathbf{D}}$ whenever $T^{k}$ converges linearly to $P_{M}$.

We now prove the following general lemma. A closely related result can be found in [5, Theorem 2.18].

Lemma 6**.**

Let $\mathcal{H}$ be a real Hilbert space, $T\in B(\mathcal{H})$ and let $M\subseteq F=\operatorname*{Fix}T$ be a nonempty, closed and linear subspace. Assume that $P_{M}=P_{M}T$ which holds, for example, if $P_{F}=P_{F}T$, or $T$ is self-adjoint, or $\|T\|\leq 1$. Then

[TABLE]

and therefore

[TABLE]

If, in addition, $T$ is normal, that is, $T^{*}T=TT^{*}$, then $T-P_{M}$ is normal too and consequently,

[TABLE]

• Proof.

Note that by assumption, $P_{M}=TP_{M}=P_{M}T$ and we can apply the binomial theorem to obtain

[TABLE]

Thus (10) follows. The operator $P_{M}$ is self-adjoint and hence $T-P_{M}$ is normal. We recall that for any normal $N\in B(\mathcal{H})$, $\|N^{k}\|=\|N\|^{k}$; see, for example, [11, Lemma 8.32]. Thus equality (11) follows from (9).

We now show that $P_{M}=P_{M}T$ follows from $P_{F}=P_{F}T$. Observe that $F$ is a closed linear subspace. Indeed, due to the continuity of $F$, for every $F\ni x^{k}\rightarrow x$, we get $x=\lim x^{k}=\lim T(x^{k})=T(x)$. Consequently, since $M\subseteq F$ are both closed linear subspaces, we have $P_{M}=P_{M}P_{F}$; see [11, Lemma 9.2]. This implies that $P_{M}T=P_{M}P_{F}T=P_{M}P_{F}=P_{M}$.

In the next step we show that $P_{F}=P_{F}T$ holds for any self-adjoint $T$. To this end, we recall that by the characterization of the orthogonal projection [11, Theorem 4.9], $y=P_{F}(x)$ if an only if $y\in F$ and $\langle x-y,z\rangle=0$ for every $z\in F$. Now note that $P_{F}T(x)\in F$ and moreover,

[TABLE]

which completes this part of the proof.

Finally, we show that when $\|T\|\leq 1$, then the identity $P_{F}=P_{F}T$ also holds. In this case $\|(I+T+\ldots+T^{k-1})/k\|\leq 1$ and $\|T^{k}(x)/k\|\leq\frac{1}{k}\rightarrow 0$. Consequently, by the mean ergodic theorem [16, Corollary VIII.5.4], we have

[TABLE]

which completes the proof. $\blacksquare$

Before formulating our next result, following [2, Definition 3.2], we recall the following generalization of the cosine of the Friedrichs angle for more than two subspaces.

Definition 7**.**

Let $M_{1},\ldots,M_{r}\subseteq\mathcal{H}$ be closed linear subspaces and let $M:=\bigcap_{i}M_{i}$. The Friedrichs number is defined by

[TABLE]

The above definition coincides in the case of $r=2$ with (4); see [2, Lemma 3.1]. Moreover, $\cos(M_{1},\ldots,M_{r})\in[0,1]$. Thus we can indeed extend the notion of the Friedrichs angle $\theta\in[0,\pi/2]$ with the implicit definition $\cos(\theta)=\cos(M_{1},\ldots,M_{r})$.

Theorem 8** (Exact norm value).**

Let $M_{1},\ldots,M_{r}\subseteq\mathcal{H}$ be closed linear subspaces and let $M:=\bigcap_{i}M_{i}$. Moreover, let $\mathbf{C},\mathbf{D}\subseteq\mathcal{H}^{r}$ be defined as in (5). Then, for every $k=1,2,\ldots$, we have

[TABLE]

• Proof.

Note that $T:=\frac{1}{r}\sum_{i=1}^{r}P_{M_{i}}$ is self-adjoint, $\|T\|\leq 1$ and $\operatorname*{Fix}T=M$. Thus the first equality follows from Lemma 6. The last equality again follows from Lemma 6, but this time applied to $\mathbf{T}:=P_{\mathbf{D}}P_{\mathbf{C}}P_{\mathbf{D}}$. Furthermore, we see that

[TABLE]

where the second equality follows from $P_{\mathbf{C}\cap\mathbf{D}}=P_{\mathbf{C}\cap\mathbf{D}}P_{\mathbf{C}}=P_{\mathbf{C}\cap\mathbf{D}}P_{\mathbf{D}}$ and the latter one follows from (3). On the other hand, $P_{\mathbf{D}}(\mathbf{B})=\mathbf{D}\cap\mathbf{B}$, where $\mathbf{B}$ is the unit ball in $\mathcal{H}^{r}$. This, when combined with (6), leads to

[TABLE]

In order to complete the proof we show that $\cos(\mathbf{C},\mathbf{D})^{2}=\frac{r-1}{r}\cos(M_{1},\ldots,M_{r})+\frac{1}{r}$, where we follow the argument from the proof of [2, Proposition 3.6]. Indeed, by the definition of $\mathbf{C}$ and $\mathbf{D}$, we have $\mathbf{x}=(x_{1},\ldots,x_{r})\in\mathbf{C}\cap(\mathbf{C}\cap\mathbf{D})^{\perp}$ if and only if each $x_{i}\in M_{i}\cap M^{\perp}$. Moreover, $\mathbf{y}=(y,\ldots,y)\in\mathbf{D}\cap(\mathbf{C}\cap\mathbf{D})^{\perp}$ if and only if $y\in M^{\perp}$. Consequently,

[TABLE]

where the last equality follows from

[TABLE]

This completes the proof. $\blacksquare$

Remark 9**.**

Observe that, by (Proof.)–(Proof.), we have

[TABLE]

The above equalities also follow from [2, Proposition 3.6 and Proposition 3.7]. However, our proof for the first equality in (21) differs from the one presented in [2].

Example 10** (Example 5 revisited).**

In the setting of Example 5, a direct application of Pierra’s formalization in a product space leads to an estimate which, in view of Theorem 8, is not the optimal one. The remedy to this problem is to consider $P_{\mathbf{D}}P_{\mathbf{C}}P_{\mathbf{D}}$ instead of $P_{\mathbf{D}}P_{\mathbf{C}}$. Indeed, for any $\mathbf{x}=(x,\ldots,x)\in\mathbf{D}$, we have $(P_{\mathbf{D}}P_{\mathbf{C}}P_{\mathbf{D}})^{k}(\mathbf{x})=(P_{\mathbf{D}}P_{\mathbf{C}})^{k}(\mathbf{x})$ and consequently,

[TABLE]

Although the above inequality recovers the optimal error bound from Theorem 8, it does not explain why this estimate is optimal.

Remark 11** (Two subspaces).**

Let $M_{1},M_{2}\subseteq\mathcal{H}$ be closed linear subspaces and let $M:=M_{1}\cap M_{2}$. By (3) and Theorem 8,

[TABLE]

where the inequalities are strict whenever $\cos(M_{1},M_{2})<1$. This somehow explains why, in general, the alternating projection method is indeed faster than its simultaneous variant whenever we have linear convergence. The numerical verification of this observation can be found, for example, in [9, Fig. 1, p. 1071].

Next, we recall the following fact.

Fact 12**.**

Let $M_{1},\ldots,M_{r}\subseteq\mathcal{H}$ be closed linear subspaces and let $M:=\bigcap_{i}M_{i}$. Moreover, let $\mathbf{C}$ and $\mathbf{D}$ be as in Example 5. The following conditions are equivalent:

1. (i)

$\sum_{i=1}^{r}M_{i}^{\perp}$* is closed.* 2. (ii)

$\cos(M_{1},\ldots,M_{r})<1$* (subspaces are not aligned).* 3. (iii)

$\left\|\frac{1}{r}\sum_{i=1}^{r}P_{M_{i}}-P_{M}\right\|<1$. 4. (iv)

$\|P_{\mathbf{D}}P_{\mathbf{C}}-P_{\mathbf{C}\cap\mathbf{D}}\|<1$. 5. (v)

$\cos(\mathbf{C},\mathbf{D})<1$. 6. (vi)

$\mathbf{C}^{\perp}+\mathbf{D}^{\perp}$* is closed in $\mathcal{H}^{r}$.*

• Proof.

By (3) applied to $\mathbf{C}$ and $\mathbf{D}$, and (21), we have (ii) $\Leftrightarrow$ (iii) $\Leftrightarrow$ (iv) $\Leftrightarrow$ (v). In order to complete the proof it suffices to show that (i) $\Leftrightarrow$ (iii) holds. The equivalence (v) $\Leftrightarrow$ (vi) will follow by using a similar argument, but in the product space $\mathcal{H}^{r}$.

Assume that (i) holds. Then, by Theorem 3 (i), there are $c>0$ and $q\in(0,1)$ such that $\|T^{k}-P_{M}\|\leq cq^{k}$, where $T:=\frac{1}{r}\sum_{i=1}^{r}P_{M_{i}}$. This implies that $\|T^{k}-P_{M}\|<1$ for some integer $k\geq 1$. Since, by Theorem 8, $\|T^{k}-P_{M}\|=\|T-P_{M}\|^{k}$, we conclude that $\|T-P_{M}\|<1$ too.

Now assume that (iii) holds and (i) does not, that is, $\sum_{i=1}^{r}M_{i}^{\perp}$ is not closed. By Theorem 3, $T^{k}$ converges arbitrarily slowly to $P_{M}$. This is in contradiction with assumption (iii), in view of which $T^{k}$ converges linearly to $P_{M}$. This completes the proof. $\blacksquare$

Remark 13**.**

The equivalence (i) $\Leftrightarrow$ (ii) from Fact 12 can be found in [2, Theorem 4.1, (3) $\Leftrightarrow$ (11)]. The argument follows from the fact that $\cos(M_{1},\ldots,M_{r})<1$ if and only if the family $\{M_{1},\ldots,M_{r}\}$ is linearly regular [2, Proposition 3.9], which, by [7, Theorem 5.19], is equivalent to (i). As we have already mentioned in the Introduction, for the case of $r=2$, an independent proof can be found in [11, Theorem 9.35].

We can now extend Theorems 3 and 4 in the case of the simultaneous projection method.

Theorem 14** (Dichotomy with optimal error bound).**

Let $M_{1},\ldots,M_{r}\subseteq\mathcal{H}$ be closed linear subspaces, $M:=\bigcap_{i}M_{i}$ and let $T:=\frac{1}{r}\sum_{i=1}^{r}P_{M_{i}}$. Then exactly one of the following two statements holds:

1. (i)

$\sum_{i=1}^{r}M_{i}^{\perp}$* is closed. Then $T^{k}$ converges linearly to $P_{M}$ and*

[TABLE]

is the smallest possible number, independent of $x$, in the set of all $q\in(0,1)$ such that

[TABLE]

for all $x\in\mathcal{H}$. 2. (ii)

$\sum_{i=1}^{r}M_{i}^{\perp}$* is not closed. Then $T^{k}$ converges arbitrarily slowly to $P_{M}$. Moreover, there is a dense linear subspace $X\subseteq\mathcal{H}$ on which $T^{k}$ converges super-polynomially fast to $P_{M}$.*

• Proof.

If $\sum_{i=1}^{r}M_{i}^{\perp}$ is closed, then linear convergence and the optimal error bound follow from Theorem 8 and Fact 12.

Assume now that $\sum_{i=1}^{r}M_{i}^{\perp}$ is not closed. The arbitrarily slow convergence is an immediate consequence of Theorem 3 (ii). In order to complete the proof, we have to show that the convergence is super-polynomially fast on some dense linear subspace $X\subseteq\mathcal{H}$. To this end, we apply the alternating projection formalization discussed in Example 5.

Indeed, by Fact 12, $\mathbf{C}^{\perp}+\mathbf{D}^{\perp}$ is not closed in $\mathcal{H}^{r}$. Consequently, by Theorem 4, there is a dense linear subspace $\mathbf{X}\subseteq\mathcal{H}^{r}$ on which $(P_{\mathbf{C}}P_{\mathbf{D}})^{k}$ converges super-polynomially fast to $P_{\mathbf{C}\cap\mathbf{D}}$. Note that since $P_{\mathbf{D}}$ is nonexpansive, for each $\mathbf{x}\in\mathcal{H}^{r}$, we have

[TABLE]

Consequently, for every $\mathbf{x}\in\mathbf{X}$ and $\alpha>0$, we have

[TABLE]

This implies that $(P_{\mathbf{D}}P_{\mathbf{C}})^{k}$ converges super-polynomially fast to $P_{\mathbf{C}\cap\mathbf{D}}$ on $P_{\mathbf{D}}(\mathbf{X})$. On the other hand, since $P_{\mathbf{D}}(\mathbf{X})\subseteq\mathbf{D}$, we can define

[TABLE]

Observe that $X$ is a linear subspace of $\mathcal{H}$ because $P_{\mathbf{D}}(\mathbf{X})$ is a linear subspace of $\mathcal{H}^{r}$, where the latter fact follows from the linearity of $P_{\mathbf{D}}$. Moreover, by (6), for each $x\in X$ and $\alpha>0$, we have

[TABLE]

where $\mathbf{x}=(x,\ldots,x)$. Consequently, $T^{k}$ converges super-polynomially fast to $P_{M}$ on $X$. It remains to prove that $X$ is dense in $\mathcal{H}$ or, equivalently, that $P_{\mathbf{D}}(\mathbf{X})$ is dense in $\mathbf{D}$. Note that the second statement follows from the continuity of the metric projection $P_{\mathbf{D}}$ as we now show. Indeed, let $\mathbf{x}\in\mathbf{D}$. Since $\mathbf{X}$ is dense in $\mathcal{H}^{r}$, there is a sequence $\{\mathbf{x}_{k}\}_{k=0}^{\infty}\subseteq\mathbf{X}$ such that $\mathbf{x}_{k}\rightarrow\mathbf{x}$ and the above-mentioned continuity yields $P_{\mathbf{D}}(\mathbf{x}_{k})\rightarrow P_{\mathbf{D}}(\mathbf{x})=\mathbf{x}$. This completes the proof. $\blacksquare$

Corollary 15** (Affine subspaces).**

Let $V_{1},\ldots,V_{r}\subseteq\mathcal{H}$ be closed affine subspaces and assume that $V:=\bigcap_{i}V_{i}\neq\emptyset$. Moreover, let $T:=\frac{1}{r}\sum_{i=1}^{r}P_{V_{i}}$. Then exactly one of the following two statements holds:

1. (i)

$\sum_{i=1}^{r}(V_{i}-V_{i})^{\perp}$* is closed. Then $T^{k}$ converges linearly to $P_{V}$ and*

[TABLE]

is the smallest possible number, independent of $x$, in the set of all $q\in(0,1)$ such that

[TABLE]

for all $x\in\mathcal{H}$, where $\cos(V_{1},\ldots,V_{r}):=\cos(V_{1}-V_{1},\ldots,V_{r}-V_{r})$. 2. (ii)

$\sum_{i=1}^{r}(V_{i}-V_{i})^{\perp}$* is not closed. Then $T^{k}$ converges arbitrarily slowly to $P_{V}$. Moreover, there is a dense affine subspace $Y\subseteq\mathcal{H}$ on which $T^{k}$ converges super-polynomially fast to $P_{V}$.*

• Proof.

The proof is based on the translation formula

[TABLE]

which holds true for every closed and convex set $C\subseteq\mathcal{H}$, $x,v\in\mathcal{H}$. Note that for any $x\in\mathcal{H}$ and $v\in V$, we have $V_{i}=M_{i}+v$ and $V=M+v$, where $M_{i}:=V_{i}-V_{i}$ and $M:=V-V$ are closed linear subspaces. This holds, in particular, for $v=P_{V}(0)$ which, when combined with an induction argument, leads to

[TABLE]

If $\sum_{i}M_{i}^{\perp}$ is closed, then, by Theorem 14 (i) and (33), estimate (31) holds with $q$ defined as in (30). If there were another $0<q<\frac{r-1}{r}\cos(V_{1},\ldots,V_{r})+\frac{1}{r}$ for which (31) holds, then, by (33), this would imply that

[TABLE]

which is impossible in view of Theorem 8.

If $\sum_{i}M_{i}^{\perp}$ is not closed, then, by Theorem 14 (ii), $\left(\frac{1}{r}\sum_{i=1}^{r}P_{M_{i}}\right)^{k}$ converges arbitrarily slowly to $P_{M}$. This implies, by (33), that $\left(\frac{1}{r}\sum_{i=1}^{r}P_{V_{i}}\right)^{k}$ also converges arbitrarily slowly to $P_{V}$. Moreover, by Theorem 14 (ii), there is a dense subspace $X\subseteq\mathcal{H}$ on which $\left(\frac{1}{r}\sum_{i=1}^{r}P_{M_{i}}\right)^{k}$ converges super-polynomially fast to $P_{M}$. It is easy to see that, by (33), $\left(\frac{1}{r}\sum_{i=1}^{r}P_{V_{i}}\right)^{k}$ converges super-polynomially fast to $P_{V}$ on $Y:=X+v$, which in its turn is a dense affine subspace of $\mathcal{H}$. $\blacksquare$

Remark 16**.**

(Cyclic projection method) Let $M_{1},\ldots,M_{r}\subseteq\mathcal{H}$ be closed linear subspaces and let $M:=\bigcap_{i}M_{i}$. By [18, Theorem 1], $P_{M_{r}\cap M^{\perp}}\ldots P_{M_{1}\cap M^{\perp}}=P_{M_{r}}\ldots P_{M_{1}}-P_{M}$. Moreover, by Lemma 6 applied to $T:=P_{M_{r}}\ldots P_{M_{1}}$, we see that

[TABLE]

which leads to the following (not necessarily optimal) error bound:

[TABLE]

If we assume that $\sum_{i=1}^{r}M_{i}^{\perp}$ is closed, then, by [2, Theorem 4.1], $\|P_{M_{r}\cap M^{\perp}}\ldots P_{M_{1}\cap M^{\perp}}\|<1$ and the convergence is indeed linear. The above estimate can be found, for example, in [4, Remark 5.5.3], [14, Lemma 11.58] and [15, Lemma 9.2]. Note that for closed affine subspaces $V_{1},\ldots,V_{r}$ with nonempty intersection $V:=\bigcap_{i}V_{i}$ we can, similarly to (33), derive the equality

[TABLE]

for every $v\in V$. This leads to the following error bound:

[TABLE]

which yields linear convergence whenever $\sum_{i=1}^{r}(V_{i}-V_{i})^{\perp}$ is closed. The above estimate can be found, for example, in [13, Theorem 6.6].

Appendix

In this section we present an argument showing that Theorem 4 which is stated here for a real Hilbert space indeed follows from [3, Theorem 4.3], which is shown to be true for a complex Hilbert space. The proof is based on a hand-written note which has kindly been provided to us by Catalin Badea and David Seifert.

• Proof.

Let $\mathcal{H}_{\mathbb{C}}=\mathcal{H}\times\mathcal{H}$ be a complexification of the real Hilbert space $\mathcal{H}$ equipped with the scalar product $\langle\cdot,\cdot\rangle_{\mathbb{C}}\colon\mathcal{H}_{\mathbb{C}}\times\mathcal{H}_{\mathbb{C}}\rightarrow\mathbb{C}$ defined by

[TABLE]

where the induced norm satisfies $\|x+iy\|^{2}_{\mathbb{C}}=\|x\|^{2}+\|y\|^{2}$. Consider $\tilde{M}:=M\times M$ and $\tilde{M}_{j}:=M_{j}\times M_{j}$, $j=1,\ldots,r$, which are closed linear subspaces of $\mathcal{H}_{\mathbb{C}}$. Moreover, define $\tilde{P}$ and $\tilde{P}_{j}$ as the complexification of $P=P_{M}$ and $P_{j}=P_{M_{j}}$, that is, $\tilde{P}(x+iy):=Px+iPy$ and $\tilde{P}_{j}(x+iy):=P_{j}x+iP_{j}y$. Observe that $\tilde{P}$ and $\tilde{P}_{j}$ are linear, idempotent and, in fact, are orthogonal projections onto $\tilde{M}$ and $\tilde{M}_{j}$, respectively. Indeed, for any $a+ib\in\mathbb{C}$ and $x+iy\in\mathcal{H}_{\mathbb{C}}$, by the linearity of $P$, we have

[TABLE]

which shows that $\tilde{P}$ is linear. Since $P=P^{2}$, it easily follows that $\tilde{P}=\tilde{P}^{2}$. Moreover, for each $u+iv\in\tilde{M}$, since $P$ is an orthogonal projection, we get

[TABLE]

which shows that $\tilde{P}$ is an orthogonal projection too. The same argument can be repeated for each $\tilde{P}_{j}$. Finally, observe that

[TABLE]

Indeed, it is easy to see that $\tilde{M}^{\perp}\supseteq M^{\perp}\times M^{\perp}$. Moreover, if $x+iy\in\tilde{M}^{\perp}$ and $m+im\in\tilde{M}$, then, by the definition of $\langle\cdot,\cdot\rangle_{\mathbb{C}}$,

[TABLE]

This implies that $x+y$, $x-y\in M^{\perp}$ and consequently, $x=\frac{1}{2}((x+y)+(x-y))\in M^{\perp}$ and $y=\frac{1}{2}((x+y)-(x-y))\in M^{\perp}$.

By assumption and Fact 12, we have

[TABLE]

We claim that $\cos(\tilde{M}_{1},\ldots,\tilde{M}_{r})=1$. Indeed, for each $\varepsilon>0$, there are $m_{j}\in M_{j}\cap M^{\perp}$ such that $\sum_{j}\|m_{j}\|^{2}\neq 0$ and

[TABLE]

Moreover, by setting $\tilde{m}_{j}:=m_{j}+i0$, we see that $\tilde{m}_{j}\in\tilde{M}_{j}\cap\tilde{M}^{\perp}$, $\langle\tilde{m}_{j},\tilde{m}_{k}\rangle=\langle m_{j},m_{k}\rangle$ and, in particular, $\sum\|\tilde{m}_{j}\|^{2}=\sum\|m_{j}\|^{2}\neq 0$. Consequently, $1-\varepsilon\leq\cos(\tilde{M}_{1},\ldots,\tilde{M}_{r})\leq 1$ holds for each $\varepsilon>0$, which shows that $\cos(\tilde{M}_{1},\ldots,\tilde{M}_{r})=1$, as claimed.

Now consider $\tilde{T}:=\tilde{P}_{r}\ldots\tilde{P}_{1}$, which is, in fact, the complexification of $T$, that is, $\tilde{T}(x+iy)=Tx+iTy$ for each $x+iy\in\mathcal{H}_{\mathbb{C}}$. By applying [3, Theorem 4.3] to $\tilde{M},\tilde{M}_{1},\ldots,\tilde{M}_{r}\subseteq\mathcal{H}_{\mathbb{C}}$ and $\tilde{T}$, we get that there is a dense subspace $\tilde{X}\subseteq\mathcal{H}_{\mathbb{C}}$ on which $\tilde{T}^{k}$ converges super-polynomially fast to $\tilde{P}$. Consequently, for each $x+iy\in\tilde{X}$ and $\alpha>0$, we have

[TABLE]

Consider $X:=\{x\in\mathcal{H}\mid x+iy\in\tilde{X}\text{ for some }y\in\mathcal{H}\}$ and observe that since $\tilde{X}$ is a dense linear subspace of $\mathcal{H}_{\mathbb{C}}$, the space $X$ is a dense linear subspace of $\mathcal{H}$. Moreover, the super-polynomial convergence of $T^{k}$ to $P$ on $X$ follows from (Proof.). $\blacksquare$

Acknowledgements. We would like to thank Catalin Badea and David Seifert for kindly providing us with the complexification argument presented in the Appendix. We are also very grateful to an anonymous referee for pertinent comments and helpful suggestions.

Funding. This research was supported in part by the Israel Science Foundation (Grant 389/12), the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund.