Errata (and improvements) for Optimization Algorithms on Matrix Manifolds

If you find an error that is not in the list below, please let us know about it.

P.-A. Absil, R. Mahony, & R. Sepulchre

• On page 8, replace the 2nd and 3rd sentences by this: "In view of the properties of matrix trace and inversion, one has the invariance property that $f(YN) = f(Y)$ for all invertible $p\times p$ matrices $N$. Each equivalence class $\{YN: N\in\mathbb{R}^{p\times p} \text{ invertible}\}$, which gathers all the $n\times p$ matrices with the same column space as $Y$, contains matrices $YN$ such that $(YN)^T\,B\,YN = I_p$. (Take $N=(Y^TBY)^{-1/2}$.) Thus, without loss of generality, we can restrict the search space to matrices $Y$ such that $Y^TBY=I_p$. Moreover, since $V$ is invertible, we can set $Y=VM$. Since $V^TBV = I_p$, the condition $Y^TBY=I_p$ is equivalent to $M^TM=I_p$."
• page 25, Proposition 3.3.2: the symbol "$x$" is used to mean two different things. To fix this, replace, e.g., "$x$" by "$\bar{x}$" in the 3rd line of the proposition.
• page 34, line -11: also require that $\varphi(x)=0$.
• page 43, line 15: "$T_{\overline{x}}\mathcal{M}$" should be "$T_{\overline{x}}\overline{\mathcal{M}}$".
• page 48, line -3 (last displayed equation of the page): delete the first "$T_{\overline{x}}$".
• page 48, line -1: insert "$=\xi_x$" after "$\mathrm{D}\pi(\bar{x})[\bar{\xi}_\bar{x}]$".
• page 59, two lines below (4.8): replace "$n \times p$" by "$p \times p$".
• page 62, Definition 4.2.1: insert "to $\{x_k\}$" after "gradient-related".
  
• pages 62-63: in Algorithm 1, without further assumptions, $t^A_k$ may not exist. In that case, Algorithm 1 produces a finite sequence. This situation is explicitly dismissed in subsequent results that assume an infinite sequence of iterates. Nevertheless, in a practical implementation of Algorithm 1, care must be taken to ensure that the inner iteration that computes $t^A_k$ terminates. This can be done as follows. In Definition 4.2.2 (Armijo point), insert the assumption that "$\eta$ is a descent direction (i.e., $\langle \operatorname*{grad}\,f(x), \eta \rangle_x < 0$)". In Algorithm 1, as first line in the for loop, insert: "0: If $\operatorname*{grad}\,f(x_k) = 0$ then Break." Still in Algorithm 1, replace line "2:" by this: "2: Pick $\eta_k$ in $T_{x_k}\mathcal M$ by a procedure guaranteeing that (i) $\langle \operatorname*{grad}\,f(x_k),\eta_k \rangle < 0$ and (ii) if the sequence $\left\{\eta_i\right\}_{i=0,1,\ldots}$ is infinite (i.e., the above Break is never reached) then it is gradient-related to $\{x_k\}$ (Definition 4.2.1). (Note that $\eta_k := -\operatorname*{grad}\,f(x_k)$ is such a procedure.)"
  
• page 64: In the proof of Theorem 4.3.1, $\alpha_k$ denotes the Armijo step size $t^A_k$ computed in Algorithm 1.
  
• page 64, line -6: insert "in $\mathcal{K}$" after "for all $k$".
  
• page 65: the "$t$" that appears in (4.15) and on the line above depends on $k$, hence "$t_k$" is a better notation.
  
• page 68, line 13: "$\delta>0$" should be "$\delta\in(0,\epsilon)$".
• page 75, table 4.1: the normal space to $\mathbb R^n$ should rather be $\{0\}$ (seeing $\mathbb R^n$ as its own embedding space).
• page 80, line 4: "(local)" can be deleted.
• page 80, line below (4.32): replace all "$\leq$" signs by "$<$".
• page 86, Algorithm 3, step 2: "$Y^T$" is missing in $-Y(Y^TY)^{-1}Y^TAY$.
• page 92, first line below Figure 5.1: replace "$\mathbb{R}^n\to\mathbb{R}^n$" by "$\mathbb{R}\to\mathbb{R}$".
• page 108, line 5: replace "$T_x\mathcal M$" by "$T_v\mathcal M$"
• page 119, line 9: delete the factor "$2$" in the first equation.
• page 123, line -6: "$\operatorname{span}(V)$" should be "the orthogonal complement of $\operatorname{span}(V)$".
• page 124, two lines below (6.35): insert "$=0$" after "$Y_k^T B_k Z_k$".
• page 130, (6.46): on the right-hand side of the equation, replace "$y$" by "$x$".
• page 143, section 7.3.2, second paragraph, 4th line: insert a minus sign, yielding $\eta_k^N = - (H_k)^{-1} \operatorname*{grad}\,f(x_k)$.
• page 144, Algorithm 10, lines 4, 9, 10: replace "$\Delta$" (but not "$\delta$") by "$\Delta_k$".
• page 147, line 6: replace "$=$" by "$\leq$".
• page 148, line -8: definition of $i_x$: remove "for all $x \in \mathcal{M}$".
• page 150, line 4: replace "$B_k$" by "$H_k$".
• page 152, line -2: replace "minimizing geodesic" by "geodesic in $\mathcal{V}$".
• page 153, second line of the proof of Lemma 7.4.8: replace "$\operatorname*{grad}\, f(v)$" by "$\operatorname*{grad}\, f(x)$".
• page 154, first line of the proof of Lemma 7.4.10: insert "$\setminus\{v\}$" at the end of the sentence.
• page 164, Algorithm 12, lines 4, 9, and 10: replace $\Delta$ by $\Delta^2$, and replace the Frobenius metric by the nonstandard metric (7.55), i.e., insert $(Y^TBY)^{-1}$ in the trace.
• page 171, equation (8.4): replace $x^T(0)$ by $u^T$ (in the middle term).
• page 174, section 8.1.3: replace the two occurences of $\mathcal{N}$ by $\mathcal{M}$.
• page 174: second displayed equation: replace the subscript $R_{\mathcal{Y}}(\eta)$ in the left-hand side by $Y+\overline{\eta}_Y$.
• page 174: at the end of the page, add: provided that the right-hand side is a bona-fide horizontal lift. Since horizontality is guaranteed by construction, it remains to require that $\mathrm{D} \pi (\bar{x}+\bar\eta_{\bar{x}}) [\mathrm{P}^h_{\bar{x}+\bar\eta_{\bar{x}}} \bar{\xi}_{\bar{x}}]$ does not depend on the choice of $\bar{x}$ in the fiber $\pi^{-1}(x)$. Equivalently stated, for all real-valued functions $f$ on the quotient manifold, $\frac{\mathrm{d}}{\mathrm{d}t} f( \bar{x}+\bar\eta_{\bar{x}} + t \mathrm{P}^h_{\bar{x}+\bar\eta_{\bar{x}}} \bar{\xi}_{\bar{x}} )|_{t=0}$ must not depend on the choice of $\bar{x}$ in the fiber $\pi^{-1}(x)$.
• page 179, line -15: "for all $p$" should be "for all $\eta$".
• page 181, line 6: remove the minus sign in the formula of $\alpha_k$. (Observe that $r_k$ is defined as $b-Ax_k$.)
• page 185, line -8: replace "$\langle \xi, (DF (x))^* [F (x)] \rangle$" by "$g(\xi, (DF (x))^* [F (x)])$".
• page 196, line -4: $\mathrm{D} \operatorname{qf}(X)[Z] = Q \rho_{\mathrm{skew}}(Q^TZR^{-1}) + (I-QQ^T)ZR^{-1}$.