| \chapter{Optimal frames inside a spherical cap} |
| \section{The finite-dimensional problem} |
| The geometry has reduced the reachable rays to a small angular region around |
| $u$. Before returning to loops, we solve the resulting frame-design problem |
| exactly. This part uses ordinary Euclidean linear algebra. The appearance of |
| a regular simplex is consistent with classical tight-frame theory |
| \cite{benedetto-fickus}; the proof below supplies the cap constraint and all |
| constants needed here. |
|
|
| Fix $0<\alpha\le\arccos(1/\sqrt d)$ and unit vectors |
| $v_1,\ldots,v_d$ with |
| \begin{equation}\label{eq:cap} |
| \ip u{v_i}\ge\cos\alpha. |
| \end{equation} |
| Let $V=[v_i]$, $S=VV^{\mathsf T}$, and $G_V=V^{\mathsf T}V$. |
| The two positive semidefinite matrices have the same eigenvalues and trace |
| $d$. Write |
| \begin{equation}\label{eq:capblock} |
| S=\begin{pmatrix}a&q^{\mathsf T}\\q&B\end{pmatrix} |
| \quad\text{relative to }\R u\oplus u^\perp. |
| \end{equation} |
| Then |
| \begin{equation}\label{eq:captraces} |
| a=\sum_i\ip u{v_i}^2\ge d\cos^2\alpha,\qquad |
| \tr B=d-a\le d\sin^2\alpha. |
| \end{equation} |
| Put |
| \begin{equation}\label{eq:capc} |
| c_\alpha=\frac d{d-1}\sin^2\alpha. |
| \end{equation} |
|
|
| \section{The least-eigenvalue optimum} |
| \begin{theorem}[Cap E-optimality and rigidity]\label{thm:capE} |
| Under \eqref{eq:cap}, |
| \begin{equation}\label{eq:capE} |
| \lambda_{\min}(G_V)\le c_\alpha. |
| \end{equation} |
| The bound is attained by |
| $v_i=\cos\alpha\,u+\sin\alpha\,z_i$, where the $z_i$ form a regular |
| simplex in $u^\perp$. If $0<\alpha<\arccos(1/\sqrt d)$, every maximizing |
| frame has this form, up to an orthogonal transformation fixing $u$ and a |
| permutation of its columns. |
| \end{theorem} |
| \begin{proof} |
| The trace of $S$ on the $(d-1)$-dimensional subspace $u^\perp$ is at most |
| $d\sin^2\alpha$. Its average Rayleigh quotient is therefore at most |
| $c_\alpha$. The least Rayleigh quotient on the entire space cannot exceed |
| this average. This proves the inequality. |
|
|
| For the displayed simplex frame, the eigenvalues are |
| $d\cos^2\alpha$ and $c_\alpha$ with multiplicity $d-1$. |
| The cap threshold ensures $d\cos^2\alpha\ge c_\alpha$, proving |
| attainment. |
|
|
| For rigidity, equality implies |
| $S-c_\alpha\Id\succeq0$ and |
| $\tr(P_{u^\perp}(S-c_\alpha\Id))=0$. |
| A positive semidefinite matrix with zero trace on a subspace vanishes on that |
| subspace, including its cross block. Hence $B=c_\alpha\Id$ and $q=0$. |
| Equality in \eqref{eq:captraces} forces every vector to be on the cap rim: |
| $\ip u{v_i}=\cos\alpha$. Write |
| $v_i=\cos\alpha\,u+\sin\alpha\,z_i$. The equations $q=0$ and |
| $B=c_\alpha\Id$ imply $\sum_i z_i=0$ and |
| $\sum_i z_iz_i^{\mathsf T}=d\Id/(d-1)$ on $u^\perp$. |
| The Gram matrix of the $z_i$ has kernel containing $\mathbf1$, rank $d-1$, |
| and all nonzero eigenvalues $d/(d-1)$. It is consequently |
| $\frac d{d-1}(\Id-\mathbf1\mathbf1^{\mathsf T}/d)$, which is exactly the |
| regular-simplex Gram matrix. Gram equality gives the stated orthogonal |
| uniqueness. |
| \end{proof} |
|
|
| \section{Determinant and inverse-trace optimality} |
| The same frame optimizes two further criteria. These are useful when total |
| uncertainty volume or average linear reconstruction error is the objective, |
| rather than the worst direction. |
|
|
| \begin{theorem}[Cap D- and A-optimality]\label{thm:capDA} |
| Under the cap assumption, |
| \begin{equation}\label{eq:capD} |
| \det G_V\le d\cos^2\alpha\left( |
| \frac d{d-1}\sin^2\alpha\right)^{d-1}. |
| \end{equation} |
| For invertible $V$, |
| \begin{equation}\label{eq:capA} |
| \tr(G_V^{-1})\ge |
| \frac1{d\cos^2\alpha}+\frac{(d-1)^2}{d\sin^2\alpha}. |
| \end{equation} |
| The simplex-rim frame attains both bounds. In the strict cap regime their |
| maximizers or minimizers have the same rigidity as in Theorem~\ref{thm:capE}. |
| \end{theorem} |
| \begin{proof} |
| If $S$ is singular the determinant statement is immediate. Otherwise the |
| Schur complement gives |
| \[ |
| \det S=a\det(B-qq^{\mathsf T}/a) |
| \le a\left(\frac{d-a}{d-1}\right)^{d-1}. |
| \] |
| The inequality uses the arithmetic--geometric mean of the positive eigenvalues |
| and discards the nonnegative quantity $|q|^2/a$ from the trace. The function |
| on the right is decreasing for $a\ge1$. Since |
| $a\ge d\cos^2\alpha\ge1$, its maximum occurs at the smallest permitted |
| $a$, which gives \eqref{eq:capD}. |
|
|
| For the inverse trace, the block inverse or Schur-complement inequalities give |
| \[ |
| \tr(S^{-1})\ge\frac1a+\tr(B^{-1}) |
| \ge\frac1a+\frac{(d-1)^2}{d-a}. |
| \] |
| The last function is increasing for $a\ge1$. Substituting |
| $a=d\cos^2\alpha$ proves \eqref{eq:capA}. Equality in either argument |
| requires $q=0$, scalar $B$, and all vectors on the cap rim; the rigidity |
| argument of the preceding theorem then applies. |
| \end{proof} |
|
|
| \section{The exterior optimum} |
| The scalar cap radius also controls a different inverse map. On skew matrices, |
| use the isometric identification with $\Lambda^2\R^d$ induced by the |
| orthonormal basis of elementary plane rotations. |
|
|
| \begin{lemma}[Singular values of skew congruence]\label{lem:wedgeSVD} |
| If $V$ has singular values $s_1\ge\cdots\ge s_d\ge0$, then the singular |
| values of $K\mapsto V^{\mathsf T}KV$ on $\so(d)$ are precisely |
| \begin{equation}\label{eq:wedgeSVD} |
| \{s_is_j:1\le i<j\le d\}. |
| \end{equation} |
| \end{lemma} |
| \begin{proof} |
| Take an ordinary singular value decomposition |
| $V=U\diag(s_i)W^{\mathsf T}$. Orthogonal conjugation on either side is an |
| isometry of the skew-matrix space. It is enough to consider diagonal $V$. |
| An elementary skew matrix in the $i,j$ plane is then multiplied by $s_is_j$. |
| These elementary matrices form an orthonormal basis for $\norm\cdot_\wedge$. |
| \end{proof} |
|
|
| \begin{theorem}[Cap-optimal skew sensing]\label{thm:capexterior} |
| For $d\ge3$ and vectors satisfying \eqref{eq:cap}, |
| \begin{equation}\label{eq:capexterior} |
| \sigma_{\min}(\sensor_V)\le c_\alpha. |
| \end{equation} |
| The simplex-rim frame attains the bound. |
| \end{theorem} |
| \begin{proof} |
| Let $\mu_1\ge\cdots\ge\mu_d$ be the eigenvalues of $S$. Since |
| $\mu_1\ge a\ge d\cos^2\alpha$, |
| $\sum_{i=2}^d\mu_i\le d\sin^2\alpha$. |
| The geometric mean of the two smallest of these $d-1$ values is at most their |
| full average. Thus |
| \[ |
| \sigma_{\min}(\sensor_V)=\sqrt{\mu_{d-1}\mu_d} |
| \le\frac1{d-1}\sum_{i=2}^d\mu_i\le c_\alpha. |
| \] |
| The simplex frame has $d-1$ equal small eigenvalues $c_\alpha$ and one |
| larger eigenvalue. Their pair products therefore attain the bound. |
| \end{proof} |
|
|
| The restriction $d\ge3$ is meaningful: two weak transverse directions are |
| needed for the least pair product to have order $\sin^2\alpha$. In dimension |
| two there is only one skew degree of freedom and the product involves one |
| large and one small singular value instead. |
|
|
| \section{A quantitative near-equality statement} |
| The equality configuration is not merely a possible design. Near optimality |
| already forces most of its covariance structure. |
|
|
| \begin{proposition}[Near-optimal cap covariance]\label{prop:nearcap} |
| Suppose \eqref{eq:cap} holds and |
| $\lambda_{\min}(S)\ge(1-\eps)c_\alpha$, with $0\le\eps<1$. |
| Then |
| \begin{align} |
| \sum_i\bigl(\sin^2\alpha-|P_{u^\perp}v_i|^2\bigr) |
| &\le d\eps\sin^2\alpha,\label{eq:near-rim}\\ |
| \norm{B-c_\alpha\Id}_\op |
| &\le\max(1,d-2)\eps c_\alpha,\label{eq:near-tight}\\ |
| |q|^2&\le\bigl(a-(1-\eps)c_\alpha\bigr) |
| (d-1)\eps c_\alpha.\label{eq:near-balance} |
| \end{align} |
| \end{proposition} |
| \begin{proof} |
| The lower spectral bound gives |
| $\tr B\ge(d-1)(1-\eps)c_\alpha$. Compare this with |
| $\tr B\le(d-1)c_\alpha$ to obtain \eqref{eq:near-rim}. Every eigenvalue |
| of $B$ is at least $(1-\eps)c_\alpha$, and the trace bound forces each |
| to be at most $(1+(d-2)\eps)c_\alpha$, proving |
| \eqref{eq:near-tight}. Finally set $m=(1-\eps)c_\alpha$. |
| The positive semidefinite block matrix $S-m\Id$ has lower block |
| $B-m\Id$ with trace at most $(d-1)\eps c_\alpha$. |
| Its two-by-two scalar compressions imply |
| $|q|^2\le(a-m)\norm{B-m\Id}_\op$, and the norm is at most that trace. |
| \end{proof} |
|
|
| These bounds quantify concentration near the rim, near isotropy in the |
| transverse space, and near vanishing of the longitudinal--transverse cross |
| covariance. They do not by themselves give a dimension-free perturbation |
| bound for every individual labeled simplex vertex. Such a stronger conclusion |
| would require an additional alignment argument and a declared norm. |
|
|
