| For a positive-definite $T\in\F^{N\times N}$, let |
| \[ |
| \mathcal T_T=\{\Phi:\Phi\Phi^*=T,\quad |
| \norm{\phi_i}^2=\tr(T)/M\text{ for every }i\}. |
| \] |
| The equal-norm Parseval space is $\mathcal T_{I_N}$. We work in its relative Euclidean topology, including singular points of the frame variety. Generic full spark \cite{FullSpark} does not by itself imply full spark at every local optimum. The following exact opening lemma provides that missing local mechanism under the two simultaneous constraints. |
|
|
| \begin{lemma}[Exact constrained rank opening]\label{enpopening} |
| Let $\Phi\in\mathcal T_T$ and $X=\Phi_J$ have rank below $\min(N,|J|)$. Let $L$ and $R$ be the orthogonal projections onto $\ker X^*$ and $\ker X$. There is a two-sided smooth feasible curve through $\Phi$ such that, for $Y=\Phi'_J(0)$, $LYR\ne0$. |
| \end{lemma} |
| \begin{proof} |
| Over $\C$, choose $a\in J$ with $Re_a\ne0$ and $b\notin J$ with $L\phi_b\ne0$. The latter exists because $T>0$ implies that all columns span the full state space. Set $h=\phi_a^*\phi_b$ and choose a unit phase $\zeta$ with $\operatorname{Re}(\zeta h)=0$. Change only the two columns: |
| \[ |
| \phi_a(t)=\cos t\,\phi_a+\zeta\sin t\,\phi_b, |
| \quad \phi_b(t)=-\overline\zeta\sin t\,\phi_a+\cos t\,\phi_b. |
| \] |
| This is a unitary coefficient change. The cross terms in the squared norms vanish, and the original column norms agree, so both constraints are preserved exactly. Its derivative satisfies $LYR=\zeta L\phi_b e_a^*R\ne0$. |
|
|
| Over $\R$, choose a minimal dependent set $C\subseteq J$ and a nonzero vector $v\in\ker\Phi_C$, extended by zero to $J$. Every coordinate of $v$ on $C$ is nonzero, and $|C|\ge2$ because every column norm is positive. Choose $b\notin J$ with $L\phi_b\ne0$. Write $P=\Phi^T\Phi$ and $g_i=P_{ib}$ for $i\in C$. If $g_a=0$ for some $a\in C$, the ordinary Givens rotation of the orthogonal equal-norm pair $(a,b)$ preserves both constraints and has $LYv=v_aL\phi_b\ne0$. |
|
|
| Otherwise all $g_i$ are nonzero. The graph on $C$ whose edges have $P_{ij}\ne0$ is connected. Indeed its disconnected components would have orthogonal spans, and projecting the circuit relation onto one component would give a proper dependent subset. There is an edge $\{a,c\}$ with $v_a/g_a\ne v_c/g_c$: if not, connectivity gives $v=\alpha g$, impossible since $v\in\ker\Phi_C$ whereas $g=\Phi_C^T\phi_b\in(\ker\Phi_C)^\perp$. |
|
|
| Near the starting matrix the three entries $P_{ab},P_{bc},P_{ca}$ remain nonzero. Define a smooth skew matrix $K(\Phi)$ on this triangle by |
| \[ |
| K_{ab}=1/P_{ab},\quad K_{bc}=1/P_{bc},\quad K_{ca}=1/P_{ca}, |
| \] |
| with reversed entries negative and all others zero. Integrate the smooth local ODE $\Phi'=\Phi K(\Phi)$. Along its two-sided local solution, |
| \[ |
| (\Phi\Phi^T)'=0,\qquad P'=PK-KP,\qquad \operatorname{diag}P'=0. |
| \] |
| At each triangle vertex the two incident products in the diagonal derivative are $1$ and $-1$, and cancel identically. Thus both constraints hold exactly, without assuming that the frame variety is a manifold at the starting point. For the selected derivative, |
| \[ |
| LYv=L\phi_b\left(-\frac{v_a}{g_a}+\frac{v_c}{g_c}\right)\ne0. |
| \] |
| Since $Rv=v$, this proves the real case. |
| \end{proof} |
|
|
| For $k\le N$, write $v_J=\sqrt{\det(\Phi_J^*\Phi_J)}$. For arbitrary $k$, write $r=\min(N,k)$ and use the $r$ relevant singular values, including zero ones. |
| \begin{theorem}[Completion under any smooth task objective]\label{smoothcompletion} |
| Let $G$ be $C^2$ on a neighborhood of $\mathcal T_T$, $\lambda>0$, and $w_J>0$ for all $k$-subsets. For $0<p<2$, every local maximizer on $\mathcal T_T$ of |
| \[ |
| G(\Phi)+\lambda\sum_{|J|=k}w_J\sum_{a=1}^{r}\sigma_a(\Phi_J)^p |
| \] |
| has every $k$-block of rank $r$. For $1\le k\le N$ and $0<q<2$, every local maximizer of |
| \[ |
| G(\Phi)+\lambda\sum_{|J|=k}w_Jv_J(\Phi)^q |
| \] |
| has every $k$-block independent. In particular at $k=N$ such local maxima are full spark. |
| \end{theorem} |
| \begin{proof} |
| Suppose first that a spectral block is deficient, and use Lemma~\ref{enpopening}. In SVD coordinates its positive singular-value block is invertible. Eliminating the off-diagonal blocks with matrices $I+O(t)$ leaves a zero-cluster block $tLYR+O(t^2)$. Since $LYR\ne0$, at least one initially zero singular value is bounded below by $c|t|$ near zero. The sum of the initially positive singular values raised to $p$ is $C^2$ along the curve: the positive spectral cluster stays separated from zero, where the scalar spectral function is smooth. Its symmetric change is $O(t^2)$. The same applies to the positive cluster in every other block; every initially zero cluster contributes a nonnegative amount. Finally $G(\Phi(t))+G(\Phi(-t))-2G(\Phi)=O(t^2)$. Summing the positive weights gives symmetric objective gain at least $c'|t|^p-Ct^2>0$ for small nonzero $t$, contradicting local maximality. |
|
|
| For volume, positive-definiteness of $T$ gives full total rank. From any dependent set of at most $k$ columns choose a minimal circuit, and extend it by columns independent modulo its span to a $k$-block of rank $k-1$. Apply Lemma~\ref{enpopening} to this block. In SVD coordinates its $k$-fold exterior product has a nonzero derivative: it is the product of the $k-1$ positive singular values times the nonzero transverse image of the single zero right-singular direction. Its $q$-volume therefore contributes at least $c|t|^q$. Every initially nonzero volume has a smooth powered value and quadratic symmetric change. Other initially zero volumes contribute nonnegatively, as does their increment from zero. Combining with the $C^2$ task term again gives $c'|t|^q-Ct^2>0$, the required contradiction. |
| \end{proof} |
|
|
| Equivalently, any local minimizer of a $C^2$ task loss minus an arbitrarily small positive multiple of the stated bonus has the rank property. The exact cusp, the feasible frame domain, and the positivity of all subset weights are essential to the claim. Smooth approximations of the cusp and additional architectural restrictions require separate arguments. The volume statement here uses a positive-definite frame operator; with arbitrary smooth task terms it must not be transferred to unrestricted fixed-norm families of potentially insufficient total rank. |
|
|
| \begin{proposition}[Sharp quadratic boundary]\label{boundary} |
| At exponent two, both unweighted objectives are constant on equal-norm Parseval frames. For $M=dN$, integer $d\ge2$, and $2\le k\le N$, the repeated-coordinate frame |
| \[ |
| \Phi=d^{-1/2}[I_N\ \cdots\ I_N] |
| \] |
| is a singular-block global maximizer of the unweighted volume objective for every $q>2$. |
| \end{proposition} |
| \begin{proof} |
| Squared singular energy sums to $N\binom{M-1}{k-1}$. Cauchy--Binet gives $\sum_J v_J^2=\binom Nk$. If $c=d^{-1/2}$, Hadamard gives $v_J\le c^k$, hence |
| \[ |
| \sum_Jv_J^q\le c^{k(q-2)}\binom Nk. |
| \] |
| All volumes in the repeated-coordinate frame are $0$ or $c^k$, so equality holds, despite repeated columns and dependent $k$-blocks. At $p=2$ the constant spectral objective already admits deficient global maxima. No additional spectral $p>2$ assertion on the simultaneous-constraint domain is needed for sharpness of the strict range $p<2$. |
| \end{proof} |
|
|
