Goal: Reconstruct the signal \(v\) from \(F^* v\)

If \(F = \begin{bmatrix} f_1 & \cdots & f_n\end{bmatrix}\), the measurement vector is \(F^*v\).

\(\Leftrightarrow FF^* = \mathrm{Id}_{d \times d}\)

Definition.

\(\{f_1,\dots, f_n\}\subset \mathbb{C}^d\) is a Parseval frame if \(\operatorname{Id}_{d\times d}=FF^*=f_1f_1^*+\dots+f_nf_n^*\).

The rows of \(F\) form an orthonormal set in \(\mathbb{C}^n\), so the space of all length-\(n\) Parseval frames in \(\mathbb{C}^d\) is the Stiefel manifold \(\operatorname{St}_d(\mathbb{C}^n)\).

real subvariety of \(\mathbb{C}^{d \times n}\)

An equal-norm Parseval frame (ENP frame) is a Parseval frame \(f_1,\dots , f_n\) with \(\|f_i\|^2=\|f_j\|^2\) for all \(i\) and \(j\).

Theorem [Casazza–Kovačevic, Goyal–Kovačevic–Kelner, Holmes–Paulsen]

ENP frames are optimal for signal reconstruction in the presence of additive white Gaussian noise and erasures.

real condition

real conditions

The spaces of frames satisfying 1, 2, or both are only real subvarieties of \(\mathbb{C}^{d \times n}\).

Proved by Cahill–Mixon–Strawn in 2017.

Generalized Frame Homotopy Conjecture

For given positive-definite, Hermitian \(S\) and \(r_1,\dots , r_n>0\), the space of frames with \(FF^*=S\) and \(\|f_i\|^2=r_i\) is path-connected.

Genericity of Full Spark Frames

With probability 1, a random ENP frame has full spark—meaning every size-\(d\) subset of the \(f_i\) spans \(\mathbb{C}^d\).

frame with fixed \(FF^*\) and fixed \(\|f_1\|^2,\dots, \|f_n\|^2\)

\((S^2,d\theta\wedge dz)\)

\((\mathbb{R}^2,dx \wedge dy) = (\mathbb{C},\frac{i}{2}dz \wedge d\bar{z})\)

\((S^2,\omega)\), where \(\omega_p(u,v) = (u \times v) \cdot p\)

\((\mathbb{R}^2,\omega)\) where \(\omega(u,v) = \langle i u, v \rangle \)

\((\mathbb{C}^n, \frac{i}{2} \sum dz_k \wedge d\overline{z}_k)\)

\((\mathbb{C}^{d \times N}, \omega)\) with \(\omega(X_1,X_2) = -\operatorname{Im} \operatorname{trace}(X_1^* X_2)\).

If \(M\) is compact, this can be normalized to give a probability measure.

Example. \(H: (S^2, d\theta\wedge dz) \to \mathbb{R}\) given by \(H(\theta,z) = z\).

\(dH = dz = \iota_{\frac{\partial}{\partial \theta}}(d\theta\wedge dz)\), so \(X_H = \frac{\partial}{\partial \theta}\).

\(H\) is constant on orbits of \(X_H\):

\(\mathcal{L}_{X_H}(H) = dH(X_H)=\omega(X_H,X_H) = 0\)

\(S^1=U(1)\) acts on \((S^2,d\theta \wedge dz)\) by

For \(r \in \mathbb{R} \simeq \mathfrak{u}(1)\), \(X_r = r\frac{\partial}{\partial \theta}\).

\(= (a,b,c) \times (x,y,z)\)

For \(V_{(a,b,c)} = \begin{bmatrix} 0 & -c & b \\ c & 0 & -a \\ -b & a & 0 \end{bmatrix} \in \mathfrak{so}(3)\),

\(X_{V_{(a,b,c)}}((x,y,z))\)

\(= (bz-cy)\frac{\partial}{\partial x} + (cx - az) \frac{\partial}{\partial y}  +(ay - bx) \frac{\partial}{\partial z}\)

\(X = \frac{\partial}{\partial \theta}\)

\(\mu(\theta,z) = z\)

\(\iota_X\omega = \iota_{\frac{\partial}{\partial \theta}} d\theta \wedge dz = dz \)

\(X_{V_{(a,b,c)}}(x,y,z) = (a,b,c) \times (x,y,z)\)

\((\iota_{X_{V_{(a,b,c)}}}\omega)_{(x,y,z)} = a dx + b dy + c dz \)

\(\mu(x,y,z)(V_{(a,b,c)}) = (x,y,z)\cdot(a,b,c)\)

Theorem [Atiyah, Guillemin–Sternberg] 

Let \((M^{2n},\omega)\) be a compact connected symplectic manifold with a Hamiltonian \(k\)-torus action with momentum map \(\mu: M \to \mathbb{R}^k\). Then

• the nonempty level sets of \(\mu\) are connected;
• the image of \(\mu\) is convex (called the moment polytope);
• the image of \(\mu\) is the convex hull of the images of fixed points of the action.

Theorem [Duistermaat–Heckman]

If \(k=n\), then the pushforward of Liouville measure to the moment polytope is a constant multiple of Lebesgue measure.

The moment map is

since \(d\mu_{\vec{v}}(Y) = \left.\frac{d}{dt}\right|_{t=0} \left(-\frac{1}{2} \langle \vec{v} + tY, \vec{v} + tY\rangle\right) = \langle -\vec{v},Y\rangle\).

The symplectic reduction over \(-\frac{1}{2} \in \mathbb{R}\) is

\(\mathbb{C}^d//_{-\frac{1}{2}} U(1) = \mu^{-1}\left(-\frac{1}{2}\right)\!/U(1) \simeq \mathbb{CP}^{d-1}\).

Parseval frames

\(\mu_{U(d)}^{-1}(\operatorname{Id}_{d \times d})\)

fixed-norm frames

\(\mu_{U(1)^N}^{-1}\left(-\frac{r_1}{2},\dots , -\frac{r_n}{2}\right)\)

\(= \operatorname{St}_d(\mathbb{C}^n)/U(d)\)

\(=\operatorname{Gr}_d(\mathbb{C}^n)\)

Theorem [Atiyah, Guillemin–Sternberg] 

Let \((M^{2n},\omega)\) be a compact connected symplectic manifold with a Hamiltonian \(k\)-torus action with momentum map \(\mu: M \to \mathbb{R}^k\). Then

• the nonempty level sets of \(\mu\) are connected;

Example for ENP frames with \(N=6, d=3\):

Must satisfy \(\ldots \geq \lambda_{i,j} \geq \lambda_{i-1,j} \geq \lambda_{i,j+1} \geq \dots\)

Proposition [with Needham; cf. Flaschka–Millson]

The eigensteps of an ENP frame are coordinates of the moment map associated to a maximal torus action on (an open, dense subset of a quotient of) ENP frame space.

Corollary [using Duistermaat–Heckman]

The eigensteps of random ENP frames are distributed according to Lebesgue measure on the eigenstep polytope.

Proof sketch

The first \(d\) columns of \(F\) fail to be full spark only if

\(f_1f_1^*+\dots+f_df_d^*\)

has a zero eigenvalue; i.e., \(\lambda_{dd}=0\).

This is the fiber over a facet of the eigenstep polytope, and hence has measure zero.

By permutation invariance of the measure, the same is true for any \(d\) columns.

Similarly for any space with fixed \(FF^*\) and \(\|f_1\|^2,\dots,\|f_n\|^2\).