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 \(\operatorname{St}_d(\mathbb{C}^n)\).

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 white noise and erasures.

choose based on noise model

detector power

Questions:

• [Admissibility] Is the space of frames with this data non-empty? Is it smooth?
• [Frame homotopy] Is it connected?
• [Sampling] Can we randomly sample it?
• [Robustness] Is a typical frame with this data well-suited to the reconstruction problem?

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

Non-Example: \((\mathbb{R}^3, dx \wedge dy)\)

For any vector \(v\), \(dx \wedge dy\left(\frac{\partial}{\partial z}, v\right) = 0\), so \(dx \wedge dy\) is degenerate.

Non-Example: \((\mathbb{R}^4, dx_1 \wedge dy_1 + y_1 dx_2 \wedge dy_2)\)

\(d(dx_1 \wedge dy_1 + y_1 dx_2 \wedge dy_2) = dy_1 \wedge dx_2 \wedge dy_2 \neq 0\)

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

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

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

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

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

\((T^* \mathbb{R}^n,\sum dq_i \wedge dp_i)\)

phase space

position

momentum

In particular, 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 \(X_H\):

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

Energy function (or Hamiltonian)

\(H(\vec{q},\vec{p}) := \frac{1}{2m}|\vec{p}|^2 + V(\vec{q})\).

\(dH = \sum\left(\frac{\partial H}{\partial p_i} dp_i + \frac{\partial H}{\partial q_i}dq_i\right)\),

\((\vec{q}(t),\vec{p}(t))\) is an integral curve for \(X_H\) if and only if

\(\frac{dq_i}{dt} = \frac{\partial H}{\partial p_i} = \frac{1}{m} p_i\)

\(\frac{dp_i}{dt} = -\frac{\partial H}{\partial q_i} = -\frac{\partial V}{\partial q_i}.\)

This is Newton’s Second Law written as a first-order system!

\(X_H = \sum\left(\frac{\partial H}{\partial p_i} \frac{\partial}{\partial q_i} - \frac{\partial H}{\partial q_i} \frac{\partial}{\partial p_i}\right)\).

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

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

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

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

\(\iota_X\omega = \iota_{\frac{\partial}{\partial \theta}} d\theta \wedge dz = (d\theta \wedge dz)\left(\frac{\partial}{\partial \theta},\cdot \right) = dz \)

Theorem [Duistermaat–Heckman]

If \(\dim(M)=2n\) and \(\dim(\mathbb{T})=k\), then the pushforward measure on the moment polytope \(\mu(M)\) is absolutely continuous w.r.t. Lebesgue measure, with continuous, piecewise-polynomial density of degree \(\leq n-k\).

In particular, if \(n=k\), then the pushforward measure is a constant multiple of Lebesgue measure.

Parseval frames

\(\mu_{U(d)}^{-1}(I_{d\times d})\)

equal-norm frames

\(\mu_{U(1)^N}^{-1}\left(-r,\dots , -r\right)\)

ENPs: \(\mu_{U(d)}^{-1}(I_{d \times d}) \cap \mu_{U(1)^n}^{-1}\left(-\frac{d}{n}, \dots , -\frac{d}{n}\right)\)

Theorem [with Needham]

Fix the spectrum of \(FF^\ast\) and fix \(\|f_1\|,\dots,\|f_n\|\). There are three possibilities:

• It is impossible for a frame to have these data
• It is impossible for a frame with these data to have full spark
• A random frame with these data will have full spark with probability 1.

Proposition [cf. Welch]

When they exist, the ENP frames are exactly the global minima of \(\operatorname{FP}\).

Theorem [Benedetto–Fickus]

As a function on equal-norm frames with fixed \(d\) and \(n\), \(\operatorname{FP}\) has no spurious local minima.

Theorem [with Needham]

Same for fusion frames.

\(\Longrightarrow\) \(\operatorname{FP}(F) = \|FF^*\|_{\operatorname{Fr}}^2\) is the squared norm of a momentum map.

Frances Kirwan

Gert-Martin Greuel [CC BY-SA 2.0 DE], from Oberwolfach Photo Collection

Image by rawpixel.com on Freepik

Theorem [Kirwan]

Reductive algebraic group action on Kähler manifold \(\Longrightarrow\) semistable points flow to global minima.

This kind of function is really nice!

\(\mathbb{C}^* \curvearrowright \mathbb{CP}^2\)

\(t \cdot [z_0:z_1:z_2] = [z_0: tz_1:\frac{1}{t}z_2]\)

Roughly: identify orbits whose closures intersect, throw away orbits on which all \(G\)-invariant polynomials vanish.

\( \mathbb{CP}^2/\!/\,\mathbb{C}^* \cong\mathbb{CP}^1\)