\(F = [f_1 \, f_2 \, \ldots \, f_N ] \in \mathbb{K}^{d \times N}\)

More generally, (unweighted) fusion frames are sequences of subspaces \(S_1, \ldots , S_N \subseteq \mathbb{K}^d\) so that \(\sum P_i\) is invertible.

Unit-norm frames \(\Longleftrightarrow\) (certain) points in \(\prod_{i=1}^N \mathbb{KP}^{d-1}\)

(certain) points in \(\prod_{i=1}^N \operatorname{Gr}(k_i,d)\)

These (fusion) frames are tight if \(\sum P_i = \lambda \mathbb{I}_{d \times d}\).

FUNTFs

TFFs

Proposition [cf. Welch]

When they exist, the FUNTFs/TFFs are exactly the global minima of \(\operatorname{FP}\).

Theorem [Benedetto–Fickus]

As a function on unit-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 polynomials vanish.

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

Example.

For UNFs, \(G = \operatorname{SL}(d)\), \(V = (\mathbb{K}^d)^{\otimes N}\),

\(g \cdot (v_1 \otimes \cdots \otimes v_N) = (gv_1) \otimes \cdots \otimes (gv_N)\)

Metatheorem [cf. Kirwan]

1. Gradient flow stays in the \(G\) orbit.
2. Gradient flow preserves semi-stability.
3. Non-minimizing critical points are unstable.

Example.