Frames, Optimization, and Geometric Invariant Theory

Clayton Shonkwiler

Colorado State University

shonkwiler.org

/cvg23

this talk!

Special Session on Harmonic Analysis and its Applications to Signals and Information, Apr. 15, 2023

Collaborators

Tom Needham

Florida State University

Funding

National Science Foundation (DMS–2107700)

Simons Foundation (#709150)

Dustin Mixon

(The) Ohio State University

Soledad Villar

Johns Hopkins University

Finite Frames

\(\mathbb{K} = \mathbb{R}\) or \(\mathbb{C}\). A (finite) frame in \(\mathbb{K}^d\) is a spanning set \(f_1, \dots, f_N \in (\mathbb{K}^d)^N\).

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

Frame operator \(FF^* = \sum f_if_i^*\)

Frames and Projectors

If \(\|f_i\|^2 = 1\) for all \(i\), the \(P_i := f_if_i^*\) are orthogonal projectors.

\(FF^* = \sum f_if_i^* = \sum P_i\)

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 \(\sum P_i = \lambda \mathbb{I}_{d \times d}\).

\(\Longleftrightarrow\)

FUNTFs

TFFs

Frame Potential

Definition [Benedetto–Fickus, Casazza–Fickus]

The (fusion) frame potential is

\(\operatorname{FP}(P_1,\ldots,P_N) = \|\sum P_i\|_{\operatorname{Fr}}^2\)

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.

Optimization

Theorem [with Mixon, Needham, and Villar]

Consider the initial value problem

\(\Gamma(F_0,0) = F_0, \qquad \frac{d}{dt}\Gamma(F_0,t) = -\operatorname{grad}\operatorname{FP}(\Gamma(F_0,t))\).

If \(F_0\) has full spark, then \(\lim_{t \to \infty} \Gamma(F_0,t)\) is a FUNTF.

Theorem [with Needham]

Same for fusion frames.

Some Symplectic Geometry

\(F \mapsto FF^*\) is the momentum map of the diagonal \(SU(d)\) action on \(\prod_{i=1}^N \mathbb{CP}^{d-1}\).

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

Frances Kirwan

Theorem [Kirwan]

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

This kind of function is really nice!

Geometric Invariant Theory (GIT)

The GIT quotient consists of group orbits which can be distinguished by \(G\)-invariant (homogeneous) polynomials.

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

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

Roughly: identify orbits whose closures intersect, throw away orbits on which all polynomials vanish.

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

Stability

Definition.

Given a reductive group \(G\) acting algebraically on a vector space \(V\), a vector \(v \in V\) is unstable if \(0 \in \overline{G \cdot v}\); otherwise, \(v\) is semi-stable.

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.

Key Technical Tool

Hilbert–Mumford Criterion.

\(v \in V\) is semi-stable \(\Longleftrightarrow\) there exists a 1-parameter subgroup \(\lambda: \mathbb{C}^* \to G\) so that 

\(\lim_{t \to 0}\lambda(t) \cdot v = 0\).

Example.

F = \begin{bmatrix} 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 1 \end{bmatrix} \mapsto e_1 \otimes e_1 \otimes e_2 \otimes e_2 \otimes e_2
\lambda(t) = \begin{bmatrix} t^{-1} & 0 \\ 0 & t \end{bmatrix}
\lambda(t) \cdot (e_1 \otimes e_1 \otimes e_2 \otimes e_2 \otimes e_2) = (t^{-1}e_1) \otimes (t^{-1}e_1) \otimes (t e_2) \otimes (t e_2) \otimes (t e_2)
= t (e_1 \otimes e_1 \otimes e_2 \otimes e_2 \otimes e_2) \to 0

Question

What other functions of interest arise as squared norms of momentum maps?

Thank you!

References

Fusion frame homotopy and tightening fusion frames by gradient descent

Tom Needham and Clayton Shonkwiler

Journal of Fourier Analysis and Applications, to appear

arXiv:2208.11045

Three proofs of the Benedetto–Fickus theorem

Dustin Mixon, Tom Needham, Clayton Shonkwiler, and Soledad Villar

Sampling, Approximation, and Signal Analysis (Harmonic Analysis in the Spirit of J. Rowland Higgins), Stephen D. Casey, M. Maurice Dodson, Paulo J. S. G. Ferreira and Ahmed Zayed, eds., to appear

arXiv:2112.02916

Frames, Optimization, and Geometric Invariant Theory

By Clayton Shonkwiler

Frames, Optimization, and Geometric Invariant Theory

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