Clayton Shonkwiler PRO
Mathematician and artist
Geometric Approaches to Frame Theory
/tu22
this talk!
Algebraic Geometry and Geometric Topology Seminar, Mar. 21, 2022
Collaborator
Florida State University
Funding
National Science Foundation (DMS–2107700)
Simons Foundation (#709150)
Finite Frames
\(\mathfrak{H} = \mathbb{R}^d\) or \(\mathbb{C}^d\). A (finite) frame in \(\mathfrak{H}\) is a spanning set \(f_1, \dots, f_N \in \mathfrak{H}\).
The Model Problem
Signal: \(v \in \mathfrak{H}\)
Design: \(f_1, \dots , f_N \in \mathfrak{H}\)
Measurements: \(\langle f_1, v \rangle, \dots , \langle f_N, v \rangle \).
Parseval’s Theorem
If \(f_1, \dots , f_d \in \mathfrak{H}\) is an orthonormal basis,
\(\|v\|^2 = \sum |\langle f_i, v \rangle |^2 = \|F^* v\|^2\)
for any \(v \in \mathfrak{H}\).
Even better,
\(v = \sum \langle f_i, v \rangle f_i = FF^* v \).
If \(F = [f_1 \, f_2 \dots f_N]\), the measurement vector is \(F^*v\).
\(\Leftrightarrow FF^* = \mathrm{Id}_{d \times d}\)
Parseval frame
Fact: For any \(n \geq d\), there exist Parseval frames \(f_1, \dots , f_N \in \mathfrak{H}\) so that \(\|f_i\|=\|f_j\|\) for all \( i,j\); these are the equal-norm Parseval frames (ENPs).
This is fragile! What if a measurement gets lost?
Gram Matrices
Both \(\operatorname{spec}(FF^\ast) = (\lambda_1, \dots , \lambda_d)\) and \((\|f_1\|^2,\dots , \|f_N\|^2) = (r_1, \dots, r_N)\) are present in the Gram matrix
\(F^\ast F = \left[\langle f_i, f_j\rangle \right]_{i,j} \in \mathscr{H}(N)\)
since \(F^\ast F\) and \(F F^\ast\) have the same rank and nonzero eigenvalues.
Unitary equivalence classes of frames have the same Gram matrix: if \(U \in O(d)\) or \(U(d)\), then
\((UF)^\ast(UF) = F^\ast U^\ast U F = F^\ast F\).
For \(\boldsymbol{\lambda} = (\lambda_1, \dots , \lambda_d, 0, \dots 0)\),
\(\mathcal{O}_{\boldsymbol{\lambda}} := \{U \operatorname{diag}(\boldsymbol{\lambda}) U^\ast | U \in U(N) \text{ or } O(N)\}\subset \mathscr{H}(N)\)
is a flag manifold.
Questions
We are often interested in specifying \(\operatorname{spec}(FF^\ast) = (\lambda_1, \dots , \lambda_d)\) and/or \((\|f_1\|^2,\dots , \|f_N\|^2) = (r_1, \dots, r_N)\).
- [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?
Lie Groups
Informal Definition. A Lie group is a group that is also a manifold (a “continuous group of symmetries”).
Examples.
- \(\mathbb{R}^n\), \(\mathbb{C}^n\)
- \(GL_n(\mathbb{R})\), \(GL_n(\mathbb{C})\)
- \(SL_n(\mathbb{R})\), \(SL_n(\mathbb{C})\)
- \(O(n)\), \(SO(n)\), \(U(n)\), \(SU(n)\) (e.g., \(SO(3)\simeq \mathbb{RP}^3\), \(SU(2) \simeq S^3\))
- Products (e.g. the compact torus \(U(1)^n\))
Lie Algebras
Informal Definition. A Lie algebra \(\mathfrak{g}\) (or \(\operatorname{Lie}(G)\)) associated to a Lie group \(G\) is the tangent space at the identity (the “infinitesimal symmetries”).
Examples.
- \(\operatorname{Lie}(\mathbb{R}^n) \simeq \mathbb{R}^n\)
- \(\mathfrak{gl}_n(\mathbb{R}) \simeq \mathbb{R}^{n \times n}\), \(\mathfrak{gl}_n(\mathbb{C}) \simeq \mathbb{C}^{n \times n}\)
- \(\mathfrak{sl}_n(\mathbb{R}) \simeq \{\text{traceless matrices}\}\), \(\mathfrak{sl}_n(\mathbb{C}) \simeq \{\text{traceless matrices}\}\)
- \(\mathfrak{o}(n) = \mathfrak{so}(n) \simeq \{\text{skew-symmetric matrices}\}\), \(\mathfrak{u}(n) \simeq \{\text{skew-Hermitian matrices}\}\), \(\mathfrak{su}(n) \simeq \{\text{traceless skew-Hermitian}\}\)
Suppose \(\gamma(t) \in U(n)\) with \(\gamma(0)=I\). Since \(I = \gamma(t)\gamma(t)^\ast\),
\(0 = \left. \frac{d}{dt}\right|_{t=0} \gamma(t)\gamma(t)^\ast = \gamma'(0)\gamma(0)^\ast + \gamma(0)\gamma'(t)^\ast = \gamma'(0) + \gamma'(0)^\ast\)
The Lie bracket \([\cdot , \cdot ]\) is the matrix commutator on matrix groups: \([A,B] = AB-BA\).
Adjoint Actions
A Lie group \(G\) acts on itself by conjugation: \(g \cdot h := g h g^{-1}\).
This fixes the identity, so linearizing gives an action of \(G\) on \(\mathfrak{g}\), called the adjoint action.
For matrix groups, the adjoint action is the conjugation action.
Example. \(\mathfrak{so}(3) \simeq \left\{\begin{pmatrix} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0\end{pmatrix}:x,y,z \in \mathbb{R}\right\}\simeq (\mathbb{R}^3, \times)\)
\begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0 \end{pmatrix} \begin{pmatrix} \cos \theta & \sin \theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}
\\
=
\begin{pmatrix} 0 & -z & y\cos \theta + x \sin \theta \\ z & 0 & -x\cos \theta + y \sin \theta \\ -y\cos \theta - x\sin \theta & x \cos \theta - y \sin \theta & 0 \end{pmatrix}
Adjoint Orbits
If \(G\) is a Lie group, each orbit of the adjoint action is a symplectic manifold.\(^1\)
1. Strictly speaking, coadjoint orbits in \(\mathfrak{g}^\ast\) are symplectic.
\(i\mathscr{H}(N) = \mathfrak{u}(N)\), so conjugation orbits of Hermitian matrices can be interpreted as adjoint orbits.
Symplectic Geometry
A symplectic manifold is a smooth manifold \(M\) together with a closed, non-degenerate 2-form \(\omega \in \Omega^2(M)\).
Example: \((\mathbb{R}^2,dx \wedge dy) = (\mathbb{C},\frac{i}{2}dz \wedge d\bar{z})\)
dx \wedge dy \left( \textcolor{12a4b6}{a \frac{\partial}{\partial x} + b \frac{\partial}{\partial y}}, \textcolor{d9782d}{c \frac{\partial }{\partial x} + d \frac{\partial}{\partial y}} \right) = ad - bc
(a,b) = a \vec{e}_1 + b \vec{e}_2 = a \frac{\partial}{\partial x} + b \frac{\partial}{\partial y}
(c,d) = c \vec{e}_1 + d \vec{e}_2 = c \frac{\partial}{\partial x} + d \frac{\partial}{\partial y}
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: \((\mathbb{R}^4, dx_1 \wedge dy_1 + y_1 dx_2 \wedge dy_2)\)
Because of the coefficient \(y_1\), this is not closed (“not divergence-free”).
Examples
\((S^2,d\theta\wedge dz)\)
\((\mathbb{R}^2,dx \wedge dy) = (\mathbb{C},\frac{i}{2}dz \wedge d\bar{z})\)
\((S^2,\omega)\), where \(\omega_p(u,v) = (u \times v) \cdot p\)
\((\mathbb{R}^2,\omega)\) where \(\omega(u,v) = \langle i u, v \rangle \)
\((\mathbb{C}^n, \frac{i}{2} \sum dz_k \wedge d\overline{z}_k)\)
\((\mathbb{C}^{m \times n}, \omega)\) 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
Volume
\(\omega^{\wedge n} = \omega \wedge \dots \wedge \omega\) is a volume form on \(M\), and induces a measure
m(U) := \int_U \omega^{\wedge n}
called Liouville measure on \(M\).
If \(M\) is compact, this can be normalized to give a probability measure.
Functions and Symplectic Gradients
If \(H: M \to \mathbb{R}\) is smooth, then there exists a unique vector field \(X_H\) so that \({dH = \iota_{X_H}\omega}\), i.e.,
dH(\cdot) = \omega(X_H, \cdot)
(\(X_H\) is the Hamiltonian vector field for \(H\) or the symplectic gradient of \(H\))
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 \(X_H\):
\(\mathcal{L}_{X_H}(H) = dH(X_H)=\omega(X_H,X_H) = 0\)
Lie Group Actions
\(S^1=U(1)\) acts on \((S^2,d\theta \wedge dz)\) by
e^{it} \cdot(\theta, z) = (\theta + t, z).
For \(r \in \mathbb{R} \simeq \mathfrak{u}(1)\), \(X_r = r\frac{\partial}{\partial \theta}\).
Let \(G\) be a Lie group, and let \(\mathfrak{g}\) be its Lie algebra. If \(G\) acts on \((M,\omega)\), then each \(V \in \mathfrak{g}\) determines a vector field \(X_V\) on \(M\) by
X_V(p) = \left.\frac{d}{dt}\right|_{t=0}\exp(t V) \cdot p
Lie Group Actions
Let \(G\) be a Lie group, and let \(\mathfrak{g}\) be its Lie algebra. If \(G\) acts on \((M,\omega)\), then each \(V \in \mathfrak{g}\) determines a vector field \(X_V\) on \(M\) by
X_V(p) = \left.\frac{d}{dt}\right|_{t=0}\exp(t V) \cdot p
\(SO(3)\) acts on \(S^2\) by rotations.
\(= (a,b,c) \times (x,y,z)\)
For \(V_{(a,b,c)} = \begin{bmatrix} 0 & -c & b \\ c & 0 & -a \\ -b & a & 0 \end{bmatrix} \in \mathfrak{so}(3)\),
\(X_{V_{(a,b,c)}}((x,y,z))\)
\(= (bz-cy)\frac{\partial}{\partial x} + (cx - az) \frac{\partial}{\partial y} +(ay - bx) \frac{\partial}{\partial z}\)
Symmetries and Conserved Quantities
Definition. A \(U(1)\) action on \((M,\omega)\) is Hamiltonian if there exists a map
\mu: M \to \mathbb{R} \simeq \mathfrak{u}(1)^*
so that \(d\mu = \iota_{X}\omega = \omega(X,\cdot)\), where \(X\) is the vector field generated by the action.
\(X = \frac{\partial}{\partial \theta}\)
\(\mu(\theta,z) = z\)
\(\iota_X\omega = \iota_{\frac{\partial}{\partial \theta}} d\theta \wedge dz = dz \)
Moment Maps
Definition. An action of \(G\) on \((M,\omega)\) is Hamiltonian if each one-parameter subgroup action is Hamiltonian. Equivalently, there exists a map
\mu: M \to \mathfrak{g}^*
so that \(\omega_p(X_V, X) = D_p \mu(X)(V)\) for each \(p \in M\), \(X \in T_pM\), and \(V \in \mathfrak{g}\).
\(X_{V_{(a,b,c)}}(x,y,z) = (a,b,c) \times (x,y,z)\)
\((\iota_{X_{V_{(a,b,c)}}}\omega)_{(x,y,z)} = a dx + b dy + c dz \)
\(\mu(x,y,z)(V_{(a,b,c)}) = (x,y,z)\cdot(a,b,c)\)
Key Tools
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.
Key Tools
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.
\(U(1)^2\) acts on \(\mathbb{CP}^2\) by
\((e^{i\theta_1},e^{i\theta_2})\cdot [z_0:z_1:z_1]:=[z_1:e^{i\theta_1}z_1:e^{i\theta_2}z_2]\)
Moment map:
\(\mu([z_0:z_1:z_2]=-\frac{1}{2(|z_0|^2+|z_1|^2+|z_2|^2)}(|z_1|^2,|z_2|^2)\)
\((0,0)=\mu([1:0:0])\)
\((-\frac{1}{2},0)=\mu([0:1:0])\)
\((0,-\frac{1}{2})=\mu([0:0:1])\)
Theorem [Duistermaat–Heckman]
If \(k=n\), then the pushforward of Liouville measure to the moment polytope is a constant multiple of Lebesgue measure.
Symplectic Geometry and Frame Homotopy
\(\mathcal{O}_{\boldsymbol{\lambda}} = \{U \operatorname{diag}(\boldsymbol{\lambda})U^\ast | U \in U(N)\} \simeq \{F \in \mathbb{C}^{d \times N} | \operatorname{spec}(FF^\ast) = (\lambda_1, \dots , \lambda_d)\}/U(d)\) is symplectic.
Proposition. The map recording diagonal entries is the momentum map of a Hamiltonian torus action.
Theorem [Atiyah and Guillemin–Sternberg]
Momentum maps of Hamiltonian torus actions have convex image and connected level sets.
Theorem [with Needham, 2021]
The space \(\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{C}^d,N}(\boldsymbol{r})\subset \mathbb{C}^{d \times N}\) of frames with \(\operatorname{spec}(FF^\ast) = (\lambda_1, \dots , \lambda_d)\) and \(\|f_i\|^2 = r_i\) for \(i=1, \dots , N\) is path-connected.
This solves the general frame homotopy problem for complex frames.
Robustness
For compressed sensing, it is desirable to require every minor of the \(d \times N\) matrix \(F\) to be invertible. Such an \(F\) is said to have full spark.
Theorem [with Needham, 2021]
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.
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.
Measures
Proof Idea
(A suitable quotient) of the space of such frames is almost toric, and spark-deficient frames lie over the boundary of the moment polytope.
Theorem [Duistermaat–Heckman]
The pushforward of Liouville measure to the moment polytope is a constant multiple of Lebesgue measure.
What about real frames?
\(\dim(\mathfrak{o}(N)) = \frac{N(N-1)}{2}\)
\(\dim(\{\text{symmetric } N \times N \text{ matrices}\})=\frac{N(N+1)}{2}\)
The analog of the unitary group for quaternionic matrices is the (compact) symplectic group \(\operatorname{Sp}(N)\); its Lie algebra \(\mathfrak{sp}(N)\) consists of quaternionic skew-Hermitian matrices.
\(\dim(\mathfrak{sp}(N)) = 2N^2+N\)
\(\dim(\{\text{Hermitian quaternionic } N \times N \text{ matrices}\}) = 2N^2-N\)
Can’t identify these!
Cartan Decomposition
An involution \(\theta: \mathfrak{g} \to \mathfrak{g}\) is a Cartan involution if \(-B(\cdot , \theta \cdot)\) is positive-definite, where \(B(\cdot , \cdot)\) is the Killing form.
Example. \(\theta: \mathfrak{sl}_n(\mathbb{R}) \to \mathfrak{sl}_n(\mathbb{R})\) defined by \(\theta(X) = -X^T\). This is the derivative of the involution on \(SL_n(\mathbb{R})\) given by \(U \mapsto (U^{-1})^T\).
\(E_{+1} =: \mathfrak{k}\) consists of skew-symmetric matrices
\(E_{-1} =: \mathfrak{p}\) consists of (traceless) symmetric matrices
\(\mathfrak{sl}_n(\mathbb{R}) = \mathfrak{k} \oplus \mathfrak{p}\) is an orthogonal decomposition — the Cartan decomposition
\([\mathfrak{k},\mathfrak{p}]\subseteq \mathfrak{p}\), so the adjoint action of \(SO(n) = \exp(\mathfrak{k})\) restricts to an action on \(\mathfrak{p}\)
\([\mathfrak{p},\mathfrak{p}]\subseteq\mathfrak{k}\), so any Lie subalgebra of \(\mathfrak{p}\) is abelian.
Theorem. \(G\) non-compact, semisimple, and \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\) a Cartan decomposition with \(K = \exp(\mathfrak{k})\) and \(P = \exp(\mathfrak{p})\), then \(G = KP\) and \(T_{[I]} G/K \simeq \mathfrak{p}\).
The adjoint action of \(K\) on \(\mathfrak{p}\) is often called the isotropy representation.
Kostant’s Convexity Theorem
Theorem [Kostant]
Let \(G\) be a Lie group with Cartan decomposition of the Lie algebra \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\).
If \(\mathfrak{a} \subset \mathfrak{p}\) is a maximal abelian subspace, \(P: \mathfrak{p} \to \mathfrak{a}\) orthogonal projection, and \(\mathcal{O}_a\) the isotropy orbit of \(a \in \mathfrak{a}\), then
\(P(\mathcal{O}_a) = \operatorname{conv}(W \cdot a)\).
\(W = N_K(\mathfrak{a})/Z_K(\mathfrak{a})\) is the Weyl group associated to \((\mathfrak{a},\mathfrak{g})\).
Example.
\(G = SL_n(\mathbb{C})\), \(K=SU(N)\), \(\mathfrak{k} = \mathfrak{su}(N)\), \(\mathfrak{p} = \mathscr{H}_0(N)\), \(a = \operatorname{diag}(a_1, \dots , a_N)\),
\(\mathcal{O}_a = \{UaU^\ast | U \in SU(N)\}\)
\(P(\mathcal{O}_a) = \operatorname{conv}(S_N \cdot a)\) is essentially Schur–Horn.
Can play the same game with real or quaternionic matrices!
Kostant’s Convexity Theorem
Theorem [Kostant]
Let \(G\) be a Lie group with Cartan decomposition of the Lie algebra \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\).
If \(\mathfrak{a} \subset \mathfrak{p}\) is a maximal abelian subspace, \(P: \mathfrak{p} \to \mathfrak{a}\) orthogonal projection, and \(\mathcal{O}_a\) the isotropy orbit of \(a \in \mathfrak{a}\), then
\(P(\mathcal{O}_a) = \operatorname{conv}(W \cdot a)\).
\(W = N_K(\mathfrak{a})/Z_K(\mathfrak{a})\) is the Weyl group associated to \((\mathfrak{a},\mathfrak{g})\).
Theorem [Mare]
Under additional technical assumptions,
\(H_{Z_K(\mathfrak{a})}^\ast (\mathcal{O}_a; \mathbb{Q}) \to H_{Z_K(\mathfrak{a})}^\ast(P^{-1}(a); \mathbb{Q})\)
is a surjection for all \(a \in \mathfrak{a}\).
In particular, \(P^{-1}(a)\) is connected if \(\mathcal{O}_a\) is.
This implies the generalized frame homotopy theorem for complex frames.
Quaternions
\(\mathbb{H} = \{a + b \boldsymbol{i} + c \boldsymbol{j} + d \boldsymbol{k}\}\)
\(\boldsymbol{i}^2 = \boldsymbol{j}^2 = \boldsymbol{k}^2 = -1\)
\(\boldsymbol{i}\boldsymbol{j}\boldsymbol{k} = -1\)
\(z+w \boldsymbol{j} \mapsto \begin{pmatrix} z & w \\ -\overline{w} & \overline{z}\end{pmatrix}\)
\(\mathbb{H}^d\) is a right vector space over the skew-field \(\mathbb{H}\).
Quaternionic Frames
A spanning set \(\{f_1, \dots , f_N\} \subset \mathbb{H}^d\) is a (finite) frame for \(\mathbb{H}^d\).
Much of standard frame theory goes through for quaternionic frames (see, e.g., Waldron).
In particular, the frame operator \(FF^\ast\) and the Gram matrix \(F^\ast F\) are well-defined, and \(\operatorname{Sp}(d)\) equivalence classes of frames in \(\mathbb{H}^{d \times N}\) are uniquely determined by their Gram matrices.
Realizing Quaternion Matrices as Complex Matrices
\(\Psi:\mathbb{H}^{N \times N} \to \mathbb{C}^{2N \times 2N}\) given by
\(\Psi(Z + W\boldsymbol{j}) = \begin{pmatrix} Z & W \\ -\overline{W} & \overline{Z} \end{pmatrix}\).
\(\Psi(\operatorname{Sp}(N)) \subset SU(2N)\) is the fixed point set of the involution
\(U \mapsto \Omega^\ast \overline{U} \Omega\),
where \(\Omega = \begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix}\).
Corresponding Cartan decomposition of the Lie algebra:
\(\mathfrak{su}(2N) = \mathfrak{k} \oplus \mathfrak{p} \simeq \Psi(\mathfrak{sp}(N)) \oplus \Psi(\mathscr{H}_0(N))\).
Admissibility and Frame Homotopy for Quaternionic Frames
Admissibility [with Needham, 2021]
\(\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{H}^d,N}(\boldsymbol{r})\neq \emptyset\) if and only if \(\boldsymbol{r} \in \operatorname{conv}(S_N \cdot (\lambda_1, \dots , \lambda_d, 0, \dots , 0))\) (cf. Casazza–Leon).
\(\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{H}^d,N}(\boldsymbol{r}):= \{F \in \mathbb{H}^{d \times N} | \operatorname{spec}(FF^\ast)=\boldsymbol{\lambda}, \|f_i\|^2 = r_i \text{ for all } i=1,\dots, N\}\)
\(\mathcal{O}_{\boldsymbol{\lambda}} = \{U \operatorname{diag}(\lambda_1, \dots , \lambda_d, 0, \dots , 0)U^\ast | U \in \operatorname{Sp}(N)\}\)
Key Idea. Terng/Kostant convexity.
Generalized Frame Homotopy [with Needham, 2021]
\(\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{H}^d,N}(\boldsymbol{r})\) is always path-connected.
Key Idea. Mare’s connectedness theorem.
Questions
Isotropy orbits of real symmetric matrices do not satisfy the technical hypothesis in Mare’s theorem; indeed, some \(\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{R}^d,N}(\boldsymbol{r})\) spaces are disconnected. Can we characterize the \((\boldsymbol{\lambda},\boldsymbol{r})\) which lead to connectedness?
Principal isotropy orbits are examples of isoparametric submanifolds; does this give any insight into the measure on real or quaternionic frame spaces? Does this lead to a sampling algorithm?
Thank you!
References
Symplectic geometry and connectivity of spaces of frames
Tom Needham and Clayton Shonkwiler
Advances in Computational Mathematics 47 (2021), no. 1, 5
Admissibility and frame homotopy for quaternionic frames
Tom Needham and Clayton Shonkwiler
Preprint, 2021
Geometric Approaches to Frame Theory
By Clayton Shonkwiler
