Applications of Grassmannians and Flag Manifolds

If \(V\) is an \(n\)-dimensional vector space and \(1 \leq d_1 < \dots < d_k = n\) is an increasing list of integers, then \(F\ell(d_1, \dots , d_k)\) is the flag manifold of nested linear subspaces \(\mathcal{S}_1 \subset \dots \subset \mathcal{S}_k = V\) with \(\dim(\mathcal{S}_i) = d_i\).

\(F\ell(d,n) = \operatorname{Gr}(d,V)\) is usually called the Grassmannian or Grassmann manifold of

\(d\)-dimensional subspaces of \(V\).

When \(d=1\), this is the projective space \(\mathbb{P}(V)\) of lines in \(V\).

Sometimes \(F\ell(n_1, \dots , n_k)\) where \(\sum_{j=1}^i n_j = d_i\).

Some Buzzwords Keywords

Flag manifolds are:

Homogeneous Spaces (differential/Riemannian geometry)
Projective Varieties (algebraic geometry)
Coadjoint/Isotropy Orbits (Lie theory)

Applications of flag manifolds include:

Ye–Wong–Lim: flag manifolds are implicit in many tasks in numerical and statistical analysis, including mesh refinement, multiresolution analysis, and canonical correlation analysis.

Spectral Decomposition

\(A\) Hermitian, \(n \times n\).

A = U \Lambda U^\ast

with \(U\) unitary (i.e., \(U U^\ast = I_{n \times n}\)), and \(\Lambda = \text{diag}(\lambda_1, \dots , \lambda_n)\).

e.g., if \(\lambda_1 = \lambda_2 = 1\) and \(\lambda_3 = 0\), then \(\{u_1,u_2\}\) is an ONB for \(E_1\), \(\{u_3\}\) is an ONB for \(E_0\), but any other choice of ONBs would give a different valid \(U\).

What is unique is the flag \(E_1 \subset E_1 \oplus E_0\), which is a point in \(F\ell(2,3) = \operatorname{Gr}(2,\mathbb{C}^3)\).

Proposition.

For fixed \(\Lambda\), \(\{U \Lambda U^\ast : U \in U(n)\}\) is a flag manifold with signature determined by the multiplicities of the \(\lambda_i\).

Not unique!

Polymers and Polygons

A polymer in solution takes on an ensemble of random shapes, with topology as the unique conserved quantity.

Modern polymer physics is based on the analogy between a polymer chain and a random walk.

– Alexander Grosberg

Protonated P2VP

Roiter/Minko

Clarkson University

Plasmid DNA

Alonso–Sarduy, Dietler Lab

EPF Lausanne

Polymers and Polygons

A polymer in solution takes on an ensemble of random shapes, with topology as the unique conserved quantity.

Knotted DNA

Wassermann et al.

Science 229, 171–174

Ansatz

Random polygon \(\Leftrightarrow\) Point in some (nice!) configuration space

Framework [with Deguchi and Cantarella]

By associating points in \(\operatorname{Gr}(2,\mathbb{C}^N)\) with piecewise-linear loops in \(\mathbb{R}^3\), get an \(E(3)\)-invariant representation. This was the first exactly-solvable model for loop random walks in \(\mathbb{R}^3\) with fixed scale.

Freely-Jointed Rings

Freely-jointed chain

\(\Longleftrightarrow\) point in \(\prod_{i=1}^N S^2 = \prod_{i=1}^N \mathbb{P}(\mathbb{C}^2)\)

Theorem [Kapovich–Millson, Hausmann–Knutson, with Cantarella]

Freely-jointed ring \(\Longleftrightarrow\) point in \(\prod_{i=1}^N \mathbb{P}(\mathbb{C}^2) /\!/SU(2) \simeq \operatorname{Gr}(2,\mathbb{C}^N)/\!/S(U(1)^N)\)

Algorithm [with Cantarella, Duplantier, and Uehara]

Sampling random freely-jointed rings in expected \(\Theta(N^{5/2})\) time.

Finite Frames

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

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

Data representation: \(\mathbb{K}^d \ni v \mapsto F^\ast v \in \mathbb{K}^N\) (the analysis operator).

Frames and Projectors

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

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

A frame is Parseval if \(v = \sum_i \langle f_i, v \rangle f_i = FF^\ast v\) for all \(v \in \mathbb{K}^d\).

i.e., \(FF^\ast = I_{d \times d}\), or equivalently \(F^\ast F\) is a rank-\(d\) orthogonal projector.

Parseval frames \(\Longleftrightarrow\) points in \(\operatorname{Gr}(d,\mathbb{K}^N)\).

In general, specifying the \(\|f_i\|^2\) determines a product of scaled projective spaces.

In general, specifying the spectrum of \(FF^\ast\) (or \(F^\ast F\)) determines a flag manifold.

Consequences

Generalized Frame Homotopy Theorem [with Needham]

The space of frames in \(\mathbb{C}^d\) (or \(\mathbb{H}^d\)) with any specified \(\|f_i\|^2\) and any specified spectrum of \(FF^\ast\) is always path-connected.

Theorem [with Needham]

In the above spaces, either there are no frames with full spark (i.e., full Kruskal rank), or the full spark frames have full measure.

Theorem [with Mixon, Needham, and Villar]

In the space of unit-norm frames, every full spark frame flows by gradient descent to a unit-norm\(^\ast\) Parseval frame. (And similarly for fusion frames.)

\(^\ast\) For technical reasons, actually equal-norm, not unit-norm.

\(\psi(t)\)

Haar Wavelet

\(\psi_{n,k}(t) := 2^{n/2}\psi(2^n t - k)\)

Wavelets and Projections

D_2(10) = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}

\(P_i(2n) = D_i(2n) D_i^T(2n)\)

\(2m \times 2n\) matrix \(A \mapsto P_i(2m) A P_j(2n)\)

\(\updownarrow\)

\(\mathcal{S} \in \operatorname{Gr}(n,\mathbb{R}^{2n})\)