Sampling Finite Unit Norm Tight Frames Using Symplectic Geometry

TL;DR

This work develops a principled framework to sample finite unit-norm tight frames (FUNTFs) uniformly by exploiting a toric symplectic structure on the FUNTF space. By performing two staged symplectic reductions, the authors obtain a toric Kähler manifold whose half-dimensional torus action yields a momentum map with a convex moment polytope, enabling a sampling scheme (Eigenlift) that first selects a point in the eigenstep polytope and then lifts to a frame fiber before random torus action. The method is implemented in Python and validated in low dimensions, with a thorough discussion of challenges in high dimensions due to polytope sampling, and observations about coherence distributions and bounds. The approach offers a novel geometric route to uniformly sample FUNTFs, with potential extensions to other prescribed spectra and norms, and suggests future directions for efficient high-dimensional sampling and applications to robust signal representations.

Abstract

Unit-norm tight frames in finite-dimensional Hilbert spaces (FUNTFs) are fundamental in signal processing, offering optimal robustness to noise and measurement loss. In this paper we introduce the Eigenlift algorithm for sampling random FUNTFs. Our approach exploits the symplectic geometry of the FUNTF space, which we characterize as a symplectic reduction of frame space by a symmetry group. We then define a Hamiltonian torus action on this reduced space whose momentum map induces a fiber bundle structure. The algorithm proceeds by sampling a point from the base space, which is a convex polytope, lifting it deterministically to a point on the corresponding fiber, then acting on this point by a random element of the torus to obtain a random FUNTF. We implement the method in Python and validate it in low-dimensional settings where it is computationally feasible to sample the base polytope via rejection sampling.

Figures (5)

• Figure 1: Eigensteps of a generic FUNTF which, by \ref{['nonincreasingSteps']} and \ref{['Weyls']}, are such that the columns, rows, and upward diagonals are nonincreasing. Moreover, the values in blue are $N/d$ and the sum of the entries in the row corresponding to $S_k$ is equal to $k$. The independent eigensteps are all those colored black, excluding the rightmost nonzero term in each row.
• Figure 2: The eigenstep polytope $\overline{\mathcal{P}}_{3,5}$; each point $(x_1,x_2)$ corresponds to a valid way of filling in the eigenstep table for FUNTFs in $\mathcal{F}^{3,5}$.
• Figure 3: The independent eigensteps of a generic FUNTF in $\mathcal{F}^{d,N}_{\boldsymbol{N/d}}(\boldsymbol{1})$. Once these values are determined, the full eigenstep table (\ref{['fig:eigensteps']}) can be constructed by setting the upper left triangle of eigensteps equal to $N/d$ and choosing $\mu_{k,\min\{k,d\}}$ to be the unique nonnegative value which satisfies the constraint $\sum_{j = 1}^{\min\{k,d\}} \mu_{k,j} = k$.
• Figure 4: Distribution of coherences of representatives of random unitary classes FUNTF's generated using Algorithm .
• Figure 5: Coherence values on the fibers of the momentum map $\Phi : \mathcal{F}_{5/3}^{3,5}(\boldsymbol{1})/(U(d) \times G) \to \mathcal{P}_{3,5}$ over the two regular values $(1.589, 1.009)$ and $(1.361, 0.711)$ corresponding to the plots on the left and right, respectively. Note that the full eigenstep table for frames on these fibers can be found by plugging the regular values in for $x_1$ and $x_2$ in the table in Example .

Theorems & Definitions (27)

• Example 2.1
• Theorem 2.2
• Theorem 2.3: Cahill et al. fickus
• Example 2.4
• Theorem 2.5: Marsden–Weinstein–Meyer Theorem for Regular Values
• Theorem 2.6: Marsden–Weinstein–Meyer Theorem for Coadjoint Orbits
• Theorem 2.7
• Lemma 3.1
• proof
• Theorem 3.2