The algebraic variety of tight frames

Abstract

Let $V=[v_1,\ldots , v_n]$ be the synthesis operator of a normalised tight frame for $\mathbb {F}^d$ , i.e., a $d\times n$ matrix with $VV^*=I$ (Proposition 2.1). Since $VV^*=I$, the collection of normalised tight frames of n vectors for a space of dimension d can be viewed as an algebraic variety (in $\mathbb {F}^{d\times n}$), as can other classes of frames, such as the equal-norm tight frames.

Appendices

Notes

The existence of equal-norm tight frames was not widely known until recently. This question was raised at Bommerholz in September 2000 (see [BF03]) and was “settled” in various ways: retrospectively [GVT98], by explicit constructions [Zim01],  [RW02] and by minimisation of the frame potential [BF03]. The connection with results such as the Schur–Horn majorisation theorem are now well known, and there are sophisticated algorithms [Str12],  [CFM12],  [CFM+13],  [FMP16] for moving over the varieties ${\mathscr {N}}_{n,\mathbb {F}^d}$ and ${\mathscr {F}}_{n,\mathbb {F}^d}$ . The corresponding algebraic varieties of spherical (t, t)-designs for $t\ne 1$ (see §6.9) are far less studied.

Exercises

7.1

Let $\mathop { U}\nolimits (\mathbb {F}^n)$, the real Lie group of $n\times n$ unitary matrices over $\mathbb {F}$, act on ${\mathscr {N}}_{n,\mathbb {F}^d}$ (the normalised tight frames of n vectors for $\mathbb {F}^d$) via right multiplication.

(a) Show that the stabiliser of $V=[I, 0]\in {\mathscr {N}}_{n,\mathbb {F}^d}$ is

(b) Since the action is irreducible, it follows that ${\mathscr {N}}_{n,\mathbb {F}^d}$ is isomorphic to . Use this to calculate its dimension.

Hint: $\dim (\mathop { U}\nolimits (\mathbb {C}^n))=n^2$, $\dim (\mathop { U}\nolimits (\mathbb {R}^n))={1\over 2}n(n-1)$.

7.2

For $v, w\in \mathbb {F}^d$ and $\sigma \in \mathbb {F}$, $|\sigma |=1$, define $f=f_\sigma :[0,1]\rightarrow \mathbb {R}$ by

$$ f(t) := \Vert t v+\sigma \sqrt{1-t^2}w\Vert ^2 = t^2\Vert v\Vert ^2+({1-t^2})\Vert w\Vert ^2 +2t\sqrt{1-t^2}\alpha , $$

where $\alpha =\alpha _\sigma :=\mathfrak {R}(\overline{\sigma }\langle v, w\rangle )$.

• (a) For a fixed $\sigma$, find a possible local maximum and minimum of $f=f_\sigma$ over [0, 1].

• (b) Optimise the possible local maximum and minimum from (a) over all $|\sigma |=1$.

• (c) By considering the end points $t=0,1$ find the maximum and minimum of $f_\sigma (t)$ over t and $\sigma$.

7.3

Suppose that $1\ge a_1\ge a_2\ge \cdots \ge a_n\ge 0$, $a_1+a_2+\cdots +a_n=d$, and there is a normalised tight frame $V^{(k)}=[v_1^{(k)},\ldots , v_n^{(k)}]\in {\mathscr {N}}_{n,\mathbb {F}^d}$ with

• (i) $\Vert v_j^{(k)}\Vert ^2=a_j$, $1\le j\le k$.

• (ii) $\Vert v_j^{(k)}\Vert ^2\ge a_{k+1}$ or $\Vert v_j^{(k)}\Vert =0$, $k+2\le j\le n$.

Show that

• (a) If $\Vert v_{k+1}^{(k)}\Vert ^2<a_{k+1}$, then $a_{k+1}\le \Vert v_{j_0}^{(k)}\Vert ^2$, for some $k+2\le j_0\le n$.

• (b) If $\Vert v_{k+1}^{(k)}\Vert ^2>a_{k+1}$, then $a_{k+1}\ge \Vert v_{j_0}^{(k)}\Vert ^2$ or $\Vert v_{j_0}^{(k)}\Vert ^2=0$, for some $k+2\le j_0\le n$.