On the Search for Tight Frames of Low Coherence

Abstract

We introduce a projective Riesz s-kernel for the unit sphere $\mathbb {S}^{d-1}$ and investigate properties of N-point energy minimizing configurations for such a kernel. We show that these configurations, for s and N sufficiently large, form frames that are well-separated (have low coherence) and are nearly tight. Our results suggest an algorithm for computing well-separated tight frames which is illustrated with numerical examples.

Appendix A

1.1 Uniform Measure

Given the hypersphere $\mathbb {S}^{d-1}$, let $C_r(x)$ be the hyperspherical cap centered at x, with r being the Euclidean distance of the furthest point to x. That is,

$$\begin{aligned} C_r(x)=\{y\in \mathbb {S}^{d-1}: \Vert x-y\Vert \le r\}. \end{aligned}$$

Recalling that $\sigma _{d-1}$ denotes the normalized surface measure, the following asymptotic formula holds:

$$\begin{aligned} \sigma _{d-1}(C_r(x))=\frac{1}{d-1}\gamma _d r^{d-1}+\mathcal {O}(r^{d+1}),\qquad (r\rightarrow 0), \end{aligned}$$

(A.1)

and also the estimate

$$\begin{aligned} \sigma _{d-1}(C_r(x))\le \frac{1}{d-1}\gamma _d r^{d-1}, \end{aligned}$$

(A.2)

where

$$\begin{aligned} \gamma _d:=\frac{\Gamma (d/2)}{\Gamma ((d-1)/2)\Gamma (1/2)}. \end{aligned}$$

(A.3)

Both estimates can be found in Sect. 3 of [33].

Fig. 7

$F^{-1}(B(p_x,r))$

Lemma A.1

When ${\mathbb {H}}={\mathbb {R}}$, the uniform measure on ${\mathcal {D}}$ is $(d-1)$-regular. Moreover, we have the estimate

$$\begin{aligned} \Phi (\sigma _{d-1})(B(p_x,r))\le \frac{2}{d-1}\gamma _dr^{d-1},\quad \text { for any }p_x\in {\mathcal {D}}, 0<r\le diam({\mathcal {D}}) \end{aligned}$$

(A.4)

Proof

${\mathcal {D}}$ is the projective space embeded in ${\mathbb {R}}{\mathbb {M}}_{d\times d}^h$. $p_x$ and $p_y$ are furthest away if $x\perp y$, so $\text {diam}({\mathcal {D}})=\sqrt{2}$.

For any point $p_x=F(x)\in {\mathcal {D}}$ and any $r\le \text {diam}({\mathcal {D}})=\sqrt{2}$, suppose in the set $B(p_x,r)\cap {\mathcal {D}}$, $p_y$ is the point that is furthest away from $p_x$. We can pick y so that $\langle x,y\rangle \ge 0$. Then $\Phi ^{-1}(B(p_x,r))$ is the union of the spherical cap centered at x with boundary point y together with its antipodal image, see Fig. 7. By (3.3),

$$\begin{aligned} |\langle x,y\rangle |^2= & {} 1-\Vert p_x-p_y\Vert ^2/2\ge 1-r^2/2,\\ ||x-y||^2= & {} 2-2\langle x,y\rangle \le 2-2\sqrt{1-r^2/2}\le 2-2(1-r^2/2)=r^2. \end{aligned}$$

By (A.2),

$$\begin{aligned} \Phi (\sigma _{d-1})(B(p_x,r))=\sigma _{d-1}(\Phi ^{-1}(B(x,r)))=2\sigma _{d-1}(C_r(x))\le \frac{2}{d-1}\gamma _dr^{d-1}. \end{aligned}$$

$\square$

1.2 Expected Value of Coherence

Let $X=\{x_i\}\in {\mathcal {S}}(d,N)$ be a random configuration on the sphere where each point is selected from a uniform distribution on the sphere. Let $\Theta =\min _{i\ne j}\arccos \langle x_i,x_j\rangle$, so

$$\begin{aligned} \xi (X)=\max _{i\ne j}|\langle x_i,x_j\rangle |\ge \max _{i\ne j}\langle x_i,x_j\rangle =\cos \Theta \ge 1-\Theta ^2/2. \end{aligned}$$

(A.5)

It is proven in [12, Theorem 2] that $F_N(t):=\Pr (N^{2/(d-1)}\Theta \le t)\rightarrow F(t)$ where $F(t)=1-\exp \Big (-\frac{\gamma _d}{2(d-1)}t^{d-1}\Big )$ is supported on $(0,\infty )$.

In order to compute the expected value of $\Theta ^2$, we define $G_N(s):=\Pr (N^{4/(d-1)}\Theta ^2\le s)=F_N(\sqrt{s})\rightarrow F(\sqrt{s})$. By a similar argument as the one in [10, Corollary 3.4], we get

$$\begin{aligned} \lim _{N\rightarrow \infty }{\mathbb {E}}(N^{4/(d-1)}\Theta ^2)&=\lim _{N\rightarrow \infty }\int _0^{\infty }(1-G_N(s))ds\\&=\int _0^{\infty }1-F(\sqrt{s})ds=\int _0^{\infty }\exp \Big (-\frac{1}{2}\kappa _{d}s^{\frac{d-1}{2}}\Big ):=C_d \end{aligned}$$

By (A.5), we have

$$\begin{aligned}{\mathbb {E}}(\xi (X))\ge {\mathbb {E}}(1-\Theta ^2/2)\sim 1-\frac{C_d}{2}N^{-\frac{4}{d-1}}.\end{aligned}$$