An Analogue of Slepian Vectors on Boolean Hypercubes

Abstract

Analogues of Slepian vectors are defined for finite-dimensional Boolean hypercubes. These vectors are the most concentrated in neighborhoods of the origin among bandlimited vectors. Spaces of bandlimited vectors are defined as spans of eigenvectors of the Laplacian of the hypercube graph with lowest eigenvalues. A difference operator that almost commutes with space and band limiting is used to initialize computation of the Slepian vectors.

Appendix: Expressions for Entries of the Matrix HTLTH

The purpose of this appendix is to derive explicit formulas for the entries of the matrix HTLTH that arises in the commutator $[\mathrm{BDO},\, {\bar{H}}[TLT,{\bar{H}}]]$. Besides providing a means to compute the commutator for large values of N, the formulas can be used as a basis for commutator estimates in terms of (N, K).

Of interest are the entries of ${\bar{H}}[TLT,\, {\bar{H}}]={\bar{H}} TLT{\bar{H}}-TLT$. As TLT is relatively uncomplicated, we focus on the entries of $H TLTH=2^N{\bar{H}} TLT{\bar{H}}$. As before since $(TLT)_{RP} =0$ unless $P=R$ or $P\sim R$, writing $|R|=r$,

$$\begin{aligned} (TLT H)_{RS}= & {} \sum _P (TLT)_{RP} H_{PS} =(TLT)_{RR} H_{RS} +\sum _{P\sim R} (TLT)_{RP} H_{PS}\\= & {} 2Nr H_{RS}-2\sqrt{r(r+1)}\sum _{P\sim R,\,p=r+1} H_{PS} -2\sqrt{r(r-1)}\sum _{P\sim R,\, p=r-1} H_{PS}\, . \end{aligned}$$

As before, if P is an addition neighbor of R whose extension element is in S or if P is a subtraction neighbor of R whose subtraction element is in S then $H_{PS}=-H_{RS}$; otherwise $H_{PS}=H_{RS}$. This gives us the following.

Lemma 4

Write $r=|R|$ and $s=|S|$. One has

$$\begin{aligned} (TLT H)_{RS}&=\left( 2Nr -2\sqrt{r(r+1)}\left[ |S^c\cap R^c|-|S\cap R^c|\right] \right. \nonumber \\&\quad \left. -2\sqrt{r(r-1)}\left[ |S^c\cap R|-|S\cap R|\right] \right) H_{RS}\, . \end{aligned}$$

(3)

Equivalently, the coefficient of $H_{RS}$ in $(TLT H)_{RS}$ is

$$\begin{aligned} c_{RS}&=2N(r-\sqrt{r(r+1)})+4s\sqrt{r(r+1)} \nonumber \\&\quad +(2r-4|R\cap S|)(\sqrt{r(r+1)}-\sqrt{r(r-1)}) \end{aligned}$$

(4)

Equation (4) follows from (3) by observing that $|S^c\cap R^c|=N-s-r+|S\cap R|$, $|S\cap R^c|=s-|S\cap R|$, etcetera. If one lets

$$\begin{aligned} c''_{RS}= -4|R\cap S|(\sqrt{r(r+1)}-\sqrt{r(r-1)}) \end{aligned}$$

and $c_{RS}'=c_{RS}-c''_{RS}$, then

$$\begin{aligned} c'_{RS}=2N(r-\sqrt{r(r+1)})+4s\sqrt{r(r+1)} +2r(\sqrt{r(r+1)}-\sqrt{r(r-1)})\, \end{aligned}$$

(5)

depends only on $r=|R|$ and $s=|S|$. Observing that $H_{RP}H_{PS}=H_{S\Delta R,P}$, one has

$$\begin{aligned} (HTLTH)_{RS}= \sum _{P} H_{S\Delta R,P} c_{PS} = \sum _{P} H_{S\Delta R,P} (c'_{PS}+c''_{PS}) \end{aligned}$$

where P runs over all subsets of $\{1,\dots , N\}$.

Now write

$$\begin{aligned} HTLTH=C'+C''; \quad C'_{RS}= \sum _{P} c'_{PS} H_{S\Delta R,P} ,\quad C''_{RS}= \sum _{P} c''_{PS} H_{S\Delta R,P} \end{aligned}$$

(6)

Lemma 5

The entries of the matrix $C'$ can be written

$$\begin{aligned} C'_{RS}=\sum _{p=0}^{N-|S\Delta R|} c'_{ps}\, \left( {\begin{array}{c}N-|S\Delta R|\\ p\end{array}}\right) \,\, { }_2F_1 (-|S\Delta R|,-p, N-|S\Delta R|-p+1;-1) \end{aligned}$$

where ${}_2 F_1(a,b;c;z)$ denotes the Gaussian hypergeometric function.

Proof

For a fixed $p\in \{0,1,\dots , S\Delta R\}$,

$$\begin{aligned} \sum _{|P|=p} H_{S\Delta R,P}=\sum _{k=0}^{\min (p,|S\Delta R|)} (-1)^k \left( {\begin{array}{c}|S\Delta R|\\ k\end{array}}\right) \left( {\begin{array}{c}N-|S\Delta R|\\ p-k\end{array}}\right) \end{aligned}$$

which follows by counting the number of ways that a p-element set can be divided into a subset of $S\Delta R$ and a subset of the complement of $S\Delta R$, and observing that $H_{S\Delta R,P}=(-1)^k$ if P has k elements in common with $S\Delta R$. The lemma then follows upon taking $n=|S\Delta R|$, $m=N-|S\Delta R|$ and $z=-1$ in the identity (7) in Lemma 6 below. $\square$

Lemma 6

Define the Gaussian hypergeometric function

$$\begin{aligned} {}_2F_1(a,b;c;z)=\sum _{k=0}^\infty \frac{(a)_k(b)_k}{(c)_k} \frac{z^k}{k!};\quad (a)_k=\frac{\Gamma (a+k)}{\Gamma (a)} \end{aligned}$$

For nonnegative integers m, n, p such that $p\le n$ one has

$$\begin{aligned} \sum _{k=0}^{\min (n,p)} \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}m\\ p-k\end{array}}\right) \, z^k =\left( {\begin{array}{c}m\\ p\end{array}}\right) {}_2F_1(-n,-p,m-p+1;z)\, . \end{aligned}$$

(7)

Proof

If n is a positive integer and $0\le k\le n$ then

$$\begin{aligned} (-n)_k= & {} (-n)(1-n)\cdots (-n+k-1)\\= & {} (-1)^k n (n-1) (n-k+1)=(-1)^k \frac{n!}{(n-k)!} \end{aligned}$$

and $(-n)_k=0$ if $k>n$. Thus, if n and p are nonnegative integers then

$$\begin{aligned} {}_2F_1(-n,-p;c;z)=\sum _{k=0}^{\min (n,p)} \frac{n!}{(n-k)!} \frac{p!}{(p-k)!}\frac{1}{(c)_k} \frac{z^k}{k!}\, . \end{aligned}$$

$\square$

In particular, if $c=m-p+1$ where $p\le m$ then $(m-p+1)_k=\frac{(m-p+k)!}{(m-p)!}$ and

$$\begin{aligned} {}_2F_1(-n,-p;m+1-p;z)= & {} \sum _{k=0}^{\min (n,p)} \frac{n!}{(n-k)!} \frac{p!}{(p-k)!}\frac{(m-p)!}{(m-(p-k))!} \frac{z^k}{k!} \\= & {} \sum _{k=0}^{\min (n,p)} \frac{n!}{k!(n-k)!} \frac{p! (m-p)!}{m!}\frac{m!}{(p-k)!(m-(p-k))!} \, z^k \\= & {} \left( {\begin{array}{c}m\\ p\end{array}}\right) ^{-1} \sum _{k=0}^{\min (n,p)} \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( {\begin{array}{c}m\\ p-k\end{array}}\right) z^k\, \, . \end{aligned}$$

This proves the lemma.

Lemma 7

Let $c''_{RS}= |R\cap S| d''_{r}$ where $d''_{r}=-4\sqrt{r(r+1)}-\sqrt{r(r-1)})$. The entries of the matrix $C''$ in (6) can then be written

$$\begin{aligned} C''_{RS}= & {} \sum _P H_{R\Delta S, P} \, c''_{PS}= \sum _{p=0}^N d_p''\sum _{k=0}^{\min (p,s)} k\, (-1)^k \left( {\begin{array}{c}s-q\\ k\end{array}}\right) ^{-1}\left( {\begin{array}{c}N-s-(r-k)\\ p-k\end{array}}\right) ^{-1} \\&\quad \times \,{}_2F_{1} (-q,-k;s-q+1-k;-1) {}_2F_{1} (q-r,k-p; N-s-(r-q)\\&\quad +1-(p-k);-1) \end{aligned}$$

where $(p,r,s)=(|P|,|R|,|S|)$, $|P\cap S|=k$, and $|R\cap S|=q$.

The argument to prove Lemma 7 is similar to that of Lemma 5 but now one must take into account the distribution of values $k=|P\cap S|$, counting the number of ways to decompose P, $|P|=p$ according to its part that intersects R inside of S (or not) and its part that intersects R outside of S (or not), and recalling that $q=|S\cap R|$ is fixed.

Up to this point we have derived formulas for entries of HTLTH in terms of binomial coefficients, values of hypergeometric functions, and coefficients involving factors $\sqrt{p(p\pm 1)}$. The latter can be estimated in terms of Laurent series, in particular

$$\begin{aligned} \pm \sqrt{x(x\pm 1)} \mp x-\frac{1}{2} =\mp \frac{1}{8x}+\frac{1}{16x^2}\mp \frac{5}{128 x^3}+\frac{7}{256 x^4}+O\bigl (\frac{1}{x^5}\bigr ),\quad x\rightarrow \infty \, . \end{aligned}$$

For example, the coefficient $c'_{RS}$ in (5) behaves as

$$\begin{aligned} 2N\Bigl (-\frac{1}{2} +\frac{1}{8r}\Bigr )+4s\Bigl (r+\frac{1}{2}-\frac{1}{8r}\Bigl )+2r +O\Bigl (\frac{1}{r^2}\Bigr ) \end{aligned}$$

for large r, while $c''_{RS}= -4|R\cap S|(\sqrt{r(r+1)}-\sqrt{r(r-1)})$ behaves as

$$\begin{aligned} -4|R\cap S| \Bigl ( 1+O\Bigl (\frac{1}{r^2}\Bigr )\Bigr )\, . \end{aligned}$$

Therefore, in the expansion

$$\begin{aligned} (HTLTH)_{RS}=\sum _P (c'_{PS}+c''_{PS}) H_{R\Delta S,P} \end{aligned}$$

each of the coefficients $c_{PS}=c'_{PS}+c''_{PS}$ is essentially bounded by the maximum of N and the product ps. The nonzero entries of the commutator [Q, HTLTH] correspond to values $(HTLTH)_{RS}$ such that $|R|\le K$ and $|S|>K$ or $|S|\le K$ and $|R|>K$. As previously indicated, norm bounds for $[Q,\, \mathrm{BDO}]$ based on these approximations of $C'$ and $C''$ will appear in future work.