In this section, we prove the following higher order Cheeger inequalities.
Theorem 5.1
There exists an absolute constant \(C>0\) such that for any finite graph G with signature s and all \(n \in [N]\), we have
$$\begin{aligned} \frac{1}{2}\lambda _n(\Delta _{\mu }^s)\le h_n^s(\mu )\le Cn^3\sqrt{d_{\mu }\lambda _n(\Delta _{\mu }^s)}. \end{aligned}$$
(5.1)
Note that in Theorem 5.1 the signature group \(\Gamma \) can be either \(S_k^1\) or U(1).
The upper bound of \(h_n^s(\mu )\) in (5.1) is the essential part of Theorem 5.1 and its proof relies on the development of a proper spectral clustering algorithm for the operator \(\Delta _{\mu }^s\). In other words, we aim to find an n-subpartition \(\{ V_p \}_{[n]}\) with small constants \(\phi _{\mu }^s(V_p)\), based on the information contained in the eigenfunctions of the operator \(\Delta _{\mu }^s\).
Let \(f_i\) be an orthonormal family of eigenfunctions corresponding to \(\lambda _i(\Delta _{\mu }^s)\) for \(i \in [n]\). We consider the following map:
$$\begin{aligned} F: V\rightarrow \mathbb {C}^n, \quad F(u) = (f_1(u), f_2(u), \ldots , f_n(u)). \end{aligned}$$
(5.2)
Since \(\lambda _n(\Delta _{\mu }^s)=\mathcal {R}_{\mu }^s(f_n)\), the Rayleigh quotient of F is also bounded by \(\lambda _n(\Delta _\mu ^s)\):
$$\begin{aligned} \mathcal {R}_{\mu }^s(F)&:= \frac{\sum _{\{u,v\}\in E}w_{uv}\Vert F(u)-s_{uv}F(v)\Vert ^2}{\sum _{u\in V}\mu (u)\Vert F(u)\Vert ^2} \nonumber \\&= \frac{\sum _{p\in [n]}\sum _{\{u,v\}\in E}w_{uv}|f_p(u)-s_{uv}f_p(v)|^2}{\sum _{p\in [n]}\sum _{u\in V}\mu (u)|f_p(u)|^2}\nonumber \\&\le \lambda _n(\Delta _\mu ^s), \end{aligned}$$
(5.3)
where \(\Vert \cdot \Vert \) stands for the standard Hermitian norm in \(\mathbb {C}^n\). Our goal is to construct n maps \(\Psi _p: V \rightarrow {\mathbb C}^n\), \(p \in [n]\), with pairwise disjoint supports such that
-
(1)
each \(\Psi _p\) can be viewed as a localization of F, i.e., \(\Psi _p\) is the product of F and a cut-off function \(\eta : V\rightarrow \mathbb {R}\) (see 5.13 below),
-
(2)
each Rayleigh quotient satisfies \(\mathcal {R}_{\mu }^s(\Psi _p)\le C(n) \mathcal {R}_{\mu }^s(F)\), where C(n) is a constant only depending on n.
Then, applying Lemmas 4.4 and 4.8 will finish the proof.
This strategy is adapted from the proof of the higher order Cheeger inequalities for unsigned graphs due to Lee et al. [28, 29]. A critical new point here is to find a proper metric on the space of points \(\{F(u)|u\in V\}\subset \mathbb {C}^n\) for the spectral clustering algorithm. In other words, we need a proper metric to localize the map F. The original algorithm in [28, 29] used a spherical metric. The second author [32] studied a spectral clustering via metrics on real projective spaces to prove higher order dual Cheeger inequalities for unsigned graphs. Later in [3], the above two algorithms and, hence, the corresponding two kinds of inequalities, were unified in the framework of Harary’s signed graphs, i.e., graphs with signatures \(s: E^{or}\rightarrow \{+1,-1\}\). In particular, the metrics on real projective spaces were shown to be the proper metrics for clustering in the framework of signed graphs. In our current more general setting of graphs with signatures \(s:E^{or}\rightarrow \Gamma \), where \(\Gamma =S_k^1\) or \(\Gamma =U(1)\), the new metrics will be defined on lens spaces and complex projective spaces.
Lens spaces and complex projective spaces
In this subsection, we provide metrics of lens spaces and complex projective spaces for the spectral clustering algorithms in the case of \(\Gamma =S_k^1\) and \(\Gamma =U(1)\), respectively. Both lens spaces and complex projective spaces are important objects in geometry and topology. See, e.g., [26, Chapter 5] for details about these spaces.
Let \(\mathbb {S}^{2n-1}:=\{\mathbf {z}\in \mathbb {C}^n\mid \Vert \mathbf {z}\Vert =1\}\) be the unit sphere in the space \(\mathbb {C}^n\). Then \(\Gamma \subset {\mathbb C}\) acts on \(\mathbb {S}^{2n-1}\) by scalar multiplication. For any two points \(\mathbf {z}_1, \mathbf {z}_2\in \mathbb {S}^{2n-1}\subset \mathbb {C}^n\), we define the following equivalence relation:
$$\begin{aligned} \mathbf {z}_1\sim \mathbf {z}_2 \Leftrightarrow \exists \,\gamma \in \Gamma \quad \text {such that}\,\mathbf {z}_1=\gamma \mathbf {z}_2. \end{aligned}$$
(5.4)
For \(\Gamma = S_k^1\), the corresponding quotient space \(\mathbb {S}^{2n-1}/\Gamma \) is the lens space \(L(k;1,\ldots ,1)\), while for \(\Gamma =U(1)\), the quotient space \(\mathbb {S}^{2n-1}/\Gamma \) is the complex projective space \(\mathbb {C}P^{n-1}\). Let \([\mathbf {z}]\) denote the equivalence class of \(\mathbf {z}\in \mathbb {S}^{2n-1}\). We consider the following metric on \(\mathbb {S}^{2n-1}/\Gamma \):
$$\begin{aligned} d([\mathbf {z}_1], [\mathbf {z}_2]):=\min _{\gamma \in \Gamma }\Vert \mathbf {z}_1-\gamma \mathbf {z}_2\Vert . \end{aligned}$$
(5.5)
The space \(\mathbb {S}^{2n-1}/\Gamma \) can also be endowed with a distance \(d_{quot}\) which is induced from the standard Riemannian metric on \(\mathbb {S}^{2n-1} \subset {\mathbb R}^{2n}\). This induced metric has positive Ricci curvature. If \(\Gamma = S_k^1\), the sectional curvature of this metric is constant equal to 1, and if \(\Gamma = U(1)\), this metric is the well-known Fubini-Study metric. The two metrics d and \(d_{quot}\) on \(\mathbb {S}^{2n-1}/\Gamma \) are equivalent, i.e., there exist two constants \(c_1, c_2 > 0\) such that for all \([z_1], [z_2] \in S^{2n-1}/\Gamma \),
$$\begin{aligned} c_1d_{quot}([\mathbf {z}_1], [\mathbf {z}_2])\le d([\mathbf {z}_1], [\mathbf {z}_2])\le c_2 d_{quot}([\mathbf {z}_1], [\mathbf {z}_2]). \end{aligned}$$
(5.6)
Recall the concept of the metric doubling constant \(\rho _{\mathbb {X}}\) of a metric space \((\mathbb {X}, d_{\mathbb {X}})\). This constant is the infimum of all numbers \(\rho \) such that every ball B in \(\mathbb {X}\) can be covered by \(\rho \) balls of half the radius of B.
Proposition 5.2
The metric doubling constant \(\rho _\Gamma \) of \((\mathbb {S}^{2n-1}/\Gamma ,d)\) satisfies
$$\begin{aligned} \log _2\rho _\Gamma \le Cn, \end{aligned}$$
(5.7)
where C is an absolute constant.
Proof
By equivalence (5.6), we only need to consider the metric space \((\mathbb {S}^{2n-1}/\Gamma ,d_{quot})\). Since \(\mathbb {S}^{2n-1}/\Gamma \) with its standard metric has nonnegative Ricci curvature, the Bishop–Gromov comparison theorem guarantees
$$\begin{aligned} \frac{\mathrm {vol}(B_r([\mathbf {z_1}]))}{\mathrm {vol}(B_{r/2}([\mathbf {z_1}]))}\le \bar{C}^n, \end{aligned}$$
(5.8)
for some absolute constant \(\bar{C}\). (Note that the real dimension of the lens space is \(2n-1\) and of the complex projective space is \(2n-2\).) A standard argument implies now the claim of the proposition. For details see, e.g., [8, p. 67] or [32, Section 2.2].\(\square \)
The metric d on \(\mathbb {S}^{2n-1}/\Gamma \) induces a pseudo metric on the space \(\mathbb {C}^n\setminus \{0\}\), which—by abuse of notation—will again be denoted by d:
$$\begin{aligned} d(\mathbf {z}_1, \mathbf {z}_2):=d\left( \left[ \frac{\mathbf {z}_1}{\Vert \mathbf {z}_1\Vert }\right] ,\left[ \frac{\mathbf {z}_2}{\Vert \mathbf {z}_2\Vert }\right] \right) . \end{aligned}$$
(5.9)
The following obvious property is the reason why we use the metric d on \(S^{2n-1}/\Gamma \) from (5.5). This reason will become clear in the next Sect. 5.2.
Proposition 5.3
For every pair \(\mathbf {z}_1,\mathbf {z}_2\in \mathbb {C}^n\setminus \{0\}\) and every \(\gamma \in \Gamma \), we have
$$\begin{aligned} d(\mathbf {z}_1,\mathbf {z}_2)=d(\mathbf {z}_1,\gamma \mathbf {z}_2). \end{aligned}$$
(5.10)
The considerations of the next two subsections prepare the ground for the study of the Rayleigh quotient \(\mathcal {R}_{\mu }^s(F)\) of the map \(F: V\rightarrow \mathbb {C}^n\) defined in (5.2).
Localization of the map F
We endow the support \(V_F:=\{u\in V|F(u)\ne 0\}\) with the pseudo metric \(d_F\) induced by d via
$$\begin{aligned} d_F(u,v):=d(F(u), F(v)). \end{aligned}$$
(5.11)
Given a subset \(S\subseteq V\) and \(\epsilon >0\), we first define a cut-off function \(\eta : V\rightarrow \mathbb {R}\) by
$$\begin{aligned} \eta (u):= {\left\{ \begin{array}{ll} 0, &{} \text {if}\, F(u)=0, \\ \max \{0, 1-\frac{1}{\epsilon }d_F(u, S\cap V_F)\}, &{} \text {otherwise} \end{array}\right. } \end{aligned}$$
(5.12)
and then localize F via \(\eta \) as
$$\begin{aligned} \Psi :=\eta F: V\rightarrow \mathbb {C}^n. \end{aligned}$$
(5.13)
Note that the \(\epsilon \)-neighborhood \(N_{\epsilon }(S\cap V_F, d_F):=\{u\in V|d_F(u, S\cap V_F)<\epsilon \}\) of \(S\cap V_F\) contains the support of the map \(\Psi \).
In the next lemma, \(G_F=(V_F,E_F)\) denotes the induced subgraph on \(V_F\) of G.
Lemma 5.4
If \(\{u,v\}\in E_F\) and \(\Vert F(v)\Vert \le \Vert F(u)\Vert \) then
$$\begin{aligned} d(F(u), F(v))\Vert F(v)\Vert \le \Vert F(u)-s_{uv}F(v)\Vert . \end{aligned}$$
(5.14)
Proof
Observe that we only need to prove
$$\begin{aligned} d(F(u), F(v))\Vert F(v)\Vert \le \Vert F(u)-F(v)\Vert \end{aligned}$$
(5.15)
for any pair of points \(F(u), F(v)\in \mathbb {C}^n\setminus \{0\}\) with \(\Vert F(v)\Vert \le \Vert F(u)\Vert \): we can replace F(v) in (5.15) by \(s_{uv}F(v)\) and use Proposition 5.3 to obtain (5.14). By the definition of the metric d, we obtain (5.15) as follows:
$$\begin{aligned} d(F(u), F(v))\Vert F(v)\Vert \le&\left\| \frac{F(u)}{\Vert F(u)\Vert }-\frac{F(v)}{\Vert F(v)\Vert }\right\| \Vert F(v)\Vert \le \Vert F(u)-F(v)\Vert , \end{aligned}$$
where we used the estimate (4.7) for the latter inequality. \(\square \)
Lemma 5.4 enables us to prove the following result.
Lemma 5.5
For any \(\{u,v\}\in E\), we have
$$\begin{aligned} \Vert \Psi (u)-s_{uv}\Psi (v)\Vert \le \left( 1+\frac{1}{\epsilon }\right) \Vert F(u)-s_{uv}F(v)\Vert . \end{aligned}$$
(5.16)
Proof
If at least one of F(u) and F(v) is equal to zero, then the estimate (5.16) holds trivially. Hence, we suppose that \(u,v\in V_F\). W.l.o.g., we can assume that \(\Vert F(u)\Vert \le \Vert F(v)\Vert \) and calculate
$$\begin{aligned} \Vert \Psi (u)-s_{uv}\Psi (v)\Vert =&\ \Vert \eta (u)F(u)-s_{uv}\eta (v)F(v)\Vert \\ \le&\ |\eta (u)|\cdot \Vert F(u)-s_{uv}F(v)\Vert +|\eta (u)-\eta (v)|\cdot \Vert F(v)\Vert \\ \le&\ \Vert F(u)-s_{uv}F(v)\Vert +\frac{d_F(u,v)\Vert F(v)\Vert }{\epsilon }. \end{aligned}$$
Applying Lemma 5.4 completes the proof. \(\square \)
Note that the inequality (5.16) is useful for the estimate of the numerator of the Rayleigh quotient of \(\Psi \).
Decomposition of the underlying space via orthonormal functions
For later purposes, we work on a general measure space \((\mathcal {V},\mu )\) in this subsection, where \(\mathcal {V}\) is a topological space and \(\mu \) is a Borel measure. Two particular cases we have in mind are a vertex set V of a finite graph with a measure \(\mu : V\rightarrow \mathbb {R}^+\), and a closed Riemannian manifold with its Riemannian volume measure. We will apply the results in this subsection to the latter case in Sect. 7.
On \((\mathcal {V},\mu )\), we further assume that there exist n measurable functions
$$\begin{aligned} f_1, f_2, \ldots , f_n: \mathcal {V}\rightarrow \mathbb {C}, \end{aligned}$$
which are orthonormal, i.e., for any \(i,j\in [n]\),
$$\begin{aligned} \langle f_i, f_j\rangle :=\int _{\mathcal {V}}f_i\overline{f_j}d\mu =\delta _{ij}. \end{aligned}$$
Then the map \(F: \mathcal {V}\rightarrow \mathbb {C}^n\) is given accordingly as in (5.2).
We consider the measure \(\mu _F\) on \(\mathcal {V}\) given by
$$\begin{aligned} d\mu _F=\Vert F\Vert ^2d\mu . \end{aligned}$$
For any two points x, y in \(\mathcal {V}_F:=\{x\in \mathcal {V}: F(x)\ne 0\}\), we have the distance between them
$$\begin{aligned} d_F(x,y):=\min _{\gamma \in \Gamma }\left\| \frac{F(x)}{\Vert F(x)\Vert }-\gamma \frac{F(y)}{\Vert F(y)\Vert }\right\| . \end{aligned}$$
(5.17)
The main result of this subsection is the following theorem.
Theorem 5.6
Let \((\mathcal {V}_F, d_F, \mu _F)\) be as above. There exist an absolute constant \(C_0\) and a nontrivial n-subpartition \(\{T_i\}_{[n]}\) of \(\mathcal {V}_F\) such that
-
(i)
\(d_F(T_p, T_q)\ge \frac{2}{C_0n^{5/2}}\), for all \(p,q\in [n]\), \(p\ne q\),
-
(ii)
\(\mu _F(T_p)\ge \frac{1}{2n}\mu _F(\mathcal {V}_F)\), for all \(p\in [n]\).
The difficulty for the construction of the above n-subpartition is to achieve the property (ii). That is, we have to find a subpartition which possesses large enough measure. When \(d_F(x,y)\) is given by the spherical distance \(\left\| \frac{F(x)}{\Vert F(x)\Vert }-\frac{F(y)}{\Vert F(y)\Vert }\right\| \), Theorem 5.6 was proved in [28, 29, Lemma 3.5]. In our situation, we have to deal with the metrics, given in (5.17), of lens spaces or complex projective spaces. We refer the reader to [18] for another interesting decomposition result.
An important ingredient of the proof is the following lemma derived from the random partition theory [20, 30]. Note that a partition of a set A can also be considered as a map \(P:A\rightarrow 2^A\), where \(x\in A\) is mapped to the unique set P(x) of the partition that contains x. A random partition \(\mathcal {P}\) of A is a probability measure \(\nu \) on a set of partitions of A. Then \(\mathcal {P}(x)\) is understood as a random variable from the probability space to subsets of A containing x.
Lemma 5.7
Let A be a subset of the metric space \((\mathbb {S}^{2n-1}/\Gamma , d)\) (for d recall 5.5). Then for every \(r>0\) and \(\delta \in (0,1)\), there exists a random partition \(\mathcal {P}\) of A, i.e., a distribution \(\nu \) over partitions of A such that
-
(i)
\(\mathrm {diam}(S)\le r\) for any S in every partition P in the support of \(\nu \),
-
(ii)
\(\mathbb {P}_{\nu }\left[ B_{r/\alpha }(x)\subseteq \mathcal {P}(x)\right] \ge 1-\delta \) for all \(x\in A\), where \(\alpha =32\log _2(\rho _{\Gamma })/\delta \).
We refer to [20, Theorem 3.2] and [30, Lemma 3.11] for the proof, see also [32, Theorem 2.4]. For convenience, we describe briefly the construction of the random partition claimed in Lemma 5.7. Let \(\{x_i\}_{[m]}\) be a r / 4-net of \(\mathbb {S}^{2n-1}/\Gamma \), that is, \(d(x_i, x_j)\ge r/4\), for any \(i\ne j\), and \(\mathbb {S}^{2n-1}/\Gamma =\bigcup _{i\in [m]}B_{r/4}(x_i)\). Since \((\mathbb {S}^{2n-1}/\Gamma , d)\) is compact, m is a finite number. For \(R\in [r/4,r/2]\), we construct a partition of \((\mathbb {S}^{2n-1}/\Gamma , d)\) as follows. A permutation \(\sigma \) of the set [m] provides an order for all points in the net which is used to define, for every \(i\in [m]\),
$$\begin{aligned} S_{i}^{R,\sigma }:=\left\{ x\in \mathbb {S}^{2n-1}/\Gamma \mid x\in B_R(x_i)\text { and }\sigma (i)<\sigma (j)\text { for all } j\in [m] \text { with } x\in B_R(x_j)\right\} . \end{aligned}$$
That is, we have \(x\in S_{i}^{R,\sigma }\) if \(\sigma (i)\) is the smallest number for which x is contained in \(B_R(x_i)\). Then \(P^{R,\sigma }=\{S_{i}^{R,\sigma }\}_{[m]}\) constitutes a partition of \(\mathbb {S}^{2n-1}/\Gamma \). Now let \(\sigma \) be a uniformly random permutation of [m], and R be chosen uniformly random from the interval [r / 4, r / 2]. These choices define a random partition \(\mathcal {P}\). If we choose R uniformly from a fine enough discretization of the interval [r / 4, r / 2], we can make \(\mathcal {P}\) to be finitely supported. In fact, this random partition fulfills the two properties in Lemma 5.7.
Remark 5.8
Lemma 5.7 holds true for any metric space. In particular, the finiteness of the r / 4-net is not necessary. This is shown in [30, Lemma 3.11].
Lemma 5.7 leads to the following result. Note that, the property (ii) in Lemma 5.7 ensures the existence of at least one subpartition which captures a large fraction of the whole measure.
Lemma 5.9
On \((\mathcal {V}_F, d_F, \mu _F)\), for any \(r>0\) and \(\delta \in (0,1)\), there exists a nontrivial subpartition \(\{\widehat{S}_i\}_{[m]}\) such that
-
(i)
\(\mathrm {diam}(\widehat{S}_i, d_F)\le r\) for any \(i\in [m]\),
-
(ii)
\(d_F(\widehat{S}_i, \widehat{S}_j)\ge 2r/\alpha \), where \(\alpha =32\log _2(\rho _\Gamma )/\delta \),
-
(iii)
\(\sum _{i\in [m]}\mu _F(\widehat{S}_i)\ge (1-\delta )\mu _F(\mathcal {V}_F)\).
Proof
Let \(\mathcal {P}\) be the random partition on \(\mathcal {V}_F\) induced from the one constructed in Lemma 5.7 via the map F. Let \(I_{B_{r/\alpha }(x)\subseteq \mathcal {P}(x)}\) be the indicator function for the event that \(B_{r/\alpha }(x)\subseteq \mathcal {P}(x)\) happens. Then we obtain from Lemma 5.7 (ii)
$$\begin{aligned} \mathbb {E}_{\mathcal {P}}\left( \int _{\mathcal {V}}I_{B_{r/\alpha }(x)\subseteq \mathcal {P}(x)}d\mu _F(x)\right) \ge (1-\delta )\mu _F(\mathcal {V}) \end{aligned}$$
(5.18)
by interchanging the expectation and the integral. On the other hand, we have
$$\begin{aligned}&\mathbb {E}_{\mathcal {P}}\left( \int _{\mathcal {V}}I_{B_{r/\alpha }(x)\subseteq \mathcal {P}(x)}d\mu _F(x)\right) \nonumber \\&\quad =\sum _{P\in \mathcal {P}}\sum _{S\in P}\int _{S}I_{B_{r/\alpha }(x)\subseteq \mathcal {P}(x)}d\mu _F(x)\mathbb {P}_{\nu }(P)\nonumber \\&\quad =\sum _{P\in \mathcal {P}}\sum _{S\in P}\int _{\widehat{S}}d\mu _F(x)\mathbb {P}_\nu (P), \end{aligned}$$
(5.19)
where \(\widehat{S}:=\{x\in S: B_{r/\alpha }(x)\subseteq S\}\). Hence, there exists a partition \(P=\{S_i\}_{[m]}\) of \(\mathcal {V}_F\) for some natural number m such that
$$\begin{aligned} \sum _{i\in [m]}\mu _F(\widehat{S}_i)\ge (1-\delta )\mu _F(\mathcal {V}). \end{aligned}$$
(5.20)
This completes the proof. \(\square \)
In order to prove Theorem 5.6, we also need the following result.
Lemma 5.10
If a subset \(S\subseteq \mathcal {V}\) satisfies \(\mathrm {diam}(S\cap \mathcal {V}_F, d_F)\le r\) for some \(r\in (0,1)\), then
$$\begin{aligned} \mu _F(S)\le \frac{1}{n(1-r^2)}\mu _F(\mathcal {V}). \end{aligned}$$
(5.21)
Proof
W.l.o.g., we can assume that \(S\subseteq \mathcal {V}_F\). Using the fact that \(f_1, \ldots , f_n\) are orthonormal, we obtain the following two properties. First, we have
$$\begin{aligned} \mu _F(\mathcal {V})=\int _{\mathcal {V}}\sum _{p\in [n]}|f_p|^2d\mu =n. \end{aligned}$$
(5.22)
Second, we have for any \(\mathbf {z}:=(z_1,z_2,\ldots ,z_n)\in \mathbb {C}^n\) with \(\Vert \mathbf {z}\Vert =1\),
$$\begin{aligned} \int _{\mathcal {V}}\left| \langle \mathbf {z}, F(x)\rangle \right| ^2d\mu (x) = \int _{\mathcal {V}}\sum _{p,q\in [n]} z_p\overline{z_q}\overline{f_p(x)}f_q(x)d\mu (x)=1. \end{aligned}$$
(5.23)
Combining (5.22) and (5.23), we conclude for any \(y\in S\),
$$\begin{aligned} \frac{\mu _F(\mathcal {V})}{n} =&\int _{\mathcal {V}}\left| \left\langle \frac{F(y)}{\Vert F(y)\Vert }, F(x) \right\rangle \right| ^2d\mu (x)\nonumber \\ =&\int _{\mathcal {V}} \left| \left\langle \frac{F(y)}{\Vert F(y)\Vert }, \frac{F(x)}{\Vert F(x)\Vert } \right\rangle \right| ^2d\mu _F(x). \end{aligned}$$
(5.24)
Since \(|z|^2\ge \left( z+\overline{z}\right) ^2/4\) for each \(z\in \mathbb {C}\), we obtain that for any \(\gamma \in \Gamma \):
$$\begin{aligned} \left| \left\langle \frac{F(y)}{\Vert F(y)\Vert }, \frac{F(x)}{\Vert F(x)\Vert }\right\rangle \right| ^2 =&\ \left| \left\langle \frac{F(y)}{\Vert F(y)\Vert }, \gamma \frac{F(x)}{\Vert F(x)\Vert } \right\rangle \right| ^2\nonumber \\ \ge&\ \frac{1}{4}\left( 2-\left\| \frac{F(y)}{\Vert F(y)\Vert }-\gamma \frac{F(x)}{\Vert F(x)\Vert }\right\| ^2\right) ^2. \end{aligned}$$
(5.25)
Recalling (5.17), the definition of \(d_F\), we arrive at
$$\begin{aligned} \frac{\mu _F(\mathcal {V})}{n}\ge \int _{S}\left( 1-\frac{1}{2}d_F(y,x)^2\right) ^2d\mu _F(x)\ge (1-r^2)\mu _F(S). \end{aligned}$$
(5.26)
\(\square \)
Proof of Theorem 5.6
With Lemma 5.9 and 5.10 at hand, Theorem 5.6 can be proved similarly as [28, 29, Lemma 3.5], see also [32, Lemma 6.2]. For convenience, we recall it here. Let \(\{\widehat{S}_i\}_{[m]}\) be the subpartition constructed in Lemma 5.9. Then by Lemma 5.10, we have for each \(i\in [m]\),
$$\begin{aligned} \mu _F(\widehat{S}_i)\le \frac{1}{n(1-r^2)}\mu _F(\mathcal {V}). \end{aligned}$$
(5.27)
We apply the following procedure to \(\{\widehat{S}_i\}_{[m]}\). If we can find two sets of the subpartition, say \(\widehat{S}_i\) and \(\widehat{S}_j\), such that
$$\begin{aligned} \mu _F(\widehat{S}_i)\le \frac{1}{2n}\mu _F(\mathcal {V}),\quad \mu _F(\widehat{S}_j)\le \frac{1}{2n}\mu _F(\mathcal {V}), \end{aligned}$$
then replace them by \(\widehat{S}_i\cup \widehat{S}_j\). Thus, when we stop, we obtain the sets \(T_1, T_2, \ldots , T_l\) for some number l, such that
$$\begin{aligned} \mu _F(T_i)\le \frac{1}{n(1-r^2)}\mu _F(\mathcal {V}), \quad \forall \, i\in [l], \end{aligned}$$
and
$$\begin{aligned} \mu _F(T_i)\ge \frac{1}{2n}\mu _F(\mathcal {V}), \quad \forall \, i\in [l-1]. \end{aligned}$$
Setting \(r=\frac{1}{3\sqrt{n}}\) and \(\delta =\frac{1}{4n}\), we check that
$$\begin{aligned} (n-1)\cdot \frac{1}{n(1-r^2)}<1-\delta -\frac{1}{2n}. \end{aligned}$$
(5.28)
This implies that \(l\ge n\). Moreover, if we redefine \(T_n:=\bigcup _{j=n}^lT_j\), we have
$$\begin{aligned} \mu _F(T_n)\ge \frac{1}{2n}\mu _F(\mathcal {V}). \end{aligned}$$
(5.29)
Thus the subpartition \(\{T_i\}_{[n]}\) satisfies the property (ii). One can then verify the property (i) by Proposition 5.2 and Lemma 5.9. \(\square \)
Proof of Theorem 5.1
We first prove the upper bound of (5.1). Let \(\{T_i\}_{[n]}\) be the subpartition of \(V_F\) obtained from Theorem 5.6. Choosing \(\epsilon =\frac{1}{C_0n^{5/2}}\), we define the cut-off functions \(\eta _{p}\) as in (5.12) (replacing the set S there by \(T_p\)). Then the maps \(\Psi _p:=\eta _{p}F\), \(p\in [n]\), have pairwise disjoint support. Recalling that \(\Psi _p|_{T_p}=F|_{T_p}\), and applying Lemma 5.5 as well as fact (ii) of Theorem 5.6, we obtain that for any \(p\in [n]\),
$$\begin{aligned} \mathcal {R}_{\mu }^s(\Psi _p)&\le \left( 1+\frac{1}{\epsilon }\right) ^2\frac{\sum _{\{u,v\}\in E}w_{uv}\Vert F(u)-s_{uv}F(v)\Vert ^2}{\sum _{u\in T_p}\mu (u)\Vert F(u)\Vert ^2}\nonumber \\&\le 2n(1+C_0n^{5/2})^2\mathcal {R}_{\mu }^s(F)\le Cn^6\mathcal {R}_{\mu }^s(F), \end{aligned}$$
(5.30)
where C is an absolute constant. For every \(p\in [n]\), the map \(\Psi _p\) has at least one coordinate function \(\psi _p\) that satisfies \(\mathcal {R}_{\mu }^s(\psi _p) \le \mathcal {R}_{\mu }^s(\Psi _p)\). In particular, we find functions \(\psi _p\), \(p\in [n]\), with pairwise disjoint support and an absolute constant C such that
$$\begin{aligned} \mathcal {R}_{\mu }^s(\psi _p)\le Cn^6 \mathcal {R}_{\mu }^s(F). \end{aligned}$$
(5.31)
Now inequality (5.3) and Lemma 4.4 for \(\Gamma = S_k^1\) or Lemma 4.8 for \(\Gamma = U(1)\) yield the desired upper bound of (5.1).
Now we prove the lower bound of (5.1). Suppose that the n-way Cheeger constant \(h_n^{s}(\mu )\) is achieved by the nontrivial n-subpartition \(\{\widetilde{V}_p\}_{[n]}\) and that the function \(\widetilde{\tau }_p: \widetilde{V}_p\rightarrow \Gamma \) achieves the frustration index \(\iota ^s(\widetilde{V}_p)\) for each \(p\in [n]\). Moreover, consider functions \(\widetilde{f}_p: V \rightarrow {\mathbb C}\) with pairwise disjoint support given for \(p\in [n]\) by:
$$\begin{aligned} \widetilde{f}_p(u):={\left\{ \begin{array}{ll} \widetilde{\tau }_p(u), &{} \text {if}\, u\in \widetilde{V}_p; \\ 0, &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$
(5.32)
By the min–max principle (2.8), we know
$$\begin{aligned} \lambda _n(\Delta _{\mu }^s)\le \max _{a_1,\ldots , a_n}\mathcal {R}_{\mu }^s(\widetilde{f}_a), \end{aligned}$$
(5.33)
where the maximum is taken over all complex numbers \(a_1, \ldots , a_n \in {\mathbb C}\) such that the linear combination \(\widetilde{f}_a:=\sum _{p\in [n]}a_p\widetilde{f}_p\) of \(\widetilde{f}_1,\ldots , \widetilde{f}_n\) is nontrivial. This implies
$$\begin{aligned} \sum _{u\in V}\mu (u)|\widetilde{f}_a(u)|^2=\sum _{p\in [n]}|a_p|^2\mathrm {vol}_{\mu }(\widetilde{V}_p). \end{aligned}$$
(5.34)
We now want to relate (5.33) and (5.34) to the frustration index and the boundary measure. To that direction, we set \(B_{uv}:=w_{uv}|\widetilde{f}_a(u)-s_{uv}\widetilde{f}_a(v)|^2\) and obtain
$$\begin{aligned} \sum _{\{u,v\}\in E} B_{uv} = \frac{1}{2} \sum _{p,q \in [n]} \sum _{\begin{array}{c} u \in \widetilde{V}_p\\ v \in \widetilde{V}_q \end{array}} B_{uv} + \sum _{p\in [n]}\sum _{\begin{array}{c} u \in \widetilde{V}_p\\ v \in V^* \end{array}} B_{uv} + \frac{1}{2}\sum _{u,v \in V^*} B_{uv}, \end{aligned}$$
where \(V^* = \left( \bigcup _{p \in [n]} \widetilde{V}_p \right) ^c\). For \(u,v \in \widetilde{V}_p\), \(p \in [n]\), we have
$$\begin{aligned} |\widetilde{f}_a(u)-s_{uv}\widetilde{f}_a(v)|^2=|a_p|^2 \cdot |\widetilde{\tau }_p(u)-s_{uv}\widetilde{\tau }_p(v)|^2, \end{aligned}$$
(5.35)
while for \(u\in \widetilde{V}_p\) and \(v\in \widetilde{V}_q\) with \(p,q \in [n]\) and \(p\ne q\) we have
$$\begin{aligned} |\widetilde{f}_a(u)-s_{uv}\widetilde{f}_a(v)|^2=|a_p\widetilde{\tau }_p(u)-s_{uv}a_q\widetilde{\tau }_q(v)|^2\le 2(|a_p|^2+|a_q|^2). \end{aligned}$$
(5.36)
Now the definitions of the frustration index and of the boundary measure yield
$$\begin{aligned} \sum _{\{u,v\}\in E} B_{uv} \le&\, \sum _{p\in [n]}|a_p|^2\, \left( 2\iota ^s(\widetilde{V}_p) \, +\, 2\big |E(\widetilde{V}_p,\bigcup _{q\ne p}\widetilde{V}_q)\big | \, +\, \left| E(\widetilde{V}_p, V^*)\right| \right) \nonumber \\ \le&\ 2\, \sum _{p\in [n]}|a_p|^2\, \left( \iota ^s(\widetilde{V}_p) \, +\, \left| E(\widetilde{V}_p,\widetilde{V}_p^c)\right| \right) . \end{aligned}$$
(5.37)
If we now combine the estimates (5.33), (5.34), and (5.37), we arrive at
$$\begin{aligned} \lambda _n(\Delta _{\mu }^s)\le 2\max _{p\in [n]}\phi _{\mu }^s(\widetilde{V}_p)=2h_n^s(\mu ). \end{aligned}$$
(5.38)