The last time I ate at a place that gives fortune cookies, I got this:

The second-to-last time I had dinner in a place that gives fortune cookies, however, I had got this:

Today it’s my favorite of Italy’s public holidays.

To keep a long story long, at the start of WW2, Italy, which was an ally of Germany, was initially neutral, in part because its armed forces were completely unprepared for war. At some point in the May of 1940, with German troops advancing into France, and British troops evacuating the continent, Italy decided to join what looked like a soon-to-end war, in order to claim some French territories and colonies.

But then, in 1941, Germany attacked Russia and Japan attacked the US, underestimating what they were getting into, and by the beginning of 1943 the tide was clearly turning against the “axis.” Italy’s king, who was definitely not the “fight until the last man” type, had Mussolini arrested, installed a general as prime minister, and started negotiating Italy’s surrender with the allies (even as Italian troops were fighting with the Germans in Russia and in Africa). Eventually, on September 8, 1943, the king announced a cease-fire. Because of the secrecy of the negotiations, nobody knew what was going in advance, and most of the Italian troops that were fighting with the Germans were taken prisoners, while the rest of the armed forces basically disbanded. German troops came into Italy from the North to occupy it, even as allied troops landed in Sicily and took control of most of Southern Italy. The king fled to the South, and the Germans freed Mussolini and installed him as head of a puppet government in the North.

With the Italian army disbanded, and with the allies neglecting the “Southern front” in Italy as they were plotting the landing in Normandy, guerilla groups were formed in Northern Italy to fight the Germans. Eventually, in April 1945 the German troops were retreating from the Eastern and Western fronts against the advancing American and Russian forces, and the allied made another push in Italy; concurrently, the resistance organizations planned an insurrection that, on April 25, liberated Torino and Milan. All the German forces in Italy surrendered on April 29.

The resistance was the training ground of some of the first generation of politicians of the new Italian Republic (a referendum to abolish the monarchy passed in 1946, and a new Republican constitution was approved in 1948), and it brought people who were willing to die for their ideals into politics. That spirit didn’t last very long, but it remains one of the few bright spots in recent Italian history.

Long in the making, the online course on expanders starts today.

In the first week of class: what are the topics of the course, and how to prove that the eigenvalues of the adjacency matrix of a regular graph tell you how many connected components there are in the graph.

[I have been asked by the office of public affairs of the Institute for Advanced Study to publicize the following press release. L.T.]

April 1, 2013. For immediate release.

Cofounders Jean Bourgain and Peter Sarnak announce today the launch of eXpandr, a new venture that aims to become the world’s leading provider of expander graphs.

“We are excited about our mission to change the way the world uses expanders.” said CEO Guli Mars, who joined eXpandr after a distinguished career in several leading technology companies. “Expanders are vital to revenue-generating logarithms, and our technology will revolutionize a multi-billion dollar market.”

“Big data, disruption”, said Juan Raman, senior vice president for marketing. “Innovation, cloud computing”, Mr. Raman continued.

“Let p be a prime congruent to 1 modulo 4″ said Jean Bourgain, cofounder and senior vice-president for analytic number theory, “and consider the irreducible representations of PSL(2,p).”

About the Institute for Advanced Study. The Institute for Advanced Study is one of the world’s leading centers for theoretical research and intellectual inquiry. The Institute exists to encourage and support fundamental research in the sciences and humanities—the original, often speculative thinking that produces advances in knowledge that change the way we understand the world. Work at the Institute takes place in four Schools: Historical Studies, Mathematics, Natural Sciences and Social Science. It provides for the mentoring of scholars by a permanent Faculty of no more than 28, and it offers all who work there the freedom to undertake research that will make significant contributions in any of the broad range of fields in the sciences and humanities studied at the Institute.

The Institute, founded in 1930, is a private, independent academic institution located in Princeton, New Jersey. Its more than 6,000 former Members hold positions of intellectual and scientific leadership throughout the academic world. Some 33 Nobel Laureates and 38 out of 52 Fields Medalists, as well as many winners of the Wolf or MacArthur prizes, have been affiliated with the Institute.

About eXpandr. eXpandr aims to disrupt the way the world uses expander graphs, and to become the leading commercial provider of expanders. eXpandr received \$2 million in angel investing and will launch its first product by Summer 2013.

Having (non-rigorously) defined the Laplacian operator in manifolds in the previous post, we turn to the proof of the Cheeger inequality in manifolds, which we restate below.

Theorem 1 (Cheeger’s inequality) Let ${M}$ be an ${n}$-dimensional smooth, compact, Riemann manifold without boundary with metric ${g}$, let ${L:= - {\rm div} \nabla}$ be the Laplace-Beltrami operator on ${M}$, let ${0=\lambda_1 \leq \lambda_2 \leq \cdots }$ be the eigenvalues of ${L}$, and define the Cheeger constant of ${M}$ to be

$\displaystyle h(M):= \inf_{S\subseteq M : \ 0 < \mu(S) \leq \frac 12 \mu(M)} \ \frac{\mu_{n-1}(\partial(S))}{\mu(S)}$

where the ${\partial (S)}$ is the boundary of ${S}$, ${\mu}$ is the ${n}$-dimensional measure, and ${\mu_{n-1}}$ is ${(n-1)}$-th dimensional measure defined using ${g}$. Then

$\displaystyle h(M) \leq 2 \sqrt{\lambda_2} \ \ \ \ \ (1)$

We begin by recalling the proof of the analogous result in graphs, and then we will repeat the same steps in the context of manifolds.

Theorem 2 (Cheeger’s inequality in graphs) Let ${G=(V,E)}$ be a ${d}$-regular graph, ${A}$ be its adjacency matrix, ${L:= I - \frac 1d A}$ be its normalized Laplacian matrix, ${0 = \lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_{|V|}}$ be the eigenvalues of ${L}$, and define ${\mu(S):= d \cdot |S|}$ for every subset of vertices ${S\subseteq V}$. Define the conductance of ${G}$ as

$\displaystyle \phi(G) := \min_{S\subseteq V: \ 0 < \mu(S) \leq \frac 12 \mu(V) } \frac{| \partial S|}{\mu(S)}$

where ${\partial S}$ is the number of edges with one endpoint in ${S}$ and one endpoint in ${\bar S}$. Then

$\displaystyle \phi(G) \leq \sqrt{2 \lambda_2} \ \ \ \ \ (2)$

1. Proof the Cheeger inequality in graphs

We will use the variational characterization of the eigenvalues of the Laplacian ${L}$ of a graph ${G}$.

$\displaystyle \lambda_1 = \min_{f \in {\mathbb R}^v} \frac {f^T Lf}{f^T f} \ \ \ \ \ (3)$

and if ${f_1}$ is a minimizer in the above expression then

$\displaystyle \lambda_2 = \min_{f \in {\mathbb R}^V: \ f \perp f_1} \frac {f^T Lf}{f^T f}$

Following the definition of ${L}$ we see that

$\displaystyle f^T L f = \frac 1d \sum_{(u,v)\in E} |f_u - f_v |^2$

and so the minimum in (3) is 0, and it is achieved for ${f = (1,\cdots,1)}$. This means that

$\displaystyle \lambda_2 = \min_{f \in {\mathbb R}^V: \ \sum_v f_v = 0} \frac {\sum_{(u,v) \in E} |f_u-f_v|^2}{d \sum_v f_v^2} \ \ \ \ \ (4)$

The expression in the right-hand-side of (4) is an important one, and it is called the Rayleigh quotient of ${f}$, which we will denote by ${R(f)}$:

$\displaystyle R(f):= \frac {\sum_{(u,v) \in E} |f_u-f_v|^2}{d \sum_v f_v^2}$

It is also useful to consider the variant of the Rayleigh quotient where there are no squares; this does not have a standard name, so let us call it the ${\ell_1}$ Rayleigh quotient and denote it by ${R_1}$:

$\displaystyle R_1(g):= \frac {\sum_{(u,v) \in E} |g_u-g_v|}{d \sum_v |g_v|}$

The proof of the graph Cheeger inequality now continues with the proof of the following three facts.

Lemma 3 (Rounding of ${\ell_1}$ embeddings) For every non-negative vector ${g\in {\mathbb R}^V_{\geq 0}}$ there is a value ${t\geq 0}$ such that

$\displaystyle \phi(\{ v: g(v) > t \}) \leq R_1(g)$

Lemma 4 (Embedding of ${\ell_2^2}$ into ${\ell_1}$) For every non-negative vector ${f\in {\mathbb R}^V_{\geq 0}}$, we have

$\displaystyle R_1(f^2) \leq \sqrt{2 R(f)}$

Lemma 5 (From an eigenvector to a non-negative vector) For every ${f\in {\mathbb R}^V}$ such that ${\sum_v f_v =0}$ there is a non-negative ${f'\in {\mathbb R}^V_{\geq 0}}$ such that ${\mu(\{ v: f'(v) >0 \}) \leq \frac 12 \mu(V)}$ and such that

$\displaystyle R(f') \leq R(f)$

Now let us start from a function ${f}$ that optimizes (4), so that ${\sum_v f_v = 0}$ and ${R(f) = \lambda_2}$, then apply Lemma 5 to find a function ${f'}$ such that the volume of the vertices having positive coordinates in ${f'}$ is at most ${\frac 12 \mu(V)}$ and such that ${R(f') \leq R(f) = \lambda_2}$. Then consider the vector ${g\in {\mathbb R}^V}$ such that ${g_v := f'^2_v}$; by Lemma 4, we have ${R_1(g) \leq \sqrt{2 R(f')} \leq \sqrt{2\lambda_2}}$, and by Lemma 3 there is a threshold ${t}$ such that the set ${S:= \{ v: g(v) > t \}}$ has conductance ${h(S) \leq \sqrt {2\lambda_2}}$. Since ${S}$ is a subset of the vertices having positive coordinates in ${g}$, we have ${\mu(S) \leq \frac 12 \mu(V)}$, and so

$\displaystyle \phi(G) \leq \sqrt{2 \lambda_2 }$

which is the Cheeger inequality for graphs. It remains to prove the three lemmas.

Proof: of Lemma 3. For each threshold ${t}$, define the set

$\displaystyle S_t := \{ v: g_v > t \}$

The idea of the proof is that if we pick ${t}$ at random then the probability that an edge belongs to ${\partial S_t}$ is proportional to ${|g_u - g_v|}$ and the probability that ${v\in S_t}$ is proportional to ${|g_v|}$, so that the expected number of edges in ${\partial S_t}$ is proportional to the numerator of ${R_1(g)}$ and the expected number of vertices in ${S_t}$ is proportional to the denominator of ${R_1(g)}$; if ${R_1(g)}$ is small, it is not possible for ${\partial S_t/\mu(S_t)}$ to always be large for every ${t}$.

To avoid having to normalize the range of ${t}$ to be between ${0}$ and ${1}$, instead of taking averages over a random choice of ${t}$, we will consider the integral over all values of ${t}$. We have

$\displaystyle \int_0^\infty |\partial (S_t) | {\rm d} t = \sum_{u,v} |g_u - g_v |$

because we can write ${|\partial (S_t)| = \sum_{(u,v) \in E} I_{u,v} (t)}$, where ${I_{u,v} (t) = 1}$ if ${(u,v) \in \partial S_t}$ and ${I_{u,v}(t) = 0}$ otherwise, and we see that only the values of ${t}$ between ${g_u}$ and ${g_v}$ make ${I_{u,v}(t) =1}$, so ${\int_{0}^\infty I_{u,v} (t) {\rm d} t = |g_u - g_v|}$.

We also have

$\displaystyle \int_0^\infty d |S_t| {\rm d} t = d \sum_v |g_v|$

and if we denote by ${t^*}$ the threshold such that ${\phi(S_{t^*})}$ is smallest among all the ${\phi(S)}$, then

$\displaystyle \sum_{u,v} |g_u - g_v | = \int_0^\infty |\partial (S_t) | {\rm d} t$

$\displaystyle \geq \int_0^\infty h(S_{t^*}) d|S_t| {\rm d} t$

$\displaystyle = h(S_{t^*}) d \sum_v |g_v|$

so that

$\displaystyle h(S_{t^*}) \leq \frac { \sum_{u,v} |g_u - g_v | }{d \sum_v |g_v| } = R_1(g)$

$\Box$

Proof: of Lemma 3. Let us consider the numerator of ${R_1(f^2)}$; it is:

$\displaystyle \sum_{(u,v)\in E} |f^2_u - f^2_v|$

$\displaystyle = \sum_{(u,v)\in E} |f_u - f_v| \cdot (f_u + f_v)$

$\displaystyle \leq \sqrt{\sum_{(u,v)\in E} |f_u - f_v|^2} \sqrt{ \sum_{(u,v)\in E}(f_u + f_v)^2 }$

(we used Cauchy-Swarz)

$\displaystyle \leq \sqrt{R(f) \cdot d\sum_v f_v^2} \sqrt{ \sum_{(u,v)\in E} 2f_u^2 + 2f_v^2 }$

(we used the definition of ${R_2}$ and Cauchy-Swarz again)

$\displaystyle = \sqrt{R(f) \cdot d\sum_v |f_v|^2} \sqrt{ 2 d\sum_{v} f_v^2 }$

$\displaystyle = \sqrt{2 R(f) } \cdot d \sum_v f_v^2$

And so

$\displaystyle R_1(f^2) = \frac {\sum_{(u,v)\in E} |f^2_u - f^2_v|}{d \sum_v f_v^2} \leq \sqrt{2 R_2(f) }$

$\Box$

Proof: of Lemma 5. Let ${m}$ be the median of ${f}$, and consider ${\bar f}$ defined as ${\bar f_v := f_v - m}$. We have

$\displaystyle R(\bar f) \leq R(f)$

because the numerators of ${R(\bar f)}$ and ${R(f)}$ are the same (the additive term ${-m}$ cancels). The denominators are such that

$\displaystyle \sum_v \bar f_v^2 = || \bar f||^2 = || f||^2 + || - m \cdot {\bf 1} ||^2 \geq || f||^2 = \sum_v f_v^2$

because ${f}$ and the vector ${{\bf 1} = (1,\ldots,1)}$ are orthogonal, and so by Pythagoras’s theorem the length-squared of ${\bar f = f - m {\bf 1}}$ equals the length-squared of ${f}$ plus the length-squared of ${-m {\bf 1}}$.

Let us define ${f^+_v := \min\{ 0, \bar f_v\}}$ and ${f^-_v := \min \{ 0, -\bar f_v\}}$ so that ${\bar f = f^+ - f^-}$. We use the following fact:

Fact 6 Let ${a,b \in {\mathbb R}^V_{\geq 0}}$ be disjointly supported non-negative vectors (“disjointly supported” means that they are non-zero on disjoint subsets of coordinates), then

$\displaystyle \min\{ R(a) , R(b) \} \leq R(a-b)$

Proof: The numerator of ${R(a-b)}$ is

$\displaystyle \sum_{(u,v)\in E} |a_v - b_v + b_u - a_v|^2 \geq \sum_{(u,v)\in E} |a_v - a_u|^2 + |b_v - b_u|^2$

and, using orthogonality and Pythagoras’s theorem, the denominator of ${R(a-b)}$ is

$\displaystyle d || a- b||^2 = d||a ||^2 + d|| b||^2$

The fact now follows from the inequality

$\displaystyle \min \left \{ \frac {n_1}{d_1} , \frac{n_2}{d_2} \right \} \leq \frac{n_1+n_2}{d_1+d_2}$

$\Box$

The lemma now follows by observing that ${f^+}$ and ${f^-}$ are non-negative and disjointly supported, so

$\displaystyle \min\{ R(f^+) , R(f^-) \} \leq R(\bar f) \leq R(f)$

and that both ${f^+}$ and ${f^-}$ have at most ${n/2}$ non-zero coordinate. $\Box$

2. Proof of the Cheeger inequality in manifolds

We will now translate the proof of the graph Cheeger inequality to the setting of manifolds.

As you may remember, we started off by saying that ${L}$ is symmetric and so all its eigenvalues are real and they are given by the variational characterization. Now we are already in trouble because the operator ${L}$ on manifolds cannot be thought of as a matrix, so what does it mean for it to be symmetric? The consequence of symmetry that is exploited in the analysis of the spectrum of symmetric matrices is the fact that if ${A}$ is symmetric, then for every ${x,y}$ we have

$\displaystyle \langle x,Ay \rangle = x^T Ay = (A^Tx)^Ty = (Ax)^Ty = \langle Ax,y \rangle$

and the property ${\langle x, Ay \rangle = \langle Ax , y \rangle}$ makes no references to coordinates, and it is well defined even for linear operators over infinite-dimensional spaces, provided that there is a notion of inner product. If we the define the inner product

$\displaystyle \langle f,g \rangle := \int_M f(x)\cdot g(x) \ {\rm d} \mu(x)$

on functions ${f: M \rightarrow {\mathbb R}}$, and more generally

$\displaystyle \langle f,g \rangle := \int_M \langle f(x), g(x)\rangle_X \ {\rm d} \mu(x)$

for functions ${f: M \rightarrow X}$, where ${X}$ is a vector space with inner product ${\langle \cdot,\cdot,\rangle_X}$, then we can say that an operator ${A}$ is self-adjoint if

$\displaystyle \langle f, Ag \rangle = \langle Af ,g \rangle$

for all (appropriately restricted) functions ${f,g}$. If ${M}$ is compact, this property is true for the Laplacian, and, in particular, ${-{\rm div}}$ and ${\nabla}$ are adjoints of each others, that is,

$\displaystyle \langle \nabla f, g \rangle = \langle f, - {\rm div} g \rangle$

(The discrete analog would be that ${C^T}$ is the transpose of ${C}$.)

Self-adjointness (and appropriate conditions on ${M}$) imply a version of the spectral theorem and of the variational characterization. In particular, all eigenvalues of ${L}$ are real, and if there is a minimum one then it is

$\displaystyle \lambda_1 = \min_{f: M \rightarrow {\mathbb R}} \frac {\langle f, Lf \rangle}{\langle f,f\rangle}$

and if ${f_1}$ is a minimizer of the above, then

$\displaystyle \lambda_2 = \min_{f: M \rightarrow {\mathbb R},\ \langle f, f_1 \rangle = 0} \frac {\langle f, Lf \rangle}{\langle f,f\rangle}$

(The minimization is quantified over all functions that are square-integrable, and the minimum is achieved because if ${M}$ is compact then the space of such functions is also compact and the cost function that we are minimizing is continuous. In this post, whenever we talk about “all functions,” it should be understood that we are restricting to whatever space of functions makes sense in the context.)

From the property that ${-{\rm div}}$ and ${\nabla}$ are adjoint, we have

$\displaystyle \langle f, Lf \rangle = \langle f, -{\rm div} \nabla f \rangle = \langle \nabla f,\nabla f \rangle$

so

$\displaystyle \lambda_1 = \min_{f:M \rightarrow {\mathbb R}} \frac{ \int_M \langle \nabla f(x),\nabla f(x) \rangle \ {\rm d} \mu(x) } {\int M f^2(x) \ {\rm d} \mu (x)}$

where the Rayleigh quotient

$\displaystyle R(f):= \frac{ \int_M \langle \nabla f(x),\nabla f(x) \rangle \ {\rm d} \mu(x) } {\int M f^2(x) \ {\rm d} \mu (x)} = \frac {\int_M ||\nabla f(x)||^2 \ {\rm d} \mu (x)}{\int_M f^2(x) \ {\rm d} \mu (x)}$

is always non-negative, and it is zero for constant ${f\equiv 1}$, so we see that ${\lambda_1=0}$ and

$\displaystyle \lambda_2 = \min_{f:M \rightarrow {\mathbb R} : \int_M f=0 \ {\rm d} \mu} R(f)$

By analogy with the graph case, we define the “${\ell_1}$ Rayleigh quotient”

$\displaystyle R_1(g) := \frac {\int_M ||\nabla g(x)|| \ {\rm d} \mu (x)}{\int_M |g(x)| \ {\rm d} \mu (x)}$

And we can prove the analogs of the lemmas that we proved for graphs.

Lemma 7 (Rounding of ${\ell_1}$ embeddings) For every non-negative function ${g: V \rightarrow {\mathbb R}_{\geq 0}}$ there is a value ${t\geq 0}$ such that

$\displaystyle h(\{ x: g(x) > t \}) \leq R_1(g)$

where the Cheeger constant ${h(S)}$ of a subset ${S\subseteq M}$ of the manifold is

$\displaystyle h(S) := \frac{\mu_{n-1} (\partial S)}{\mu(S)}$

Lemma 8 (Embedding of ${\ell_2^2}$ into ${\ell_1}$) For every non-negative function ${f; m \rightarrow {\mathbb R}_{\geq 0}}$, we have

$\displaystyle R_1(f^2) \leq \sqrt{2 R(f)}$

Lemma 9 (From an eigenfunction to a non-negative function) For every function ${f: M \rightarrow R}$ such that ${\int_M f \ {\rm d} \mu =0}$ there is a non-negative ${f': M \rightarrow {\mathbb R}_{\geq 0}}$ such that ${\mu(\{ x: f'(x) >0 \}) \leq \frac 12 \mu(V)}$ and such that

$\displaystyle R(f') \leq R_2(f)$

Let us see the proof of these lemmas.

Proof: of Lemma 7. For each threshold ${t}$, define the set

$\displaystyle S_t := \{ x: g(x) > t \}$

Let ${t*}$ be a threshold for which ${h(S_{t*})}$ is minimized

We will integrate the numerator and denominator of ${h(S_t)}$ over all ${t}$. The coarea formula for nonnegative functions is

$\displaystyle \int_M || \nabla g || {\rm d} \mu = \int_{0}^\infty \mu_{n-1} ( \partial \{ x: g(x) > t \}) {\rm d} t$

and we also have

$\displaystyle \int_M |g| {\rm d} \mu = \int_{0}^\infty \mu ( \{ x: g(x) > t \}) {\rm d} t$

which combine to

$\displaystyle \int_M || \nabla g || {\rm d} t = \int_0^\infty \mu_{n-1} ( \partial S_t) {\rm d} t$

$\displaystyle = \int_0^\infty h(S_t) \mu(S_t) {\rm d} t$

$\displaystyle \geq h(S_{t^*}) \int_0^\infty \mu(S_t) {\rm d} t$

$\displaystyle = h(S_{t^*}) \int_M g {\rm d} t$

so that

$\displaystyle h(S_{t^*}) \leq \frac{\int_M || \nabla g || {\rm d} t }{ \int_M g {\rm d} t} = R_1(g)$

$\Box$

Proof: of Lemma 8. Let us consider the numerator of ${R_1(f^2)}$; it is:

$\displaystyle \int_M || \nabla f^2|| {\rm d} \mu$

We can apply the chain rule, and see that

$\displaystyle \nabla f^2(x) = 2f(x) \cdot \nabla f(x)$

which implies

$\displaystyle \int_M || \nabla f^2 || {\rm d} \mu$

$\displaystyle = \int_M || 2f(x) \nabla f(x)|| {\rm d} \mu(x)$

$\displaystyle = \int_M 2|f(x)| \cdot ||\nabla f(x)|| {\rm d} \mu (x)$

and, after applying Caucy-Swarz,

$\displaystyle \leq \sqrt{\int_M 4 f^2(x) {\rm d} \mu(x)} \cdot \sqrt{\int_M ||\nabla f(x)||^2 {\rm d} \mu(x)}$

$\displaystyle = 2 \cdot \left( \int_M f^2 {\rm d} \mu \right) \cdot \sqrt{ R(f)}$

And so

$\displaystyle R_1(f^2) \leq 2 \sqrt{R_2(f) }$

$\Box$

Proof: of Lemma 9. Let ${m}$ be a median of ${f}$, and consider ${\bar f}$ defined as ${\bar f(x) := f(x) - m}$. We have

$\displaystyle R(\bar f) \leq R(f)$

because the numerators of ${R(\bar f)}$ and ${R(f)}$ are the same (the derivatives of functions that differ by a constant are identical) and the denominators are such that

$\displaystyle \int_M \bar f^2{\rm d} \mu$

$\displaystyle = \int_M f^2 - 2mf + m^2 {\rm d} \mu$

$\displaystyle = \int_M f^2 + m^2 {\rm d} \mu$

$\displaystyle \geq \int_M f^2 {\rm d} \mu$

where we used the fact the integral of ${f}$ is zero.

Let us define ${f^+(x) := \min\{ 0, \bar f(x)\}}$ and ${f^-_v := \min \{ 0, -\bar f(x)\}}$ so that ${\bar f(x) = f^+(x) - f^-(x)}$. We use the following fact:

Fact 10 Let ${a,b : M \rightarrow {\mathbb R}_{\geq 0}}$ be disjointly supported non-negative functions (“disjointly supported” means that they are non-zero on disjoint subsets of inputs), then

$\displaystyle \min\{ R(a) , R(b) \} \leq R(a-b)$

Proof: We begin with the following observation: if ${a}$ is a non-negative function, and ${a(x)=0}$, then ${\nabla a(x) = \{ \bf 0 \}}$, because ${x}$ has to be a local minimum.

Consider the expression ${||\nabla (a-b)||^2}$ occurring in the numerator of ${R(a-b)}$. We have

$\displaystyle ||\nabla (a-b)||^2$

$\displaystyle = || \nabla a - \nabla b ||^2$

$\displaystyle = || \nabla a||^2 + || \nabla b||^2 - 2 \langle \nabla a,\nabla b \rangle$

But

$\displaystyle \langle \nabla a,\nabla b \rangle = 0$

because for every ${x}$ at least one of ${a(x)}$ or ${b(x)}$ is zero, and so at least one of ${\nabla a(x)}$ or ${\nabla b(x)}$ is zero.

Using this fact, we have that the numerator of ${R(a-b)}$ is equal to the sum of the numerators of ${R(a)}$ and ${R(b)}$:

$\displaystyle \int_M ||\nabla a-b||^2 {\rm d} \mu = \int_M ||\nabla a-b||^2 {\rm d} \mu + \int_M ||\nabla a-b||^2 {\rm d}$

and the denominator of ${R(a-b)}$ is also the sum of the denominators of ${R(a)}$ and ${R(b)}$:

$\displaystyle =\int_M (a-b)^2 {\rm d} \mu$

$\displaystyle =\int_M a^2 {\rm d} \mu + \int_M b^2 {\rm d} \mu - 2 \int_M ab {\rm d} \mu$

$\displaystyle =\int_M a^2 {\rm d} \mu + \int_M b^2 {\rm d} \mu$

because ${a(x)b(x)=0}$ for every ${x}$. The fact now follows from the inequality

$\displaystyle \min \left \{ \frac {n_1}{d_1} , \frac{n_2}{d_2} \right \} \leq \frac{n_1+n_2}{d_1+d_2}$

$\Box$

The lemma now follows by observing that ${f^+}$ and ${f^-}$ are non-negative and disjointly supported, so

$\displaystyle \min\{ R(f^+) , R(f^-) \} \leq R(\bar f) \leq R(f)$

and that both ${f^+}$ and ${f^-}$ have a support of volume at most ${\frac 12 \mu(M)}$. $\Box$

If anybody is still reading, it is worth observing a couple of differences between the discrete proof and the continuous proof.

The ${\ell_1}$ Rayleigh quotient is defined slightly differently in the continuous case. It would correspond to defining it as

$\displaystyle \frac {\sum_{u\in V} \sqrt{\sum_{v: (u,v)\in E} (f_u-f_v)^2 } }{d \sum_v |f_v|}$

in the discrete case.

If ${a,b \in {\mathbb R}^V}$ are disjointly supported and nonnegative, the sum of the numerators of the Rayleigh quotients ${R(a)}$ and ${R(-b)}$ can be strictly smaller than the numerator of ${R(a-b)}$, while we always have equality in the continuous case. In the discrete case, the sum of the numerators of ${R(a)}$ and ${R(b)}$ can be up to twice the numerator of ${R(a+b)}$ (this fact is useful, but it did not come up in this proof), while again we have exact equality in the continuous case.

The chain rule calculation

$\displaystyle \nabla f^2 = 2f \nabla f$

corresponds to the step

$\displaystyle f_v^2 - f_u^2 = (f_v + f_u) \cdot (f_v - f_u)$

In the continuous case, ${f_v}$ and ${f_u}$ are “infinitesimally close”, so we can approximate ${f_v + f_u}$ by ${2f_v}$.

Readers of in theory have heard about Cheeger’s inequality a lot. It is a relation between the edge expansion (or, in graphs that are not regular, the conductance) of a graph and the second smallest eigenvalue of its Laplacian (a normalized version of the adjacency matrix). The inequality gives a worst-case analysis of the “sweep” algorithm for finding sparse cuts, it shows a necessary and sufficient for a graph to be an expander, and it relates the mixing time of a graph to its conductance.

Readers who have heard this story before will recall that a version of this result for vertex expansion was first proved by Alon and Milman, and the result for edge expansion appeared in a paper of Dodzuik, all from the mid-1980s. The result, however, is not called Cheeger’s inequality just because of Stigler’s rule: Cheeger proved in the 1970s a very related result on manifolds, of which the result on graphs is the discrete analog.

So, what is the actual Cheeger’s inequality?

Theorem 1 (Cheeger’s inequality) Let ${M}$ be an ${n}$-dimensional smooth, compact, Riemann manifold without boundary with metric ${g}$, let ${L:= - {\rm div} \nabla}$ be the Laplace-Beltrami operator on ${M}$, let ${0=\lambda_1 \leq \lambda_2 \leq \cdots }$ be the eigenvalues of ${L}$, and define the Cheeger constant of ${M}$ to be

$\displaystyle h(M):= \inf_{S\subseteq M : \ 0 < \mu(S) \leq \frac 12 \mu(M)} \ \frac{\mu_{n-1}(\partial(S))}{\mu(S)}$

where the ${\partial (S)}$ is the boundary of ${S}$, ${\mu}$ is the ${n}$-dimensional measure, and ${\mu_{n-1}}$ is ${(n-1)}$-th dimensional measure defined using ${g}$. Then

$\displaystyle h(M) \leq 2 \sqrt{\lambda_2} \ \ \ \ \ (1)$

The purpose of this post is to describe to the reader who knows nothing about differential geometry and who does not remember much multivariate calculus (that is, the reader who is in the position I was in a few weeks ago) what the above statement means, to describe the proof, and to see that it is in fact the same proof as the proof of the statement about graphs.

In this post we will define the terms appearing in the above theorem, and see their relation with analogous notions in graphs. In the next post we will see the proof.

The foundations of cryptography were laid down in 1982, the annus mirabilis that saw the publications of the work of Blum and Micali on pseudorandom generators, of Goldwasser and Micali on rigorous definitions of encryption, and of Yao, who gave a more general definitional approach. The paper of Shafi Goldwasser and Silvio Micali, in particular, introduced the incredibly influential concept of indistinguishability of distributions, and the idea of defining security in terms of simulation of an ideal model in which the security requirements are self-evident. (For example, because in the ideal model an adversary is not able to access the channel that we use to send encrypted data.) Almost every definition of security in cryptography follows the simulation approach, which also guides proofs of security. Shafi and Silvio both went on to do foundational work in cryptography, complexity theory, and algorithms, including their work on zero knowledge, secure multiparty computation, and property testing.

So it was with much joy, this early morning in Japan, that I heard the news that Shafi Goldwasser and Silvio Micali have been named as recipients of this year’s Turing award.

Omer Reingold has more information about their work. With no offense to colleagues around my age and younger, Shafi and Silvio are also representative of a time when leading theoretical computer scientists were more interesting people. They both have incredible charisma.

My favorite memory of Shafi and Silvio is from the time I interviewed for a faculty job at MIT. Shafi was in the last weeks of her pregnancy and did not make an appointment to see me, but then the day of my interview she changed her mind and showed up in Silvio’s office halfway through my meeting with him.

Silvio had been looking at my schedule and was giving me advice on how to talk to various people. Shafi asked what we were talking about, and then proceeded to give the opposite advice that Silvio had been giving me. The two of them spent the rest of the meeting arguing with each other.

Long in the planning, my online course on graph partitioning algorithms, expanders, and random walks, will start next month.

The course page is up for people to sign up. A friend of mine has compared my camera presence to Sheldon Cooper’s in “Fun with Flags,” which is sadly apt, but hopefully the material will speak for itself.

Meanwhile, I will be posting about some material that I have finally understood for the first time: the analysis of the Arora-Rao-Vazirani approximation algorithm, the Cheeger inequality in manifolds, and the use of the Selberg “3/16 theorem” to analyze expander constructions.

If you are not a fan of recorded performances, there will be a live show in Princeton at the end of June.

Click for full size