This year we have a targeted search in all areas of quantum computing, with a particular emphasis on quantum algorithms and quantum complexity theory. Candidates interested in a faculty position should apply here.
Notes for tomorrow’s lecture on Approximation by ultrametrics are now up. (Yes, notes from the preceding two lectures still coming.)
(The combination of power and simplicity in this lecture is pretty incredible.)
Notes for the third lecture Metrical task systems on a weighted star are up.
After not generating any new posts for a long time, I am happy to have finally switched platforms. The new site is hosted on gitpages at tcsmath.github.io. While the domain name tcsmath.org should point there, I am trying to work around well-known https certificate issues that cause Chrome to have a nervous breakdown.
The motivation for switching platforms is a course I’ve just started co-teaching with Seb Bubeck on Competitive analysis via convex optimization. The notes for the first lecture are live: Regret minimization and competitive analysis.
1. Construction of Föllmer’s drift
In a previous post, we saw how an entropy-optimal drift process could be used to prove the Brascamp-Lieb inequalities. Our main tool was a result of Föllmer that we now recall and justify. Afterward, we will use it to prove the Gaussian log-Sobolev inequality.
where is a progressively measurable drift and such that has law .
where the minima are over all processes of the form (1).
thus we need only exhibit a drift achieving equality.
where is the Brownian semigroup defined by
We are left to show that has law and .
We will prove the first fact using Girsanov’s theorem to argue about the change of measure between and . As in the previous post, we will argue somewhat informally using the heuristic that the law of is a Gaussian random variable in with covariance . Itô’s formula states that this heuristic is justified (see our use of the formula below).
The following lemma says that, given any sample path of our process up to time , the probability that Brownian motion (without drift) would have “done the same thing” is .
Remark 1 I chose to present various steps in the next proof at varying levels of formality. The arguments have the same structure as corresponding formal proofs, but I thought (perhaps naïvely) that this would be instructive.
then under the measure given by
the process has the same law as .
Proof: We argue by analogy with the discrete proof. First, let us define the infinitesimal “transition kernel” of Brownian motion using our heuristic that has covariance :
We can also compute the (time-inhomogeneous) transition kernel of :
Here we are using that and is deterministic conditioned on the past, thus the law of is a normal with mean and covariance .
To avoid confusion of derivatives, let’s use for the density of and for the density of Brownian motion (recall that these are densities on paths). Now let us relate the density to the density . We use here the notations to denote a (non-random) sample path of :
where the last line uses .
Now by “heuristic” induction, we can assume , yielding
In the last line, we used the fact that is the infinitesimal transition kernel for Brownian motion.
From Lemma 2, it will follow that has the law where is the law of . In particular, has the law which was our first goal.
Given our preceding less formal arguments, let us use a proper stochastic calculus argument to establish (3). To do that we need a way to calculate
Notice that this involves both time and space derivatives.
Itô’s lemma. Suppose we have a continuously differentiable function that we write as where is a space variable and is a time variable. We can expand via its Taylor series:
Normally we could eliminate the terms , etc. since they are lower order as . But recall that for Brownian motion we have the heuristic . Thus we cannot eliminate the second-order space derivative if we plan to plug in (or , a process driven by Brownian motion). Itô’s lemma says that this consideration alone gives us the correct result:
This generalizes in a straightforward way to the higher dimensional setting .
With Itô’s lemma in hand, let us continue to calculate the derivative
For the time derivative (the first term), we have employed the heat equation
where is the Laplacian on .
Note that the heat equation was already contained in our “infinitesimal density” in the proof of Lemma 2, or in the representation , and Itô’s lemma was also contained in our heuristic that has covariance .
Using Itô’s formula again yields
giving our desired conclusion (3).
Our final task is to establish optimality: . We apply the formula (3):
where we used . Combined with (2), this completes the proof of the theorem.
2. The Gaussian log-Sobolev inequality
First, we discuss the correct way to interpret this. Define the Ornstein-Uhlenbeck semi-group by its action
This is the natural stationary diffusion process on Gaussian space. For every measurable , we have
The log-Sobolev inequality yields quantitative convergence in the relative entropy distance as follows: Define the Fisher information
One can check that
thus the Fisher information describes the instantaneous decay of the relative entropy of under diffusion.
So we can rewrite the log-Sobolev inequality as:
This expresses the intuitive fact that when the relative entropy is large, its rate of decay toward equilibrium is faster.
Martingale property of the optimal drift. Now for the proof of (5). Let be the entropy-optimal process with . We need one more fact about : The optimal drift is a martingale, i.e. for .
Let’s give two arguments to support this.
Argument one: Brownian bridges. First, note that by the chain rule for relative entropy, we have:
But from optimality, we know that the latter expectation is zero. Therefore -almost surely, we have
This implies that if we condition on the endpoint , then is a Brownian bridge (i.e., a Brownian motion conditioned to start at and end at ).
This implies that , as one can check that a Brownian bridge with endpoint is described by the drift process , and
That seemed complicated. There is a simpler way to see this: Given and any bridge from to , every “permutation” of the infinitesimal steps in has the same law (by commutativity, they all land at ). Thus the marginal law of at every point should be the same. In particular,
Argument two: Change of measure. There is a more succinct (though perhaps more opaque) way to see that is a martingale. Note that the process is a Doob martingale. But we have and we also know that is precisely the change of measure that makes into Brownian motion.
The latter quantity is . In the last equality, we used the fact that is precisely the change of measure that turns into Brownian motion.
After using it in the last post on non-positively curved surfaces, I thought it might be nice to give a simple proof of the Okamura-Seymour theorem in the dual setting. This argument arose out of conversations in 2007 with Amit Chakrabarti and former UW undergrad Justin Vincent. I was later informed by Yuri Rabinovich that he and Ilan Newman discovered a similar proof.
Note that establishing a node-capacitated version of the Okamura-Seymour theorem was an open question of Chekuri and Kawarabayashi. Resolving it positively is somewhat more difficult.
By rational approximation and subdivision of edges, we may assume that is unweighted. The following proof is constructive, and provides an explicit sequence of cuts on whose characteristic functions form the coordinates of the embedding . Each such cut is obtained by applying the following lemma. Note that for a subset of vertices, we use the notation , for the graph obtained from by contracting the edges across
Proof: Fix a plane embedding of that makes the boundary of the outer face. Since is -connected, is a cycle, and since is bipartite and has no parallel edges, .
Consider an arbitrary pair of distinct vertices on . There is a unique path from to that runs along counterclockwise; call this path . Consider a path and a vertex . We say that lies below if, in the plane embedding of , lies in the closed subset of the plane bounded by and .
(Note that the direction of is significant in the definition of “lying below,” i.e., belowness with respect to a path in is not the same as belowness with respect to the reverse of the same path in .)
We say that lies strictly below if lies below and . We use this notion of “lying below” to define a partial order on the paths in : for we say that is lower than if every vertex in lies below .
We now fix the pair and a path so that the following properties hold:
- If are distinct vertices with preceding and , then and .
- If is lower than and , then .
Note that a suitable pair exists because . Finally, we define the cut as follows: does not lie strictly below .
For the rest of this section, we fix the pair , the path and the cut as defined in the above proof.
Proof: If the lemma holds trivially, so assume . Also assume without loss of generality that precedes in the path . The conditions on imply that all vertices in lie strictly below . Therefore, the path is lower than and distinct from
By property (3), we have , which implies . Since is bipartite, the cycle formed by and must have even length; therefore, .
Proof: If there is nothing to prove. If not, we can write
for some where, for all , and . Let and denote the endpoints of with preceding in the path . Further, define and .
By Lemma 1, we have for . Since is a shortest path, we have for . Therefore
The latter quantity is precisely which completes the proof since .
Proof of Claim 1: Let be arbitrary and distinct. It is clear that , so it suffices to prove the opposite inequality. We begin by observing that
Let be a path in that achieves the minimum in the above expression. First, suppose . Then we must have . Now, , which implies and we are done.
Next, suppose . Then, there exists at least one vertex in that lies on . Let be the first such vertex and the last (according to the ordering in ) and assume that precedes in the path . Let . Note that may be trivial, because we may have . Now, , whence
where the first line follows from Eq. (1) and the definition of and the third line is obtained by applying Lemma 2 to the path . If at least one of lies in , then and we are done.
Therefore, suppose . Let . For a path and vertices on , let us use as shorthand for . By property (1), we have and since is bipartite, this means . By property (2), we have and . Using these facts, we now derive
Using this in (**) above and noting that , we get . This completes the proof.
Proof of Theorem 1: Assume that is -connected. We may also assume that is bipartite. To see why, note that subdividing every edge of by introducing one new vertex per edge leaves the metric essentially unchanged except for a scaling factor of .
We shall now prove the stronger statement that for every face of there exists a sequence of cuts of such that for all on , we have and that for , . We prove this by induction on .
The result is trivial in the degenerate case when is a single edge. For any larger and any cut , the graph is either a single edge or is -connected. Furthermore, contracting a cut preserves the parities of the lengths of all closed walks; therefore is also bipartite.
Apply the face-preserving cut lemma (Lemma 1) to obtain a cut . By the above observations, we can apply the induction hypothesis to to obtain cuts of corresponding to the image of in . Each cut induces a cut of . Clearly for any . Finally, for any , we have
where the first equality follows from the property of and the second follows from the induction hypothesis. This proves the theorem.