(The combination of power and simplicity in this lecture is pretty incredible.)

]]>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.

Consider with , where is the standard Gaussian measure on . Let denote an -dimensional Brownian motion with . We consider all processes of the form

where is a progressively measurable drift and such that has law .

Theorem 1 (Föllmer)It holds that

where the minima are over all processes of the form (1).

*Proof:* In the preceding post (Lemma 2), we have already seen that for any drift of the form (1), it holds that

thus we need only exhibit a drift achieving equality.

We define

where is the Brownian semigroup defined by

As we saw in the previous post (Lemma 2), the chain rule yields

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 1I 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.

Lemma 2Let denote the law of . If we definethen 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 **

Consider again a measurable with . Let us define . Then the classical log-Sobolev inequality in Gaussian space asserts that

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

or equivalently

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.

**Proof of the log-Sobolev inequality.** In any case, now we are ready for the proof of (5). It also comes straight from Lehec’s paper. Since is a martingale, we have . So by Theorem 1:

The latter quantity is . In the last equality, we used the fact that is precisely the change of measure that turns into Brownian motion.

]]>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.

Theorem 1Given a weighted planar graph and a face of , there is a non-expansive embedding such that is an isometry.

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

Lemma 2 (Face-preserving cut lemma)Let be a -connected bipartite planar graph and be the boundary of a face of . There exists a cut such that

*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.

Lemma 1Let and be distinct vertices on and be such that . Then .

*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

This gives

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.

]]>As I have mentioned before, one of my favorite questions is whether the shortest-path metric on a planar graph embeds into with distortion. This is equivalent to such graphs having an -approximate multi-flow/min-cut theorem. We know that the distortion has to be at least 2. By a simple discretization and compactness argument, this is equivalent to the question of whether every simply-connected surface admits a bi-Lipschitz embedding into .

In a paper of Tasos Sidiropoulos, it is proved that every simply-connected surface of *non-positive curvature* admits a bi-Lipschitz embedding into . A followup work of Chalopin, Chepoi, and Naves shows that actually such a surface admits an *isometric* emedding into . In this post, we present a simple proof of this result that was observed in conversations with Tasos a few years ago—it follows rather quickly from the most classical theorem in this setting, the Okamura-Seymour theorem.

Suppose that is a geodesic metric space (i.e. the distance between any pair of points is realized by a geodesic whose length is ). One says that has *non-positive curvature* (in the sense of Busemann) if for any pair of geodesics and , the map given by

is convex.

Theorem 1Suppose that is homeomorphic to and is endowed with a geodesic metric such that has non-positive curvature. Then embeds isometrically in .

We will access the non-positive curvature property through the following fact. We refer to the book Metric spaces of non-positive curvature.

Lemma 2Every geodesic in can be extended to a bi-infinite geodesic line.

*Proof of Theorem 1:* By a standard compactness argument, it suffices to construct an isometric embedding for a finite subset . Let denote the convex hull of . (A set is *convex* if for every we have for every geodesic connecting to .)

It is an exercise to show that the boundary of is composed of a finite number of geodesics between points . For every pair , let denote a geodesic line containing and which exists by Lemma 2. Consider the collection of sets , and let denote the set of intersection points between geodesics in . Since is a collection of geodesics, and all geodesics intersect at most once (an easy consequence of non-positive curvature), the set is finite.

Consider finally the set . The geodesics in naturally endow with the structure of a planar graph , where two vertices are adjacent if they lie on a subset of some geodesic in and the portion between and does not contain any other points of . Note that is the outer face of in the natural drawing, where is the boundary of (the union of the geodesics in ).

We can put a path metric on this graph by defining the length of an edge as . Let denote the induced shortest-path metric on the resulting graph. By construction, we have the following two properties.

- If or for some , then .
- For every , there is a shortest path between two vertices in such that .

Both properties follow from our construction using the lines .

Now let us state the geometric (dual) version of the Okamura-Seymour theorem.

Theorem 3 (Okamura-Seymour dual version)For every planar graph and face , there is a -Lispchitz mapping such that is an isometry.

Let us apply this theorem to our graph and face . Consider and . By property (1) above, we have . Since , from Theorem 3, we have . But property (2) above says that and lie on a – shortest-path in . Since is -Lipschitz, we conclude that it maps the whole path isometrically, thus , showing that is an isometry, and completing the proof.

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*path space*. After finding an appropriate entropy-maximizer, the Brascamp-Lieb inequality will admit a gorgeous one-line proof. Our argument is taken from the beautiful paper of Lehec.

For simplicity, we start first with an entropy optimization on a discrete path space. Then we move on to Brownian motion.

**1.1 Entropy optimality on discrete path spaces **

Consider a finite state space and a transition kernel . Also fix some time .

Let denote the space of all paths . There is a natural measure on coming from the transition kernel:

Now suppose we are given a starting point , and a target distribution specified by a function scaled so that . If we let denote the law of , then this simply says that is a density with respect to . One should think about as the natural law at time (given ), and describes a perturbation of this law.

Let us finally define the set of all measures on that start at and end at , i.e. those measures satisfying

and for every ,

Now we can consider the entropy optimization problem:

One should verify that, like many times before, we are minimizing the relative entropy over a polytope.

One can think of the optimization as simply computing the most likely way for a mass of particles sitting at to end up in the distribution at time .

The optimal solution exists and is unique. Moreover, we can describe it explicitly: is given by a time-inhomogeneous Markov chain. For , this chain has transition kernel

where is the *heat semigroup* of our chain , i.e.

Let denote the time-inhomogeneous chain with transition kernels and and let denote the law of the random path . We will now verify that is the optimal solution to (1).

We first need to confirm that , i.e. that has law . To this end, we will verify inductively that has law . For , this follows by definition. For the inductive step:

We have confirmed that . Let us now verify its optimality by writing

where the final equality uses the fact we just proved: has law . Continuing, we have

where the final inequality uses the definition of in (2). The latter quantity is precisely by the chain rule for relative entropy.

**Exercise:** One should check that if and are two time-inhomogeneous Markov chains on with respective transition kernels and then indeed the chain rule for relative entropy yields

We conclude that

and from this one immediately concludes that . Indeed, for any measure , we must have . This follows because is the law of the endpoint of a path drawn from and is the law of the endpoint of a path drawn from . The relative entropy between the endpoints is certainly less than along the entire path. (This intuitive fact can again be proved via the chain rule for relative entropy by conditioning on the endpoint of the path.)

** 1.2. The Brownian version **

Let us now do the same thing for processes driven by Brownian motion in . Let be a Brownian motion with . Let be the standard Gaussian measure and recall that has law .

We recall that if we have two measures and on such that is absolutely continuous with respect to , we define the *relative entropy*

Our “path space” will consist of drift processes of the form

where denotes the drift. We require that is progressively measurable, i.e. that the law of is determined by the past up to time , and that . Note that we can write such a process in differential notation as

with .

Fix a smooth density with . In analogy with the discrete setting, let us use to denote the set of processes that can be realized in the form (4) and such that and has law .

Let us also use the shorthand to represent the entire path of the process. Again, we will consider the entropy optimization problem:

As in the discrete setting, this problem has a unique optimal solution (in the sense of stochastic processes). Here is the main result.

Theorem 1 (FĂ¶llmer)If is the optimal solution to (5), then

Just as for the discrete case, one should think of this as asserting that the optimal process only uses as much entropy as is needed for the difference in laws at the endpoint. The RHS should be thought of as an integral over the expected relative entropy generated at time (just as in the chain rule expression (3)).

The reason for the quadratic term is the usual relative entropy approximation for infinitesimal perturbations. For instance, consider the relative entropy between a binary random variable with expected value and a binary random variable with expected value :

I am going to delay the proof of Theorem 1 to the next post because doing it in an elementary way will require some discussion of Ito calculus. For now, let us prove the following.

Lemma 2For any process given by a drift , it holds that

*Proof:* The proof will be somewhat informal. It can be done easily using Girsanov’s theorem, but we try to keep the presentation here elementary and in correspondence with the discrete version above.

Let us first use the chain rule for relative entropy to calculate

Note that has the law of a standard -dimensional of covariance .

If is an -dimensional Gaussian with covariance and , then

Therefore:

where the latter expectation is understood to be conditioned on the past up to time .

In particular, plugging this into (6), we have

** 1.3. Brascamp-Lieb **

The proof is taken directly from Lehec. We will use the entropic formulation of Brascamp-Lieb due to Carlen and Cordero-Erausquin.

Let be a Euclidean space with subspaces . Let denote the orthogonal projection onto . Now suppose that for positive numbers , we have

By (8), we have for all :

The latter equality uses the fact that each is an orthogonal projection.

Let denote a standard Gaussian on , and let denote a standard Gaussian on for each .

Theorem 3 (Carlen & Cordero-Erausquin version of Brascamp-Lieb)For any random vector , it holds that

*Proof:* Let with denote the entropy-optimal drift process such that has the law of . Then by Theorem 1,

where the latter inequality uses Lemma 2 and the fact that has law .

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I wanted to post here a draft of the lecture notes. These extend and complete the series of posts here on non-negative and psd rank and lifts of polytopes. They also incorporate many corrections, and have exercises of varying levels of difficulty. The bibliographic references are sparse at the moment because I am posting them from somewhere in the Adriatic (where wifi is also sparse).

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