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.
Isometric embedding of non-positively curved surfaces into
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
We will access the non-positive curvature property through the following fact. We refer to the book Metric spaces of non-positive curvature.
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.
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.
After Boaz posted on the mother of all inequalities, it seemed about the right time to get around to the next series of posts on entropy optimality. The approach is the same as before, but now we consider entropy optimality on a 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 ,
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 .
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.
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
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
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 .
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 (Föllmer) 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.
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.
Note that has the law of a standard -dimensional of covariance .
If is an -dimensional Gaussian with covariance and , then
where the latter expectation is understood to be conditioned on the past up to time .
In particular, plugging this into (6), we have
The proof is taken directly from Lehec. We will use the entropic formulation of Brascamp-Lieb due to Carlen and Cordero-Erausquin.
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 .
Recall that our goal is to sketch a proof of the following theorem, where the notation is from the last post. I will assume a knowledge of the three posts on polyhedral lifts and non-negative rank, as our argument will proceed by analogy.
In this post, we will see how John’s theorem can be used to transform a psd factorization into one of a nicer analytic form. Using this, we will be able to construct a convex body that contains an approximation to every non-negative matrix of small psd rank.
1.1. Finite-dimensional operator norms
Let denote a finite-dimensional Euclidean space over equipped with inner product and norm . For a linear operator , we define the operator, trace, and Frobenius norms by
Let denote the set of self-adjoint linear operators on . Note that for , the preceding three norms are precisely the , , and norms of the eigenvalues of . For , we use to denote that is positive semi-definite and for . We use for the set of density operators: Those with and .
One should recall that is an inner product on the space of linear operators, and we have the operator analogs of the Hölder inequalities: and .
1.2. Rescaling the psd factorization
As in the case of non-negative rank, consider finite sets and and a matrix . For the purposes of proving a lower bound on the psd rank of some matrix, we would like to have a nice analytic description.
To that end, suppose we have a rank- psd factorization
where and . The following result of Briët, Dadush and Pokutta (2013) gives us a way to “scale” the factorization so that it becomes nicer analytically. (The improved bound stated here is from an article of Fawzi, Gouveia, Parrilo, Robinson, and Thomas, and we follow their proof.)
Proof: Start with a rank- psd factorization . Observe that there is a degree of freedom here, because for any invertible operator , we get another psd factorization .
Let and . Set . We may assume that and both span (else we can obtain a lower-rank psd factorization). Both sets are bounded by finiteness of and .
where denotes the unit ball in the Euclidean norm. Let us now set and .
Eigenvalues of : Let be an eigenvector of normalized so the corresponding eigenvalue is . Then , implying that (here we use that for any ). Since , (2) implies that . We conclude that every eigenvalue of is at most .
Finally, observe that for any and , we have
1.3. Convex proxy for psd rank
Again, in analogy with the non-negative rank setting, we can define an “analytic psd rank” parameter for matrices :
Note that we have implicit equipped and with the uniform measure. The main point here is that can be arbitrary. One can verify that is convex.
And there is a corresponding approximation lemma. We use and .
Lemma 3 For every non-negative matrix and every , there is a matrix such that and
Using Lemma 2 in a straightforward way, it is not particularly difficult to construct the approximator . The condition poses a slight difficulty that requires adding a small multiple of the identity to the LHS of the factorization (to avoid a poor condition number), but this has a correspondingly small effect on the approximation quality. Putting “Alice” into “isotropic position” is not essential, but it makes the next part of the approach (quantum entropy optimization) somewhat simpler because one is always measuring relative entropy to the maximally mixed state.