Optimization


I blogged about the Douglas-Rachford method before and in this post I’d like to dig a bit into the history of the method.

As the name suggests, the method has its roots in a paper by Douglas and Rachford and the paper is

Douglas, Jim, Jr., and Henry H. Rachford Jr., “On the numerical solution of heat conduction problems in two and three space variables.” Transactions of the American mathematical Society 82.2 (1956): 421-439.

At first glance, the title does not suggest that the paper may be related to monotone inclusions and if you read the paper you’ll not find any monotone operator mentioned. So let’s start and look at Douglas and Rachford’s paper.

1. Solving the heat equation numerically

So let us see, what they were after and how this is related to what is known as Douglas-Rachford splitting method today.

Indeed, Douglas and Rachford wanted to solve the instationary heat equation

\displaystyle \begin{array}{rcl} \partial_{t}u &=& \partial_{xx}u + \partial_{yy}u \\ u(x,y,0) &=& f(x,y) \end{array}

with Dirichlet boundary conditions (they also considered three dimensions, but let us skip that here). They considered a rectangular grid and a very simple finite difference approximation of the second derivatives, i.e.

\displaystyle \begin{array}{rcl} \partial_{xx}u(x,y,t)&\approx& (u^{n}_{i+1,j}-2u^{n}_{i,j}+u^{n}_{i-1,j})/h^{2}\\ \partial_{yy}u(x,y,t)&\approx& (u^{n}_{i,j+1}-2u^{n}_{i,j}+u^{n}_{i,j-1})/h^{2} \end{array}

(with modifications at the boundary to accomodate the boundary conditions). To ease notation, we abbreviate the difference quotients as operators (actually, also matrices) that act for a fixed time step

\displaystyle \begin{array}{rcl} (Au^{n})_{i,j} &=& (u^{n}_{i+1,j}-2u^{n}_{i,j}+u^{n}_{i-1,j})/h^{2}\\ (Bu^{n})_{i,j} &=& (u^{n}_{i,j+1}-2u^{n}_{i,j}+u^{n}_{i,j+1})/h^{2}. \end{array}

With this notation, our problem is to solve

\displaystyle \begin{array}{rcl} \partial_{t}u &=& (A+B)u \end{array}

in time.

Then they give the following iteration:

\displaystyle Av^{n+1}+Bw^{n} = \frac{v^{n+1}-w^{n}}{\tau} \ \ \ \ \ (1)

 

\displaystyle Bw^{n+1} = Bw^{n} + \frac{w^{n+1}-v^{n+1}}{\tau} \ \ \ \ \ (2)

 

(plus boundary conditions which I’d like to swipe under the rug here). If we eliminate {v^{n+1}} from the first equation using the second we get

\displaystyle (A+B)w^{n+1} = \frac{w^{n+1}-w^{n}}{\tau} + \tau AB(w^{n+1}-w^{n}). \ \ \ \ \ (3)

 

This is a kind of implicit Euler method with an additional small term {\tau AB(w^{n+1}-w^{n})}. From a numerical point of it has one advantage over the implicit Euler method: As equations (1) and (2) show, one does not need to invert {I-\tau(A+B)} in every iteration, but only {I-\tau A} and {I-\tau B}. Remember, this was in 1950s, and solving large linear equations was a much bigger problem than it is today. In this specific case of the heat equation, the operators {A} and {B} are in fact tridiagonal, and hence, solving with {I-\tau A} and {I-\tau B} can be done by Gaussian elimination without any fill-in in linear time (read Thomas algorithm). This is a huge time saver when compared to solving with {I-\tau(A+B)} which has a fairly large bandwidth (no matter how you reorder).

How do they prove convergence of the method? They don’t since they wanted to solve a parabolic PDE. They were after stability of the scheme, and this can be done by analyzing the eigenvalues of the iteration. Since the matrices {A} and {B} are well understood, they were able to write down the eigenfunctions of the operator associated to iteration (3) explicitly and since the finite difference approximation is well understood, they were able to prove approximation properties. Note that the method can also be seen, as a means to calculate the steady state of the heat equation.

We reformulate the iteration (3) further to see how {w^{n+1}} is actually derived from {w^{n}}: We obtain

\displaystyle (-I + \tau(A+B) - \tau^{2}AB)w^{n+1} = (-I-\tau^{2}AB)w^{n} \ \ \ \ \ (4)

 

2. What about monotone inclusions?

What has the previous section to do with solving monotone inclusions? A monotone inclusion is

\displaystyle \begin{array}{rcl} 0\in Tx \end{array}

with a monotone operator, that is, a multivalued mapping {T} from a Hilbert space {X} to (subsets of) itself such that for all {x,y\in X} and {u\in Tx} and {v\in Ty} it holds that

\displaystyle \begin{array}{rcl} \langle u-v,x-y\rangle\geq 0. \end{array}

We are going to restrict ourselves to real Hilbert spaces here. Note that linear operators are monotone if they are positive semi-definite and further note that monotone linear operators need not to be symmetric. A general approach to the solution of monotone inclusions are so-called splitting methods. There one splits {T} additively {T=A+B} as a sum of two other monotone operators. Then one tries to use the so-called resolvents of {A} and {B}, namely

\displaystyle \begin{array}{rcl} R_{A} = (I+A)^{-1},\qquad R_{B} = (I+B)^{-1} \end{array}

to obtain a numerical method. By the way, the resolvent of a monotone operator always exists and is single valued (to be honest, one needs a regularity assumption here, namely one need maximal monotone operators, but we will not deal with this issue here).

The two operators {A = \partial_{xx}} and {B = \partial_{yy}} from the previous section are not monotone, but {-A} and {-B} are, so the equation {-Au - Bu = 0} is a special case of a montone inclusion. To work with monotone operators we rename

\displaystyle \begin{array}{rcl} A \leftarrow -A,\qquad B\leftarrow -B \end{array}

and write the iteration~(4) in terms of monotone operators as

\displaystyle \begin{array}{rcl} (I + \tau(A+B) + \tau^{2}AB)w^{n+1} = (I+\tau^{2}AB)w^{n}, \end{array}

i.e.

\displaystyle \begin{array}{rcl} w^{n+1} = (I+\tau A+\tau B+\tau^{2}AB)^{-1}(I+\tau AB)w^{n}. \end{array}

Using {I+\tau A+\tau B + \tau^{2}A = (I+\tau A)(I+\tau B)} and {(I+\tau^{2}AB) = (I-\tau B) + (I + \tau A)\tau B} we rewrite this in terms of resolvents as

\displaystyle \begin{array}{rcl} w^{n+1} & = &(I+\tau B)^{-1}[(I+\tau A)^{-1}(I-\tau B) + \tau B]w^{n}\\ & =& R_{\tau B}(R_{\tau A}(w^{n}-\tau Bw^{n}) + \tau Bw^{n}). \end{array}

This is not really applicable to a general monotone inclusion since there {A} and {B} may be multi-valued, i.e. the term {Bw^{n}} is not well defined (the iteration may be used as is for splittings where {B} is monotone and single valued, though).

But what to do, when both and {A} and {B} are multivaled? The trick is, to introduce a new variable {w^{n} = R_{\tau B}(u^{n})}. Plugging this in throughout leads to

\displaystyle \begin{array}{rcl} R_{\tau B} u^{n+1} & = & R_{\tau B}(R_{\tau A}(R_{\tau B}u^{n}-\tau B R_{\tau B}u^{n}) + \tau B R_{\tau B}u^{n}). \end{array}

We cancel the outer {R_{\tau B}} and use {\tau B R_{\tau B}u^{n} = u^{n} - R_{\tau B}u^{n}} to get

\displaystyle \begin{array}{rcl} u^{n+1} & = & R_{\tau A}(2R_{\tau B}u^{n} - u^{n}) + u^{n} - R_{\tau B}u^{n} \end{array}

and here we go: This is exactly what is known as Douglas-Rachford method (see the last version of the iteration in my previous post). Note that it is not {u^{n}} that converges to a solution, but {w^{n} = R_{\tau B}u^{n}}, so it is convenient to write the iteration in the two variables

\displaystyle \begin{array}{rcl} w^{n} & = & R_{\tau B}u^{n}\\ u^{n+1} & = & R_{\tau A}(2w^{n} - u^{n}) + u^{n} - w^{n}. \end{array}

The observation, that these splitting method that Douglas and Rachford devised for linear problems has a kind of much wider applicability is due to Lions and Mercier and the paper is

Lions, Pierre-Louis, and Bertrand Mercier. “Splitting algorithms for the sum of two nonlinear operators.” SIAM Journal on Numerical Analysis 16.6 (1979): 964-979.

Other, much older, splitting methods for linear systems, such as the Jacobi method, the Gauss-Seidel method used different properties of the matrices such as the diagonal of the matrix or the upper and lower triangluar parts and as such, do not generalize easily to the case of operators on a Hilbert space.

Consider a convex optimization problem of the form

\displaystyle \begin{array}{rcl} \min_{x}F(x) + G(Ax) \end{array}

with convex {F} and {G} and matrix {A}. (We formulate everything quite loosely, skipping over details like continuity and such, as they are irrelevant for the subject matter). Optimization problems of this type have a specific type of dual problem, namely the Fenchel-Rockafellar dual, which is

\displaystyle \begin{array}{rcl} \max_{y}-F^{*}(-A^{T}y) - G^{*}(y) \end{array}

and under certain regularity conditions it holds that the optimal value of the dual equals the the objective value of the primal and, moreover, that a pair {(x^{*},y^{*})} is both primal and dual optimal if and only if the primal dual gap is zero, i.e. if and only if

\displaystyle \begin{array}{rcl} F(x^{*})+G(Ax^{*}) + F^{*}(-A^{T}y^{*})+G^{*}(y^{*}) = 0. \end{array}

Hence, it is quite handy to use the primal dual gap as a stopping criteria for iterative methods to solve these problems. So, if one runs an algorithm which produces primal iterates {x^{k}} and dual iterates {y^{k}} one can monitor

\displaystyle \begin{array}{rcl} \mathcal{G}(x^{k},y^{k}) = F(x^{k})+G(Ax^{k}) + F^{*}(-A^{T}y^{k})+G^{*}(y^{k}). \end{array}

and stop if the value falls below a desired tolerance.

There is some problem with this approach which appears if the method produces infeasible iterates in the sense that one of the four terms in {\mathcal{G}} is actually {+\infty}. This may be the case if {F} or {G} are not everywhere finite or, loosely speaking, have linear growth in some directions (since then the respective conjugate will not be finite everywhere). In the rest of the post, I’ll sketch a general method that can often solve this particular problem.

For the sake of simplicity, consider the following primal dual algorithm

\displaystyle \begin{array}{rcl} x^{k+1} & = &\mathrm{prox}_{\tau F}(x^{k}-A^{T}y^{k})\\ y^{k+1} & = &\mathrm{prox}_{\sigma G^{*}}(y^{k}+\sigma A(2x^{k+1}-x^{k})) \end{array}

(also know as primal dual hybrid gradient method or Chambolle-Pock’s algorithm). It converges as soon as {\sigma\tau\leq \|A\|^{-2}}.

While the structure of the algorithm ensures that {F(x^{k})} and {G^{*}(y^{k})} are always finite (since always {\mathrm{prox}_{F}(x)\in\mathrm{dom}(F)}), is may be that {F^{*}(-A^{T}y^{k})} or {G(Ax^{k})} are indeed infinite, rendering the primal dual gap useless.

Let us assume that the problematic term is {F^{*}(-A^{T}y^{k})}. Here is a way out in the case where one can deduce some a-priori bounds on {x^{*}}, i.e. a bounded and convex set {C} with {x^{*}\in C}. In fact, this is often the case (e.g. one may know a-priori that there exist lower bounds {l_{i}} and upper bounds {u_{i}}, i.e. it holds that {l_{i}\leq x^{*}_{i}\leq u_{i}}). Then, adding these constraints to the problem will not change the solution.

Let us see, how this changes the primal dual gap: We set {\tilde F(x) = F(x) + I_{C}(x)} where {C} is the set which models the bound constraints. Since {C} is a bounded convex set and {F} is finite on {C}, it is clear that

\displaystyle \begin{array}{rcl} \tilde F^{*}(\xi) = \sup_{x\in C}\,\langle \xi,x\rangle - F(x) \end{array}

is finite for every {\xi}. This leads to a finite duality gap. However, one should also adapt the prox operator. But this is also simple in the case where the constraint {C} and the function {F} are separable, i.e. {C} encodes bound constraints as above (in other words {C = [l_{1},u_{1}]\times\cdots\times [l_{n},u_{n}]}) and

\displaystyle \begin{array}{rcl} F(x) = \sum_{i} f_{i}(x_{i}). \end{array}

Here it holds that

\displaystyle \begin{array}{rcl} \mathrm{prox}_{\sigma \tilde F}(x)_{i} = \mathrm{prox}_{\sigma f_{i} + I_{[l_{i},u_{i}]}}(x_{i}) \end{array}

and it is simple to see that

\displaystyle \begin{array}{rcl} \mathrm{prox}_{\sigma f_{i} + I_{[l_{i},u_{i}]}}(x_{i}) = \mathrm{proj}_{[l_{i},u_{i}]}\mathrm{prox}_{\tau f_{i}}(x_{i}), \end{array}

i.e., only uses the proximal operator of {F} and project onto the constraints. For general {C}, this step may be more complicated.

One example, where this makes sense is {L^{1}-TV} denoising which can be written as

\displaystyle \begin{array}{rcl} \min_{u}\|u-u^{0}\|_{1} + \lambda TV(u). \end{array}

Here we have

\displaystyle \begin{array}{rcl} F(u) = \|u-u^{0}\|_{1},\quad A = \nabla,\quad G(\phi) = I_{|\phi_{ij}|\leq 1}(\phi). \end{array}

The guy that causes problems here is {F^{*}} which is an indicator functional and indeed {A^{T}\phi^{k}} will usually be dual infeasible. But since {u} is an image with a know range of gray values one can simple add the constraints {0\leq u\leq 1} to the problem and obtains a finite dual while still keeping a simple proximal operator. It is quite instructive to compute {\tilde F} in this case.

Here is a small signal boost for the

Workshop on the Interface of Statistics and Optimization

to  be held at Duke University, Durham, North Carolina from Feb 8 to Feb 10 2017. The workshop is part of the long-year program on optimization currently taking place at the Statistical and Applied Mathematical Sciences Institute (SAMSI).

There will be a lineup of invited speakers from the forefront of Statistics and Optimization each of which has made influential contributions to the other field as well. The planning is still ongoing, and hence, the list of speakers will grow some.

If you can’t make it to North Carolina next February, still mark the date since the talks will be broadcasted via the net and (if tech works out) you may even participate in the Q&A sessions after the talks via your computer.

Last week Christoph Brauer, Andreas Tillmann and myself uploaded the paper A Primal-Dual Homotopy Algorithm for {\ell_{1}}-Minimization with {\ell_{\infty}}-Constraints to the arXiv (and we missed being the first ever arXiv-paper with a non-trivial five-digit identifier by twenty-something papers…). This paper specifically deals with the optimization problem

\displaystyle \begin{array}{rcl} \min_{x}\|x\|_{1}\quad\text{s.t.}\quad \|Ax-b\|_{\infty}\leq \delta \end{array}

where {A} and {b} are a real matrix and vector, respecively, of compatible size. While the related problem with {\ell_{2}} constraint has been addressed quite often (and the penalized problem {\min_{x}\|x\|_{1} + \tfrac1{2\lambda}\|Ax-b\|_{2}^{2}} is even more popular) there is not much code around to solve this specific problem. Obvious candidates are

  • Linear optimization: The problem can be recast as a linear program: The constraint is basically linear already (rewriting it with help of the ones vector {\mathbf{1}} as {Ax\leq \delta\mathbf{1}+b}, {-Ax\leq \delta\mathbf{1}-b}) and for the objective one can, for example, perform a variable split {x = x^{+}-x^{-}}, {x^{-},x^{+}\geq 0} and then write {\|x\|_{1} = \mathbf{1}^{T}x^{+}+ \mathbf{1}^{T}x^{-}}.
  • Splitting methods: The problem is convex problem of the form {\min_{x}F(x) + G(Ax)} with

    \displaystyle \begin{array}{rcl} F(x) & = & \|x\|_{1}\\ G(y) & = & \begin{cases} 0 & \|y-b\|\leq\delta\\ \infty & \text{else.} \end{cases} \end{array}

    and hence, several methods for these kind of problem are available, such as the alternating direction method of multipliers or the Chambolle-Pock algorithm.

The formulation as linear program has the advantage that one can choose among a lot of highly sophisticated software tools and the second has the advantage that the methods are very easy to code, usually in just a few lines. Drawbacks are, that both methods do exploit too much specific structure of the problem.

Application of the problem with {\ell_{\infty}} constraint are, for example:

  • The Dantzig selector, a statistical estimation technique, were the problem is

    \displaystyle \begin{array}{rcl} \min_{x}\|x\|_{1}\quad\text{s.t.}\quad \|A^{T}(Ax-b)\|_{\infty}\leq\delta. \end{array}

  • Sparse dequantization as elaborated here by Jacques, Hammond and Fadili and applied here to de-quantizaton of speech signals by Christoph, Timo Gerkmann and myself.

We wanted to see if one of the most efficient methods known for sparse reconstruction with {\ell_{2}} penalty, namely the homotopy method, can be adapted to this case. The homotopy method for {\min_{x}\|x\|_{1} + \tfrac1{2\lambda}\|Ax-b\|_{2}^{2}} builds on the observation that the solution for {\lambda\geq \|A^{T}b\|_{\infty}} is zero and that the set of solutions {x_{\lambda}}, parameterized by the parameter {\lambda}, is piecewise linear. Hence, on can start from {\lambda_{0}= \|A^{T}b\|_{\infty}}, calculate which direction to go, how far the breakpoint is away, go there and start over. I’ve blogged on the homotopy method here already and there you’ll find some links to great software packages, but also the fact that there can be up to exponentially many breakpoints. However, in practice the homotopy method is usually blazingly fast and with some care, can be made numerically stable and accurate, see, e.g. our extensive study here (and here is the optimization online preprint).

The problem with {\ell_{\infty}} constraint seems similar, especially it is clear that for {\delta = \|b\|_{\infty}}, {x=0} is a solution. It is also not so difficult to see that there is a piecewise linear path of solutions {x_{\delta}}. What is not so clear is, how it can be computed. It turned out, that in this case the whole truth can be seen when the problem is viewed from a primal-dual viewpoint. The associated dual problem is

\displaystyle \begin{array}{rcl} \max_{y}\ -b^{T}y - \delta\|y\|_{1}\quad\text{s.t.}\quad\|A^{T}y\|_{\infty}\leq\infty \end{array}

and a pair {(x^{*},y^{*})} is primal and dual optimal if and only if

\displaystyle \begin{array}{rcl} -A^{T}y^{*}&\in&\text{Sign}(x^{*})\\ Ax^{*}-b & \in & \delta\text{Sign}(y^{*}) \end{array}

(where {\text{Sign}} denotes the sign function, multivalued at zero, giving {[-1,1]} there). One can note some things from the primal-dual optimality system:

  • You may find a direction {d} such that {(x^{*}+td,y^{*})} stays primal-dual optimal for the constraint {\leq\delta-t} for small {t},
  • for a fixed primal optimal {x_{\delta}} there may be several dual optimal {y_{\delta}}.

On the other hand, it is not that clear which of the probably many dual optimal {y_{\delta}} allows to find a new direction {d} such that {x_{\delta}+td} with stay primal optimal when reducing {\delta}. In fact, it turned out that, at a breakpoint, a new dual variable needs to be found to allow for the next jump in the primal variable. So, the solution path is piecewise linear in the primal variable, but piecewise constant in the dual variable (a situation similar to the adaptive inverse scale space method).

What we found is, that some adapted theorem of the alternative (a.k.a. Farkas’ Lemma) allows to calculate the next dual optimal {y} such that a jump in {x} will be possible.

What is more, is that the calculation of a new primal or dual optimal point amounts to solving a linear program (in contrast to a linear system for {\ell_{2}} homotopy). Hence, the trick of up- and downdating a suitable factorization of a suitable matrix to speed up computation does not work. However, one can somehow leverage the special structure of the problem and use a tailored active set method to progress through the path. Our numerical tests indicated is that the resulting method, which we termed {\ell_{1}}-Houdini, is able to solve moderately large problems faster than a commercial LP-solver (while also not only solving the given problem, but calculating the whole solution path on the fly) as can be seen from this table from the paper:

085_runtime_l1houdini

The code of \ell_1-Houdini is on Christoph’s homepage, you may also reproduce the data in the above table with your own hardware.

Yesterday I uploaded the paper “Linear convergence of the Randomized Sparse Kaczmarz Method” by Frank Schöpfer and myself to the arXiv.

Recall that the Kaczmarz method for linear systems

\displaystyle \begin{array}{rcl} Ax&=&b \end{array}

iterates

\displaystyle \begin{array}{rcl} x^{k+1} &=& x^{k} - \frac{\langle a_{i},x_{k}\rangle-b_{i}}{\|a_{i}\|^{2}}a_{i} \end{array}

where {a_{i}} is the {i}-th row of {A}, {b_{i}} is the {i}th entry of {b} and the index {i} is chosen according to some rule. We could, e.g. choose the rows in a cyclic manner, i.e. starting with the first row, proceeding to the next row and once we came to the last row we would start over from the first again. It is known (and probably proved by Kaczmarz himself) that the method converges to a solution of {Ax=b} whenever the system has a solution. Moreover, it easy to see that we converge to the minimum norm solution in case of underdetermined systems when the method is initialized with zero. This is due to the fact that the whole iteration takes place in the range space of {A^{T}}.

In this and this paper we proposed a simple modification of the Kaczmarz method, that makes it converge to sparse solutions. The modification is simply

\displaystyle \begin{array}{rcl} z^{k+1} & = & z^{k} - \frac{\langle a_{i},x_{k}\rangle-b_{i}}{\|a_{i}\|^{2}}a_{i}\\ x^{k+1}& = & S_{\lambda}(z^{k+1}) \end{array}

where {S_{\lambda}(x) = \max(|x|-\lambda,0)\text{sign}(x)} is the soft thresholding function. In this paper we proved that this variant converges, when initialized with zero and for a consistent system, to the solution of

\displaystyle \begin{array}{rcl} \min_{x}\lambda\|x\|_{1} + \tfrac12\|x\|_{2}^{2},\quad\text{s.t.}\quad Ax=b. \end{array}

For not too small values of {\lambda} this is indeed a sparse solution of {Ax=b} and Frank also proved that there is a threshold such that for {\lambda} larger than this threshold the solution is also the minimum {\ell^{1}} solution.

In general, convergence rates for the Kaczmarz method (and its sparse variant) are hard to prove. To convince oneself of this fact note that the convergence speed can drastically change if the rows of the system are reordered. The situation changes if one uses a randomization of the sparse Kaczmarz method where, in each iteration, a row is chose at random. Strohmer and Vershynin proved that this randomization leads to linear convergence rate. In the above mentioned paper we were able to prove the same result for the randomized sparse Kaczmarz method. While this sounds like an obvious generalization, the methods we use are totally different. While the linear convergence of the randomized Kaczmarz method can be proven in a few lines(well, probably one page) using only very basic tools, we need, among other things, quite intricate error bounds for Bregman projections w.r.t. piecewise linear-quadratic convex functions.

In fact, the linear convergence can be observed in practice and we illustrate the usefulness of the randomization and also the “sparsification” on some examples in the paper. For example the following figure shows the decay of the residual for the the randomized Kaczmarz method (black), the randomized sparse Kaczmarz method (red) and the randomized sparse Kaczmarz method with exact steps (green) which I did not explain here.

083_randomizedkaczmarz-figure0

More details are in the paper…

Do you remember free time as a kid that you wasted with connecting dots? If you actually liked it, here’s some good news: There are dot-to-dot books for grown-ups! Most notable, there are the books by Thomas Pavitte with great pictures with 1000 dots.

So, these are some of the books

and here is some video:

 

Actually, it takes some time to connect 1000 dots; I need ten minutes or so, depending a bit an the picture.

For the fun of it, I coded some lines in MATLAB to connect the dots automatically. And since I am a lazy programmer, I did not bother to connect the dots in the manner that was prescribed by the artist but more efficiently:

1. Greedy paths

For the greedy path, we start at some randomly chosen dot and connect the dot where we are with the closest possible dot where we haven’t been already.

Here’s how this looks for one of Thomas’ pictures:

201602040822-greedy-crop

2. Shortest paths

The greedy path sometimes makes very large jumps (when it runs into some corner, using all the dots in the vicinity). This leads to some spurious large jumps in the picture. In used some simple heuristics to find some “locally shortest paths” through the thousand dots. (And “locally shortest” means that there are no two edges for which swapping improves the total lengths of the paths.) Actually, I started out with the goal to solve the travelling salesman problem over the thousand dots, i.e., to find the shortest path of all. Then it turned out that

  1. Solving the travelling salesman problem is not that simple to solve – well it’s one of the best investigated NP-hard optimization problems and there is no doubt that it would take my laptop only little time to solve it with 1000 dots if fed with the right code.
  2. The locally shortest path already looks quite appealing and I am not sure how the shortest path would look any different.

Here is the locally shortest path:

201602040822-tspheuristic-crop.png

Oh, by the way: the image is a pug dog, the one that you can party see on this cover

Here are some more pictures (not by Thomas Pavitte). Middle is the greedy, right the locally shortest path:

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Since the Wikipedia page of the travelling salesman problem contains a formulation as an integer linear program to solve it, I may give it a try in the future…

Optimal transport in a discrete setting goes as follows: Consider two vectors {p,q\in{\mathbb R}^N} with non-negative entries and {\|p\|_1 = \|q\|_1}. You may also say that {p} and {q} are two probability distributions or discrete measures with same total mass. A transport plan for {p} and {q} is a matrix {\pi\in{\mathbb R}^{N\times N}} with non-negative entries such that

\displaystyle \sum_i \pi_{i,j} = p_j,\quad\sum_j\pi_{i,j} = q_i.

The interpretation of {\pi} is, that {\pi_{i,j}} indicates how much of the mass {p_j} sitting in the {j}-th entry of {p} is transported to the {i}-th entry of {q}. To speak about optimal transport we add an objective function to the constraints, namely, the cost {c_{i,j}} that says how much it costs to transport one unit from the {j}-th entry in {p} to the {i}-th entry in {q}. Then the optimal transport problem is

\displaystyle \min_{\pi\in{\mathbb R}^{N\times N}} \sum_{i,j}c_{i,j}\pi_{i,j}\quad\text{s.t.}\quad\sum_i \pi_{i,j} = p_j,\quad\sum_j\pi_{i,j} = q_i.

The resulting {\pi} is an optimal transport plan and the resulting objective value is the minimal cost at which {p} can be transported to {q}. In fact the minimization problem is a linear program and not only that, it’s one of the best studied linear programs and I am sure there is a lot that I don’t know about the structure of this linear program (you may have a look at these slides by Jesus De Loera to get an impression what is known about the structure of the linear program)

So it looks like discrete optimal transport is a fairly standard problem with standard solvers available. But all solvers have one severe drawback when it comes to large {N}: The optimization variable has {N^2} entries. If {N^2} is too large to store {\pi} or keep {\pi} in memory, it seems that there is not much one can do. This is the memory bottleneck for optimal transport problems.

1. Kantorovich-Rubinstein duality

In the case when the cost has the special form {c_{i,j} = |i-j|} on can reduce the memory-burden. This special cost makes sense if the indices {i} and {j} correspond to spatial locations, since then the cost {c_{i,j} = |i-j|} is just the distance from location {i} to {j}. It turns out that in this case there is a simple dual optimal transport problem, namely

\displaystyle \max_{f\in{\mathbb R}^N} f^T(p-q)\quad\text{s.t.}\quad |f_i-f_{i-1}|\leq 1,\ 2\leq i\leq N.

(This is a simple form of the Kantorovich-Rubinstein duality and works similarly if the cost {c} is any other metric on the set of indices.) The new optimization problem is still linear but the memory requirements is only {N} and not {N^2} anymore and moreover there are only {O(N)} constraints for {f}. This idea is behind the method from the paper Imaging with Kantorovich-Rubinstein discrepancy by Jan Lellmann, myself, Carola Schönlieb and Tuomo Valkonen.

2. Entropic regularization

In this post I’d like to describe another method to break through the memory bottleneck of optimal transport. This method works for basically any cost {c} but involves a little bit of smoothing/regularization.

We go from the linear program to a non-linear but still convex one by adding the negative entropy of the transport plan to the objective, i.e. we consider the objective

\displaystyle \sum_{i,j}\Big[c_{i,j}\pi_{i,j} + \gamma \pi_{i,j}(\log(\pi_{i,j}) -1)\Big]

for some {\gamma>0}.

What’s the point of doing so? Let’s look at the Lagrangian: For the constraints {\sum_i \pi_{i,j} = p_j} and {\sum_j\pi_{i,j} = q_i} we introduce the ones vector {{\mathbf 1}\in{\mathbb R}^n}, write them as {\pi^T{\mathbf 1} = p} and {\pi{\mathbf 1}=q}, add Lagrange multipliers {\alpha} and {\beta} and get

\begin{array}{rl}\mathcal{L}(\pi,\alpha,\beta) = & \sum_{i,j}\Big[c_{i,j}\pi_{i,j} + \pi_{i,j}(\log(\pi_{i,j}) -1)\Big]\\ & \quad+ \alpha^T(\pi^T{\mathbf 1}-p) + \beta^T(\pi{\mathbf 1}-q)\end{array}

The cool thing happens with the optimality condition when deriving {\mathcal{L}} with respect to {\pi_{i,j}}:

\displaystyle \partial_{\pi_{i,j}}\mathcal{L} = c_{i,j} + \gamma\log(\pi_{i,j}) + \alpha_j + \beta_i \stackrel{!}{=} 0

We can solve for {\pi_{i,j}} and get

\displaystyle \pi_{i,j} = \exp(-\tfrac{c_{i,j}}{\gamma})\exp(-\tfrac{\alpha_j}\gamma)\exp(-\tfrac{\beta_i}\gamma).

What does that say? It says that the optimal {\pi} is obtained from the matrix

\displaystyle M_{i,j} = \exp(-\tfrac{c_{i,j}}{\gamma})

with rows and columns rescaled by vectors {u_j = \exp(-\tfrac{\alpha_j}\gamma)} and {v_i = \exp(-\tfrac{\beta_i}\gamma)}, respectively, i.e.

\displaystyle \pi = \mathbf{diag}(v)M\mathbf{diag}(u).

The reduces the memory requirement from {N^2} to {2N}! The cost for doing so is the regularization by the entropy.

Actually, the existence of the vectors {u} and {v} also follows from Sinkhorn’s theorem which states that every matrix {A} with positive entries can be written as {A = D_1MD_2} with diagonal matrices and a doubly stochastic matrix {M} (i.e. one with non-negative entries and unit row and column sums). The entropic regularization for the transport plan ensures that the entries of the transport plan has indeed positive (especially non-zero) entries.

But there is even more to the story:. To calculate the vectors {u} and {v} you simply have to do the following iteration:

\displaystyle \begin{array}{rcl} u^{n+1} &=& \frac{p}{Mv^n}\\ v^{n+1} &=& \frac{q}{M^Tu^{n+1}} \end{array}

where the fraction means element-wise division. Pretty simple.

What the iteration does in the first step is simply to take the actual {v} and calculates a column scaling {u} such that the column sums match {p}. In the second step it calculates the row scaling {v} such that the row sums match {q}. This iteration is also known as Sinkhorn-Knopp algorithm.

This is pretty simple to code in MATLAB. Here is a simple code that does the above iteration (using c_{i,j} = |i-j|^2):

%Parameters
gamma = 10; %reg for entropy
maxiter = 100; % maxiter
map = colormap(gray);

N = 100; % size
x = linspace(0,1,N)';%spatial coordinate

% marginals
p = exp(-(x-0.2).^2*10^2) + exp(-abs(x-0.4)*20);p=p./sum(p); %for colums sum
q = exp(-(x-0.8).^2*10^2);q = q./sum(q); % for row sum

[i,j] = meshgrid(1:N);
M = exp(-(i-j).^2/gamma); % exp(-cost/gamma)

% intialize u and v
u = ones(N,1);v = ones(N,1);

% Sinkhorn-Knopp
% iteratively scale rows and columns
for k = 1:maxiter
    % update u and v
    u = p./(M*v);
    v = q./(M'*u);
    % assemble pi (only for illustration purposes)
    pi = diag(v)*M*diag(u);
    % display pi (with marginals on top and to the left)
    imagesc([p'/max(p) 0;pi/max(pi(:)) q/max(q)])
    colormap(1-map)
    drawnow
end

Here are some results:

079_sinkhorn40 079_sinkhorn7 079_sinkhorn20 079_sinkhorn10

(From the left to the right: \gamma=40,20,10,7. The first row of pixels is {p}, the last column is {q} and in between there is {\pi}, all things normalized such that black is the maximal value in {p}, {q} and {\pi}, respectively.)

You see that for large {\gamma}, the plan is much more smooth and not so well localized as it should be for an optimal plan.

Oh, and here is an animation of 100 iterations of Sinkhorn-Knopp showing the result after both u and v have been updated:

079_sinkhorn

There is a catch with this regularization: For small {\gamma} (in this example about {\gamma\leq 6}) the method runs into problem with under- and overflow: the entries in {Mv^n} and {M^Tu^n} become very small. One can fight this effect a bit but I don’t have a convincing numerical method to deal with this, yet.It seems that the entries of the optimal u and v really have to be incredibly small and large, and I mean really small and large (in the order of 10^{300} and 10^{-300} in both u and v).

While the Sinkhorn-Knopp algorithm is already from the 1960s, its application to optimal transport seems fairly new – I learned about from in talks by Gabriel Peyré and Bernhard Schmitzer and the reference is Sinkhorn Distances: Lightspeed Computation of Optimal Transport (presented at NIPS 2013) by Marco Cuturi.

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