### Sparsity

In sparse recovery one aims to leverage the sparsity of some vector ${u}$ to get a good reconstruction of this vector from a noisy and indirect measurement ${u^{0}}$. Sparsity can be expressed in two different ways: The analysis way and the synthesis way. For both of them you need a system of other vectors ${v_{k}}$.

• The analysis way: Here you consider the inner products ${\langle u,v_{k}\rangle}$ and the sparsity assumption is, that this sequence is sparse.
• The synthesis way: Here you assume that there is sparse sequence of coefficients ${\psi_{k}}$ such that ${u=\sum_{k} \psi_{k}v_{k}}$.

The first approach is called “analysis” since here one analyzes the vector ${u}$ with the system of vectors. In the second approach, you use the vectors ${v_{k}}$ to synthesize the vector ${u}$.

We will keep things very simple here and only consider a noisy observation (and not an indirect one). The recovery approach in the analysis way is then to find ${u}$ as a solution of

$\displaystyle \tfrac12\|u-u_{0}\| + \alpha R(\langle u,v_{k}\rangle)$

while for the synthesis way you write ${u = \sum_{k} \psi_{k}v_{k} = V\psi}$ (where we collected the vectors ${v_{k}}$ as the columns of the matrix ${V}$) and solve

$\displaystyle \tfrac12\|V\psi-u_{0}\| + \alpha R(\psi)$

The functional ${R}$ should enforce the sparsity of the vector of coefficients in the former case and the sparsity of ${\psi}$ in the latter case. With the matrix ${V}$ the analysis way reads as

$\displaystyle \tfrac12\|u-u_{0}\| + \alpha R(V^{T}u)$

One important observation is, that both approaches coincide if the matrix ${V}$ is orthonormal: Since ${V^{-1} = V^{T}}$ in this case, ${u=V\psi}$ is equivalent to ${\psi = V^{T}u}$, i.e. ${\psi_{k} = \langle u,v_{k}\rangle}$. For other sets of vectors ${v_{k}}$, this is not true and it is not so clear, how the analysis and the synthesis approach are related. In this post I’d like to show one instance where both approaches do lead to results that are not even close to being similar.

1. The “ROF analysis” approach to denoising

If ${u^{0}}$ is an image, the famous Rudin-Osher-Fatemi denoising model fits under the “analysis” framework: The vectors ${v_{k}}$ are all the finite differences of neighboring pixels such that ${V^{T}u = \nabla u}$ with the discrete gradient ${\nabla u}$. As sparsifying functional you take a mixed ${2}$${1}$ norm, i.e. ${R(\nabla u) = \|\nabla u\|_{2,1} = \sum_{ij}\sqrt{(\partial_{1}u_{ij})^{2} + (\partial_{2}u_{ij})^{2}}}$. The denoised ${u}$ is the solution to

$\displaystyle \min_{u} \tfrac12\|u-u^{0}\|^{2} + \alpha\|\nabla u\|_{2,1}$

Thus, the model will lead to some denoised ${u}$ with a sparse gradient. This sparse gradient is partly the reason for the success of the model in that it keeps edges, but is also responsible for the drawback of this approach: The denoised image will be mostly piesewise constant.

2. The “ROF synthesis” approach

As ${V^{T} = \nabla}$, the corresponding synthesis approach would be to solve

$\displaystyle \min_{\psi}\tfrac12\|\nabla^{T}\psi - u^{0}\|^{2} + \alpha \|\psi\|_{2,1}$

and the set ${u = \nabla^{T}\psi}$. In this approach, the denoised image will be a sparse linear combination of particular two-pixel images. If you think about that, it does not really sound like a good idea to approximate an image by a sparse linear combination of two-pixel images. (One technicality: The operator ${\nabla^{T}}$ is not onto – ${\nabla}$ has a non-trivial kernel, consisting of the constant images and hence, all images in the range of ${\nabla^{T}}$ have zero mean. Thus, to get something remotely close to ${u^{0}}$ one should subtract the mean of ${u^{0}}$ before the reconstruction and add it back afterwards.)

Here are some results:

Note that the results actually fit to what we would have predicted: For larger ${\alpha}$ we get a “sparser gradient” (less edges, less variation) for the analysis approach, and “sparser two-pixel combinations” for the synthesis approach. In fact, for the synthesis approach to give something that is remotely close to the image, one needs quite a lot two-pixel images, i.e. a very low sparsity.

In conclusion, one should not consider the analysis and synthesis approach to be something similar.

Incidentally, the dual of the analysis approach

$\displaystyle \min_{u} \tfrac12\|u-u^{0}\|^{2} + \alpha\|\nabla u\|_{2,1}$

can be written as

$\displaystyle \min_{\|\phi\|_{2,\infty}\leq\alpha}\tfrac12\|\nabla^{T}\phi-u^{0}\|^{2} = \min_{\phi}\tfrac12\|\nabla^{T}\phi-u^{0}\|^{2} + I_{\|\cdot\|_{2,\infty}\leq\alpha}(\phi)$

which looks related to the synthesis approach (one basically replaces the ${2,1}$-norm by its conjugate function). Actually, a similar relation between the dual of the analysis approach and the synthesis approach always holds.

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:

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.

I fell a little bit behind on reporting on my new preprints. In this posts I’ll blog on two closely related ones; one of them already a bit old, the other one quite recent:

The papers are

As clear from the titles, both papers treat a similar method. The first paper contains all the theory and the second one has few particularly interesting applications.

In the first paper we propose to view several known algorithms such as the linearized Bregman method, the Kaczmarz method or the Landweber method from a different angle from which they all are special cases of another algorithm. To start with, consider a linear system

$\displaystyle Ax=b$

with ${A\in{\mathbb R}^{m\times n}}$. A fairly simple and old method to solve this, is the Landweber iteration which is

$\displaystyle x^{k+1} = x^k - t_k A^T(Ax^k-b).$

Obviously, this is nothing else than a gradient descent for the functional ${\|Ax-b\|_2^2}$ and indeed converges to a minimizer of this functional (i.e. a least squares solution) if the stepsizes ${t_k}$ fulfill ${\epsilon\leq t_k\leq 2\|A\|^{-2} - \epsilon}$ for some ${\epsilon>0}$. If one initializes the method with ${x^0=0}$ it converges to the least squares solution with minimal norm, i.e. to ${A^\dag b}$ (with the pseudo-inverse ${A^\dag}$).

A totally different method is even older: The Kaczmarz method. Denoting by ${a_k}$ the ${k}$-th row of ${A}$ and ${b_k}$ the ${k}$-th entry of ${b}$ the method reads as

$\displaystyle x^{k+1} = x^k - a_{r(k)}^T\frac{a_{r(k)}\cdot x^k - b_k}{\|a_{r(k)}\|_2^2}$

where ${r(k) = (k\mod m) +1}$ or any other “control sequence” that picks up every index infinitely often. This method also has a simple interpretation: Each equation ${a_k\cdot x = b_k}$ describes a hyperplane in ${{\mathbb R}^n}$. The method does nothing else than projecting the iterates orthogonally onto the hyperplanes in an iterative manner. In the case that the system has a solution, the method converges to one, and if it is initialized with ${x^0=0}$ we have again convergence to the minimum norm solution ${A^\dag b}$.

There is yet another method that solves ${Ax=b}$ (but now it’s a bit more recent): The iteration produces two sequences of iterates

$\displaystyle \begin{array}{rcl} z^{k+1} & = &z^k - t_k A^T(Ax^k - b)\\ x^{k+1} & = &S_\lambda(z^{k+1}) \end{array}$

for some ${\lambda>0}$, the soft-thresholding function ${S_\lambda(x) = \max(|x|-\lambda,0)\mathrm{sgn}(x)}$ and some stepsize ${t_k}$. For reasons I will not detail here, this is called the linearized Bregman method. It also converges to a solution of the system. The method is remarkably similar, but different from, the Landweber iteration (if the soft-thresholding function wouldn’t be there, both would be the same). It converges to the solution of ${Ax=b}$ that has the minimum value for the functional ${J(x) = \lambda\|x\|_1 + \tfrac12\|x\|_2^2}$. Since this solution of close, and for ${\lambda}$ large enough identical, to the minimum ${\|\cdot\|_1}$ solution, the linearized Bregman method is a method for sparse reconstruction and applied in compressed sensing.

Now we put all three methods in a joint framework, and this is the framework of split feasibility problems (SFP). An SFP is a special case of a convex feasibility problems where one wants to find a point ${x}$ in the intersection of multiple simple convex sets. In an SFP one has two different kinds of convex constraints (which I will call “simple” and “difficult” in the following):

1. Constraints that just demand that ${x\in C_i}$ for some convex sets ${C_i}$. I call these constraints “simple” because we assume that the projection onto each ${C_i}$ is simple to obtain.
2. Constraints that demand ${A_ix\in Q_i}$ for some matrices ${A_i}$ and simple convex sets ${Q_i}$. Although we assume that projections onto the ${Q_i}$ are easy, these constraints are “difficult” because of the presence of the matrices ${A_i}$.

If there were only simple constraints a very basic method to solve the problem is the methods of alternating projections, also known as POCS (projection onto convex sets): Simply project onto all the sets ${C_i}$ in an iterative manner. For difficult constraints, one can do the following: Construct a hyperplane ${H_k}$ that separates the current iterate ${x^k}$ from the set defined by the constraint ${Ax\in Q}$ and project onto the hyperplane. Since projections onto hyperplanes are simple and since the hyperplane separates we move closer to the constraint set and this is a reasonable step to take. One such separating hyperplane is given as follows: For ${x^k}$ compute ${w^k = Ax^k-P_Q(Ax^k)}$ (with the orthogonal projection ${P_Q}$) and define

$\displaystyle H_k = \{x\ : (A^Tw^k)^T\cdot x \leq (A^Tw^k)^T\cdot x^k - \|w^k\|_2^2\}.$

Illustration of projections onto convex sets and separating hyperplanes

Now we already can unite the Landweber iteration and the Kaczmarz method as follows: Consider the system ${Ax=b}$ as a split feasibility problem in two different ways:

1. Treat ${Ax=b}$ as one single difficult constraint (i.e. set ${Q=\{b\}}$). Some calculations show that the above proposed method leads to the Landweber iteration (with a special stepsize).
2. Treat ${Ax=b}$ as ${m}$ simple constraints ${a_i\cdot x = b_i}$. Again, some calculations show that this gives the Kaczmarz method.

Of course, one could also work “block-wise” and consider groups of equations as difficult constraints to obtain “block-Kaczmarz methods”.

Now comes the last twist: By adapting the term of “projection” one gets more methods. Particularly interesting is the notion of Bregman projections which comes from Bregman distances. I will not go into detail here, but Bregman distances are associated to convex functionals ${J}$ and by replacing “projection onto ${C_i}$ or hyperplanes” by respective Bregman projections, one gets another method for split feasibility problems. The two things I found remarkable:

• The Bregman projection onto hyperplanes is pretty simple. To project some ${x^k}$ onto the hyperplane ${H = \{x\ :\ a^T\cdot x\leq \beta\}}$, one needs a subgradient ${z^k\in\partial J(x^k)}$ (in fact an “admissible one” but for that detail see the paper) and then performs

$\displaystyle x^{k+1} = \nabla J^*(z^k - t_k a)$

(${J^*}$ is the convex dual of ${J}$) with some appropriate stepsize ${t_k}$ (which is the solution of a one-dimensional convex minimization problem). Moreover, ${z^{k+1} = z^k - t_k a}$ is a new admissible subgradient at ${x^{k+1}}$.

• If one has a problem with a constraint ${Ax=b}$ (formulated as an SFP in one way or another) the method converges to the minimum-${J}$ solution of the equation if ${J}$ is strongly convex.

Note that strong convexity of ${J}$ implies differentiability of ${J^*}$ and Lipschitz continuity of ${\nabla J}$ and hence, the Bregman projection can indeed be carried out.

Now one already sees how this relates to the linearized Bregman method: Setting ${J(x) = \lambda\|x\|_1 + \tfrac12\|x\|_2^2}$, a little calculation shows that

$\displaystyle \nabla J^*(z) = S_\lambda(z).$

Hence, using the formulation with a “single difficult constraint” leads to the linearized Bregman method with a specific stepsize. It turns out that this stepsize is a pretty good one but also that one can show that a constant stepsize also works as long as it is positive and smaller that ${2\|A\|^{-2}}$.

In the paper we present several examples how one can use the framework. I see one strengths of this approach that one can add convex constraints to a given problem without getting into any trouble with the algorithmic framework.

The second paper extends a remark that we make in the first one: If one applies the framework of the linearized Bregman method to the case in which one considers the system ${Ax=b}$ as ${m}$ simple (hyperplane-)constraints one obtains a sparse Kaczmarz solver. Indeed one can use the simple iteration

$\displaystyle \begin{array}{rcl} z^{k+1} & = &z^k - a_{r(k)}^T\frac{a_{r(k)}\cdot x^k - b_k}{\|a_{r(k)}\|_2^2}\\ x^{k+1} & = &S_\lambda(z^{k+1}) \end{array}$

and will converge to the same sparse solution as the linearized Bregman method.

This method has a nice application to “online compressed sensing”: We illustrate this in the paper with an example from radio interferometry. There, large arrays of radio telescopes collect radio emissions from the sky. Each pair of telescopes lead to a single measurement of the Fourier transform of the quantity of interest. Hence, for ${k}$ telescopes, each measurement gives ${k(k-1)/2}$ samples in the Fourier domain. In our example we used data from the Very Large Array telescope which has 27 telescopes leading to 351 Fourier samples. That’s not much, if one want a picture of the emission with several ten thousands of pixels. But the good thing is that the Earth rotates (that’s good for several reasons): When the Earth rotates relative to the sky, the sampling pattern also rotates. Hence, one waits a small amount of time and makes another measurement. Commonly, this is done until the earth has made a half rotation, i.e. one complete measurement takes 12 hours. With the “online compressed sensing” framework we proposed, one can start reconstructing the image as soon the first measurements have arrived. Interestingly, one observes the following behavior: If one monitors the residual of the equation, it goes down during iterations and jumps up when new measurements arrive. But from some point on, the residual stays small! This says that the new measurements do not contradict the previous ones and more interestingly this happened precisely when the reconstruction error dropped down such that “exact reconstruction” in the sense of compressed sensing has happened. In the example of radio interferometry, this happened after 2.5 hours!

Reconstruction by online compressed sensing

You can find slides of a talk I gave at the Sparse Tomo Days here.

I recently updated my working hardware and now use a tablet pc for work (namely a Nexus 10). In consequence, I also updated the software I used to have things more synchronized across devices. For my RSS feeds I now use feedly and the gReader app. However, I was not that happy with the method to store and mark paper I found but found the sharing interfaces between the apps pretty handy. I adopted the workflow that when I see a paper that I want to remember I sent them to my Evernote account where I tag them. Then, from time to time I go over the papers I marked and have a more detailed look. If I think, they deserve to be kept for future reference, they get a small entry here. Here’s the first take with just two papers from the last weeks (there are more in my backlog…):

On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems by Radu Ioan Boţ, Ernö Robert Csetnek, André Heinrich, Christopher Hendrich (Math Prog): As first sight, I found this work pretty inaccessible but the title sounded interesting. I was a bit scared by the formula for the kind of problems they investigated: Solve the following inclusion for ${x}$

$\displaystyle 0 \in z + Ax + \sum_{i=1}^m L_i^*((B_i\square D_i)(L_ix -r_i)) + Cx$

where ${A}$, ${B_i}$ and ${D_i}$ are maximally monotone, ${D_i}$ also ${\nu_i}$ strongly monotone, ${C}$ is ${\eta}$-coercive, ${L_i}$ are linear and bounded and ${\square}$ denotes the parallel sum, i.e. ${A\square B = (A^{-1}+B^{-1})^{-1}}$. Also the proposed algorithm looked a bit like a monster. Then, on later pager, things became a bit more familiar. As an application, they considered the optimization problem

$\displaystyle \min_x f(x) + \sum_{i=1}^m (g_i\square l_i)(L_ix - r_i) + h(x) - \langle x,z\rangle$

with convex ${f}$, ${g_i}$, ${l_i}$ (${l_i}$ ${\nu_i^{-1}}$ strongly convex), ${h}$ convex with ${\eta}$-Lipschitz gradient and ${L_i}$ as above. By noting that the parallel sum is related to the infimal convolution of convex functions, things became clearer. Also, the algorithm looks more familiar now (Algorithm 18 in the paper – I’m too lazy to write it down here). They have an analysis of the algorithms that allow to deduce convergence rates for the iterates (usually ${\mathcal{O}(1/n)}$) but I haven’t checked the details yet.

Sparse Regularization: Convergence Of Iterative Jumping Thresholding Algorithm by Jinshan Zeng, Shaobo Lin, Zongben Xu: At first I was excited but then I realized that they simple tackled

$\displaystyle \min F + \lambda \Phi$

with smooth ${F}$ and non-smooth, non-convex ${\Phi}$ by “iterative thresholding”, i.e.

$\displaystyle x^{n+1} = \mathrm{prox}_{\mu\lambda\Phi}(x^n - \mu \nabla F(x^n)).$

The paper really much resembles what Kristian and I did in the paper Minimization of non-smooth, non-convex functionals by iterative thresholding (at least I couldn’t figure out the improvements…).

Another few notes to myself:

Let ${\Omega}$ be a compact subset of ${{\mathbb R}^d}$ and consider the space ${C(\Omega)}$ of continuous functions ${f:\Omega\rightarrow {\mathbb R}}$ with the usual supremum norm. The Riesz Representation Theorem states that the dual space of ${C(\Omega)}$ is in this case the set of all Radon measures, denoted by ${\mathfrak{M}(\Omega)}$ and the canonical duality pairing is given by

$\displaystyle \langle\mu,f\rangle = \mu(f) = \int_\Omega fd\mu.$

We can equip ${\mathfrak{M}(\Omega)}$ with the usual notion of weak* convergence which read as

$\displaystyle \mu_n\rightharpoonup^* \mu\ \iff\ \text{for every}\ f:\ \mu_n(f)\rightarrow\mu(f).$

We call a measure ${\mu}$ positive if ${f\geq 0}$ implies that ${\mu(f)\geq 0}$. If a positive measure satisfies ${\mu(1)=1}$ (i.e. it integrates the constant function with unit value to one), we call it a probability measure and we denote with ${\Delta\subset \mathfrak{M}(\Omega)}$ the set of all probability measures.

Example 1 Every non-negative integrable function ${\phi:\Omega\rightarrow{\mathbb R}}$ with ${\int_\Omega \phi(x)dx}$ induces a probability measure via

$\displaystyle f\mapsto \int_\Omega f(x)\phi(x)dx.$

Quite different probability measures are the ${\delta}$-measures: For every ${x\in\Omega}$ there is the ${\delta}$-measure at this point, defined by

$\displaystyle \delta_x(f) = f(x).$

In some sense, the set ${\Delta}$ of probability measure is the generalization of the standard simplex in ${{\mathbb R}^n}$ to infinite dimensions (in fact uncountably many dimensions): The ${\delta}$-measures are the extreme points of ${\Delta}$ and since the set ${\Delta}$ is compact in the weak* topology, the Krein-Milman Theorem states that ${\Delta}$ is the weak*-closure of the set of convex combinations of the ${\delta}$-measures – similarly as the standard simplex in ${{\mathbb R}^n}$ is the convex combination of the canonical basis vectors of ${{\mathbb R}^n}$.

Remark 1 If we drop the positivity assumption and form the set

$\displaystyle O = \{\mu\in\mathfrak{M}(\Omega)\ :\ |f|\leq 1\implies |\mu(f)|\leq 1\}$

we have the ${O}$ is the set of convex combinations of the measures ${\pm\delta_x}$ (${x\in\Omega}$). Hence, ${O}$ resembles the hyper-octahedron (aka cross polytope or ${\ell^1}$-ball).

I’ve taken the above (with almost similar notation) from the book “ A Course in Convexity” by Alexander Barvinok. I was curious to find (in Chapter III, Section 9) something which reads as a nice glimpse on semi-continuous compressed sensing: Proposition 9.4 reads as follows

Proposition 1 Let ${g,f_1,\dots,f_m\in C(\Omega)}$, ${b\in{\mathbb R}^m}$ and suppose that the subset ${B}$ of ${\Delta}$ consisting of the probability measures ${\mu}$ such that for ${i=1,\dots,m}$

$\displaystyle \int f_id\mu = b_i$

is not empty. Then there exists ${\mu^+,\mu^-\in B}$ such that

1. ${\mu^+}$ and ${\mu^-}$ are convex combinations of at most ${m+1}$ ${\delta}$-measures, and
2. it holds that for all ${\mu\in B}$ we have

$\displaystyle \mu^-(g)\leq \mu(g)\leq \mu^+(g).$

In terms of compressed sensing this says: Among all probability measures which comply with the data ${b}$ measured by ${m}$ linear measurements, there are two extremal ones which consists of ${m+1}$ ${\delta}$-measures.

Note that something similar to “support-pursuit” does not work here: The minimization problem ${\min_{\mu\in B, \mu(f_i)=b_i}\|\mu\|_{\mathfrak{M}}}$ does not make much sense, since ${\|\mu\|_{\mathfrak{M}}=1}$ for all ${\mu\in B}$.

ISMP is over now and I’m already home. I do not have many things to report on from the last day. This is not due the lower quality of the talks but due to the fact that I was a little bit exhausted, as usual at the end of a five-day conference. However, I collect a few things for the record:

• In the morning I visited the semi-planary by Xiaojun Chenon non-convex and non-smooth minimization with smoothing methods. Not surprisingly, she treated the problem

$\displaystyle \min_x f(x) + \|x\|_p^p$

with convex and smooth ${f:{\mathbb R}^n\rightarrow{\mathbb R}}$ and ${0. She proposed and analyzed smoothing methods, that is, to smooth the problem a bit to obtain a Lipschitz-continuous objective function ${\phi_\epsilon}$, minimizing this and then gradually decreasing ${\epsilon}$. This works, as she showed. If I remember correctly, she also treated “iteratively reweighted least squares” as I described in my previous post. Unfortunately, she did not include the generalized forward-backward methods based on ${\text{prox}}$-functions for non-convex functions. Kristian and I pursued this approach in our paper Minimization of non-smooth, non-convex functionals by iterative thresholding and some special features of our analysis include:

• A condition which excludes some (but not all) local minimizers from being global.
• An algorithm which avoids this non-global minimizers by carefully adjusting the steplength of the method.
• A result that the number of local minimizers is still finite, even if the problem is posed in ${\ell^2({\mathbb N})}$ and not in ${{\mathbb R}^n}$.

Most of our results hold true, if the ${p}$-quasi-norm is replaced by functions of the form

$\displaystyle \sum_n \phi_n(|x_n|)$

with special non-convex ${\phi}$, namely fulfilling a list of assumptions like

• ${\phi'(x) \rightarrow \infty}$ for ${x\rightarrow 0}$ (infinite slope at ${0}$) and ${\phi(x)\rightarrow\infty}$ for ${x\rightarrow\infty}$ (mild coercivity),
• ${\phi'}$ strictly convex on ${]0,\infty[}$ and ${\phi'(x)/x\rightarrow 0}$ for ${x\rightarrow\infty}$,
• for each ${b>0}$ there is ${a>0}$ such that for ${x it holds that ${\phi(x)>ax^2}$, and
• local integrability of some section of ${\partial\phi'(x) x}$.

As one easily sees, ${p}$-quasi-norms fulfill the assumptions and some other interesting functions as well (e.g. some with very steep slope at ${0}$ like ${x\mapsto \log(x^{1/3}+1)}$).

• Jorge Nocedalgave a talk on second-order methods for non-smooth problems and his main example was a functional like

$\displaystyle \min_x f(x) + \|x\|_1$

with a convex and smooth ${f}$, but different from Xiaojun Chen, he only considered the ${1}$-norm. His talked is among the best planary talks I have ever attended and it was a great pleasure to listen to him. He carefully explained things and put them in perspective. In the case he skipped slides, he made me feel that I either did not miss an important thing, or understood them even though he didn’t show them He argued that it is not necessarily more expensive to use second order information in contrast to first order methods. Indeed, the ${1}$-norm can be used to reduce the number of degrees of freedom for a second order step. What was pretty interesting is, that he advocated semismooth Newton methods for this problem. Roland and I pursued this approach some time ago in our paper A Semismooth Newton Method for Tikhonov Functionals with Sparsity Constraints and, if I remember correctly (my notes are not complete at this point), his family of methods included our ssn-method. The method Roland and I proposed worked amazingly well in the cases in which it converged but the method suffered from non-global convergence. We had some preliminary ideas for globalization, which we could not tune enough to retain the speed of the method, and abandoned the topic. Now, that the topic will most probably be revived by the community, I am looking forward to fresh ideas here.

Next Page »