### Regularization

Some remark before you read this post: It is on a very specialized topic and only presents a theoretical insight which seems to be of no practical value whatsoever. Continue at your own risk of wasting your time.

Morozov’s discrepancy principle is a means to choose the regularization parameter when regularizing inverse ill-posed problems. To fix the setting, we consider two Hilbert spaces ${X}$ and ${Y}$ and a bounded linear operator ${A:X\rightarrow Y}$. We assume that the range of ${A}$ is a non-closed subset of ${Y}$. As a consequence of the Bounded Inverse Theorem the pseudo-inverse ${A^\dag}$ (defined on ${{\mathrm rg} A \oplus ({\mathrm rg} A)^\bot}$) is unbounded.

This is the classical setting for linear inverse problems: We have a process, modeled by ${A}$, such that the quantity ${x^\dag\in X}$ we are interested in gives rise to on output ${y^\dag = Ax^\dag}$. We are able to measure ${y}$ but, as it is always the case, the is some noise introduced in the measurement process and hence, we have access to a measured quantity ${y^\delta\in Y}$ which is a perturbation of ${y}$. Our goal is, to approximate ${x^\dag}$ from the knowledge of ${y^\delta}$. Note that simply taking ${A^\dag y^\delta}$ is not an option, since firstly ${y^\delta}$ is not guaranteed to be in the domain of the pseudo-inverse and, somehow even more severely and also more practical, the unboundedness of ${A^\dag}$ will lead to a severe (in theory infinitely large) amplification of the noise, rendering the reconstruction useless.

The key idea is to approximate the pseudo-inverse ${A^\dag}$ by a family of bounded operators ${R_\alpha:Y\rightarrow X}$ with the hope that one may have

$\displaystyle R_\alpha y^\delta \rightarrow x^\dag\ \text{when}\ y^\delta\rightarrow y\in{\mathrm rg} A\ \text{and}\ \alpha\rightarrow 0 \ \text{appropriately.} \ \ \ \ \ (1)$

(Note that the assumption ${\alpha\rightarrow 0}$ is just a convention. It says that we assume that ${R_\alpha}$ is a closer approximation to ${A^\dag}$ the closer ${\alpha}$ is to zero.) Now, we have two tasks:

1. Construct a good family of regularizing operators ${R_\alpha}$ and
2. devise a parameter choice, i.e. a way to choose ${\alpha}$.

The famous Bakushinskii veto says that there is no parameter choice that can guarantee convergence in~(1) in the worst case and only uses the given data ${y^\delta}$. The situation changes if one introduces knowledge about the noise level ${\delta = \|y-y^\delta\|}$. (There is an ongoing debate if it is reasonable to assume that the noise level is known – my experience when working with engineers is that they are usually very good in quantifying the noise present in their system and hence, in my view the assumption that noise level is known is ok.)

One popular way to choose the regularization parameter in dependence on ${y^\delta}$ and ${\delta}$ is Morozov’s discrepancy principle:

Definition 1 Morozov’s discrepancy principle states that one shall choose the regularization parameter ${\alpha(\delta,y^\delta)}$ such that

$\displaystyle \|AR_{\alpha(\delta,y^\delta)}y^\delta - y^\delta\| = c\delta$

for some fixed ${c>1}$.

In other words: You shall choose ${\alpha}$ such that the reconstruction ${R_\alpha y^\delta}$ produces a discrepancy ${\|AR_{\alpha}y^\delta - y^\delta\|}$ which is in the order of and slightly larger than the noise level.

Some years ago I wrote a paper about the use of Morozov’s discrepancy principle when using the augmented Lagragian method (aka Bregman iteration) as an iterative regularization method (where one can view the inverse of the iteration counter as a regularization parameter). The paper is Morozov’s principle for the augmented Lagrangian method applied to linear inverse problems (together with , the arxiv link is here). In that paper we derived an estimate for the (squared) error of ${R_\alpha y^\delta}$ and ${x^\dag}$ that behaves like

$\displaystyle C \frac{c(\sqrt{c}+1)}{\sqrt{c-1}}\delta$

for some ${C>0}$ and the ${c>1}$ from Morozov’s discrepancy principle. The somehow complicated dependence on ${c}$ was a bit puzzling to me. One can optimize ${c>1}$ such that the error estimate is optimal. It turns out that ${c\mapsto \frac{c(\sqrt{c}+1)}{\sqrt{c-1}}}$ attains the minimal value of about ${4.68}$ for about ${c=1.64}$. I blamed the already quite complicated analysis of the augmented Lagragian method for this obviously much too complicated values (and in practice, using ${c}$ much closer to ${1}$ usually lead to much better results).

This term I teach a course on inverse problems and also covered Morozov’s discrepancy principle but this time for much simpler regularization methods, namely for linear methods such as, for example, Tikhonov regularization, i.e.

$\displaystyle R_\alpha y^\delta = (A^* A + \alpha I)^{-1} A^* y^\delta$

(but other linear methods exist). There I arrived at an estimate for the (squared) error of ${R_\alpha y^\delta}$ any ${x^\dag}$ of the form

$\displaystyle C(\sqrt{c} + \tfrac1{\sqrt{c-1}})^2\delta.$

Surprisingly, the dependence on ${c}$ for this basic regularization method is also not very simple. Optimizing the estimate over ${c>1}$ leads to an optimal value of about ${c= 2.32}$ (with a minimal value of the respective constant of about ${5.73}$). Again, using ${c}$ closer to ${1}$ usually lead to much better results.

Well, these observations are not of great importance… I just found it curious to observe that the analysis of Morozov’s discrepancy principle seems to be inherently a bit more complicated than I thought…

Today I’d like to collect some comments one a few papers I stumbled upon recently on the arXiv.

1. TGV minimizers in 1D

First, about a month ago two very similar paper appeared in the same week:

Both papers treat the recently proposed “total generalized variation” model (which is a somehow-but-not-really-higher-order generalization of total variation). The total variation of a function ${u\in L^1(\Omega)}$ (${\Omega\subset{\mathbb R}^d}$) is defined by duality

$\displaystyle TV(u) = \sup\Big\{\int_\Omega \mathrm{div} \phi\, u\,dx\ :\ \phi\in C^\infty_c(\Omega,{\mathbb R}^d), |\phi|\leq 1\Big\}.$

(Note that the demanded high regularity of the test functions ${\phi}$ is not essential here, as we take a supremum over all these functions under the only, but important, requirement that the functions are bounded. Test functions from ${C^1_c(\Omega,{\mathbb R}^d)}$ would also do.)

Several possibilities for extensions and generalization of the total variation exist by somehow including higher order derivatives. The “total generalized variation” is a particular successful approach which reads as (now using two non-negative parameter ${\alpha,\beta}$ which do a weighting):

$\displaystyle TGV_{\beta,\alpha}^2(u) = \sup\Big\{\int_\Omega \mathrm{div}^2 \phi\, u\,dx\ :\ \phi\in C^\infty_c(\Omega,S^{d\times d}),\ |\phi|\leq \beta,\ |\mathrm{div}\phi|\leq \alpha\Big\}.$

To clarify some notation: ${S^{d\times d}}$ are the symmetric ${d\times d}$ matrices, ${\mathrm{div}^n}$ is the negative adjoint of ${\nabla^n}$ which is the differential operator that collects all partial derivatives up to the ${n}$-th order in a ${d\times\cdots\times d}$-tensor. Moreover ${|\phi|}$ is some matrix norm (e.g. the Frobenius norm) and ${|\mathrm{div}\phi|}$ is some vector norm (e.g. the 2-norm).

Both papers investigate so called denoising problems with TGV penalty and ${L^2}$ discrepancy, i.e. minimization problems

$\displaystyle \min_u \frac12\int_\Omega(u-u^0)^2\, dx + TGV_{\alpha,\beta}^2(u)$

for a given ${u^0}$. Moreover, both papers treat the one dimensional case and investigate very special cases in which they calculate minimizers analytically. In one dimension the definition of ${TGV^2}$ becomes a little more familiar:

$\displaystyle TGV_{\beta,\alpha}^2(u) = \sup\Big\{\int_\Omega \phi''\, u\,dx\ :\ \phi\in C^\infty_c(\Omega,{\mathbb R}),\ |\phi|\leq \beta,\ |\phi'|\leq \alpha\Big\}.$

Some images of both papar are really similar: This one from Papafitsoros and Bredies

and this one from Pöschl and Scherzer

Although both paper have a very similar scopes it is worth to read both. The calculations are tedious but both paper try to make them accessible and try hard (and did a good job) to provide helpful illustrations. Curiously, the earlier paper cites the later one but not conversely…

Another paper I found very interesting was

This paper shows a nice duality which I haven’t been aware of, namely the one between the subgradient descent methods and conditional gradient methods. In fact the conditional gradient method which is treated is a generalization of the conditional gradient method which Kristian and I also proposed a while ago in the context of ${\ell^1}$-minimization in the paper Iterated hard shrinkage for minimization problems with sparsity constraints: To minimize the sum

$\displaystyle F(u) + \Phi(u)$

with a differentiable ${F}$ and a convex ${\Phi}$ for which the subgradient of ${\Phi}$ is easily invertible (or, put differently, for which you can minimize ${\langle u,a\rangle + \Phi(u)}$ easily), perform the following iteration:

1. At iterate ${u^n}$ linearize ${F}$ but not ${\Phi}$ and calculate a new point ${v^n}$ by

$\displaystyle v^n = \mathrm{argmin}_v \langle F'(u^n),v\rangle + \Phi(v)$

2. Choose a stepsize ${s^n\in [0,1]}$ and set the next iterate as a convex combination of ${u^n}$ and ${v^n}$

$\displaystyle u^{n+1} = u^n + s_n(v^n - u^n).$

Note that for and indicator function

$\displaystyle \Phi(u) = \begin{cases} 0 & u\in C\\ \infty & \text{else} \end{cases}$

you obtain the conditional gradient method (also known as Frank-Wolfe method). While Kristian and I derived convergence with an asymptotic rate for the case of ${F(u) = \tfrac12\|Ku-f\|^2}$ and ${\Phi}$ strongly coercive, Francis uses the formulation ${F(u) = f(Au)}$ the assumption that the dual ${f^*}$ of ${f}$ has a bounded effective domain (which say that ${f}$ has linear growth in all directions). With this assumption he obtains explicit constants and rates also for the primal-dual gap. It was great to see that eventually somebody really took the idea from the paper Iterated hard shrinkage for minimization problems with sparsity constraints (and does not think that we do heuristics for ${\ell^0}$ minimization…).

A quick post to keep track of several things:

• Christian Leonard  has lecture notes on convex optimization with an application to optimal transport on his website.
• The paper Variational Properties of Value Functions by  Aravkin, Burke, and Friedlander discuss how the value of minimization problems like $\min \rho(Ax-b)\quad \mbox{s.t}\quad \phi(x)\leq tau$ depend on $\tau$ and $\latex b$. In inverse problems, the value function seems to contain important information on the the regularization process and hence, the results in this paper maybe helpful in designing and analyzing parameter choice rules.
• The paper Accelerated and Inexact Forward-Backward Algorithms by Villa,Salzo, Baldassarre, and Verri looks like an interesting development in the fiel of splitting methods.
• The paper  Consistency of the posterior distribution in generalized linear inverse problems by Natalia Bochkina is another contribution on “probabilitic inverse problems” where one does not only try to infer a regularized solution to an ill posed problems but also how the uncertainty in the data in propagated through the regularization process.

Another few notes to myself:

If you are working on optimization with partial differential equations as constraints, you may be interested in the website

“OPTPDE – A Collection of Problems in PDE-Constrained Optimization”, http://www.optpde.net.

If you have developed an algorithm which can handle a certain class of optimization problems you need to do evaluations and tests on how well the method performs. To do so, you need well constructed test problems. This could be either problems where the optimal solution is known analytically our problems where the solution is known with a rigorous error bound obtained with a bullet-proof solver. Both things are not always easy to obtain and OPTPDE shall serve as a resource for such problems. It has been designed by Roland Herzog, Arnd Rösch, Stefan Ulbrich and Winnifried Wollner.

The generation of test instance for optimization problems seems quite important to me and indeed, several things can go wrong if this is not done right. Frequently, one sees tests for optimization routines on problems where the optimal solution is not known. Since there are usually different ways to express optimality conditions it is not always clear how to check for optimality; even more so, if you only check for “approximate optimality”, e.g. up to machine precision. A frequently observed effect is a kind of “trusted method bias”. By this I mean that an optimal solution is calculated by some trusted method and comparing the outcome of the tested routine with this solution. However, the trusted method uses some stopping criterion usually based on some specific set of formulations of optimality conditions and these can be different from what the new method has been tuned to. And most often, the stopping criteria do not give a rigorous error bound for the solution or the optimal objective value.

For sparse reconstruction problems, I dealt with this issue in “Constructing test instance for Basis Pursuit Denoising” (preprint available here) but I think this methodology could be used for other settings as well.

Today there are several things I could blog on. The first is the planary by Rich Baraniuk on Compressed Sensing. However, I don’t think that I could reflect the content in a way which would be helpful for a potential reader. Just for the record: If you have the chance to visit one of Rich’s talk: Do it!

The second thing is the talk by Bernd Hofmann on source conditions, smoothness and variational inequalities and their use in regularization of inverse problems. However, this would be too technical for now and I just did not take enough notes to write a meaningful post.

As a third thing I have the talk by Christian Clason on inverse problems with uniformly distributed noise. He argued that for uniform noise it is much better to use an ${L^\infty}$ discrepancy term instead of the usual ${L^2}$-one. He presented a path-following semismooth Newton method to solve the problem

$\displaystyle \min_x \frac{1}{p}\|Kx-y^\delta\|_\infty^p + \frac{\alpha}{2}\|x\|_2^2$

and showed examples with different kinds of noise. Indeed the examples showed that ${L^\infty}$ works much better than ${L^2}$ here. But in fact it works even better, if the noise is not uniformly distributed but “impulsive” i.e. it attains bounds ${\pm\delta}$ almost everywhere. It seems to me that uniform noise would need a slightly different penalty but I don’t know which one – probably you do? Moreover, Christian presented the balancing principle to choose the regularization parameter (without knowledge about the noise level) and this was the first time I really got what it’s about. What one does here is, to choose ${\alpha}$ such that (for some ${\sigma>0}$ which only depends on ${K}$, but not on the noise)

$\displaystyle \sigma\|Kx_\alpha^\delta-y^\delta\|_\infty = \frac{\alpha}{2}\|x_\alpha^\delta\|_2^2.$

The rational behind this is, that the left hand side is monotonically non-decreasing in ${\alpha}$, while the right hand side is monotonically non-increasing. Hence, there should be some ${\alpha}$ “in the middle” which make both somewhat equally large. Of course, we do neither want to “over-regularize” (which would usually “smooth too much”) nor to “under-regularize” (which would not eliminate noise). Hence, balancing seems to be a valid choice. From a practical point of view the balancing is also nice because one can use the fixed-point iteration

$\displaystyle \alpha^{n+1} = 2\sigma\frac{\|Kx_{\alpha^n}^\delta - y^\delta\|_\infty}{\|x_{\alpha_n}^\delta\|_2^2}$

which converges in a few number of iterations.

Then there was the talk by Esther Klann, but unfortunately, I was late so only heard the last half…

Last but not least we have the talk by Christiane Pöschl. If you are interested in Total-Variation-Denoising (TV denoising), then you probably have heard many times that “TV denoising preserves edges” (have a look at the Wikipedia page – it claims this twice). What Christiane showed (in a work with Vicent Caselles and M. Novaga) that this claim is not true in general but only for very special cases. In case of characteristic functions, the only functions for which the TV minimizer has sharp edges are these so-called calibrated sets, introduced by Caselles et el. Building on earlier works by Caselles and co-workers she calculated exact minimizers for TV denoising in the case that the image consists of characteristic functions of two convex sets or of a single star shaped domain, that is, for a given set $B$ she calculated the solution of

$\displaystyle \min_u\int (u - \chi_B)^2dx + \lambda \int|Du|.$

This is not is as easy as it may sound. Even for the minimizer for a single convex set one has to make some effort. She presented a nice connection of the shape of the obtained level-sets with the morphological operators of closing and opening. With the help of this link she derived a methodology to obtain the exact TV denoising minimizer for all parameters. I do not have the images right now but be assured that most of the time, the minimizers do not have sharp edges all over the place. Even for simple geometries (like two rectangles touching in a corner) strange things happen and only very few sharp edges appear. I’ll keep you posted in case the paper comes out (or appears as a preprint).

Christiane has some nice images which make this much more clear:

For two circles edges are preserved if they are far enough away from each other. If they are close, the area “in between” them is filled and, moreover, obey this fuzzy boundary. I remember myself seeing effects like this in the output of TV-solvers and thinking “well, it seems that the algorithm is either not good or not converged yet – TV should output sharp edges!”.

For a star-shaped shape (well, actually a star) the output looks like this. The corners are not only rounded but also blurred and this is true both for the “outer” corners and the “inner” corners.

So, if you have any TV-minimizing code, go ahead and check if your code actually does the right things on images like this!
Moreover, I would love to see similar results for more complicated extensions of TV like Total Generalized Variation, I treated here.

In this post I gladly announce that three problems that bothered me have been solved: The computational complexity of certifying RIP and NSP and the number of steps the homotopy method needs to obtain a solution of the Basis Pursuit problem.

1. Complexity of RIP and NSP

On this issue we have two papers:

The first paper has the more general results and hence, we start with the second one: The main result of the second paper is this:

Theorem 1 Let a matrix ${A}$, a positive integer ${K}$ and some ${0<\delta<1}$ be given. It is hard for NP under randomized polynomial-time reductions to check if ${A}$ satisfies the ${(K,\delta)}$ restricted isometry property.

That does not yet say that it’s NP-hard to check if ${\delta}$ is an RIP constant for ${K}$-sparse vectors but it’s close. I think that Dustin Mixon has explained this issue better on his blog than I could do here.

In the first paper (which is, by the way, on outcome of the SPEAR-project in which I am involved…) the main result is indeed the conjectured NP-hardness of calculating RIP constants:

Theorem 2 For a given matrix ${A}$ and a positive integer ${K}$, it is NP-hard to compute the restricted isometry constant.

Moreover, this is just a corollary to the main theorem of that paper which reads as

Theorem 3 For a given matrix ${A}$ and a positive integer ${K}$, the problem to decide whether ${A}$ satisfies the restricted isometry property of order ${K}$ for some constant ${\delta<1}$ is coNP-complete.

They also provide a slightly strengthened version of Theorem~1:

Theorem 4 Let a matrix ${A}$, a positive integer ${K}$ and some ${0<\delta<1}$ be given. It is coNP-complete to check if ${A}$ satisfies the ${(K,\delta)}$ restricted isometry property.

Moreover, the paper by Pfetsch and Tillmann also proves something about the null space property (NSP):

Definition 5 A matrix ${A}$ satisfies the null space property of order ${K}$ if there is a constant ${\alpha>0}$ such that for all elements ${x}$ in the null space of ${A}$ it holds that the sum of the ${K}$ largest absolute values of ${x}$ is smaller that ${\alpha}$ times the 1-norm of ${x}$. The smallest such constant ${\alpha}$ is called the null space constant of order ${K}$.

Their main result is as follows:

Theorem 6 F or a given matrix ${A}$ and a positive integer ${K}$, the problem to decide whether ${A}$ satisfies the null space property order ${K}$ for some constant ${\alpha<1}$ is coNP-complete. Consequently, it is NP-hard to compute the null space constant of ${A}$.

2. Complexity of the homotopy method for Basis Pursuit

The second issue is about the basis pursuit problem

$\displaystyle \min_x \|x\|_1\quad\text{s.t.}\ Ax=b.$

which can be approximated by the “denoising variant”

$\displaystyle \min_x \lambda\|x\|_1 + \tfrac12\|Ax-b\|_2^2.$

What is pretty interesting about the denoising variant is, that the solution ${x(\lambda)}$ (if it is unique throughout) depends on ${\lambda}$ in a piecewise linear way and converges to the solution of basis pursuit for ${\lambda\rightarrow 0}$. This leads to an algorithm for the solution of basis pursuit: Start with ${\lambda=\|A^Tb\|_\infty}$ (for which the unique solution is ${x(\lambda)=0}$), calculate the direction of the “solution path”, follow it until you reach a “break point”, calculate the next direction and so on until ${\lambda}$ hits zero. This is for example implemented for MATLAB in L1Homotopy (the SPAMS package also seems to have this implemented, however, I haven’t used it yet). In practice, this approach (usually called homotopy method) is pretty fast and moreover, only detects a few break points. However, an obvious upper bound on the number of break point is exponential in the number of entries in ${x}$. Hence, it seemed that one was faced with a situation similar to the simplex method for linear programming: The algorithms performs great an average but the worst case complexity is bad. That this is really true for linear programming is known since some time by the Klee-Minty example, an example for which the simplex method takes an exponential number of steps. What I asked myself for some time: Is there a Klee-Minty example for the homotopy method?

Now the answer is there: Yes, there is!

The denoising variant of basis pursuit is also known as LASSO regularization in the statistics literature and this explains the title of the paper which comes up with the example:

Julien and Bin investigate the number of linear segments in the regularization path and first observe that this is upper bounded by ${(3^p+1)/2}$ is ${p}$ is the number of entries in ${x}$ (i.e. the number of variables of the problem). Then they try to construct an instance that matches this upper bound. They succeed in a clever way: For a given instance ${(A,b)}$ with a path with ${k}$ linear segments they try to construct an instance which has one more variable such that the number of linear segments in increased by a factor. Their result goes like this:

Theorem 7 Let ${A\in{\mathbb R}^{n\times p}}$ have full rank and let ${b\in{\mathbb R}^n}$ be in the range of ${A}$. Assume that the homotopy path has ${k}$ linear segments and denote by ${\lambda_1}$ the regularization parameter which corresponds to the smallest kink in the path. Now choose ${b_{n+1}\neq 0}$ and ${\alpha}$ such that

$\displaystyle 0<\alpha < \frac{\lambda_1}{2\|b\|_2^2 + b_{n+1}^2} \ \ \ \ \ (1)$

and define ${\tilde b\in{\mathbb R}^{n+1}}$ and ${\tilde A\in{\mathbb R}{(n+1)\times (p+1)}}$ by

$\displaystyle \tilde b = \begin{bmatrix} b\\ b_{n+1} \end{bmatrix}, \quad \tilde A = \begin{bmatrix} A & 2\alpha b\\ 0 & \alpha b_{n+1} \end{bmatrix}.$

Then the homotopy path for the basis pursuit problem with matrix ${\tilde A}$ and right hand side ${\tilde b}$ has ${3k-1}$ linear segments.

With this theorem at hand, it is straightforward to recursively build a “Mairal-Yu”-example which matches the upper bound for the number of linear segments. The idea is to start with a ${1\times 1}$ example and let it grow by one row and one column according to Theorem~7. We start with the simplest ${1\times 1}$ example, namely ${A = [1]}$ and ${b=[1]}$. To move to the next bigger example you can choose the next entry ${b_{n+1}}$ and we always choose ${1}$ for convenience. Moreover, you need the next ${\alpha}$ and you need to know the smallest kink in the path. I calculated the paths and kinks with L1Packv2 by Ignace Loris because it is written in Mathematica and can use exact arithmetics with rational numbers (and you will see, that accuracy will be an issue even for small instances) and seemed bullet proof for me. Let’s see where this idea brings us:

Example 1 (Mairal-Yu example)

• Stage 1: We start with ${n=p=1}$, ${b=[1]}$ and ${A=[1]}$. The homotopy path has one kink at ${\lambda_1=1}$ (with corresponding solution ${[0]}$) and hence, two linear segments. Now let’s go to the next larger instance:
• Stage 2: We can choose the entry ${b_2}$ as we like and choose it equals to 1, i.e. our new ${b}$ is

$\displaystyle b = \begin{bmatrix} 1\\1 \end{bmatrix}.$

Now we have to choose ${\alpha}$ according to (1), i.e

$\displaystyle 0 < \alpha < \frac{\lambda_1}{2\|b\|_2^2 + b_{n+1}^2} = \frac{1}{2+1} = \frac{1}{3}$

and we can choose, e.g., ${\alpha = 1/4}$ which gives our new matrix

$\displaystyle A = \begin{bmatrix} 1 & \frac12\\ 0 & \frac14 \end{bmatrix}.$

The calculation of the new regularization path shows that it has exactly the announced number of 5 segments and the parameter of the smallest kink is ${\lambda_1 = \frac{1}{13}}$.

• Stage 2: Again we choose ${b_{n+1} = 1}$ giving

$\displaystyle b = \begin{bmatrix} 1\\1\\1 \end{bmatrix}$

For the choice of ${\alpha}$ we need that

$\displaystyle 0<\alpha < \frac{1}{13(4+1)} = \frac{1}{75}$

and we may choose

$\displaystyle \alpha = \frac1{80}.$

which gives the next matrix

$\displaystyle A = \begin{bmatrix} 1 & \frac12 & \tfrac{1}{40}\\ 0 & \frac14 & \tfrac{1}{40}\\ 0 & 0 & \tfrac{1}{80} \end{bmatrix}.$

We calculate the regularization path, observe that it has the predicted 14 segments and that the parameter of the smallest kink is ${\lambda_1 = \frac{1}{193}}$.

• Stage 3: Again we choose ${b_{n+1} = 1}$ giving

$\displaystyle b = \begin{bmatrix} 1\\1\\1\\1 \end{bmatrix}$

For the choice of ${\alpha}$ we need that

$\displaystyle 0<\alpha < \frac{1}{193(6+1)} = \frac{1}{1351}$

and we see that things are getting awkward here…

Proceeding in this way we always increase the number of linear segments ${k_n}$ for the ${n\times n}$-case from ${k_n}$ to ${k_{n+1} = 3k_n-1}$ in each step and one checks easily that this leads to ${k_n = (3^n+1)/2}$ which is the worst case! If you are interested in the regularization path: I produced picture for the first three dimensions (well, I could not draw a 4d ${\ell^1}$-ball) and here they are:

1d Mairal-Yu example

2d Mairal-Yu example

3d Mairal-Yu example

It is not really easy to perceive the whole paths from the pictures because the magnitude of the entries vary strongly. I’ve drawn the path in red, each kink marked with a small circle. Moreover, I have drawn the according ${\ell^1}$-balls of the respective radii to provide more geometric information.

The paper by Mairal and Yu has more results of the paths if one looks for approximate solutions of the linear system but I will not go into detail about them here.

At least two questions come to mind:

• The Mairal-Yu example is ${n\times n}$. What is the worst case complexity for the true rectangular case? In other words: What is the complexity for ${p\times n}$ in terms of ${p}$ and ${n}$?
• The example and the construction leads to matrices that does not have normed columns and moreover, they are far from being equal in norm. But matrices with normed columns seem to be more “well behaved”. Does the worst case complexity lowers if the consider matrices with unit-norm columns? Probably one can construct a unit-norm example by proper choice of ${b}$

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