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The '''Landweber iteration''' or '''Landweber algorithm''' is an algorithm to solve [[ill-posed]] linear [[inverse problems]], and it has been extended to solve non-linear problems that involve constraints. The method was first proposed in the 1950s,<ref name="Landweber"/> and it can be now viewed as a special case of many other more general methods.<ref name="Combettes"/>
Shares in LVMH (LVMH.PA), the world's biggest luxury goods group, fell sharply on Wednesday after an unexpected slowdown in sales growth at its fashion and leather business, which includes the Louis Vuitton, [http://www.pcs-systems.co.uk/Images/celinebag.aspx Celine Bag Online] and Dior brands.<br><br>The share price was down 6.4 percent at 135.50 euros by 1140 GMT, a six-week low and wiping around 4.8 [http://En.wiktionary.org/wiki/billion+euros billion euros] ($6.5 billion)off the market value of France's fourth-biggest listed company.<br><br>The group, which also owns Ruinart champagne and Hennessy cognac, saw sales growth at the fashion and leather division slide to 3 percent in the third quarter, against expectations of 7 to 8 percent.<br>In a conference call, Chief Financial Officer Jean-Jacques Guiony blamed price increases in Japan for the slowdown, as well as softer demand for some brands.<br>However, one London-based analyst noted that Japan accounted for only around 15 percent of LVMH's fashion and leather sales, "so we did not get a full explanation".<br><br>LVMH has been trying to stem a decline in Louis Vuitton's sales growth by introducing new and pricier leather bags, which analysts expected would lead to short-term losses in sales.<br>"I understand that the repositioning of Louis Vuitton takes time and may be a bumpy ride," said Exane BNP Paribas analyst Luca Solca.<br>Before the results were announced, LVMH shares were up 4.4 percent since January 1, underperforming the overall luxury sector, which was up more than 20 percent.<br><br>Analysts said concerns about the future growth of Louis Vuitton had been exacerbated by the announcement this month that it was parting company with its star designer, Marc Jacobs.<br>SUPPLY CONSTRAINTS<br>Guiony said the recent launch of new leather collections had not come in time to have a meaningful impact on sales, and acknowledged that production was constrained by a lack of high-quality leather.<br><br>"Without these supply constraints, we would produce more than what we do," Guiony said.<br>Louis Vuitton shop assistants polled by Reuters last month said they had been provided with only a small number of new handbags, such as the Capucines model, priced at 3,500 euros, which had flown off the shelves.<br>LVMH has been buying tanneries to secure supplies but experts say the market is under pressure partly because the number of calves raised and slaughtered is driven more by demand for meat -- which has been in decline -- than by demand for quality hides.<br><br>In addition, China, the luxury industry's main driver since the late 2000s, has started to run out of steam in the last year due to an economic slowdown and a government crackdown on gift-giving.<br>Guiony said Vuitton's sales in mainland China were "flattish" but, thanks to sales to Chinese tourists, overall sales growth to Chinese customers was in the "mid-single digits plus".<br>Guiony said trends in watches and jewelry had slightly improved in China, but not in fashion and leather.<br><br>He said trading remained difficult in Europe, particularly for perfume and cosmetics, where sales were "flattish".<br>LVMH's overall sales grew 2 percent in the third quarter. Organic growth was 8 percent, of which 6 percentage points were accounted for by the relative weakness of the U.S. dollar and the Japanese yen against the euro.<br>($1=0.7406 euros)<br>(Additional reporting by James Regan; Editing by James Jukwey and Kevin Liffey)
 
== Basic algorithm ==
The original Landweber algorithm <ref name="Landweber"/> attempts to recover a signal ''x'' from measurements ''y''. The linear version assumes that <math>y=Ax</math> for a [[linear operator]] ''A''. When the problem is in finite [[dimensions]], ''A'' is just a matrix.
 
When ''A'' is [[nonsingular]], then an explicit solution is <math> x = A^{-1} y</math>. However, if ''A'' is [[ill-conditioned]], the explicit solution is a poor choice since it is sensitive to any errors made on ''y''. If ''A'' is [[Mathematical singularity|singular]], this explicit solution doesn't even exist. The Landweber algorithm is an attempt to [[Regularization (mathematics)|regularize]] the problem, and is one of the alternatives to [[Tikhonov regularization]]. We may view the Landweber algorithm as solving:
 
: <math> \min_x 0.5 \|Ax-y\|_2^2 </math>
 
using an iterative method. For [[ill-posed]] problems, the iterative method may be purposefully stopped before convergence.
 
The algorithm is given by the update
 
: <math> x_{k+1} = x_{k} - A^*(Ax_k - y). </math>
 
If we write <math> f(x) = 0.5 \|Ax-y\|_2^2 </math>, then the update can be written in terms of the [[gradient]]
 
: <math> x_{k+1} = x_k - \nabla f(x_k) </math>
 
and hence the algorithm is a special case of [[gradient descent]].
 
Discussion of the Landweber iteration as a [[regularization (mathematics)|regularization]] algorithm can be found in.<ref>Louis, A.K. (1989): Inverse und schlecht gestellte Probleme. Stuttgart, Teubner</ref><ref>Vainikko, G.M., Veretennikov, A.Y. (1986): Iteration Procedures in Ill-Posed Problems. Moscow, Nauka (in Russian)</ref>
 
== Nonlinear extension ==
In general, the updates generated by
<math> x_{k+1} = x_{k} - \tau \nabla f(x_k) </math>
will generate a sequence <math>f(x_k)</math> that [[convergence (mathematics)|converges]] to a minimizer of ''f'' whenever ''f'' is [[convex function|convex]]
and the stepsize <math>\tau</math> is chosen such that <math> 0 < \tau < 2/( \|A\|^2 ) </math> where <math> \|\cdot \| </math> is the [[spectral norm]].
 
Since this is special type of gradient descent, there currently is not much benefit to analyzing it on its own as the nonlinear Landweber, but such analysis was performed historically by many communities not aware of unifying frameworks.
 
The nonlinear Landweber problem has been studied in many papers in many communities; see, for example,.<ref>A convergence analysis of the Landweber iteration for nonlinear ill-posed problems
Martin Hanke, Andreas Neubauer and Otmar Scherzer. NUMERISCHE MATHEMATIK
Volume 72, Number 1 (1995), 21-37, DOI: 10.1007/s002110050158</ref>
 
== Extension to constrained problems ==
If ''f'' is a [[convex function]] and ''C'' is a [[convex set]], then the problem
 
: <math> \min_{x \in C} f(x) </math>
 
can be solved by the constrained, nonlinear Landweber iteration, given by:
 
: <math> x_{k+1} = \mathcal{P}_C( x_{k} - \tau \nabla f(x_k) )</math>
 
where <math>\mathcal{P}</math> is the [[projection (mathematics)|projection]] onto the set ''C''. Convergence is guaranteed when <math> 0 < \tau < 2/( \|A\|^2 ) </math>.<ref>Eicke, B.: Iteration methods for convexly constrained ill-posed problems in Hilbert space. Numer. Funct. Anal. Optim. 13, 413–429 (1992)</ref> This is again a special case of [[projected gradient descent]] (which is a special case of the [[Forward–backward algorithm (operator splitting)|forward–backward algorithm]]) as discussed in.<ref name="Combettes"/>
 
== Applications ==
Since the method has been around since the 1950s, it has been adopted by many scientific communities, especially those studying ill-posed problems. In particular, the [[computer vision]] community <ref>
Johansson, B., Elfving, T., Kozlovc, V., Censor, Y., Forssen, P.E., Granlund, G.; "The application of an oblique-projected Landweber method to a model of supervised learning", Math. Comput. Modelling, vol 43, pp 892–909 (2006)</ref> and the signal restoration community.<ref>Trussell, H.J., Civanlar, M.R.: The Landweber iteration and projection onto convex sets. IEEE Trans. Acoust., Speech, Signal Process. 33, 1632–1634 (1985)</ref> It is also used in [[image processing]], since many image problems, such as [[deblurring]], are ill-posed. Variants of this method have been used also in sparse approximation problems and compressed sensing settings.<ref>{{cite web
|url        = http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=6136024&tag=1
|title      = Recipes for hard thresholding methods
|author      = Anastasios Kyrillidis and Volkan Cevher
}}</ref>
 
== References ==
<references>
<ref name="Landweber">Landweber, L. (1951): An iteration formula for Fredholm integral equations of the first kind.
Amer. J. Math. 73, 615–624</ref>
<ref name="Combettes">P. L. Combettes and J.-C. Pesquet, "Proximal splitting methods in signal processing," in: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, (H. H. Bauschke, R. S. Burachik, P. L. Combettes, V. Elser, D. R. Luke, and H. Wolkowicz, Editors), pp. 185–212. Springer, New York, 2011. [http://www.ann.jussieu.fr/~plc/prox.pdf PDF]</ref>
</references>
 
[[Category:Image processing]]
[[Category:Inverse problems]]
[[Category:Gradient methods]]

Revision as of 11:48, 12 February 2014

Shares in LVMH (LVMH.PA), the world's biggest luxury goods group, fell sharply on Wednesday after an unexpected slowdown in sales growth at its fashion and leather business, which includes the Louis Vuitton, Celine Bag Online and Dior brands.

The share price was down 6.4 percent at 135.50 euros by 1140 GMT, a six-week low and wiping around 4.8 billion euros ($6.5 billion)off the market value of France's fourth-biggest listed company.

The group, which also owns Ruinart champagne and Hennessy cognac, saw sales growth at the fashion and leather division slide to 3 percent in the third quarter, against expectations of 7 to 8 percent.
In a conference call, Chief Financial Officer Jean-Jacques Guiony blamed price increases in Japan for the slowdown, as well as softer demand for some brands.
However, one London-based analyst noted that Japan accounted for only around 15 percent of LVMH's fashion and leather sales, "so we did not get a full explanation".

LVMH has been trying to stem a decline in Louis Vuitton's sales growth by introducing new and pricier leather bags, which analysts expected would lead to short-term losses in sales.
"I understand that the repositioning of Louis Vuitton takes time and may be a bumpy ride," said Exane BNP Paribas analyst Luca Solca.
Before the results were announced, LVMH shares were up 4.4 percent since January 1, underperforming the overall luxury sector, which was up more than 20 percent.

Analysts said concerns about the future growth of Louis Vuitton had been exacerbated by the announcement this month that it was parting company with its star designer, Marc Jacobs.
SUPPLY CONSTRAINTS
Guiony said the recent launch of new leather collections had not come in time to have a meaningful impact on sales, and acknowledged that production was constrained by a lack of high-quality leather.

"Without these supply constraints, we would produce more than what we do," Guiony said.
Louis Vuitton shop assistants polled by Reuters last month said they had been provided with only a small number of new handbags, such as the Capucines model, priced at 3,500 euros, which had flown off the shelves.
LVMH has been buying tanneries to secure supplies but experts say the market is under pressure partly because the number of calves raised and slaughtered is driven more by demand for meat -- which has been in decline -- than by demand for quality hides.

In addition, China, the luxury industry's main driver since the late 2000s, has started to run out of steam in the last year due to an economic slowdown and a government crackdown on gift-giving.
Guiony said Vuitton's sales in mainland China were "flattish" but, thanks to sales to Chinese tourists, overall sales growth to Chinese customers was in the "mid-single digits plus".
Guiony said trends in watches and jewelry had slightly improved in China, but not in fashion and leather.

He said trading remained difficult in Europe, particularly for perfume and cosmetics, where sales were "flattish".
LVMH's overall sales grew 2 percent in the third quarter. Organic growth was 8 percent, of which 6 percentage points were accounted for by the relative weakness of the U.S. dollar and the Japanese yen against the euro.
($1=0.7406 euros)
(Additional reporting by James Regan; Editing by James Jukwey and Kevin Liffey)

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