Rules of thumb for L A TEX

Transcription

1 Thesis Questions and Suggestions A review for Kourosh Modaressi, February 8, 27 Rules of thumb for L A TEX A significant difficulty for readers has been that the 141 pages of L A TEX are about twice as long as necessary (because of too much white space on every page. Some general suggestions follow. 1. Specify less than 12pt font. 11pt and even 1pt are allowed by the university. 2. Use the default \parindent (a positive amount. If indentation is suppressed, it is hard to know when a new paragraph begins. 3. Use something less than double-spacing. The university allows a factor of about 1.2 (see \setstretch below. 4. Use the first page of the Preface and each chapter they shouldn t be blank. 5. Make sure there is no blank line preceding a displayed equation. Also remove blank lines after the display unless you want the tet to start a new paragraph. 6. A sentence that contains a displayed equation should be regarded as a normal (complete sentence. Imagine the display to be a single character in the middle of the sentence. 7. Ordinary words like and or reg inside math epressions look like algebraic variables. Use \tet{and} or \tetit{reg}. 8. At present, all equations are numbered whether they are referred to or not. Certainly use \label{} and \ref{} where relevant, but remove unreferenced equation numbers. 9. Refer to a previous equation if it eists, rather than repeating the same epression multiple times. 1. Norms should not be typeset with. Make use of macros like \norm{...} defined below. 11. Replace horizontal, diagonal, vertical dots (... by \ldots, \ddots, \vdots. 12. Abbreviations like e.g. and i.e. look like the end of sentences if they are followed by a space. They should be typeset as e.g.\ or e.g., to generate an ordinary space. 13. Use bibte for the bibliography. 1

2 Thesis format The following eample contains some useful commands. %\documentclass[twoside,1pt,final]{report} \documentclass[twoside,1pt,draft]{report} \includeonly{chapter, Chapter1, Chapter2, Chapter3, Chapter4} %\includeonly{chapter, Chapter2} \usepackage{amsfonts,amsmath,amssymb,graphic} \usepackage{vmargin} \setpapersize{usletter} \setmarginsrb{1.6in}{.7in}{1.1in}{.6in} % left, top, right, bottom {.2in}{.2in}{.2in}{.5in} % head height, head sep, % foot height, foot skip \usepackage{setspace} \setstretch{1.213} \newcommand{\clearemptydoublepage} {\newpage{\thispagestyle{plain}\cleardoublepage}} \newcommand{\bmat}[1]{\begin{bmatri}#1\end{bmatri}} % Needs {amsmath} \newcommand{\pmat}[1]{\begin{pmatri}#1\end{pmatri}} \newcommand{\matlab}{{\sc Matlab}} \newcommand{\norm}[1]{\ #1\ } % for ordinary vectors \newcommand{\normm}[1]{\left\ #1\right\ } % for larger things \begin{document} \pagenumbering{roman} \pagestyle{plain} \include{chapter} \include{chapter1} \include{chapter2} \include{chapter3} \include{chapter4} \clearemptydoublepage \bibliographystyle{abbrv} \bibliography{refs} \end{document} 2

3 The mathematical problems being studied The aim of the thesis is to solve two structured least-squares problems of the form Nonoverlapping Local Tikhonov Regularization: K 1... K q d λ 1 L 1 1 i...., NLTR. λ q L q q where the matrices K i and L i are given (they have n i columns and i is an n i -vector, and Overlapping Local Tikhonov Regularization: K d λ 1 L 1., OLTR. λ q L q where K and each L i are given. The problems are nontrivial even if the λ i are known. The full aim is to find good values for each λ i by computing many solutions i or and optimizing some function. The strategy has been to work with least-squares subproblems of the form ( ( ( ( Ki di i λ i L i and K d i λ i L i respectively, because effective methods are allegedly known for such subproblems (although in reality only the case L i = I seems to allow significant economy. Technical comments and questions 1. Chapter 1 is 3 pages of tetbook background material, only some of which is used later. Thought could be given to which parts are really necessary. 2. At a lower level, p19 repeats most of p18, and half of p2 repeats part of p On p22, ( and ( are the same comple equation. 4. What is p in ( ? 5. Chapter 1 could be the place to summarize classical methods for solving regularized least-squares (RLS problems; that is, the above subproblems with subscript i s deleted: ( K λl ( d. RLS Alternatively there could be a new chapter devoted to the classical methods. 6. Chapter 2 is 6 pages. This is too many for a reader to discern the structure. 3

4 7. On p31 and elsewhere, the word priori is used without definition. Probably some other term is needed. 8. Section 2.2 starts with general L in (2.2.2 (2.2.3 but then sets L = I. For later reference, this section needs to continue with methods for a general L. 9. On p35, K is said to be a nonsingular matri, but elsewhere it is assumed to be rectangular. Do you mean that it has full column rank? 1. Again on p35, if L does not have full row rank, applying an orthogonal transformation cannot alter its rank. 11. On p37, what is the optimal value of eact reg? Ideally it should be zero. 12. In (2.3.16, is S ( intentional (is it a derivative?, or should it be just S(? 13. Choose one of ( and ( On the same page (bottom, what are s i and s j? 15. ( sits in mid-air. It should be part of some sentence. 16. On p42, i is said to be a block matri when it is simply a vector. 17. Choose one of ( and ( The NLTR problem on p41 seems reasonable, but p43 gives essentially no motivation for the OLTR formulation. Why would it be useful? 19. Section 2.4 reintroduces the structured NLTR problem, but then section 2.5 (pp describes several standard methods that ignore the structure, and gives incorrect operation counts. As mentioned, these methods belong earlier in the Introduction or in a new Chapter 2. Note that for Cholesky on the normal equations, there is no need to spell out the steps in such detail; they are a basic part of numerical linear algebra. (The same steps are listed again on pp The beginning of section 2.6 (pp describes Elden s method for transforg a problem RLS involving L to one in which L = I. Again, since structure is ignored, this should be given in an earlier chapter. At the end, give a concise method for recovering the original (without repeating the epression R T. 21. On p55, V is a confusing name for a triangular matri, given that it usually refers to an orthogonal matri from an SVD. 22. The remainder of section 2.6 (pp tries to reduce the NLTR problem to one in which all L i = I. It does so by applying Elden s method to each subproblem (unnecessarily repeating the algebra with lots of subscript i s, ecept λ i is kept separate from L i. This section is long and confused for the following reasons. 4

5 23. Starting at step (2-4 there is a single sentence that continues to the end of p6. A key transformed subproblem involving K i, i, di appears in the middle. Unfortunately it is not a coherent sentence, and the subproblem simply appears without eplanation. This is where errors enter unnoticed. 24. Step (2 loops for i = 1,..., q, yet step (3 says to repeat step (2 for i + 1. Step (3 is really part of step (2 and just needs to define d i The definition d i+1 = d i cannot be correct because d i has reduced length. The right-hand side for the K i+1 subproblem would have to have length m. There is no obvious correction for this error. 26. Similarly the full transformed NLTR problem is stated without a coherent derivation, using a matri K = ( K1... Kq in which the submatrices Ki have differing numbers of rows. The transformed NLTR problem is therefore not well defined. Again, there is no obvious correction. 27. Section 2.7 assumes that the subproblems are well defined and essentially the only items of interest. It begins (pp with a description of four direct methods for solving problem RLS with L = I (and λ = 1 for some reason. More rotations are needed to eliate I in each case, so the operation counts are more than those stated. 28. Section 2.7 continues (pp with a description of Elden s method for RLS problems in which K is bidiagonal and L = I. All the subscripts i are redundant. It would be simpler to describe the method for the generic RLS problem, not for a subproblem. 29. The SVD part of section 2.7 (pp deals with problem NLTR in socalled standard form with each L i = I. The derivation uses an SVD of all of K = ( K 1... K q, so it is really dealing with an RLS problem in which λl has become a diagonal matri Λ. These pages are the most confused of all, and it is not clear why they are present. Equation (2.7.1 defines z = (K T K +Λ 2 1 v i (where v i is a singular vector of K and subsequently overlooks the fact that z depends on i. Then come the following remarkable equations: Kv i = σ i u i, ( K T u i = σ i v i, ( K T = σ i v i u 1 i, ( K T Kv i = (σ i v i u 1 i (σ i u i = σi 2 v i, ( where u i and v i are vectors. This astoundingly faulty algebra helps eplain why there has been such difficulty in writing a thesis. 3. The preceding error has no impact because ( is true without the help of ( Nevertheless, the final line of section 2.7 is highly suspect: n n Kv i = σ i (u T i d (σi 2 + λ 2 j 1 v i. ( j=1 5

6 31. Sections (pp move on to generalized cross-validation (GCV. Since optimization of the λ i values is intended to be an important part of the thesis, these sections could well be in a chapter of their own. (However, as yet there is no original work on optimizing λ. 32. Section 2.8 should start by clarifying which problem is being discussed. Where is Λ defined? How is its initial value chosen? How is the net value chosen? How can one plot a graph of CV versus a vector Λ? 33. Section 2.9 (pp 8 86 tries to eplain how GCV works. There are algebraic errors and the section is too hard to check. In the end it avoids the issue by referring to Golub and von Matt. It would be more helpful to state clearly which problem the Golub/von Matt method solves. 34. Section 2.1 (pp deals with problem OLTR (another topic that could be in its own chapter. Most of the section (pp repeats earlier material without respecting the OLTR structure. The remaining original material (pp tries to work with the second type of subproblem above (involving a single K and λ i L i with solution λi. The OLTR solution is assumed to be of the form q Λ = a i λi, ( with q a i = 1. (A plausible idea, but some justification is certainly desirable. The subsequent method for computing a i is quite baffling; e.g, in { } q q Λ Λ = a i F 1 (λ 2 i I λ 2 i I λi ( the sums can t both be over i. The matri F 1 is later ignored for convenience. There is no proof of convergence, and no eperimental results. It is hard to conclude that the OLTR algorithm is a useful original contribution. However, simplified eperiments might help (see last page. 35. Chapter 3 (pp gives details of two applications requiring regularization, and numerical results comparing classical Tikhonov with NLTR. Some questions arise: In ( and (3.3.32, L is written as [ L 1 L 1 ] [ L 2 and L1 L 2 ] L 1. Isn t L supposed to be block-diagonal? How was q = 3 chosen to be best? How were λ i and L i selected? (There is no mention of GCV. For the second eample, what are the dimensions of K, L 1 and L 2? Figures 3.5 and 3.6 suggest n = 5. The figures are said to be for noise level std =.5, but the tables don t include this level. Is.5 intentional? Since L 1 and L 2 have different row sizes, how could NLTR be implemented? What can be said about the software written or used? (Presumably Matlab? 6

7 36. Section 3.4 (pp gives some Contributions and Conclusions. Again these would logically be another chapter, but at present they are too brief and vague. It is strange to mention GCV in the Conclusions but not in the eperimental results. 37. In choosing between GCV and the L-curve, there s a big difference between There were few reasons for preferring GCV and There were a few reasons for preferring GCV (!. 38. Evidently GCV must be used multiple times to choose each λ i. The numerical results should give an impression of how many cycles are required. Essential theory and eperiments To detere if problem NLTR can be treated in the desired manner, it will be essential to study the case q = 2. A simplified form of the proposed approach is to solve the partitioned problem,y using local subproblems of the form ( ( A B y b k A k c k, y k By k d k, k = 1, 2,... NLTR If such an algorithm can be devised, it would be in the spirit of the thesis. Convergence must be proved, and Matlab scripts will be needed for numerical verification. There will then be some hope that the proposed NLTR algorithm can be corrected and tested. Note that a sequence of subproblems is likely to be needed (k 2 because it looks like block-gauss-seidel on the normal equations for NLTR. Similarly for problem OLTR, it is necessary to study the case A b B C OLTR using local subproblems of the form ( ( A ck y k y B k, z k ( A C z k ( dk, k = 1, 2,.... Would α k y k + β k z k for suitable α k, β k? This is what the current OLTR algorithm assumes. 7