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Editorial The previous issue of our newsletter featuring Alan Turing was quite well-received. There were requests to translate or reproduce some of the articles. Your suggestions and contributions to feature well-known mathematicians in our future issues will be most welcome. This issue carries two interviews, one with Srinivasa Varadhan, and the other with Gus Lehrer. Srinivasa Varadhan is a highly regarded probabilist who was honoured with the Abel Prize in 2007 and also presented with the National Medal of Science by President Obama in 2010. The second interviewee was Gus Lehrer, an algebraist that has experienced the wartime horrors of Europe. He is known for developing the Howlett–Lehrer theory which is very useful in different areas of mathematics. The contribution by Professor Lê Tuấn Hoa, Managing Director of the Vietnam Institute for Advanced Study in Mathematics (VIASM) and his co-author, Trần Văn Nhung, traces the genesis of VIASM in detail. The article “Indo-French Cooperation in Mathematics” describes how French and Indian mathematicians built up a close collaboration in various topics in mathematics. We hope to publish similar articles on regional and international cooperation in research and education in mathematics. No solutions were sent in for the problems in the Problem Corner of the previous issue of APMN. As a result, the problems from the previous issue are still open to readers to send in their solutions. A book token would be awarded to readers who send in the correct answers. We are happy to say that Professor Peter Hall (University of Melbourne) has kindly and graciously agreed to help us arrange interviews with well-known Australian mathematicians. We have started to publish these interviews since our last issue, and more will appear in future issues of APMN. It is our hope that each mathematician interviewed would also contribute an article to highlight his/her work. One way for us to overcome the shortage of expository articles for the newsletter is to translate articles that have appeared in various national mathematics newsletters and bulletins. So far, we have published articles translated from Chinese, Japanese and Korean. We hope to be able to do so for articles published in other languages in the Asia Pacific region. Of course, it would be nice to have original articles in English sent directly to APMN. Swee Cheng Lim Editor Asia Pacific Mathematics Newsletter welcomes contributions on the following items: • Expository articles on mathematical topics of general interest • Articles on mathematics education • Introducing centres of excellence in mathematical sciences • News of mathematical societies in the Asia Pacific region • Introducing well-known mathematicians from the Asia Pacific region • Book reviews • Conference reports and announcements held in Asia Pacific countries • Letters from readers on relevant topics and issues • Other items of interest to the mathematics community
Compressive Sensing and Applications Myungjoo Kang and Myeongmin Kang Kang Asia Pacific Mathematics Newsletter 1 Compressive Sensing and Application 1. Compressive Sensing Consider a one-dimensional, finite-length signal x ∈ CN . We will vectorise a two-dimensional image or higher-dimensional data into a long onedimensional vector. Many real-world signals can be well-approximated by sparse or compressible under a suitable basis. Let Ψ=[ψ1|ψ2|···|ψN] be an orthonormal basis. Then a signal x can be N� expressed as x = 〈x, ψn〉ψn. We say that x is n=1 k-sparse under Ψ if {fn = 〈x, ψn〉}n=1,...,N has only k-nonzero coefficients, and that x is compressible under Ψ if {〈x, ψn〉}n=1,...,N has a few large coefficients. Compressive sensing is that a sparse signal can be recovered from what was previously believed to be incomplete information. Consider Φ = [φ1|φ2|···|φN] ∈ C M×N for some M < N. Then, we can obtains b =Φx = ΦΨf = Af where A = ΦΨ. The measurements Φ is fixed and does not depend on the signal x and then A is selected independent of f . A is referred to as the encoder and obviously encoder is linear. In the encoder, we need to design a good sensing matrix A. The decoder is the attempted recovery of f from its sensing matrix A and b. We define ||f ||0 := |supp f | for a signal f . The quantity || · ||0 is often called ℓ0-norm although it is actually not a norm. With a sparsity prior, a natural decoder is to search for the sparsest vector f that b = Af : min || f ||0 subject to b = Af . (1) We need to check that if the problem (1) has a solution, the solution is unique. For given matrix A, spark(A) is the smallest number of columns that are linearly dependent. Using this concept, we get a condition of uniqueness. Let x0 be a ksparse N-dimensional vector, let A be a matrix of M × N, and let y = Ax0. If k < spark(A) 2 , then x0 is a unique solution of problem (1). Conversely, if k ≥ spark(A) 2 , then (1) does not have x0 as its unique solution. Since the decoder is welldefined for small k, we need an efficient reconstruction algorithm. Unfortunately, the problem (1) is combinatorial problem and NP-hard in general. Essentially two approaches have mainly been pursued: greedy algorithm and convex relaxation. We will introduce greedy algorithms and convex relaxation for solving (1). 2. Greedy Algorithm A greedy algorithm computes the support of signal iteratively, at each step finding one or more new elements and subtracting their contribution from the measurement vector. Examples include Matching Pursuit (MP), Orthogonal Matching Pursuit (OMP), stagewise OMP, regularised OMP, weak OMP. We introduce MP and OMP here. In 1993, Matching Pursuit is proposed by S Mallat and Z Zhang [17]. Matching Pursuit is an algorithm that decomposes any signal into a linear expansion of atoms that are selected from a redundant dictionary. Let ai be i-th column of A and xi be i-th component of x. Assume that ||ai||2 = 1 for all i. Equation Ax = b is equivalent to b = x1a1 + ···+ xNaN. We want to compute a linear expansion of b over a set {ai : i = 1, 2, ..., N} and their coefficients are sparse. The idea of Matching Pursuit is choosing column of A, in order to best match its inner product structures. For a signal f , f can be decomposed to f = 〈 f , ai〉ai + Rf , where Rf is the residual vector. Clearly, ai is orthogonal to Rf . So, || f || 2 2 = |〈 f , ai〉| 2 + ||Rf || 2 2 by Pythagoras theorem. We have to choose ai such that |〈 f , ai〉| is maximum in order to minimise ||Rf ||. Using this idea, we iteratively choose the column of A that has highest absolute inner product with current residual vector ri = b − Axi and inner product of selected column aj is added to coefficient xj. Orthogonal Matching Pursuit [6, 20] is improved Matching Pursuit by orthogonalising the direct projection with a Gram–Schmidt procedure. OMP algorithm iteratively selects the column of A in the same way like MP. The difference is April 2012, Volume 2 No 2 1
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Page 9 and 10: [18] T. Cai, G. Xu and J. Zhang, On
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Page 15 and 16: Indo-French Cooperation Asia Pacifi
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Page 27 and 28: M Kashiwara, N B Chau, T V Nhung, J
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Page 39 and 40: taken up in many centres, but I am
Page 41 and 42: Endre Szemerédi Receives 2012 Abel
Page 43 and 44: News from Australia Winner of the 2
Page 45 and 46: Professor Zhou has worked at Nankai
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Page 53 and 54: Other News Chinese Undergraduate So
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Conference CALENDAR Conferences in
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11 - 13 Jun 2012 2012 Workshop on G
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17 - 27 Jul 2012 The 5th Mathematic
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4 - 6 Dec 2012 New Zealand Mathemat
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4 - 6 Jul 2012 Mathematical Analysi
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Mathematical Societies in Asia Paci
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Mathematical Society of the Philipp
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