Chapter 3 Sparse Signal Recovery
Chapter 3 Sparse Signal Recovery
3.1 Sparsity:Applications and Development What is sparse? 1.Many data mining tasks can be represented using a vector or a matrix. 2.Sparsity implies many zeros in a vector or a matrix
3.1 Sparsity: Applications and Development What is sparse? 1. Many data mining tasks can be represented using a vector or a matrix. 2. Sparsity implies many zeros in a vector or a matrix
3.1 Sparsity:Applications and Development As seen in the last chapter in linear regression,we are actually solving this problem: y Φ w y(x,w)=wrφ(x)+n Where n is noise. 十 We have learned that,if p >n,there will be serious over-fitting. n×1 nxp pxI n×1 To suppress over-fitting,we can add a p>>n regularizer.We can add a sparse regularizer (LASSO)to render the target vector sparse,to select a small number of basis function
3.1 Sparsity: Applications and Development As seen in the last chapter in linear regression, we are actually solving this problem: Where is noise. We have learned that, if , there will be serious over-fitting. To suppress over-fitting, we can add a regularizer. We can add a sparse regularizer (LASSO) to render the target vector sparse, to select a small number of basis function. 𝒘 𝒏
3.1 Sparsity:Applications and Development In image processing,to compress a image,we first do a transformation to the pixel matrix to render it sparse,such transformations are 1 Singular Value Decomposition 2 Discrete Cosine Transform 3 Wavelet Transform... Note that the black pixels indicate the matrix values are close to zero ft corner thus making the matrix easy to compress 2 layer discrete cosine transfornpmed in with Haar wavelet basis
3.1 Sparsity: Applications and Development In image processing, to compress a image, we first do a transformation to the pixel matrix to render it sparse, such transformations are 1 Singular Value Decomposition 2 Discrete Cosine Transform 3 Wavelet Transform… Note that the black pixels indicate the matrix values are close to zero thus making the matrix easy to compress 2 layer discrete cosine transform with Haar wavelet basis U S V SVD Up left corner zoomed in DCT Original
3.1 Sparsity:Applications and Development Some times,on Weibo,interesting news originate from certain users and is forwarded many times by other users.We now know who forwards the messages and when the messages are forwarded. Now we want to construct a relationship (who friended whose Weibo) network from the above information.This can be abstracted as a topological graph. Sparsity:each node is linked to a small number of neighbors. Equivalent matrix representation
3.1 Sparsity: Applications and Development Some times, on Weibo, interesting news originate from certain users and is forwarded many times by other users. We now know who forwards the messages and when the messages are forwarded. Now we want to construct a relationship (who friended whose Weibo) network from the above information. This can be abstracted as a topological graph. Sparsity: each node is linked to a small number of neighbors
3.1 Sparsity:Applications and Development Collaborative filtering: Items ? ? ? ? ? ? ? ? ? ? ? ? ? Customers ? ? ? ? ? ? ? Customers are asked to rank items ? ? ? ? ? ? ? ? ? ? ？ ? ? ? Not all customers ranked all items ? ? ? ? ? Predict the missing rankings ? ？ ? ? ? ?
3.1 Sparsity: Applications and Development Collaborative filtering: Customers are asked to rank items Not all customers ranked all items Predict the missing rankings
3.1 Sparsity:Applications and Development Movies The Netflix prize: ? ? ? ? ? ? ? ? ? ? ? ? ? ? Users ? ? ? ? ? ? ? ? ? ? ? ? About a million users and ? ? ? ? ? ? ? ? ? ? 25000 movies ? Known rankings are sparsely distributed Predict unknown ratings
3.1 Sparsity: Applications and Development The Netflix prize: About a million users and 25000 movies Known rankings are sparsely distributed Predict unknown ratings
3.1 Sparsity:Applications and Development In 2006,monumental papers of compressive sensing were published: Emmanuel Candes,Justin Romberg,and Terence Tao,Robust uncertainty principles:Exact signal reconstruction from highly incomplete frequency information.(IEEE Trans.on Information Theory,52(2)pp.489-509,February 2006) David Donoho,Compressed sensing.(IEEE Trans.on Information Theory,52(4), pp.1289-1306,April2006) Emmanuel Candes and Terence Tao,Near optimal signal recovery from random projections:Universal encoding strategies?(IEEE Trans.on Donoho返a时etet目ghdeshao prize Information Theory,52(12),pp.5406-5425,December 2006)
3.1 Sparsity: Applications and Development In 2006, monumental papers of compressive sensing were published: Emmanuel Candès, Justin Romberg, and Terence Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. (IEEE Trans. on Information Theory, 52(2) pp. 489 - 509, February 2006) David Donoho, Compressed sensing. (IEEE Trans. on Information Theory, 52(4), pp. 1289 - 1306, April 2006) Emmanuel Candès and Terence Tao, Near optimal signal recovery from random projections: Universal encoding strategies? (IEEE Trans. on Information Theory, 52(12), pp. 5406 - 5425, December 2006) Donoho was awarded the Shao prize Emmanuel Candès Terrace Tao
3.2 Sparsity Rendering Algorithms The very important problem in compressive sensing is solving this problem: Given a sparse s,and do this compressiony=Φw,Φis a underdetermined matrix.Now the target is from y,to recover s Bad news is:is underdetermined and we know, normally,y =w has infinite solutions. Good news is:we have a prior information:w is sparse
3.2 Sparsity Rendering Algorithms The very important problem in compressive sensing is solving this problem: Given a sparse , and do this compression , is a underdetermined matrix. Now the target is from , to recover Bad news is: is underdetermined and we know, normally, has infinite solutions. Good news is: we have a prior information: is sparse 𝑦 𝛷 𝑤
3.2 Sparsity Rendering Algorithms Here are two concerns: 1:How sparse should w be so that it can be accurately recovered. 2:Is there any requisition fordΦ？ For question 1,we know that y =w has infinite solutions,thus,we have to attach some conditions to s to this solution unique. As s is sparse,we should make it the sparsest solution for y =w. For question 2,we have the following lemma. Suppose a m x n matrix is such that every set of 2S columns are of are linearly independent.Then an S-sparse (the vector w has s non- zero elements)vector w can be reconstructed uniquely from y =w
3.2 Sparsity Rendering Algorithms Here are two concerns: 1: How sparse should be so that it can be accurately recovered. 2: Is there any requisition for ? For question 1, we know that has infinite solutions, thus, we have to attach some conditions to to this solution unique. As is sparse, we should make it the sparsest solution for . For question 2, we have the following lemma. Suppose a matrix is such that every set of 2S columns are of are linearly independent. Then an S-sparse (the vector has S nonzero elements) vector can be reconstructed uniquely from
