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Data Analytics Courses in Pune.

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Data Analytics Courses in Pune.

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  1. Singular Value Decomposition (SVD)

  2. Matrix Transpose

  3. Matrix Multiplication

  4. Matrix Inverse If, A B I , identity matrix × = A1 - Then, B = Identity matrix :   0 1 0 0 ...   1 0 0 0 ...     . . . . . . ... .   0 0 ...1 0       0 0 1 0 ...

  5. X= UΣVT U and V are Orthonormal matrices rxr diagonal matrix with r non-zero diagonal elements

  6. VT X U ∑ 1 0 0 0 0 3 0 0 0 0 4 0 0 2 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 -1 1 0 0 0 4 0 0 3 0 0 2.24 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0.45 0 0 0 0.89 0 0 0 1 0 0 = x x 0 0 0 OPTIONAL OPTIONAL OPTIONAL OPTIONAL R Code > M=matrix(c(1,0,0,0,0,0,0,4,0,3,0,0,0,0,0,0,2,0,0,0),nrow=4,ncol=5) > X=svd(M) > X$u > X$d > X$v > X$u%*%diag(X$d)%*%t(X$v)

  7. Applications of SVD in image processing – closest rank-k approximation for a matrix X u X i ∑ = 1 k T k v Σ = i i i – Each term in the summation expression above is called principal image

  8. Original matrix (X) 1 0 0 0 2 0 0 3 0 0 0 0 0 0 0 0 4 0 0 0 Original size 4*5=20 bytes VT X U ∑ 1 0 0 0 0 3 0 0 0 0 4 0 0 2 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 -1 1 0 0 0 4 0 0 3 0 0 2.24 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0.45 0 0 0 0.89 0 0 0 1 0 0 = x x 0 0 0 k=1 0 x 4 x 0 0 1 0 1 0 0 0 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 Compressed size 4*1+1+1*5=10 bytes

  9. k=2 0 0 x 0 1 0 0 1 0 4 0 x 0 3 0 1 0 0 0 = 0 0 1 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 4 0 0 0 Compressed size 4*2+2+2*5=20 bytes k=3 0 0 1x 0 1 0 0 0 0 1 0 0 4 0 0 3 0 0 2.24 0x 0 0 1 0 0 0 1 0 0 0 0 0.89 0= 0 1 0 0 0 0 0 0 4 0 3 0 0 0 2 0 0 0 0 0 0 Compressed size 4*3+3+3*5=30 bytes 0.45 0 k=4 0 0 1 0 0 1 0 0 0 0 0 -1 1 0 0 0 4 0 0 0 0 3 0 2.24 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0.89 1 0= 0 1 0 0 0 0 0 0 4 0 3 0 0 0 2 0 0 0 0 0 0 Compressed size 4*4+4+4*5=40 bytes x x 0.45 0 0 0

  10. The image compression example in http://journal.batard.info/post/2009/04/08/svd- fun-profit • Original size = 384*384 bytes = 147,456 bytes • k=1: 384*1+1+1*384=769 bytes • k=10: 384*10+10+10*384=7,690 bytes • k=20: 384*20+20+20*384=15,380 bytes • k=50: 384*50+50+50*384=38,450 bytes • k=100: 384*100+100+100*384=76,900 bytes • k=200: 384*200+200+200*384=153,800 bytes

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