Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Singular Value DecompositionLinear Alegebra, Course 124A, Fall, 2007

Prof. Peter Dodds

Department of Mathematics & StatisticsUniversity of Vermont

Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Outline

Approximating matrices with SVDThe basic ideaGuess who?Bonus example 1Bonus example 2Bonus example 3Bonus example 4Bonus PuzzleColbert on Equations

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Outline

Approximating matrices with SVDThe basic ideaGuess who?Bonus example 1Bonus example 2Bonus example 3Bonus example 4Bonus PuzzleColbert on Equations

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

I Idea: use SVD to approximate images.I Interpret elements of matrix A as color values of an

image.I Truncate series SVD representation of A:

A = UΣV T =r∑

i=1

σi ui vTi

I Use fact that σ1 > σ2 > . . . > σr > 0.

I For color: approximate 3 matrices (RGB).

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

I Idea: use SVD to approximate images.I Interpret elements of matrix A as color values of an

image.I Truncate series SVD representation of A:

A = UΣV T =r∑

i=1

σi ui vTi

I Use fact that σ1 > σ2 > . . . > σr > 0.

I For color: approximate 3 matrices (RGB).

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

I Idea: use SVD to approximate images.I Interpret elements of matrix A as color values of an

image.I Truncate series SVD representation of A:

A = UΣV T =r∑

i=1

σi ui vTi

I Use fact that σ1 > σ2 > . . . > σr > 0.

I For color: approximate 3 matrices (RGB).

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

I Idea: use SVD to approximate images.I Interpret elements of matrix A as color values of an

image.I Truncate series SVD representation of A:

A = UΣV T =r∑

i=1

σi ui vTi

I Use fact that σ1 > σ2 > . . . > σr > 0.

I For color: approximate 3 matrices (RGB).

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

I Idea: use SVD to approximate images.I Interpret elements of matrix A as color values of an

image.I Truncate series SVD representation of A:

A = UΣV T =r∑

i=1

σi ui vTi

I Use fact that σ1 > σ2 > . . . > σr > 0.

I For color: approximate 3 matrices (RGB).

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Outline

Approximating matrices with SVDThe basic ideaGuess who?Bonus example 1Bonus example 2Bonus example 3Bonus example 4Bonus PuzzleColbert on Equations

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =1∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =2∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =3∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =4∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =5∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =6∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =7∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =8∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =9∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =10∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =20∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =30∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =40∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =50∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (80x60)

A =60∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Decay of sigma values: Einstein

0 20 40 600

2000

4000

6000

8000

10000

k

σ k

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Outline

Approximating matrices with SVDThe basic ideaGuess who?Bonus example 1Bonus example 2Bonus example 3Bonus example 4Bonus PuzzleColbert on Equations

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =1∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =2∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =3∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =4∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =5∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =6∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =7∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =8∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =9∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =10∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =20∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =30∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =40∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =50∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =60∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =100∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =200∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =320∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =480∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Outline

Approximating matrices with SVDThe basic ideaGuess who?Bonus example 1Bonus example 2Bonus example 3Bonus example 4Bonus PuzzleColbert on Equations

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =1∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =2∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =3∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =4∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =5∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =6∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =7∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =8∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =9∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =10∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =20∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =30∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =40∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =50∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =60∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =100∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =200∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =320∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =480∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Outline

Approximating matrices with SVDThe basic ideaGuess who?Bonus example 1Bonus example 2Bonus example 3Bonus example 4Bonus PuzzleColbert on Equations

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =1∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =2∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =3∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =4∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =5∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =6∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =7∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =8∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =9∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =10∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =20∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =30∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =40∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =50∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =60∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =100∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =200∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =320∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =480∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Outline

Approximating matrices with SVDThe basic ideaGuess who?Bonus example 1Bonus example 2Bonus example 3Bonus example 4Bonus PuzzleColbert on Equations

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =1∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =2∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =3∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =4∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =5∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =6∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =7∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =8∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =9∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =10∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =20∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =30∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =40∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =50∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =60∑

i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =100∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =200∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =320∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Image approximation (480x640)

A =480∑i=1

σi ui vTi

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Outline

Approximating matrices with SVDThe basic ideaGuess who?Bonus example 1Bonus example 2Bonus example 3Bonus example 4Bonus PuzzleColbert on Equations

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

The bonus puzzle

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

The bonus puzzle

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

Outline

Approximating matrices with SVDThe basic ideaGuess who?Bonus example 1Bonus example 2Bonus example 3Bonus example 4Bonus PuzzleColbert on Equations

Singular ValueDecomposition

Approximatingmatrices with SVDThe basic idea

Guess who?

Bonus example 1

Bonus example 2

Bonus example 3

Bonus example 4

Bonus Puzzle

Colbert on Equations

The truth about mathematics

The Colbert Report on Math (Feburary 2, 2007)