CS475 / CS675 Lecture 20: July 7, 2016 Bidiagonalization SVD Image - - PowerPoint PPT Presentation

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CS475 / CS675 Lecture 20: July 7, 2016 Bidiagonalization SVD Image Compression Reading: [TB] Chapter 31 CS475/CS675 (c) 2016 P. Poupart & J. Wan 1 Alternative SVD Technique Assume is square, i.e., Consider the symmetric matrix:

SLIDE 1

CS475 / CS675 Lecture 20: July 7, 2016

Bidiagonalization SVD Image Compression Reading: [TB] Chapter 31

SLIDE 2

Alternative SVD Technique

Assume

is square, i.e.,

Consider the

symmetric matrix:

Since

,

then
Λ

SLIDE 3

Alternative SVD Technique

Hence,

eigendeomposition of

Algorithm:

– Compute eigendecomposition of . – Set

– Extract , from

Stable algorithm

SLIDE 4

Two‐phase SVD

Idea: First reduce the matrix to bidiagonal form, then

diagonalize it.

Picture:

SLIDE 5

Golub‐Kahan Bidiagonalization

Apply Householder reflectors on the left and the right
reflectors on the left,
n the right
CS475/CS675 (c) 2016 P. Poupart & J. Wan

SLIDE 6

Low‐Rank Approximation

Theorem:

is the sum of rank‐one matrices:

Proof:

SLIDE 7

Low‐Rank Approximation

Theorem: For any ,

, define

Then
Proof: first note that
…
⋱
⋮
It is the SVD of
Hence:
CS475/CS675 (c) 2016 P. Poupart & J. Wan

SLIDE 8

Low‐Rank Approximation

Suppose

with such that

Then

‐dim subspace such that

Note

. Then

CS475/CS675 (c) 2016 P. Poupart & J. Wan

SLIDE 9

Low‐Rank Approximation

‐dim subspace such that

– E.g.,

, , … ,

– Note:

But
contradiction

SLIDE 10

Low‐Rank Approximation

Notes

1. …

is the best rank‐ approximation of . The error of approximation is (in ‐norm)

SLIDE 11

Application: Image Compression

image can be represented by matrix where

pixel value at
Compress the image by storing less than

entries

Let
, the best rank‐

approx of

Keep the first

singular values and use

approximate ; i.e.,

compressed image

SLIDE 12

Application: Image Compression

Example:

,

To store

, only need to store and

– This requires only

words

In contrast, to store
ne needs

words

Compression ratio:
CS475/CS675 (c) 2016 P. Poupart & J. Wan

CS475 / CS675 Lecture 20: July 7, 2016

Bidiagonalization SVD Image Compression Reading: [TB] Chapter 31

Alternative SVD Technique

is square, i.e.,

symmetric matrix:

,

Alternative SVD Technique

eigendeomposition of

– Compute eigendecomposition of . – Set

– Extract , from

Two‐phase SVD

diagonalize it.

Golub‐Kahan Bidiagonalization

Low‐Rank Approximation

is the sum of rank‐one matrices:

Low‐Rank Approximation

, define

Low‐Rank Approximation

with such that

‐dim subspace such that

. Then

Low‐Rank Approximation

‐dim subspace such that

– Note:

Low‐Rank Approximation

1. …

is the best rank‐ approximation of . The error of approximation is (in ‐norm)

Application: Image Compression

image can be represented by matrix where

entries

approx of

singular values and use

approximate ; i.e.,

Application: Image Compression

,

, only need to store and

words

words

Application: Image Compression

k Relative error

3 10 20