Matrix Factorizations: Singular Value Decomposition

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Matrix Factorizations: Singular Value Decomposition. Presented by Nik Clark MTH 421. Introduction. In the exciting world of numerical analysis, one may wonder “Why? Why do I study matrices and their factorizations?”

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### Matrix Factorizations:Singular Value Decomposition

Presented by Nik Clark

MTH 421

Introduction
• In the exciting world of numerical analysis, one may wonder “Why? Why do I study matrices and their factorizations?”
• Such a simple answer, “Because matrices and their decompositions can take you anywhere!”
Choleski

Jordan

LU

Polar

Proper Orthogonal

QR

Schur

Singular Value

Spectral (Eigendecomposition)

Matrix Decompositions

Some interesting ways to decompose a matrix

Singular Value Decomposition

We’d like to more formally introduce you to Singular Value Decomposition (SVD) and some of its applications.

What is SVD?
• SVD is a type of factorization for a rectangular, real or complex, matrix.
• THEOREM:

All matrices Amxn have a singular value decomposition.

Why should we care?
• All math is awesome.
• Also because SVD can be applied to several situations:
• Data Compression
• Analyzing DNA Gene Expression Data
• Solving Least Squares Problems
• Information Retrieval
More Caring
• Image Processing
• De-blurring
• Seismology
• Digital Signal Processing
• Noise Reduction
• Data Hiding
• Cryptography
• Watermarks
• Researching Databases
Back to the SVD
• Each mxn matrix A decomposes into the product of three matrices A=UΣVT
• U is orthogonal and mxm. Its columns span col(A).
• V is also orthogonal, but is nxn. Its columns span row(A)
•  is a diagonal matrix where the singular values of A are along the main diagonal, and all other values are zero.
More on SVD
• The m columns of matrix U are the eigenvectors of AAT
• The n columns of matrix V are the eigenvectors of ATA
• The entries of the main diagonal of matrix  are the singular values of A, denoted by i
Another Way to Write
• An additional way to write A, apart from A=UΣVT :

A = 1u1vT1 +2u2vT2 + …+rurvTr ,

Where r = rank(A), and is defined to be the number of linearly independent columns of A.

Importance
• Each term of the expansion is already in order of importance.
• 1≥ 2 ≥ … ≥ k ≥ 0
Rank k Approximation
• Ak=UkΣkVTk
• Where Ak is the first k terms of the SVD factorization of A
• i.e. A = 1u1vT1 +2u2vT2 + …+kukvTk
More on Rank
• A = 1u1vT1 +2u2vT2 + …+kukvTk
• Where each term of the expansion is a rank 1 matrix.
An Example: Code breaking
• Cipher- a method for encrypting a message.
• Cryptography – The art of creating a coded message using a cipher.
• Cryptanalysis – The art of breaking a code by finding its weakness.
• Cryptology – the study of the aforementioned definitions.
How do we decode the code?
• Most commonly, a cryptogram is created by substituting one letter for another.
• When decoding a code (in english) it is easiest to first decode the vowels.
• Vowel pairs are less frequent than consonant-vowel pairs.
• More frequently, vowels follow consonants, vfc.
vcf
• When vowels follow consonants, there is a mathematical proportion:

number of vowel pairs

number of vowels

number consonant-vowel pairs

number of consonants

<

The matrix
• We define matrix A to be a digram frequency of the letter combinations of our text.
• aij is the number of times the ith letter is followed by the jth letter.
• We define vector wi to equal 1 when the letter is a vowel and 0 otherwise.
• Vector ci is defined to equal 1 when the letter is a consonant and 0 otherwise.
Thus
• Our vectors w and c are orthogonal.
• wTAw is the number of vowel pairs.
• cTAw is the number of consonant vowel pairs.
• We can now rewrite our equation:

wTAw

wTA(w+c)

cTAw

cTA(w+c)

<

Singular Vectors
• A ~ A1 = 1u1vT1
• f = k1* u1
• f = k2* v1
• Where f represents the frequency of the words in the text.
Rank Two
• A ~ A2 = 1u1vT1 +2u2vT2
• Rank two is the frequency of vowel pairs.
• Each vowel (except for u) corresponds to a (+, -) vector pair.
• Each consonant corresponds to a (-, +) vector pair.
• There are some exceptions
Neuter Letters
• Those letters that correspond to a (+, +) vector pair or a (-, -) vector pair are called neuter.
• These letter patterns correspond to any text we consider.
An Encrypted Example
• The following matrix is the digram matrix of an encrypted text (cryptogram) similar to the sentence below.
• What does this mean??
• Gsviv rh ml zkkorw nzgsvnzgrxk ru gsviv rh ml nzgsvnzgrxh gl zkkob.