Singular Value Decomposition (SVD) CS 663 Ajit Rajwade

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1 Sngular Value Decomposton (SVD) CS 663 jt Rajwade

2 Sngular value Decomposton For any m x n matrx the followng decomposton always exsts: USV U V U S V R mn UU VV I m I n R mn U V R R mm nn Dagonal matrx wth nonnegatve entres on the dagonal called sngular values. Columns of U are the egenvectors of (called the left sngular vectors). Columns of V are the egenvectors of (called the rght sngular vectors). he non - zero sngular values are the postve square roots of non - zero egenvalues of or.

3 Sngular value Decomposton For any m x n real matrx the SVD conssts of matrces USV whch are always real ths s unlke egenvectors and egenvalues of whch may be complex even f s real. he sngular values are always non-negatve even though the egenvalues may be negatve. Whle wrtng the SVD the followng conventon s assumed and the left and rght sngular vectors are also arranged accordngly:... m m 3

4 Sngular value Decomposton If only r < mn(mn) sngular values are nonzero the SVD can be represented n reduced form as follows: r r r n r m n m R S R V R U R USV 4

5 Sngular value Decomposton USV r S u v t hs m by n matrx u v s the product of a column vector u and the transpose of column vector v. It has rank. hus s a weghted summaton of r rank- matrces. Note: u and v are the -th column of matrx U and V respectvely. 5

6 Sngular value decomposton USV ( USV hus the left sngular vectors of (.e. columns of U) are the egenvectors of )( USV. ) USV he sngular values of (.e. dagonal elements of S) are square - rootsof the egenvalues of VSU. US U ( USV hus the rght sngular vectors of (.e. columns of V) are the egenvectors of ) ( USV. ) VSU USV he sngular values of (.e. dagonal elements of S) are square - rootsof the egenvalues of. VS V 6

7 pplcaton: SVD of Natural Images n mage s a D array each entry contans a grayscale value. he mage can be treated as a matrx. It has been observed that for many mage matrces the sngular values undergo rapd decay (note: they are always non-negatve). n mage can be approxmated wth the k largest sngular values and ther correspondng sngular vectors: k S u v t k mn( m n) 7

8 Sngular values of the Mandrll Image: notce the rapd exponental decay of the values! hs s characterstc of MOS natural mages.

9 Left to rght top to bottom: Reconstructed mage usng the frst = sngular values and sngular vectors. Last mage: orgnal 9

10 Left to rght top to bottom we dsplay: u v where = Note each mage s ndependently rescaled to the 0- range for dsplay purpose. Note: the spatal frequences ncrease as the sngular values decrease 0

11 SVD: Use n Image Compresson Instead of storng mn ntensty values we store (n+m+)r ntensty values where r s the number of stored sngular values (or sngular vectors). he remanng m-r sngular values (and hence ther sngular vectors) are effectvely set to 0. hs s called as storng a low-rank (rank r) approxmaton for an mage.

12 Propertes of SVD: Best low-rank reconstructon SVD gves us the best possble rank-r approxmaton to any matrx (t may or may not be a natural mage matrx). In other words the soluton to the followng optmzaton problem: mn ˆ ˆ F where rank( ˆ ) s gven usng the SVD of as follows: ˆ r S u v t where USV rr mn( m n) Note: We are usng the sngular vectors correspondng to the r largest sngular values. hs property of the SVD s called the Eckart Young heorem.

13 mn Propertes of SVD: Best low-rank reconstructon ˆ ˆ F where rank( ˆ ) rr mn( m n) F m n j j Frobenus norm of the matrx (fancy way of sayng you square all matrx values add them up and then take the square root!) Note : ˆ r r... n F Why? 3

14 Geometrc nterpretaton: Eckart- Young theorem he best lnear approxmaton to an ellpse s ts longest axs. he best D approxmaton to an ellpsod n 3D s the ellpse spanned by the longest and second-longest axes. nd so on! 4

15 Propertes of SVD: Sngularty square matrx s non-sngular (.e. nvertble or full-rank) f and only f all ts sngular values are non-zero. he rato σ /σ n tells you how close s to beng sngular. hs rato s called condton number of. he larger the condton number the closer the matrx s to beng sngular. 5

16 Propertes of SVD: Rank Inverse Determnant he rank of a rectangular matrx s equal to the number of non-zero sngular values. Note that rank() = rank(s). SVD can be used to compute nverse of a square matrx: USV VS R U nn bsolute value of the determnant of square matrx s equal to the product of ts sngular values. det( ) det( USV ) det( U)det(S)de t (V ) det( S) n 6

17 Propertes of SVD: Pseudo-nverse SVD can be used to compute pseudo-nverse of a rectangular matrx: 0 otherwse. ) ( non - zero sngular values and for all ) ( ) ( ) ( where R n m S S S S U VS USV 7

18 Propertes of SVD: Frobenus norm he Frobenus norm of a matrx s equal to the square-root of the sum of the squares of ts sngular values: F m n j j trace( ) trace(( USV ) ( USV )) trace( V S V) trace( VV S ) trace( S ) 8

19 Geometrc nterpretaton of the SVD ny m x n matrx transforms a hypersphere Q of unt radus (called as unt sphere) n R n nto a hyperellpsod n R m (assume m >= n). Q Q 9

20 Geometrc nterpretaton of the SVD But why does transform the hypersphere nto a hyperellpsod? hs s because = USV. V transforms the hypersphere nto another (rotated/reflected) hypersphere. S stretches the sphere nto a hyperellpsod whose semaxes concde wth the coordnate axes as per V. U rotates/reflects the hyperellpsod wthout affectng ts shape. s any matrx has an SVD decomposton t wll always transform the hypersphere nto a hyperellpsod. If does not have full rank then some of the sem-axes of the hyperellpsod wll have length 0! 0

21 Geometrc nterpretaton of the SVD ssume has full rank for now. he sngular values of are the lengths of the n prncpal sem-axes of the hyperellpsod. he lengths are thus σ σ σ n. he n left sngular vectors of are the drectons u u u n (all unt-vectors) algned wth the n sem-axes of the hyperellpsod. he n rght sngular vectors of are the drectons v v v n (all unt-vectors) n hypersphere Q whch the matrx transforms nto the sem-axes of the hyperellpsod.e. v u

22 Geometrc nterpretaton of the SVD Expandng on the prevous equatons we get the reduced form of the SVD n x n orthonormal matrx V m x n matrx (m >> n) wth orthonormal columns - U n x n dagonal matrx - S

23 Computaton of the SVD We wll not explore algorthms to compute the SVD of a matrx n ths course. SVD routnes exst n the LPCK lbrary and are nterfaced through the followng MLB functons: s = svd( X) returns a vector of sngular values. [USV] = svd( X) produces a dagonal matrx S of the same dmenson as X wth nonnegatve dagonal elements n decreasng order and untary matrces U and V so that X = U*S*V'. [USV] = svd( X0) produces the "economy sze" decomposton. If X s m-by-n wth m > n then svd computes only the frst n columns of U and S s n-by-n. [USV] = svd(x'econ') also produces the "economy sze" decomposton. If X s m- by-n wth m >= n t s equvalent to svd( X0). For m < n only the frst m columns of V are computed and S s m-by-m. s = svds(k) computes the k largest sngular values and assocated sngular vectors of matrx. 3

24 SVD Unqueness If the sngular values of a matrx are all dstnct the SVD s unque up to a multplcaton of the correspondng columns of U and V by a sgn factor. Why? r S S (-u u v t )(-v S t ) u v S t u S v t u v t... S... S (-u rr rr r u r v t r )(-v t r ) 4

25 SVD Unqueness matrx s sad to have degenerate sngular values f t has the same sngular value for or more pars of left and rght sngular vectors. In such a case any normalzed lnear combnaton of the left (rght) sngular vectors s a vald left (rght) sngular vector for that sngular value. 5

26 ny other applcatons of SVD? Face recognton the popular egenfaces algorthm can be mplemented usng SVD too! Pont matchng: Consder two sets of ponts such that one pont set s obtaned by an unknown rotaton of the other. Determne the rotaton! Structure from moton: Gven a sequence of mages of a object undergong rotatonal moton determne the 3D shape of the object as well as the 3D rotaton at every tme nstant! 6

27 PC lgorthm usng SVD. Compute the mean of the gven ponts:. Deduct the mean from each pont: 3. Compute the covarance matrx of these mean-deducted ponts: d d N R R N x x x x x x x N d d d N R R Note N N ] x... x [x X C XX x x C N :

28 PC lgorthm usng SVD 4. Instead of fndng the egenvectors of C we fnd the left sngular vectors of X and ts sngular values X USV U R dd U contans the egenvectors of XX 5. Extract the k egenvectors n U correspondng to the k largest sngular values to form the extracted egenspace: Uˆ k U(:: k). USV are obtaned by computng the SVD of X. here s an mplct assumpton here that the frst k ndces ndeed correspond to the k largest egenvalues. If that s not true you would need to pck the approprate ndces.

29 References Scentfc Computng Mchael Heath Numercal Lnear lgebra reftehen and Bau ecomposton 9

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