Singular Value Decomposition: Theory and Applications
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1 Sngular Value Decomposton: Theory and Applcatons Danel Khashab Sprng 2015 Last Update: March 2, Introducton A = UDV where columns of U and V are orthonormal and matrx D s dagonal wth postve real values. The dagonal elements of D are called sngular values. The m rows of U are called left-sngular vectors and d rows of V are called rght-sngular vectors. The SVD of A gves the best rank k approxmaton to A wth respect to squared-norm, for any k. Remark 1. SVD s defned for all matrces, whereas the more commonly used Egenvector Decomposton requres the matrx A be square and certan other condtons on the matrx to ensure orthogonalty of the egenvectors. The left-sngular vectors of A are egenvectors of AA. The rght-sngular vectors of A are egenvectors of A A. The non-zero sngular values of A (found on the dagonal entres of D) are the square roots of the non-zero egenvalues of both A A and AA. Lemma 1. Suppose v 1, v 2,..., v r are left sngular vectors as a result of SVD. It can be shown that these vectors satsfy the followng maxmzatons: A n d = U n r D r r V T r d Fgure 3.2: The SVD decomposton of an n d matrx. 3.3 Best Rank k Approxmatons Let A be an n d matrx and let r A = σ u v T =1 be the SVD of A. For k {1, 2,..., r}, let k 1
2 v 1 = arg max Av v 2 = arg max Av v 3 = arg max Av. v v 2 v r = arg max Av... v v r 1 It can be shown that σ 1 (A) = Av 1 s the dagonal element of D (or the frst sngular value). Smlarly for other sngular values. The lemma gets translated nto the followng Theorem. Theorem 1. For 1 k r, let V k be the subspace spanned by v 1, v 2,..., v k. For each k, V k s the best-ft k-dmensonal subspace for matrx A. Lemma 2. For any matrx A, the sum of squares of the sngular values equals the square of the Frobenus norm. That s, σ 2 (A) = a 2,j = A 2 F,j Lemma 3. It can be shown the followng relaton between the sngular values and rght/left-sngular vectors: u = 1 σ (A) Av Theorem 2. The left-sngular vectors are parwse orthogonal. Theorem 3. For any matrx B of rank at most k: A A k F A B F Lemma 4. A A k 2 2 = σ2 k+1 Example 1 (Interpretng an SVD on revews matrx). Suppose matrx A s matrx of costumerrestaurant ratngs (rows beng persons and columns beng restaurants). Left sngular vectors, u have sze the number of people, and can be seen as the orthogonal drectons of revews by people. Specfcally, the frst left sngular vector, corresponds to the most popular drecton/pattern of revews by people. Smlarly, rght sngular vectors, v have sze the number of restaurants, and can be seen as the orthogonal drectons of revews gven to restaurants. For example, the frst rght sngular vector, corresponds to the most popular drecton of revews to restaurants. 2
3 The sngular values show the popularty of drectons/revew patterns. If σ 1 σ 2 t shows that there s a consensus n the revews by people for restaurants. If σ 1 σ 2 σ 3, t shows that there s two major scorng patterns. The bgger the sngular gap σ 1 σ 2 s, the more consensus exsts n the revews. The defnton of consensus here s delcate. The consensus here s defned based on the drectonalty of the revews. In other words, SVD does not care whether the revews are bg or small (or postve or negatve). Instead t values the consensus n terms how coherent the revews are n one specfc drecton (n customer-restaurant space). 2 Power Method Consder an arbtrary matrx B. The power teraton algorthm starts wth a vector x 0, and gves an approxmaton to the domnant egenvector, f converges at all (otherwse a random vector). Gven the ntalzaton x 0, the updates of the algorthm are the followngs: x k+1 = Bx k Bx k The convergence s guaranteed under the followng two condtons: B has an egenvalue strctly greater n magntude than ts other egenvalues. The startng vector x 0 has a nonzero component n the drecton of an domnant egenvector. The next lemma explans how the Power Iteratons s useful for SVD. Lemma 5. Gven a matrx B = AA, the power teraton algorthm on converges to u 1, the frst sngular vector of A, f converges at all. Proof. Consder an SVD of A: A = σ u v Then for B we have: ( B = AA = σ u v ) ( σ u v ) = σ σ j u u,j Repeatng the multplcaton: B 2 = σ σ j u u,j,j σ σ j u u =,j σ 2 σ 2 j u ( u u j ) u j = σ 4 u u Snce u u j = 0, for j. Smlarly we have: B k = σ 2 u u 3
4 If σ 1 > σ 2, as k grows we have the followng convergence: B k σ 2k 1 u 1 u 1 By proper normalzaton of the rank-1 matrx u 1 u 1 and a lttle algebra, we can fnd u 1. However the ssue wth the above method s that we need to handle the matrx u 1 u 1 whch can be very large n practce. The trck s that, nstead of computng B k σ 2k 1 u 1 u 1 we choose a random pont x 0 = α u and calculate B k x 0. ( ) B k x 0 σ1 2k u 1 u 1 α u = α 1 σ1 2k u 1 Whch gves u 1, after a normalzaton over a vector. Note that startng from a random vector calculaton of powers need a matrx vector operaton, whch computatonally s a moderate operaton. Example 2. We show a sample run of power teraton. Suppose: A = and we are startng from x = [11] and k = 3. We repeat the power method for three steps: 1. x 0 = [1, 1] 2. for k = {1, 2, 3} 3. x k = Ax k 1 / Ax k 1 The results are: x 0 = [1, 1] x 1 = [0.24, 0.97] x 2 = [0.06, 0.99] x 3 = [0.01, 0.99] We wll see that the drect calculatons wll gve u = [0, 1] whch s very close to x 3. We can computng the exact values drectly: [ ] B = A 4 0 A = 0 16 We fnd the solutons to Bv = λv. The egenvalues and egenvectors of B are: [ ] 1 0 [u 1, u 2 ] = 0 1 [λ 1, λ 2 ] = [16, 4] 4
5 The egenvectors of A A are the rght sngular vectors of A. Also the sngular values equal to the egenvalues. Snce v = 1 σ (A) Au, the two other left sngular vectors are: [v 1, v 2 ] = Lemma 6. For any gven A, u 1 s the frst sngular vector of A (suppose σ 1 > σ 2 ) Proof. 2.1 Further topcs Mssng data n SVD Consder the matrx of revews n example 1. If some people refuse to gve revews for some restaurants, we wll have some mssng values n our matrx. The queston s, how to handle the mssng values when dong SVD. In many practcal applcatons, the mssng values are replaced wth zeros. But stll there seems to be a need for methods whch gve grantees on the results, whle beng practcal and fast. One other trck s usng regularzer (say an l 2 regularzer) n the objectve functon. In other words, nstead of usng SVD drectly whch mnmzes the Frobenus norm type object, we can augment the objectve wth a regularzer and mnmze t drectly (say wth gradent descent). In ths descrpton, ths corresponds to mnmzng the components of the norm whch can be measured,.e. those whch have known values. The regularzaton term can be seen as a Bayesan pror on the components of the feature vectors, wth the SVD calculatng the maxmum lkelhood estmator, subject to ths pror and the known values. 3 Bblographcal notes 5
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