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CS 395/495-26: Spring 2004

CS 395/495-26: Spring 2004. IBMR: Singular Value Decomposition (SVD Review) Jack Tumblin jet@cs.northwestern.edu. Matrix Multiply: A Change-of-Axes. Matrix Multiply: Ax = b x and b are column vectors A has m rows, n columns. a 11 … a 1n … … a m1 … a mn. x 1 … x n. b 1 … b m.

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CS 395/495-26: Spring 2004

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  1. CS 395/495-26: Spring 2004 IBMR: Singular Value Decomposition(SVD Review) Jack Tumblin jet@cs.northwestern.edu

  2. Matrix Multiply: A Change-of-Axes • Matrix Multiply: Ax = b • x and b are column vectors • A has m rows, n columns a11 … a1n … … am1 … amn x1 … xn b1 … bm = b2 x2 x b Ax=b b1 b3 x1 Input (N dim) (2D) Output (M-dim) (3D)

  3. Matrix Multiply: A Change-of-Axes • Matrix Multiply: Ax = b • x and b are column vectors • A has m rows, n columns • Matrix multiply is just a set of dot-products;it ‘changes coordinates’ for vector x, makes b. • Rows of A = A1,A2,A3,... = new coordinate axes • Ax = a dot product for each new axis a11 … a1n … … am1 … amn x1 … xn b1 … bm = A2 b2 x2 x b Ax=b b1 A1 A3 b3 x1 Input (N dim) (2D) Output (M-dim) (3D)

  4. How Does Matrix ‘stretch space’? • Sphere of all unit-length x ellipsoid of b • Output ellipsoid’s axes are always perpendicular (true for any ellipsoid) **THUS** we can make ┴, unit-length axes… ellipsoid (disk) b2 x2 b x Ax=b b1 1 1 x1 b3 u2 u1 Input (N dim.) Output (M dim.)

  5. SVD finds: ‘Output’ Vectors U, and… • Sphere of all unit-length x ellipsoid of b • Output ellipsoid’s axes form orthonormal basis vectors Ui: • Basis vectors Ui are columns of U matrix SVD(A) = USVTcolumns of U = OUTPUT basis vectors ellipsoid (disk) b2 x2 b x Ax=b b1 1 1 x1 b3 u2 u1 Input (N dim.) Output (M dim.)

  6. SVD finds: …‘Input’ Vectors V, and… • For each Ui, make a matching Vi that: • Transforms to Ui (with scaling si) : si (A Vi) = Ui • Forms an orthonormal basis of input space SVD(A) = USVTcolumns of V = INPUT basis vectors b2 x2 x b v1 Ax=b v2 b1 x1 b3 u2 u1 Input (N dim.) Output (M dim.)

  7. SVD finds: …‘Scale’ factors S. • We have Unit-length Output Ui, vectors, • Each from a Unit-length Input Vi vector, so • So we need ‘scale factor’ (singular values)si to link input to output: si (A Vi) = Ui SVD(A) = USVT s1 b2 x2 x b v1 Ax=b v2 b1 x1 b3 s2 u2 u1 Input (N dim.) Output (M dim.)

  8. SVD Review: What is it? • Finish: SVD(A) = USVT • add ‘missing’ Ui or Vi, define these si=0. • Singular matrix S: diagonals are si : ‘v-to-u scale factors’ • Matrix U, Matrix V have columns Ui and Vi b2 x2 x b v1 u3 Ax=b v2 b1 x1 b3 u2 u1 Input (N dim.) Output (M dim.) A = USVT ‘Find Input & Output Axes, Linked by Scale’

  9. ‘Let SVDs Explain it All for You’ • cool! U and V are Orthonormal! U-1 = UT, V-1 = VT • Rank(A)? == # of non-zero singular values si • ‘ill conditioned’? Some si are nearly zero • ‘Invert’ a non-square matrix A ? Amazing! pseudo-inverse: A=USVT; A VS-1UT = I; so A-1 =VS-1 UT b2 x2 x v1 u3 Ax=b v2 b1 x1 b3 u2 u1 Input (N dim.) Output (M dim.) A = USVT ‘Find Input & Output Axes, Linked by Scale’

  10. ‘Solve for the Null Space’ Ax=0 • Easy! x is the Vi axis that doesn’t affect the output (si = 0) • are allsi nonzero? then the only answer is x=0. • Null ‘space’ because Vi is a DIMENSION--- x = Vi * a • More than 1 zero-valued si? null space >1 dimension... x = aVi1 + bVi2 + … b2 x2 x v1 u3 Ax=b v2 b1 x1 b3 u2 u1 Input (N dim.) Output (M dim.) A = USVT ‘Find Input & Output Axes, Linked by Scale’

  11. More on SVDs: Handout • Zisserman Appendix 3, pg 556 (cryptic) • Linear Algebra books • Good online introduction: (your handout)Leach, S. “Singular Value Decomposition – A primer,” Unpublished Manuscript, Department of Computer Science, Brown University. http://www.cs.brown.edu/research/ai/dynamics/tutorial/Postscript/ • Google on Singular Value Decomposition...

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