11-755/ Machine Learning for Signal Processing. Algebra. Class Sep Instructor: Bhiksha Raj

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1 -755/8-797 Mchine Lerning for Signl Processing Fundmentls of Liner Alger Clss -3 Sep Instructor: Bhiksh Rj Sep -755/8-797

2 Administrivi Registrtion: Anyone on witlist still? Homework : Will e hnded out with clss 3 Liner lger Sep -755/8-797

3 Overview Vectors nd mtrices Bsic vector/mtrix t opertions Vector products Mtrix products Vrious mtrix types Mtrix inversion Mtrix interprettion Eigennlysis Singulr vlue decomposition Sep -755/

4 Book Fundmentls of Liner Alger, Gilert Strng Importnt to e very comfortle with liner lger Appers repetedly in the form of Eigen nlysis, SVD, Fctor nlysis Appers through vrious properties of mtrices tht re used in mchine lerning, prticulrly when pplied to imges nd sound Tody s lecture: Definitions iti Very smll suset of ll tht s used Importnt suset, intended to help you recollect Sep -755/

5 Incentive to use liner lger Pretty nottion! x T A y j y j i x i ij Esier intuition Relly convenient geometric interprettions Opertions esy to descrie verlly Esy code trnsltion! for i=:n for j=:m c(i)=c(i)+y(j)*x(i)*(i,j) end end C=x*A*y Sep -755/

6 And other things you cn do From Bch s Fugue in Gm Fre equency Rottion + Projection + Scling Time Decomposition (NMF) Mnipulte Imges Mnipulte Sounds Sep -755/

7 Sclrs, vectors, mtrices, A sclr is numer =, = 34, = -, etc A vector is liner rrngement of collection of sclrs 3, 34 3 A mtrix A is rectngulr rrngement of collection of vectors 3 A MATLAB syntx: =[ 3], A=[ ;3 4] Sep -755/

8 Vector/Mtrix types nd shpes Vectors re either column or row vectors c, r c c, s A sound cn e vector, series of dily tempertures cn e vector, etc Mtrices cn e squre or rectngulr S c d, R c d e, M f Imges cn e mtrix, collections of sounds cn e mtrix, etc Sep -755/

9 Dimensions of mtrix The mtrix size is specified y the numer of rows nd columns c c r c, c = 3x mtrix: 3 rows nd column r = x3 mtrix: row nd 3 columns f e d c d c S R, S = x mtrix R = x 3 mtrix Pcmn = 3 x 399 mtrix -755/8-797 Pcmn = 3 x 399 mtrix Sep 9

10 Representing n imge s mtrix 3 pcmen A 3x399 mtrix Row nd Column = position A 3x879 mtrix Triples of x,y nd vlue A x879 vector Y X v 5 6 Vlues only; X nd Y re implicit Unrveling the mtrix Note: All of these cn e recst s the mtrix tht forms the imge Representtions nd 4 re equivlent The position is not represented Sep -755/8-797

11 Exmple of vector Vectors usully hold sets of numericl ttriutes X, Y, vlue [,, ] Ernings, losses, suicides [$ $ 3] Etc Consider reltive Mnhttn vector Provides reltive position y giving numer of venues nd streets to cross, eg [3v 33st] [-5v 6st] [v 4st] [v 8st] Sep -755/8-797

12 Vectors Ordered collection of numers Exmples: [3 4 5], [ c d], [3 4 5]!= [4 3 5] Order is importnt Typiclly viewed s identifying (the pth from origin to) loction in n N-dimensionl spce (3,4,5) 5 (4,3,5) z 3 y 4 x Sep -755/8-797

13 Vectors vs Mtrices (3,4,5) A vector is geometric nottion for how to get from (,) ) to some loction in the spce A mtrix is simply collection of destintions! Properties of mtrices re verge properties of the trveller s pth to these destintions Sep -755/

14 Bsic rithmetic opertions Addition nd sutrction Element-wise opertions A B MATLAB syntx: + nd - Sep -755/

15 Vector Opertions (3,4,5) 3 5 (3,-,-3) (6,,) 3 Opertions tell us how to get from ({}) to the result of the vector opertions (345)+(3 (3,4,5) (3,-,-3) 3)=(6) (6,,) Sep -755/

16 Opertions exmple Rndom(3,columns(M)) 6 5 Adding rndom vlues to different representtions of the imge -755/8-797 representtions of the imge Sep 6

17 Vector norm Mesure of how ig vector is: Notted s x In Mnhttn vectors mesure of distnce MATLAB syntx: norm(x) [-v 7st] [-6v st] Sep -755/

18 Vector Norm Length = sqrt( ) (3,4,5) Geometriclly the shortest distnce to trvel from the origin to the destintion As the crow flies Assuming Eucliden Geometry Sep -755/

19 Trnsposition A trnsposed row vector ecomes column (nd vice vers) x, x T c c y c, y T A trnsposed mtrix gets ll its row (or column) vectors trnsposed in order c d X c d e f, X T e M, M T c f MATLAB syntx: Sep -755/

20 Vector multipliction Multipliction is not element-wise! Dot product, or inner product Vectors must hve the sme numer of elements Row vector times column vector = sclr d c e d e c f f Cross product, outer product or vector direct product Column vector times row vector = mtrix d e f d e f d e f c c d ce c f MATLAB syntx: * Sep -755/8-797

21 Vector dot product in Mnhttn Multiplying the yrd vectors Insted of venue/street we ll use yrds = [ 6], = [77 3] The dot product of the two vectors reltes to the length of projection How much of the first vector hve we covered y following the second one? The nswer comes ck s unit of the first vector so we divide y its length T yd [yd 6yd] norm 6 norm 393yd [77yd 3yd] norm 86 Sep -755/8-797

22 Vector dot product D S D Sqrt(ene ergy) frequency Vectors re spectr frequency frequency Energy t discrete set of frequencies Actully x496 X xis is the index of the numer in the vector Represents frequency Y xis is the vlue of the numer in the vector Represents mgnitude Sep -755/8-797

23 Vector dot product D S D Sqrt(ene ergy) frequency frequency frequency How much of D is lso in S How much cn you fke D y plying n S DS / D S = Not very much How much of D is in D? DD / D / D = 5 Not d, you cn fke it To do this, D, S, nd D must e the sme size Sep -755/

24 Vector cross product The column vector is the spectrum The row vector is n mplitude modultion The crossproduct is spectrogrm Shows how the energy in ech frequency vries with time The pttern in ech column is scled version of the spectrum Ech row is scled version of the modultion Sep -755/

25 Mtrix multipliction Generliztion of vector multipliction Dot product of ech vector pir A B Dimensions i must mtch!! Columns of first mtrix = rows of second Result inherits the numer of rows from the first mtrix nd the numer of columns from the second mtrix MATLAB syntx: * Sep -755/

26 Multiplying Vector y Mtrix Y (,:) 9 Y (,:) 8 9 YX Y X 6 Multipliction li of vector X y mtrix Y expresses the vector X in terms of projections of X on the row vectors of the mtrix Y It scles nd rottes the vector Alterntely t l viewed, it scles nd rottes t the spce the underlying plne Sep -755/

27 Mtrix Multipliction Y The mtrix rottes nd scles the spce Including its own vectors Sep -755/

28 Mtrix Multipliction The normls to the row vectors in the mtrix ecome the new xes X xis = norml to the second row vector Scled y the inverse of the length of the first row vector Sep -755/

29 Mtrix Multipliction is projection The k-th xis corresponds to the norml to the hyperplne represented y the k-,k+n-th row vectors in the mtrix Any set of K- vectors represent hyperplne of dimension K- or less The distnce long the new xis equls the length of the projection on the k-th row vector Expressed in inverse-lengths of the vector Sep -755/

30 Mtrix Multipliction: Column spce c z y x y x c So much for spces wht does multiplying f e y d z y f e d So much for spces wht does multiplying mtrix y vector relly do? It mixes the column vectors of the mtrix It mixes the column vectors of the mtrix using the numers in the vector The column spce of the Mtrix is the The column spce of the Mtrix is the complete set of ll vectors tht cn e formed y mixing its columns -755/8-797 y mixing its columns Sep 3

31 Mtrix Multipliction: Row spce d c f x y x c y d e f e Left multipliction mixes the row vectors of the mtrix The row spce of the Mtrix is the complete set of ll vectors tht cn e formed y mixing its rows Sep -755/

32 Mtrix multipliction: Mixing vectors X Y = 7 A physicl exmple The three column vectors of the mtrix X re the spectr of three notes The multiplying column vector Y is just mixing vector The result is sound tht is the mixture of the three notes Sep -755/

33 Mtrix multipliction: Mixing vectors x x x 5 75 x 4 x 4 x Mixing two imges The imges re rrnged s columns position vlue not included The result of the multipliction is rerrnged s n imge Sep -755/

34 Mtrix multipliction: nother view k kk k k k k NK N N B A Wht does this men? k kk Mk k k Mk NK N MN M Wht does this men? N N NK N MN K M K M NK N NK MN M N MN M M MN M The outer product of the first column of A nd the first fb+ t d t fth d l fa d -755/8-797 row of B + outer product of the second column of A nd the second row of B + Sep 34

35 Why is tht useful? Y X Sounds: Three notes modulted independently Sep -755/

36 Mtrix multipliction: Mixing modulted spectr Y X Sounds: Three notes modulted independently Sep -755/

37 Mtrix multipliction: Mixing modulted spectr Y X Sounds: Three notes modulted independently Sep -755/

38 Mtrix multipliction: Mixing modulted spectr X Sounds: Three notes modulted independently Sep -755/

39 Mtrix multipliction: Mixing modulted spectr X Sounds: Three notes modulted independently Sep -755/

40 Mtrix multipliction: Mixing modulted spectr Sounds: Three notes modulted independently Sep -755/

41 Mtrix multipliction: Imge trnsition i j i j Imge fdes out linerly Imge fdes in linerly Sep -755/

42 Mtrix multipliction: Imge trnsition i j i j Ech column is one imge i 9 i 8 i i 9 9 i 8 8 i in 9 in 8 in The columns represent sequence of imges of decresing intensity Imge fdes out linerly Sep -755/

43 Mtrix multipliction: Imge trnsition i j i j Imge fdes in linerly Sep -755/

44 Mtrix multipliction: Imge trnsition i j i j Imge fdes out linerly Imge fdes in linerly Sep -755/

45 The Identity Mtrix Y An identity mtrix is squre mtrix where All digonl elements re All off-digonl elements re Multipliction y n identity mtrix does not chnge vectors Sep -755/

46 Digonl Mtrix Y All off-digonl elements re zero Digonl elements re non-zero Scles the xes My flip xes Sep -755/

47 Digonl mtrix to trnsform imges How? Sep -755/

48 Stretching 5 6 Loction-sed representtion Scling mtrix only scles the X xis The Y xis nd pixel vlue re scled y identity Not good wy of scling Sep -755/

49 Stretching D = A 5 5 Newpic 5 5 EA ( NxN) Better wy Sep -755/

50 Modifying color B G R P P Newpic Scle only Green -755/8-797 y Sep 5

51 Permuttion Mtrix x y 5 y z Y Z z x 4 (3,4,5) Z(oldX) Y (old Z) X 3 X (old Y) A permuttion mtrix simply rerrnges the xes The row entries re xis vectors in different order The result is comintion of rottions nd reflections The permuttion mtrix effectively permutes the rrngement of the elements in vector Sep -755/

52 Permuttion Mtrix P P 6 5 Reflections nd 9 degree rottions of imges nd ojects -755/8-797 imges nd ojects Sep 5

53 Permuttion Mtrix P P P P N x x x Reflections nd 9 degree rottions of imges nd ojects N N z z z y y y Reflections nd 9 degree rottions of imges nd ojects Oject represented s mtrix of 3-Dimensionl position vectors Positions identify ech point on the surfce -755/8-797 Positions identify ech point on the surfce Sep 53

54 Rottion Mtrix Y x' y' x cos xsin y sin y cos (x,y) R cos sin sin cos x X y x' X new y' y y Y R X X new (x,y ) (x,y) X X x x A rottion mtrix rottes the vector y some ngle Alterntely viewed, it rottes the xes The new xes re t n ngle to the old one Sep -755/

55 Rotting picture 45 cos 45 sin 45 sin 45 cos R Note the representtion: 3-row mtrix Note the representtion: 3 row mtrix Rottion only pplies on the coordinte rows The vlue does not chnge Why is pcmn griny? -755/8-797 Why is pcmn griny? Sep 55

56 3-D Rottion Xnew Ynew Z Y Znew X degrees of freedom seprte ngles Wht will the rottion mtrix e? Sep -755/

57 Projections Wht would we see if the cone to the left were trnsprent if we looked t it long the norml to the plne The plne goes through the origin Answer: the figure to the right How do we get this? Projection Sep -755/

58 Projection Mtrix 9degrees W W projection Consider ny plne specified y set of vectors W, W Or mtrix [W W ] Any vector cn e projected onto this plne The mtrix A tht rottes nd scles the vector so tht it ecomes its projection is projection mtrix Sep -755/

59 Projection Mtrix 9degrees W W projection Given set of vectors W, W, which form mtrix W = [W W ] The projection mtrix tht trnsforms ny vector X to its projection on the plne is P = W (W T W) - W T We will visit mtrix inversion shortly Mgic ny set of vectors from the sme plne tht re expressed s mtrix will give you the sme projection mtrix P = V (V T V) - V T Sep -755/

60 Projections HOW? Sep -755/

61 Projections Drw ny two vectors W nd W tht lie on the plne ANY two so long s they hve different ngles Compose mtrix W = [W W] Compose the projection mtrix P = W (W T W) - W T Multiply every point on the cone y P to get its projection View it I m missing step here wht is it? Sep -755/

62 Projections The projection ctully projects it onto the plne, ut you re still seeing the plne in 3D The result of the projection is 3-D vector P = W (W T W) - W T = 3x3, P*Vector = 3x The imge must e rotted till the plne is in the plne of the pper The Z xis in this cse will lwys e zero nd cn e ignored How will you rotte it? (rememer you know W nd W) Sep -755/

63 Projection mtrix properties The projection of ny vector tht is lredy on the plne is the vector itself Px = x if x is on the plne If the oject is lredy on the plne, there is no further projection to e performed The projection of projection is the projection P (Px) = Px Tht is ecuse Px is lredy on the plne Projection mtrices re idempotent t P = P Sep Follows from the ove -755/

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