Lecture 4 spectral method Basic perturbation Theory related Aswe willsee in plantedcliqueandmany planted problems the about The first few eigenvectors of FLEX Contain information the undenYL planted structures Hee x denotethedatamatrix ingeneral e digue g Xcouldbe adjacencymath A in eg planted ELA STERN 1 I planted clique But we only observeX and do not know EEX and hope that these we may compute the Kay eigenvectors of are closeto leady eigenvectors of EE X Write X FLEXI t X EEX We willshow that the error of estray the Kady eigenvectorsof FIX canbe bounded in terms of the size ofperturbation X ELXI Reviewof Linear algebra suppose X EIR is symmetric matrix Def eigenvalue and eigenvector C and ElRh is an The pain X V with A IR V eigen pair of X if Xv Xv j eigenvector eigenvalue X we can order the eigenvalues of by their sizes XlZX2Z Zan with the conespondy eigenvectors Vi V2 Vn Men we can write the egadeapos of X as X VAVT di Vivi where u k Vn a t I in Also notethat rank X h 7 exactly r nonzerodi's Howabout rectangle matrix Def singular value decomposition I suppose XE 112mm is The singular value decopositionof X X U 2 VT E Gi Ui Vi hthogoygf.mn U Elli Uh EIR II gr t K I n Vn Ellen U leftsingular vector vi right Giza singular value and XTX We can obtain SVD of X from EVD of XXT Indeed 2 T XXT U UT XT X 2 so si xxT Fixx Matrix norms a Frobeniusnorm View as mn dim vector anddefine llxlt EHveccxslk fz.FI operator'onorm view as linear operator X IR H Hp IRM 11.11g with operator norm HXHp q sup 11 Xv Hq 452 Forthis course the most relevant one is p L 2 H X Hop E 11 112 32 known as spectral norm as sometimes I'll write 11 12 22 simply 1 1112 Remark HXHop 6maxlX UIVIINE Pt HXHop2 sup INUK sup HE 11412 1 1142 2 2 Ui'sare 7 sup city orthogonal HUH Foi 6max4X H Hop is a norm and HXHop NXTHop HXYHop E Hilltop 11THop If X X is a vector then 11 119 11 112 11 Hop is orthogonal invariant ie forany Reocn RR4342 2 and R C 0 m wehave HR XRHop HXHop Columns If X Xi Xn has orthogonal rows then 11 Followsfrom Gmax X L 11 op L Remark Note Matrix Inner product x 17 E Vecchi veil's Xijtij Tr X'T Thus LX nut sup CHA HXHop Gmax X sup Hulk HIEI AHAH rankCAFI Likewise If X is symmetric then minimax.im SupcXiWT7 llXfop 5maxlX1 nixpdmaxlXII supkxvrlVlkUNT 1 NIKI I U ter perturbation eigenstructures of whereVE Ui f Xiao i perturbation Assume X and f Xt Z whee Z is FX and close Question Are eigenvalues and eigenvectors of X T when Z is small Answet No in general consider and Xe Then d CX 421 1 0 det XI Xe It AllXetF.dzHe Je deff A E Moregenerally consider too Imax Ail Xl 0 but Xi Xe f 1 E'T e2 here denotes the imiginay point NotLipschitzcontinuous in E Even in the symmetric matrix case eigenvector perturbationsmay fail dramatically Ii and t i det f HEE del f ex 1 The eigenvalues arethe same 112171 1 E AIX Nlt HE 1121 1 apart 1lover the eigenvectors are far KHE WH f WH Y but hat f between the eigenvalues a Spectre Lessie we need separation leigen gap perturbatinboundfreigenvalues Let X f Z be real s c matrices in RMn Let f Xt Z wehave that VVT di IX t Antz XCX t if CE t 1142 1 histhe L X vV'T CZ wht leagejuvofX C Xt't wt E 87k ACT saw E EYE W t di X tdi IZ ANH Edith dik e att 71211 INN AHH E Mah Hilal Hnat op Moregenerally we hue the foamy Weyl's inequality Hw 1hm Weyl's inequality LidsKii's inquality x E 11211 Ail dik op perturbation bounds for eigenspaces matrices in Let X 7 Z be real symmetric IR and f Xt Z In Suppose X Idi Vi Vi with dizdz f Pi ViVIT with A Ipa 2 2 Pn bound UE u We want to prove a perturbation for and r E v and more generally for U Un Un and vi Vz Vr Caveat o u and V are determined upto theirsign Similarly V and V are determinedupto orthogonal transformation There are two possible workarounds Consider spectral distance dslay E Min llntsvlk SETH F.nl J2 2050 d2 25m20s 251nF Moregenerally dslUYE inf HU VRHop REOCH Consider projectiondistance 2 since Hunt w HE 2 l LUN In the generalcase consider 11001WTVF or Hult VVTHop Thm Davis Kahan Let coso KUN I Then spectrol 112419 gap since s ma gap role PI WLOG assure pizdz otherwise switch of X and T Let us start from the eigenvalue equation Xu Jiu and TV pi V Denote Us uz un CIR then uix IYE.lx IIII.tt IIII Hence Viv Utter put UI 2 n Ruin in Join utter Vivi 7 VIZ n Ta l oy the Il Hop norm on both sides gives Hop Ig Hop Hop HUI Hutter HE a HUI ViHz 11211op P1 da Finallynote that vilkuih vifI yut.lv HUIhlli 2 Te d Lu v SIRO R h al 1 11 1112112 ll orthogonal invariant 11 11 1112 12 any norm ll Hope generally for 11211 VIL E HUI matar rtJ cased I AHHH I spectralgap case2 T T T Prez Phi Nr Ary X known one can generalize this to singular rectos by a technique as self adjoint dilation Let X U E VT Y TE TT Xlmth consider plinth Oy Xo J E and similarly for f obserlethat Hana III It ri I o III I Thus we can apply Davis Kahan Thm to obtain 25M since smeifu I Yj Hop 2 HEE El mail.to a e2HEIok fix 04411.