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WEAK Supervision INDEP CASE A simple truck for LATENT VARIABLES Correlations Inverse Covariance Graphs Chinossians Graphical morsels Nuggets Method of moments OSED IN CROWD Sooney WS Classical Stars Ctm
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Page 1: WEAKcs229.stanford.edu/notes2020fall/notes2020fall/live... · 2020. 11. 12. · WEAK Supervision INDEP CASE A simpletruck for LATENT VARIABLES Correlations Inverse Covariance Graphs

WEAK Supervision

INDEP CASE A simple truck for LATENT VARIABLESCorrelations Inverse Covariance Graphs Chinossians

Graphical morsels

Nuggets Method of momentsOSED IN CROWD Sooney WS ClassicalStars Ctm

Page 2: WEAKcs229.stanford.edu/notes2020fall/notes2020fall/live... · 2020. 11. 12. · WEAK Supervision INDEP CASE A simpletruck for LATENT VARIABLES Correlations Inverse Covariance Graphs

Gw x x cRdXi Rd 1,13 o ABSTAIN for 5 1 m

find Ply 5 Xiii

Given Observe Ground truth Cunosservers

DATA 1 12 13 4x I l l l U

X l l ya

X t l i yCD

IDEAL hi Are Noisy voters function Inaccurate

Page 3: WEAKcs229.stanford.edu/notes2020fall/notes2020fall/live... · 2020. 11. 12. · WEAK Supervision INDEP CASE A simpletruck for LATENT VARIABLES Correlations Inverse Covariance Graphs

Mobil No Abstains Xi X E I I INDEOENT Errors

CAct LABELERhas A hiddenacuray makes AN Enron w 1 Pprobs

for Eaey Example X Aro AjXjlxY y at prob pg 1 es aight

1 x y w prob 1Pj M is ornery

WE NEED TO ESTIMATE PjPf X j x l yet P Ajax il y D

Gone compote pcy II Xfore m 73

Agree disagree

DE II L Y Pj 1 t CiPj fD 2ps I

detwe aj 2Py I

E Aill t I 4 cEy

ow

I DiPj t ClPi l Pj Agree

t 1 CiPi Pj t Pi CiPj disagree

Ai Aj

form a matrix M E lRmm

Mj E X 1

NI WE CAN Estimatefrom DATA Unlike y

CRAZYAlgon.euOBSERVE MyMyx AiAjax

2MyM as ComptetriplesMik

Page 4: WEAKcs229.stanford.edu/notes2020fall/notes2020fall/live... · 2020. 11. 12. · WEAK Supervision INDEP CASE A simpletruck for LATENT VARIABLES Correlations Inverse Covariance Graphs

WE HAVE Solved 000 Sign Bases on Observers My

Sign Mj sign ai sign a

ONCE WE KNOW Even A single lasselor's syn

tower a a Are solutions

wEAkd tou ai o breaks symmetry

THIS IS A Simple solution to an cm like Problem D

Page 5: WEAKcs229.stanford.edu/notes2020fall/notes2020fall/live... · 2020. 11. 12. · WEAK Supervision INDEP CASE A simpletruck for LATENT VARIABLES Correlations Inverse Covariance Graphs

WHAT if hi Are correlated

In general hopeless

AyyAND12 AreconelAA

A graphicalmonels Az y Az Y

INDER'r

E 42431y Ethelg Elk lyforges

G lyE 4.0HE I F Eh ABNorges

Page 6: WEAKcs229.stanford.edu/notes2020fall/notes2020fall/live... · 2020. 11. 12. · WEAK Supervision INDEP CASE A simpletruck for LATENT VARIABLES Correlations Inverse Covariance Graphs

Nugget Covariance Structure

X N lo lA

Xz X t Ez Ez N NLoD o oXz B

Xz X t Es Es N Nlo1

MEANIEx 3 0 ECxi EC E o ELxD

COVARIANCE1 I

E x 3 1 IEEE ECCx 5 Efx 3 t LET

2t Xz

IECx Xz IEC I E I EEK xDE Xz lE Cx Ee Chatted

oIEExt t IEEX.E.ITCxiEs3t5EfGtD1

L II No obvious

E

II I n

theCnet is awesome let's prove Gaossians

Wainwright LOH 2014 About discrete cases

RATNER 2018

r

Page 7: WEAKcs229.stanford.edu/notes2020fall/notes2020fall/live... · 2020. 11. 12. · WEAK Supervision INDEP CASE A simpletruck for LATENT VARIABLES Correlations Inverse Covariance Graphs

r

A propbahdy distribution p Md co D factorizeson agrees w a graph G CUE of

Phx Coec.ptPeCxiixD IeuPilxiNorm constant

pix C exp x x Gaussian

define A _I carotation

dog pay Los C xtAxAijxix

If p fnetons wat f CUE

log peg leg co t LosPekin t Z logPichiedJEE c

Consider Ci j EE 22k2glog Rx

72 I Air 1 Agi 2Aij covariance symmetric

22x2y O 2Aij

s I o whereon Ci DEE

IT'S not quite As clean for discrete r v

Page 8: WEAKcs229.stanford.edu/notes2020fall/notes2020fall/live... · 2020. 11. 12. · WEAK Supervision INDEP CASE A simpletruck for LATENT VARIABLES Correlations Inverse Covariance Graphs

BnektooonPnoblem

OBSEcuEsZ4iXzksl.n

UI

ASSUME WE know graphicalstructure GIV EHEN from Above Ccij HE EI 0

IDEAL if Tve Aren't too many Edges SKOULD BEADLEAcernay AND Connelnow strength

KNOW0Entrees Do not 0Entrees But I canmeasure

E o O wtf Block Inversions lemma

D 1,1437

Co out I zzT z

E if Ii z

An

Cc j EEo O Zi 2 linear system

Lg 5 lose thoseLinearSgEN Iv loss

Page 9: WEAKcs229.stanford.edu/notes2020fall/notes2020fall/live... · 2020. 11. 12. · WEAK Supervision INDEP CASE A simpletruck for LATENT VARIABLES Correlations Inverse Covariance Graphs

y s

e WE CAN Recover whenever this full RANK

This technique Addy unmanles what WE newlyIs Powerful Hyler RANK Solves

LEARN Graph Stroctone guess w SDP

RECAIMETHOD of moments

Nugget Graphs probably graphicalmorsels

Application WEAK Sopenuise


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