[PDF] [PDF] Machine Learning Notation - GitHub Pages

$ I_ {\rm SDE} $+—An Indicator for Multi and Many-Objective

29 mar 2019 · In aggregation-based EMO algorithms, scalarizing functions are employed to map the objective values of a multiobjective problem into a single 

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Note: LaTeX template courtesy of UC Berkeley EECS dept The subdifferential of the indicator function at x is known as the normal cone, NC(x), of C:

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Note: LaTeX template courtesy of UC Berkeley EECS dept By generalizing X from an indicator function to any random variable we can get the definition of the

[PDF] Machine Learning Notation - GitHub Pages

1(x; cond) The indicator function of x: 1 if the condition is true, 0 otherwise g[f; x] A functional that maps f to f(x) Sometimes we use a function f whose argument is 

[PDF] Probabilities and random variables

sum to one) • The indicator function of an event A is the random variable defined Monthly; I was seeing how closely LATEX could reproduce the original text

List of Symbols - SpringerLink

Locally convex cone of P-valued functions on X, Characteristic function of a subset E ⊂ X, II 1 1 Symbols Related to Continuous Cone-valued Functions of the manuscript authors will be asked to prepare the final LaTeX source files ( and 

[PDF] The Comprehensive LaTeX Symbol List

8 oct 2002 · This document lists 2590 symbols and the corresponding LATEX commands that produce them Some of these symbols corresponding LATEX command to the right of each symbol A table's caption Characteristic Value

[PDF] Math 431 – Spring 2014 Homework 9 Hand in the following problems:

Let Xi denote the indicator function of the first time i is seen on one of the five die rolls Then, by linearity of expectation and exchangeability, E[X] =E[X1] + ··· + 

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MachineLearningNota tion

Shan-HungWu

1Num bers&Arrays

aAscalar(integerorreal)

AAscalarconstant

aAvector

AAmatrix

AAtensor

I n

Then⇥nidentitymatrix

DAdiagonalmatrix

diag(a)Asq uare,diagonalmatrixwi thdiagonal entriesgivenbya aAscalarrandomvariable aAvector-valuedrandomvariable

AAmatrix-valuedrandomvariable

2Sets &Graphs

AAset

RTheset ofrealnum bers

{0,1}Theset containin g0and1 {0,1,···,n}Theseto fallinte gersbe tween0and n [a,b]Thereal intervalincl udingaandb (a,b]Thereal intervalexc ludingabutincl uding b

A\BSetsubtr action,i.e.,thesetcontainingthe

elementsofAthatarenotin B

GAg raphwhoseeachver texx

(i) denotesa randomvariableande dgedenotes conditionaldependency(directed) or correlation(undirected) Pa(x (i) )Thepar entsofavertexx (i) inG

3Index ing

a i

Elementiofvector a,wi thindexin g

startingat1 a i

Alle lementsofvectoraexceptforelemen t

i A i,j

Element(i,j)ofmatrixA

A i,:

RowiofmatrixA

A :,i

Columniofmatrix A

A i,j,k

Element(i,j,k)ofa3-D tensorA

A :,:,i

2-Dslice ofa3-D tensor

a i

Elementiofthe randomvectora

4Func tions

f:A!BAfunctionfwithdomain AandrangeB fgCompositionoffunctionsfandg omittedsometimes) lnxNaturallogarithmofx (x)Logisticsigmoid, i..e,(1+e xp(x)) 1 ⇣(x)Softplus,ln(1+exp( x)) kxk p L p normofx kxkL 2 normofx x

Positivepartofx,i.e.,max(0,x)

1(x;cond)Theind icatorfunctionofx:1ifthe

conditionistrue,0otherwise g[f;x]Afunctionalthatmapsftof(x) Sometimesweuseafunction fwhoseargumen tisascalar,but applyittoavector, matrix,orten sor:f(x),f(X),orf(X). Thismeanst oapplyftothe arrayelem ent-wise.Forexa mple, ifC=(X),thenC i,j,k =(X i,j,k )forall i,jandk.

5Ca lculus

f 0 (a)or df dx (a)Derivativeoff:R!Ratinp ut pointa f xi (a)Partialderivativeo ff:R n !Rwith respecttox i atinp uta rf(a)2R n

Gradientoff:R

n !Ratin puta rf(A)2R m⇥n

Matrixderivativesof f:R

m⇥n !R atin putA rf(A)Tensorderivativ esoffatinp utA

J(f)(a)2R

m⇥n

TheJaco bianmatrixoff:R

n !R m atinp uta r 2 f(a)or

H(f)(a)2R

n⇥n

TheHess ianmatrixoff:R

n !Rat inputpointa f(x)dxDefiniteintegraloverth eentire domainofx S f(x)dxDefiniteintegralwithres pecttox overthes etS 1

6Li nearAlgebra

A

TransposeofmatrixA

A

Moore-Penrosepseudo-inverseofA

ABElement-wise(Hadamard)productofA

andB det(A)DeterminantofA tr(A)TraceofA e (i)

Thei-thstandardbasi svector(aone-hot

vector)

7Pr obability&Info.Theory

a?bRandomvariablesaandbareind ependent a?b|cTheyarecon ditionall yindependentgivenc

Pr(a|b)or

Pr(a|b)

Shorthandfortheprobabili ty

Pr(a=a|b=b)

P a (a)Aprobabilitymassfunctionofthediscrete randomvariablea p a (a)Aprobabilitydensityfunctionofthe continuousrandomvariablea

P(a= a)EitherP

a (a)orp a (a) N(µ,⌃)TheGaus siandistributionwithmean µ andcovariance matrix⌃ x⇠P(✓)Randomvariablexhasdistri butionP E x⇠P [f(x)]Expectationoff(x)withrespec ttoP

Var[f(x)]Varianceoff(x)

Cov[f(x),g(x)]Covarianceoff(x)andg(x)

H(x)Shannonentropy oftherandomvariablex

D KL (PkQ)Kullback-Leibler(KL)divergencefrom distributionQtoP

8Mac hineLearning

XTheseto ftrainin gexamp les

NSizeofX

(x (i) ,y (i) )Thei-thexamplepai rinX(supervised learning) x (i)

Thei-thexample inX(unsupervised

learning)

DDimensionofadatapointx

(i)

KDimensionofalabely

(i) X2R

N⇥D

Designmatrix,where X

i,: denotesx (i)

P(x,y)Adatageneratingdistribution

FHypothesisspaceoffunction stobelearnt,

i.e.,amode l

C[f]Acostfunctionaloff2F

(x 0 ,y 0 )Atestingpair

ˆyLabelpredicted byafunctionf,i.e.,

ˆy=f(x

0 )(supervisedlearning)

9Ty pesetting

Section*Sectionthat canbeskippedforthe first

timeread ing

Section**Sectionforreferen ceonly(willnotbe

taught) [Proof]Proveityours elf [Homework]Youhav ehomework 2quotesdbs_dbs13.pdfusesText_19