IFAC World Congress 2014 Randomized Methods for Analysis ... · Tutorial on Randomized Methods, Cape Town @RT 2014 12:15 Application to channel equalization (12:15 - 13:45 Lunch 13:45

Randomized Methods for Analysis and Design of Control Systems

M C i F b i i D bb Si G i

IFAC World Congress 2014

Tutorial on Randomized Methods, Cape Town @RT 2014

Marco Campi, Fabrizio Dabbene, Simone Garatti, Maria Prandini, Roberto Tempo

Schedule

8:30 - 8:45 Welcome and presentation of the day 8:45 - 10:15 Introduction to the scenario approach (Campi)

10:15 - 10:45 Coffee break10:45 - 11:30 The fundamental theorems of the scenario ---------------- approach (Garatti)11:30 - 12:15 Application to channel equalization (Prandini)


11:30 12:15 Application to channel equalization (Prandini)12:15 - 13:45 Lunch13:45 - 14:45 Probabilistic methods for analysis of --------------------- -- uncertain systems (Tempo)14:45 - 15:00 Coffee break15:00 - 16:00 Sequential randomized algorithms for -------------------- design (Dabbene) 16:00 - 16:30 Conclusions and discussion

Key References – Tutorials and Books

S. Garatti and M.C. Campi, “ModulatingRobustness in Control Design: Principlesand Algorithms,” IEEE CSM, 2013


R. Tempo, G. Calafiore and F. Dabbene,“Randomized Algorithms for Analysisand Control of Uncertain Systems, withApplications,” Second Edition,Springer-Verlag, London, 2013

Key References – Tutorials and Surveys

M.C. Campi, “Why Is Resorting to Fate Wise? A CriticalLook at Randomized Algorithms in Systems andControl,” European Journal of Control, 2010

G. Calafiore, F. Dabbene and R. Tempo “Research onP b bili ti D i M th d ” A t ti 2011


Probabilistic Design Methods,” Automatica, 2011 R. Tempo and H. Ishii, “Monte Carlo and Las Vegas

Randomized Algorithms for Systems and Control: AnIntroduction,” European Journal of Control, 2007

M. Vidyasagar, “Statistical Learning Theory andRandomized Algorithms for Control,” IEEE CSM, 1998

Key References – Recent Tutorials

I.R. Petersen and R. Tempo, “Robust Control ofUncertain Systems: Classical Results and RecentDevelopments,” Automatica, May 2014

H. Ishii and R. Tempo, “The PageRank


Problem, Multiagent Consensus andWeb Aggregation: A Systems andControl Viewpoint,” IEEE CSM, 2014

F. Dabbene and R. Tempo, “Randomized Methods forControl,” Encyclopedia of Systems and Control,Springer-Verlag, London, 2015 (to appear)

Software (Open Source)

R-RoMulOC: Randomized and Robust Multi-ObjectiveControl toolbox

http://projects.laas.fr/OLOCEP/rromuloc/


RACT: Randomized Algorithms Control Toolbox forMatlab

http://ract.sourceforge.net

Randomized Algorithms (RAs)

Randomized algorithms are frequently used in manyareas of engineering, computer science, physics,finance, optimization,…


Main objective of this tutorial: Introduction to rigorousstudy of randomized methods for uncertain systems andcontrol

The theory is ready for specific applications

Randomized Algorithms (RAs)

Computer science (RQS for sorting, data structuring)

Robotics (motion and path planning problems)

Mathematics of finance (path integrals)


Bioinformatics (string matching problems)

Computer vision (computational geometry)

PageRank computation (distributed algorithms)

Opinion dynamics in social networks

A Success Story: Randomization in


yComputer Science

A Success Story in CS

Problem: Sorting N real numbers

Algorithm: RandQuickSort (RQS)

RQS is implemented in a C library of Linux for sortingnumbers[1-2]


numbers[1 2]

[1] C.A.R. Hoare (1962)[2] D.E. Knuth (1998)

A Success Story in CS

Problem: Sorting N real numbers

Algorithm: RandQuickSort (RQS)

RQS is implemented in a C library of Linux for sortingnumbers


numbers

Sorting Problem

given N real x1 x2 x3 sort them in

numbers x4 x5 x6 increasing order

S1

RandQuickSort (RQS)

The idea is to divide the original set S1 into two setshaving (approximately) the same cardinality

This requires finding the median of S1 (which may bedifficult)


This operation is performed using randomization

RandQuickSort (RQS)

RQS is a recursive algorithm consisting of two phases

1. randomly select a number xi (e.g. x4)2. deterministic comparisons between xi and other (N-1) numbers


x2 x3 x1 x5x6

numbers smaller than x4 numbers larger than x4

S2 S3

4x

RQS: Binary Tree Structure

We use randomization at each step of the (binary) tree


Running Time of RQS

Because of randomization, running time may bedifferent from one run of the algorithm to the next one

RQS is very fast: Average running time is O(N log(N)) This is a major improvement compared to brute force


approach (e.g. when N = 2M) Average running time holds for every input with

probability at least 1-1/N (i.e. it is highly probable) The so-called Chernoff bound can be used to prove this Improvements for RQS to avoid achieving the worst

case running time O(N 2)

Find Algorithm

Find Algorithm: Find the k-th smallest number in a set Basically it is a RQS but it terminates when the number

is found Average running time of Find is O(N)


Another Success Story: Randomization


yin Mathematical Finance

(Quasi) Monte Carlo Methods for Computational Finance

QMC methods to estimate the prize of collaterizedmortgage obligations

The problem is to approximate the average mortgage

( ) df u u


taking N samples for each variable, but we need Nn

total number of points

Curse of dimensionality: n = 360!

[0,1]( ) d

nf u u

Probabilistic Methods for Analysis of


yUncertain Systems


Example: H Performance

Consider the linear system

Example: Frequency Response

0 1

0 1 0 01 1

x x u wa a

1 0z x


with (nominal) parametersa0 = 1 a1 = 0.8

The transfer function z = G(s) w is given by

2

1( )0.8 1

G ss s

disturbanceserrors

wz

H performance||G(s)|| = sup |G(j)| ≤ γ

Performance is satisfied for γ = 1 35

Example: H Norm


Performance is satisfied for γ 1.35

Bode plot (magnitude)

System Performance with Uncertainty

Consider an uncertain stable transfer function G(s,q)

z = G(s,q) wG(s,q) w z


where w and z are disturbances and errors and qrepresents uncertainty bounded in a set Q of radius ρ > 0

Consider the uncertain linear system

Example[1]: System Performance with Uncertainty

0 1

0 1 0 01 1

x x u wa a

1 0z x


with parametersa0 = 1 + q0 a1 = 0.8 + q1

and bounding setQ = {q = [q0 q1 ]T : ||q|| }

[1] R. Tempo, G. Calafiore, F. Dabbene (2013)

Given performance level the objective is tocompute the maximal radius of Q such that

G(s,q) is stable and ||G(s,q)|| for all q Q

Example: Radius of Uncertainty

ργ= 2


G(s,q) is stable and ||G(s,q)|| if and only if

< 0.8 and2(0 .8 ρ ) 1 ρ

2 2

Example: Radius of Uncertainty

Largest radius of Q suchthat performance is satisfied is = 0.025ρ


Conclusion: Stability and performance are satisfied for all q Qwith radius = 0.025 ρ

Objective of Robustness

Objective of robustness: To guarantee stability andperformance for all

q Q


We may also use the notation to denote uncertaintyand B for bounding set


Probabilistic Robustness

Different Paradigm Proposed

Different paradigm based on a probabilistic model ofuncertainty which leads to randomized algorithms foranalysis and synthesis

Within this setting a different notion of problem


Within this setting a different notion of problemtractability is needed

Benefits and pitfalls of risk analysis

Objective: Breaking the curse of dimensionality[1]

[1] R. Bellman (1957)


Probabilistic Methods

Probabilistic Model of Uncertainty

Assume that q is a random vector with given density

function and support set Q

Probability density function associated to q


Examples: Uniform

or Gaussian pdf

Uniform Density U [Q]

Univariate uniform density

b

1/(b-a)

[ , ]a bU


Multivariate uniform density U [Q]

1 if

vol( )0 otherwise

q QQQ

U

a b

Probability of Performance

Define a performance function

J(q): Q → R

Given level , probability of performance (reliability) is


PJ = Prob{q Q: J(q) }

Example: If G(s,q) is stable and J(q) = ||G(s,q)||

PJ = Prob{q Q: ||G(s,q)|| }

Measure of Performance Violation

Objective: Achieve probabilistic performancePJ = Prob{q Q: J(q) } ≥ 1 -

where (0,1) is a probabilistic parameter calledaccuracy


accuracy

Computation of Probability of Performance

ComputingPJ = Prob{q Q: J(q) }

requires to solve a difficult integration problem Taking uniform density U [Q]


Taking uniform density U [Q]

In some special cases we can easily compute thisprobability

( ) γd

Prob : ( ) γvol( )J q

qq Q J q

Q


Worst Case vs Probabilistic Approaches


Example: H Performance

Recall Performance Violation

Increase the radius

Observation: If we allow a small violation


allow a small violationof performance we may increase the radius significantly

Computation of Performance Violation

Take uniform pdf in Q

Allowing 5% violationwe increase of 54%


we increase of 54% obtaining 0.038 (instead of 0.025)

For several values of we compute PJ ()

Performance Degradation Function

If a 5% violation is allowed we increase of 54%

PJ (ρ)


0.038

increase of 54%obtaining 0.038

Radius 0.038 compared to = 0.025ρ

0.038=0.025ρ=0.025 ρ=0.038


Probabilistic Robustness Analysis

Probabilistic Model

Probability density function associated to B

We assume that is a random matrix (vector) with given

density function and support B


y pp

Example: Uniform density in B

Uniform Density

Consider uniform density U[B] within B

th i0

if)(vol

1 BBBU


In this case, for a subset S B

otherwise0

)(vol)(vol

)(vol

dProb

BS

BS S

Good and Bad Sets

We define two subsets of B

Bgood = {: J( } BBbad = {: J( } B


bad { ( }

Bgood is the set of satisfying performance Measure of robustness is

good

dvol good ΒB

Probability of Performance

Given a performance level , we define the probability ofperformance

Prob{J() }


Violation and Reliability

We define the violation probability

V = 1 - Prob{J() } = Prob{J() > }

Probability of performance is also denoted as reliability


R = Prob{J() } = 1 – V

Computation of Violation and Reliability

Computing V and R requires to solve a difficultintegration problem

In some special cases we can easily compute violationand reliability


Otherwise use randomized algorithms to determineprobabilistic estimates of V and R

Monte Carlo and Las Vegas


Monte Carlo and Las Vegas Randomized Algorithms

Monte Carlo and Las Vegas

Monte Carlo was invented by Metropolis, Ulam, vonNeumann, Fermi, … (Manhattan project)


Metropolis Fermi Ulam, Feymann, von Neumann

Las Vegas first appeared in computer science in the lateseventies

Randomized Algorithm: Definition

Randomized Algorithm (RA): An algorithm that makesrandom choices during its execution to produce a result

Example of a “random choice” is a coin toss


heads or tails



Example: Matlab codeset r =1:0.01:3;


set_r 1:0.01:3;for k =1:length(set_r)

if (rand > 0.5) then a_opt(k) = hel(k);else a_opt(k) = 3.7;end if

a_lin(k) =(e/(e-1))*r;a_sub(k) =(a/(a-1))*(r+log(a)-1);

end



For hybrid systems, “random choices” could be


o yb d sys e s, do c o ces cou d beswitching between different states or logical operations

For uncertain systems, “random choices” require (vectoror matrix) random sample generation


Monte Carlo Randomized Algorithm


Monte Carlo Randomized Algorithm (MCRA): Arandomized algorithm that may produce incorrect results,but with bounded probability of error



Monte Carlo Randomized Algorithm (MCRA): Arandomized algorithm that may produce incorrect results,but with bounded probability of “error”


Prob{“error” > } < 2e(-2N2) Hoeffding inequality

where is the probabilistic accuracy of the estimate, N isthe sample size (sample complexity) and e is the Eulernumber

Example of Monte Carlo: Area/Volume Estimation

Estimate the volume of the red area: Generate N samplesuniformly in the rectangle; count how many (M) fallwithin the red area, then the estimated area = M/N


Las Vegas Randomized Algorithm

Las Vegas Randomized Algorithm (LVRA): Arandomized algorithm that always produces correctresults, the only variation from one run to another is therunning time


Las Vegas Randomized Algorithm

Las Vegas Randomized Algorithm (LVRA): Arandomized algorithm that always produces correctresults, the only variation from one run to another is therunning time


Example: Randomized Quick Sort (RQS)


Randomized Algorithms for Control

Ingredients for RAs

Assume that is random with given pdf and support B Accuracy (0,1) and confidence (0,1) be assigned Performance function for analysis and level

↓ ↓


↓ ↓

J = J()

Randomized Algorithms for Analysis

Different classes of randomized algorithms forprobabilistic analysis to estimate

Probability of performance


Probability of performance Probability of failure

They are based on uncertainty randomization of

Sample complexity is obtained

Estimating the Probability of


Estimating the Probability of Performance

Estimate of the Probability of Performance

Objective: Construct a probabilistic estimate usingMonte Carlo randomized algorithms of reliability(probability of performance)


R = Prob{J() }

Monte Carlo Experiment

We draw N i.i.d. random samples of according to thegiven probability measure

), 2), …, ) B

The multisample within B is


The multisample within B is

1,…,N = {(1), ... , N)}

We evaluateJ()), J()), …, J(N))

Example

J


Example

J


1 2 3 4 5 6

Example

J(3)

J


1 2 3 4 5 6

J(1)

J(2) J(4)

J(5)J(6)

Empirical Reliability

We construct the empirical reliability

where I (·) denotes the indicator function

N

i

iN J

NR

1

)( )1ˆ I


where I ( ) denotes the indicator function

Notice that

where Ngood is the number of samples such that J(i))

( )

( ) 1 if ( )( )

0 otherwise

ii J γ

J

I

NN

RNgoodˆ

Sample Complexity

We need to compute the size of the Monte Carloexperiment (sample complexity)

This requires to introduce probabilistic accuracy (0,1) and confidence (0,1)


( ) ( ) Given , (0,1), we want to determine N such that the

probability event(this is the “error”

previously discussed)

holds with probability at least 1-

εˆ NRR

A Good Estimate

If the probability event

holds with probability at least 1- , the we say that the

εˆ NRR


holds with probability at least 1 , the we say that theempirical reliability is a “good” estimate of thereliability R

(Additive) Chernoff Bound[1]

(Additive) Chernoff BoundGiven , (0,1), if

2

δ2

ch 2logNN


then the probability inequality

holds with probability at least 1-

2ch ε2

[1] H. Chernoff (1952)

εˆ NRR

Remarks

Chernoff bound improves upon other bounds such asthe Law of Large Numbers (Bernoulli)

Dependence is logarithmic on 1/ and quadratic on 1/ Sample size is independent on the number of


Sample size is independent on the number ofcontroller and uncertain parameters

1-

Nch

Accuracy vs Confidence

Confidence is “cheap” because of the logarithmicdependence

Accuracy is computationally more expensive becauseof quadratic dependence


of quadratic dependence Can we improve the quadratic dependence? The answer to this question is provided by the

(multiplicative) Chernoff Bound

Hoeffding Inequality and Chernoff Bound - 1

Given (0,1), from the Hoeffding inequality we obtain

Prob{1,…,N : } ≤ 2e(-2N2)

where e denotes the Euler number

ˆ- εNR R


To guarantee confidence (0,1), we need to take Nsamples such that 2e(-2N2) ≤ holds

We obtain the (additive) Chernoff bound

N ≥ 1/ (22) log(2/ )


The Hoeffding inequality provides a bound on the taildistribution

2e(-2N2)

From the computational point of view computing the


From the computational point of view, computing theminimum value of N that 2e(-2N2) ≤ is immediate(given and it is a one-parameter problem)

The Chernoff bound provides a fundamental explicitrelation (sample complexity) N = N(, ) showing that1/ enters quadratically and 1/ logarithmically


Chernoff bound and the Hoeffding inequality hold onlyfor fixed performance function J

Some results are available for a finite number ofperformance functions


performance functions

For an infinite number of performance functions we needto use statistical learning theory

Parallel and Distributed Simulations

Samples q(1), q(2), …, q(N) are i.i.d. Contrary to MCMC or sequential Monte Carlo, this

approach leads to parallel and distributed simulations


IBM Blue Gene Cray-1 vector processor

Parallel and Distributed Simulations

Samples q(1), q(2), …, q(N) are i.i.d. Contrary to Markov Chain Monte Carlo (MCMC) or

sequential Monte Carlo, this approach leads to paralleland distributed simulations


Sample generation requires tools from importantsampling techniques

Connections with the theory of random matrices[1]

[1] G. Calafiore, F. Dabbene, R. Tempo (2000)


Bounds on the Binomial Distribution

Bounds on the Binomial Distribution

The so-called probability of failure is studied in thescenario approach and in statistical learning theory

This required bounding the binomial distribution


This required bounding the binomial distribution

0

B( ,ε, ) ε 1 εm

N ii

i

NN m

i

Bounding the Binomial Distribution and Sample Complexity

Theorem[1]: Given , (0,1) and m 0, if

h

1

1 1inf log log( )ε 1 δa

aN m aa


then

[1] T. Alamo, R. Tempo and A. Luque (2010)

0

B( ,ε, ) ε 1 ε δm

N ii

i

NN m

i

Bounding the Binomial Distribution and Sample Complexity

Suboptimal value of a is the Euler number e

Sample complexity is given by1 1logε 1 δ

eN me


Sample complexity is linear in

- 1/ (not quadratic!)

- m

-

ε 1 δe

1logδ

Probabilistic Methods:Benefits and Drawbacks

Benefits Drawbacks

very general method with immediatepractical applications, for example inaircraft design and process control industry

the results obtained provide no“deterministic certificate” of propertysatisfaction, for example H-infinityperformance

specific sample generation methods have for recursive methods the number of


specific sample generation methods havebeen developed (e.g. for norm bounded sets,hit-and-run for convex sets, particlefiltering, importance sampling, MCMC)

for recursive methods the number ofrequired experiments is generally notspecified a priori

sample size bounds are available for non-recursive methods

the method does not cover the entire samplespace, but only a finite subset of it

Monte Carlo methods are very effective indealing with the “curse of dimensionality”;the probability of error is bounded

crucial points of the safety region can bemissed, this may lead to erroneousconclusions


Computational Complexity of RAs

Computational Complexity of RAs

RAs are efficient (polynomial-time) because

1. Random sample generation of i) can be performedin polynomial-time


2. Cost associated with the evaluation of J(i)) forfixed i) is polynomial-time

3. Sample size is polynomial in the problem size andprobabilistic levels and

1. Bounds on the Sample Size

Chernoff bound is independent on the size of B, on theuncertainty structure, on the pdf and on the number ofuncertainty blocks

It depends only on probabilistic accuracy and


It depends only on probabilistic accuracy andconfidence

Same comments can be made for other bounds (suchas Bernoulli)

2. Cost of Checking Stability

Consider a polynomial

To check left half plane stability we can use the Routhtest The number of multiplications needed is

nnsasaaasp 10),(


test. The number of multiplications needed is

The number of divisions and additions is equal to thisnumber

We conclude that checking stability is O(n2)

odd for 4

1 even for 4

22

nnnn

3. Random Sample Generation

Random number generation (RNG): Linear andnonlinear methods for uniform generation in [0,1) suchas Fibonacci, feedback shift register, BBS, MT, …

Non uniform univariate random variables: Suitable


Non-uniform univariate random variables: Suitablefunctional transformations (e.g., the inversion method)

Much harder problem: Multivariate generation ofsamples of with given pdf and support B

.It can be resolved in polynomial-time


Choice of the Probability Distribution

Choice of the Probability Distribution - 1

The probability Prob{S}depends on the underlyingpdf

I b 0 d 1


It may vary between 0 and 1depending on the pdf

Choice of the ProbabilityDistribution - 2

The bounds discussed are independent on the choiceof the distribution but for computing an estimate ofProb{J() } we need to know the distribution

Research has been done in order to find the worst-case


Research has been done in order to find the worst casedistribution in a certain class[1]

Uniform distribution is the worst-case if a certaintarget is convex and centrally symmetric

[1] B. R. Barmish and C. M. Lagoa (1997)

Choice of the ProbabilityDistribution - 3

Minimax properties of the uniform distribution havebeen shown[1]


[1] E. W. Bai, R. Tempo and M. Fu (1998)


Random Sample Generation

True Random Number Generators

Hardware sources of trulystatistically random numbers

High-voltage reverse-biasedk


P-N semiconductor junctions Reverse-biased Zener diodes Radioactive Decay Lava-rand Mechanical systems

entropy key

Random Generation

(Pseudo) random number generation (RNG): Variousmethods are available for generation in the interval [0,1)

Linear and nonlinear RNGs, Fibonacci, feedback shiftregister BBS MT


register, BBS, MT, …

Non-uniform univariate random variables: Suitablefunctional transformations (e.g., the inversion method)

Multivariate random variables: Rejection and conditionaldensity methods


Multivariate Random Vector Generation

Parametric Uncertainty

We study parametric uncertainty q in ℓp norm balls

Objective: Sample generation in the ball


B {q : ||q||p 1}

We are interested in uniformsample generation within B

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1step 3

ℓp Vector Norms

Recall the ℓp vector norm of xFn

and the ℓ vector norm

1/

1|| || | | for [1, )

pnp

p ii

x x p


and the ℓ vector norm

|| || max | |iix x

Rejection Methods

Goal: to generate uniform samples in a set B (e.g. anorm ball)

Idea: If we have a “simpler" set Bd that contains B, we


can generate uniform samples in Bd, and then reject

those that fall outside B The rejection rate of the method is

Note: generation in Bd should be easy, membership of

B should be efficiently checkable

dvol( )vol( )

η BB

Rejection Methods

B Find a bounding set Bd


Bd

Generate points x(i) in Bd

Keep the points in B

and reject the others

Rejection Methods:Curse of Dimensionality

Rejection rate for generation ofuniform samples in the sphereusing an hypercube as boundingset

B


We obtain

)12/()π/2()(η n n n

n=1 n=2 n=3 n=4 n=10 n=20 n=30

1 1.2732 1.9099 3.2423 401.54 4.·107 5· 1013

Bd

Gamma Density

A random variable x has (unilateral) Gamma densitywith parameters (a,b) if

1 /1( ) 0( )

a x bx af x x e x

a b


where · is the Gamma function

We write x G(a,b)

There exist standard and efficient methods for randomgeneration according to G(a,b)

1 ξ

0

( ) ξ dξ 0xx e x

Generalized Gamma Density

A random variable has (unilateral) Generalized Gammadensity with parameters (a,c) if

-1 -( ) , 0( )

cca xx

cf x x e x


( )x a

We write x Gg(a,c)

Generalized Gamma Density

11

G ,( )

pxg p

p

pp e


p=1p=2p=4p=10p=100

Comments

Using power transformation method, a random variablex ~ Gg(a,c) is simply obtained as

x =z1/c

where z ~ G(a,1)


Samples distributed according to a (univariate) bilateraldensity x ~ fx(x) can be easily obtained from a(univariate) unilateral density z ~ fz(z)

Take x = sz, where s is an independent random signwith values +1 and -1 with equal probability

Joint Density

Let x=[x1,…,xn]T with components independentlydistributed according to the (bilateral) GeneralizedGamma density with parameters 1/p and p

The joint density of x is


|| ||

1

( ) 2 (1/ ) 2 (1/ )

p ppi

nnxx

x n ni

p pf x e ep p

Example: Multivariate Laplace

Recall that (1)=1 Multivariate (bilateral) Laplace

density

1

| |1( )

n

ii

x

f x e


is a Generalized Gammadensity with parameters 1 and 1

( )2x nf x e

Example: Multivariate Normal

Multivariate (bilateral) normal Nwith mean 0 and covariance

T

2I


is a Generalized Gamma densitywith parameters 1/2 and 2

T/2( ) π n x xxf x e

Uniform Multivariate Generation in B

TheoremLet xi be random variables distributed according to the(bilateral) Generalized Gamma density

px pgi ,G~ 1


Let w[0,1] be a random variable uniformly distributed

Then the vector

is uniformly distributed in B

ppgi

T1 ,,,1

np

xxxxxwy n

Algorithm Vector Uniform Generation

Input: n, p Output: uniform random sample y

• Generate n independent real scalars i ~Gg(1/p,p)


• Construct vector x of components xi=si i where si are randomsigns

• Generate w uniform in [0, 1]

• Return1/n

p

y w xx

Uniform Random Generation in ℓ2 - Step 1

1

2

3

4step 1

Generate n iid randomreal scalars

ξ ~ G 1( / , )g p p


-4 -3 -2 -1 0 1 2 3 4-4

-3

-2

-1

0 Construct xRn of

components

(si iid random signs)ξi i ix s


0 2

0.4

0.6

0.8

1step 2 Construct the

normalized vector

xzx

‖ ‖


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

The vector z isuniformly distributedon the surface of thep-norm ball

px‖ ‖

0 2

0.4

0.6

0.8

1step 3

Generate w uniform in[0,1], and return

.


1/ 1/n ny wzw xx


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

The vector y isuniformly distributedinside the p-norm ball.

px

Uniform Random Generation in B for p=1

0 2

0.4

0.6

0.8

1


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Uniform Random Generation in B for p=0.7

0 2

0.4

0.6

0.8

1


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Uniform Random Generation in B for p=4

0 2

0.4

0.6

0.8

1


-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

Documents

IFAC World Congress 2014 Randomized Methods for Analysis ... · Tutorial on Randomized Methods, Cape Town @RT 2014 12:15 Application to channel equalization (12:15 - 13:45 Lunch 13:45