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A Random Polynomial-Time Algorithm for Approximating the Volume of Convex Bodies

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### A Random Polynomial-Time Algorithm for Approximatingthe Volume of Convex Bodies

### The Algorithm

### Random Walk

### Proof of Correctness

### Rapidly Mixing Markov Chain

By Group 7

The Problem Definition

- The main result of the paper is a randomized algorithm for finding an approximation to the volume of a convex body ĸ in n-dimensional Euclidean space
- The paper is a joint work by Martin Dyer, Alan M. Frieze and Ravindran Kannan presented in 1991.
- This is done by assuming the existence of a membershiporacle which returns yes if a query point lies inside the convex body or not.
- n is definitely ≥3

What is a convex body?

- In Euclidean space, an object is defined as convex
- if for every pair of points within the object,
- every point on the straight line segment that joins the pair of points also lies within the object.

Convex Body

Non- Convex body

Well Roundedness?

- The algorithm mentions well rounded convex body which means the dimensions of the convex body are fixed and finite.
- Well roundedness is defined as a property of a convex body which lies between two spheres having the radii:-

1 & √ (n)x(n+1)

(where n= no. of dimensions)

The running time of the algorithm

- This algorithm takes time bounded by a polynomial in n, the dimension of the body ĸ and 1/ε where ε is relative bound error.
- The expression for the running time is:-

O(n23(log n)5ε-2 log[1/ε])

Motivation

- There is no deterministic approach of finding the volume of an n-dimensional convex body in polynomial time, therefore itwas a major challenge for the authors.
- The authors worked on a probabilistic approach to find the volume of the n-dimensional convex body using the concept of rapidly mixing markov chains.
- They reduced the probability of error by repeating the same technique multiple number of times.
- It was also the FIRST polynomial time bound algorithm of its kind.

Deterministic approach and why it doesn’t work?

- Membership oracle answers in the following way: It says yes, if a point lies inside the unit sphere and says no otherwise.
- After polynomial no of. queries, we have a set of points, which we call P, from which must form the hull of the actual figure.
- But possible candidates for the figure can range from the convex hull of P to the unit sphere.

Deterministic approach and why it doesn’t work contd.

- The ratio of convex hull (P) and unit sphere is at least
- poly(n)/2^n.
- So, there is no deterministic approximation algorithm that runs in polynomial time.

Overview of today’s presentation

- The algorithm itself will be covered by Chen Jingyuan
- Chen Min will introduce the concept of Random walk.
- Proof of correctness and the complexity of algorithm is covered by Chin Hau
- Tuan Nguyen will elaborate on the concept of Rapidly Mixing Markov’s Chains(RMMC).
- Zheng Leong will elaborate on the proof of why the markov’s chain in rapidly mixing.
- Anurag will conclude by providing the applications and improvements to the current algorithm

Chen Jingyuan

The Dilation of a Convex Body

- For any convex body K and a nonnegative real number ɑ,
- The dilation of K by a factor of ɑ is denoted as

The Problem Definition

- Input: A convex body
- Goal: Compute the volume of , .
- Here, n is the dimension of the body K.

How to describe K?

Well-guaranteed Membership Oracle&Well-rounded

- A sphere contained in the body: B.
- B is the unit ball with the origin as center.
- A sphere containing the body: rB.
- Here , n is the dimension of the body.
- A black box
- which presented with any point x in space, either replies that x is in the convex body or that it is not.

The Algorithm

- How to generate a group dilations of K?
- Let , and .
- For i=1, 2, …, k, the algorithm will generate a group dilations of K, and the ratios equals to

The Algorithm

- How to find an approximation to the ratio
- The ratio will be found by a sequence of "trials" using random walk.
- In the following discussion, let

Sample uniformly at random from Ki !

The Algorithm

After τ steps...

…

- Proper trial: if , we call it a proper trial.
- Success trial: if , we call it a success trial.

The Algorithm

- Repeat until we have made proper trials.
- And of them are success trials.
- The ratio, , will be a good approximation to the ratio of volumes that we want to compute.

Chen Min

Natural random walk

Some notations

x

Y/N

Oracle

1.Oracle:

A black box tells you whether a point x belongs

to K or not (e.g, a convex body is given by an oracle)

K

2. For any set in and a nonnegative real number , we denote by the set of points at distance at most from K.

is smoother than K

…

3.cubes:

We assume that space () is divided into cubes of side . Formally, a cube is defined as:

Where are integers

…

Any convex body can be filled

with cubes

Natural random walk

K

Steps:

Starts at any cube intersecting

It chooses a facet of the present cube each with

probability 1/(2n), where n is the dimension of the

space.

- if the cube across the chosen facet intersects K,

the random walk moves to that cube

- else, it stays in the present cube

….

….

…

j

m

i

n

k

Prob:

i j : ¼

i n : ¼

i k : ¼

i m :0

ii : ¼

Technical random walk

Why need technical random walk?

Only given K by an oracle.

How to decide whether

Cube ?

is smoother

K

Walk through

1.

is smoother than K

Prove rapidly mixing

Satisfy the constraint:

Random walk has ½ probability stay in the same cube.

Apply the theorem of Sinclair and Jerrum

2.

Technical random walk

- Q: We want to walk through . But we are only given K by an oracle, and this will not let us decide precisely whether a particular cube .

-modification

random walk is executed includes all of those cubes that intersect plus some other cubes each of which intersects , where .

Ellipsoid algorithm

offers a terminate condition

x

Terminates:

contains

C weakly intersects )

The walk will go to cube C

The walk will not go to cube C

Technical random walk

2nd modification made on natural random walk

….

…

New rules:

j

m

i

n

1. The walk has ½ probability stays in the

present cube

2. With probability 1/(4n) each, it picks one of

the facets to move across to an adjacent cube

k

Prob:

i j : 1/8

i n : 1/8

i k : 1/8

i m :0

i i : 5/8

In sum:

Discrete-time Markov Chain

A Markov Chain is a sequence of random variables

With Markov Property.

Markov Property:

The future states only depend on current state.

A simple two-state Markov Chain

Formally:

Technical random walk is a Markov Chain

Irreducible

A state j is said to be accessible from a state i if:

j

i

j is accessible from i

i is not accessible from j

A state i is said to communicate with state j

if they are mutually accessible.

j

i

A Markov chain is said to be irreducible if its state space is a single

communicating class.

The graph of random walk is connected

Markov chain for technical random walk is

irreducible

Periodicity vs. Aperiodic

A state i has period k if any return to state i

must occur in multiples of k.

i

j

If k=1, then the state is said to be aperiodic,

which means that returns to state i can occur

at irregular times.

i

j

A Markov chain is aperiodic if very state is aperiodic.

Each cube has a self loop

Markov chain for technical random walk is

aperiodic

Stationary distribution

The stationary distribution π is a vector, whose entries are non-negative and add up to 1. π is unchanged by the operation of transition matrix P on it, and is defined by:

Property of Markov chain:

If the Markov chain is irreducible and aperiodic, then there is a unique stationary distribution π .

Uniformly random generator

Markov chain for technical random walk has a

stationary distribution

Since P is symmetric for technical random walk, it is easy to see that all ’s are equal.

0.4

E.g,

i

j

0.6

0.6

0.4

Hoo Chin Hau

Overview

- Relate to
- Show that approximates within a certain bound with a probability of at least ¾

Probability of error of a single volume estimate

Based on Hoeffding’s inequality , we can relate the result of the algorithm () and p as follows:

: Number of successes

: Number of proper trials

Previously,

Nguyen Duy Anh Tuan

Recap Random walk – Markov chain

- A random walk is a process in which at every step we are at a node in an undirected graph and follow an outgoing edge chosen uniformly at random.
- A Markov chain is similar, except the outgoing edge is chosen according to an arbitrary distribution.

Ergodic Markov Chain

- A Markov chain is ergodic if it is:
- Irreducible, that is:
- Aperiodic, that is:

Markov Chain Steady-state

- Lemma:

Any finite, ergodic Markov chain converges to a unique stationary distribution π after an infinite number of steps, that is:

Markov Chain Mixing time

- Mixing time is the time a Markov chain takes to converge to its stationary distribution
- It is measured in terms of the total variation distance between the distribution at time s and the stationary distribution

Total variation distance

- Letting denotes the probability of going from i to j after s steps, the total variation distance at time s is:

Ω is the set of all states

Bounded Mixing Time

- Since it is not possible to obtain the stationary distribution by running infinite number of steps, a small value ε > 0 is introduced to relax the convergent condition.
- Hence, the mixing time τ(ε) is defined as:

Rapidly Mixing

- A Markov chain is rapidly mixing if the mixing time τ(ε) is O(poly(log(N/ε))) with N is the number of states.
- If N is exponential in problem size n, τ(ε) would be only O(poly(n)).

Rapidly Mixing

- In our case:
- n is the dimension of the convex body
- and the number of states would be (3r/δ)n (δis the size of the cube, r is the radius of the bound ball).

Rapidly Mixing

- If the value of τ is substituted to the inequality in Theorem 1 of the paper

Rapidly Mixing

- Then, we take the summation of all the states to calculate the total variation distance:

Applications

- Shows that P ≠ BPP relative to this oracle. This means the implementation of oracle cannot be in polynomial time. Further, its surprising since P=BPP is believed to be true.
- Technique can be used to integrate well behaved and bounded functions over a convex body.
- Improvements in running time of algorithm would require improvement in mixing time of random walk. This is useful because the random walk introduced in paper is frequently studied in literature.

Conclusion

- Lets revisit the algorithm, briefly.
- Given a well rounded figure K, we consider a series of rescaled figures, such that the ratio of volume for consecutive ones is a constant fraction.
- We perform a technical random walk on each figure, and look for the ‘success’, which gives us the ratio of volumes between consecutive figures to good approximation.
- We use it to obtain the volume of K, given that we know the volume of bounding sphere.
- Technical challenge is to prove convergence of markov process.

Improvements in Algorithm

- A novel technique of using Markov process to approximate the volume of a convex body.
- In current analysis, the diameter of random walk was O(n^4). So algorithm could not have been improved beyond O(n^8), without improving the diameter.
- Algorithm improved to O(n^7) by Lovasz and Simonovitz in “Random walks in a convex body and an improved volume algorithm”.
- Current algorithms reach up to O(n^4), as noted here.

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