Locally Testable Codes and Expanders - PowerPoint Presentation

Slide1

Locally Testable Codes and Expanders

Tali Kaufman

Joint work with Irit DinurSlide2

Codes as CSPs

A constraint satisfaction problem (CSP) is a sequence of constraints over variables:

f

1

(x

1

,x

2

,x

5

), f

2

(x

5

,x

1

,x

15

), …

For example, max-3SAT, max-3col, …

CSP describes a code if every two satisfying assignments are far apart (in Hamming distance). The minimum distance between a pair of sat assignments is the

distance

of the code.

Today : CSPs that yield codes.

Challenge: CSPs that yield large and robust codes.Slide3

Robust Codes

CSP describes a code if every two satisfying assignments are far apart (in Hamming distance).

CSP describes a

robust code

if a vector that is far from the code falsifies many constraints.

Robust codes are called LTCs (strong relation to PCP).

Want to understand: What makes a code robust? Slide4

Structure

One can associate a “constraint hyper-graph” with a CSP by placing a vertex per variable, and a hyper-edge per clause.

Two “separate” aspects of a CSP are

The graph structure.

The constraint type : what functions f

i

do we allow.

Today we focus on the structure.Slide5

Constraint (Hyper graph)

f

1

(x

1

,x

2

,x

3

), f2 (x3,x4,x5), …

x

1

x

3

x

4

x5

x2

x

6Slide6

Constraint graph

f

1

(x

1

,x

2

,x

3

), f2 (x3,x4,x5), …

We replace every hyper-edge with a clique between its vertices, so that we deal with graphs instead of hyper-graphs.

x

1

x

3

x

4

x5

x

2

x

6

x

1

x

3

x

4

x

5

x

2

x

6Slide7

Locally Testable Codes (LTCs)

An LTC is given by a constraint graph G on n vertices.

The valid codewords C(G)

⊆{0,1}

n

are all assignments to [n] that satisfy all constraints.

Two required properties of the constraint graph:

1. Distance

: Every two codewords differ on

Ω

(n) coordinates. The minimum distance is the code distance.2. Robustness: Assignment that is γ-far (in Hamming distance) from the code, falsifies at least ργ fraction of the constraints (for some fixed

ρ > 0 ).Slide8

Our question(s) for today

What is the relation between the structure of the constraint graph and the code obtained.

What graphs lead to codes?

[

Sipser-Spielman

]

What graphs lead to robust codes (LTCs)?Slide9

Motivation

How can we construct LTCs?

Understand whether good LTCs exist.

Construct new, simpler LTCs, PCPs.

Understand the structure of general CSPs; CSPs that are hard to approximate. Slide10

What graphs lead to codes

[

Sipser-Spielman

]: Expanders!

Expanders are graphs without small cuts.

G=(V,E) is an r-expander if for every

S

⊆

V, |S|

≤ |V|/2 E(S,V-S) > r |S|

SV-S

In good expanders small setsexpands even more.Slide11

Expanders lead to codes - The Idea

Constraint graph: good expander.

Constraints: local view; all zeros, or many non zeros.

Satisfying assignment must have a distance.

x

4

x

5

x

9

x

7x2

x

15

x1

x

3

x10

x

6

x

8

x

12

x

11

x

14

x

13

Non zero variables

zero neighborsSlide12

Satisfying assignment can not have low support:

- Small set of non-zeros: by expansion its zero neighborhood is large.

- Contradicts the requirement that each local constraint assigned many non zeros.

x

4

x

5

x

9

x

7

x2

x

15

x1

x3

x

10

x

6

x

8

x

12

x

11

x

14

x

13

Non zero variables

zero neighbors

Expanders lead to codes - The Idea

Other issue:

How to ensure large code?Slide13

What graphs lead to robust

codes?

[

Ben-Sasson, Harsha, Raskhodnikova

]: Very good expanders yield codes that are NOT robust/LTC.

Idea:

Remove one constraint from the constraint graph.

The resulted graph is still an expander.

All satisfying assignments of the new graph are far apart. Some sat assignments of the new graph do not sat the original graph.Conclusion: there is a vector far from code that falsifies only one constraint from the constraint graph.Slide14

What graphs lead to robust

codes?

[

BHR

]: Very good expanders yield codes that are not robust/LTC.

There is a vector far from code that falsifies only one constraint from the constraint graph.

But: Maybe can test this code using *other* short constraint. I.e. constraints that are

implied

by original constraints.

Main point: Due to the high expansion, there are no short constraints that are implied by the original short constraints.Example: f1 (x1,x2,x3

): x1 + x2 + x3 = 0f2 (x4,x5,x6): x4 + x5 + x6 = 0

f3

(x7,x8,x9): x7 + x

8 + x9 = 0f4 (x3,x

6,x9): x3 + x6 + x

9 = 0f5 (x2,x5,x

8): x2 + x5 + x8 = 0

Imply:

f

6

(x

1

,x

4

,x

7

): x

1

+ x

4

+ x7 = 0Slide15

Our Conclusions so far

High expansion of the constraint graph implies code (distance).

High expansion implies NON robustness.

What can we say about the structure of the constraint graphs of LTCs?Slide16

Structure theorem for LTCs

C is

ρ

-LTC : w

γ

-away from C rejected with probability >

ρ γ

Theorem [Dinur-K]

: The constraint graph of an LTC is a small set expander; all sets up to linear size expands. Theorem: In ρ-LTC with r. distance

Δ and with constraint graph G=(V,E), every S ⊆ V ; |S|≤ 0.75Δ|V| expands: E(S,V-S) > ρ

d/3

|S| d = |E|/|V|.Slide17

LTC decomposes on sparse cuts

C is

ρ

-LTC

: vector

γ

-away from C rejected with probability >

ρ γ

Decomposition on sparse cuts: Let |S| = γ|V|. If E(S,V-S) is a sparse cut, i.e., E(S,V-S) < ρ

d/3|S|= ργ/3|E| then C =~ γ/3 CS

x C

V-S = {xSyV-S

| xS 

C|S , yV-S

 C|V-S}

Note C

⊆

C

S

x C

V-S

Every c



C

S

x C

V-S

is

γ

/3-close to some codeword from C.

Main point: Such c can be rejected only by edges (constraints) that cross the cut (S,V-S) and since the cut is sparse, there are too few of those edges.Slide18

Sparse cuts occur only on large sets

C is

ρ

-LTC

: vector

γ

-away from C rejected with probability >

ρ γ

E(S,V-S) is a sparse cut: |S| = γ|V|= γ n , E(S,V-S) < ρd/3 |S|= ργ/3 |E|

C =~ γ/3 CS x CV-S

Pick a ≠ a’ in CS

with distance(a,a’) > |S|/3 = γ/3 n :

Consider aS bV-S 

C ; a’S bV-S  C

S x CV-S Distance of a’b from C is < γ

/3n since C=~ γ/3

C

S

x C

V-S

The closest codeword of C to

a’

S

b

V-S

is w

≠

a

S

b

V-S

By triangle-inequality distance (a’b,ab) = distance (a,a’) > (Δ - γ

/3) nIn particular |S|= γ n > (Δ

- γ/3) n ; |S| > 3/4 Δ

na’b

ab  C

> γ/3 n

w

 C

<

γ

/3

n by LTC

>

Δ

nSlide19

Structure theorem for LTCs

We have shown: The constraint graph of an LTC is a small set expander.

Our main theorem shows that the constraint graph can be decomposed into

constant-

many (regular) expanders with few edges in between.Slide20

Main Theorem [

Dinur-K

]

Let C be an LTC with constraint graph G=(V,E). Then V can be partitioned into V

1

,…V

t

, such that

C =~ C

1 x C2 x … x Ct and such that each G(Vi) is an expander, with few edges between expanders. Moreover, each Ci is an LTC.t is at most 1/r.distanceExpansion depends on testability paramsSlide21

Summary and Questions

Expansion of a constraint graph implies distance.

Very high expansion of the constraint graph implies non-robustness.

The constraint graph of a robust code (LTC) is a small set expander for linear size sets. It can be decomposed into constant many (regular) expanders with few edges in between.

Q: What “amount” of expansion implies LTC?

Q: What structure of the constraint graph implies LTC?

Recent work (

Dinur-K

, and

Ben-Sasson-Viderman) shows that the constraint graph cannot be dense for LTCs with good rate.