Affine Invariant LCCs and LTCs Sivakanth Gopi Joint work with Arnab - - PowerPoint PPT Presentation

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Affine Invariant LCCs and LTCs Sivakanth Gopi Joint work with Arnab Bhattacharya (Indian Institute of Science) Error Correcting Code : finite alphabet, : set of coordinates of size $ : set of all functions from

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SLIDE 1

Affine Invariant LCCs and LTCs

Sivakanth Gopi

Joint work with Arnab Bhattacharya (Indian Institute of Science)

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SLIDE 2

Error Correcting Code

  • Ξ£ : finite alphabet, Ξ§ : set of coordinates of size 𝑂
  • Ξ£$ : set of all functions from Ξ§ β†’ Ξ£
  • Hamming distance, Ξ” 𝑔, 𝑕 = Pr
  • ∈$ 𝑔 𝑦 β‰  𝑕 𝑦
  • 𝐷 βŠ‚ Ξ£$ : Error correcting code with minimum distance πœ€ if

Ξ” 𝑔, 𝑕 β‰₯ πœ€ for all 𝑔, 𝑕 ∈ 𝐷

Corruptions Corrector A Codeword 𝑔: Ξ§ β†’ Ξ£ Corrupted word 𝑔 6:Ξ§ β†’ Ξ£ Ξ” 𝑔, 𝑔 6 < πœ€ 2

What if I am interested in correcting only

  • ne coordinate of 𝑔

6?

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SLIDE 3

Locally Correctable Code (LCC)

  • Can correct any coordinate of a corrupted codeword by

querying only 𝑠 locations How do we know if Ξ” 𝑔 6,𝐷 ≀ πœ€, locally?

Corruptions Local Corrector A 𝑠 queries 𝑦 ∈ Ξ§ 𝑔 𝑦 w.h.p Codeword 𝑔: Ξ§ β†’ Ξ£ Corrupted word 𝑔 6:Ξ§ β†’ Ξ£ Ξ” 𝑔, 𝑔 6 ≀ πœ€

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SLIDE 4

Locally Testable Code (LTC)

  • Can test closeness to the code by querying only 𝑠 locations

Local Tester A 𝑠 queries

Accepts w.p > =

> if 𝑔

6 ∈ 𝐷

𝑔 6:Ξ§ β†’ Ξ£

Rejects w.p > =

> if Ξ” 𝑔

6,𝐷 > πœ€/4

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SLIDE 5

What’s known?

  • In this talk, constant query: 𝑠 = 𝑃(1), constant alphabet:

Ξ£ = O(1),

  • Let Ξ§ = N, the length of messages we can encode is

log|𝐷|

Bounds on log|𝐷| Lower Bound Upper Bound 2-query LCC log 𝑂 [Hadamard Code] 𝑃 log𝑂 [KdW04] 𝑠-query LCC (𝑠 β‰₯ 3) (log𝑂)MNO [Reed Muller Codes] 𝑂ONO/ M/P [KT00,KdW04,Woo07] 𝑠-query LTC (𝑠 β‰₯ 2) 𝑂/π‘žπ‘π‘šπ‘§π‘šπ‘π‘•(𝑂) [BS05,Din07] 𝑃(𝑂) [Trivial]

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SLIDE 6

Local codes from invariance

  • LCCs and LTCs need to satisfy many local constraints
  • Let 𝐻 be a group acting on Ξ§ and so 𝐻 also acts on

functions 𝑔: Ξ§ β†’ Ξ£ as 𝛿 𝑔 (𝑦) = 𝑔 ∘ 𝛿(𝑦)

  • Let code 𝐷 βŠ‚ Ξ£$ be invariant under this action i.e.
  • Local constraint on 𝑦O, β‹― , 𝑦M β‡’ Local constraint on

𝛿 𝑦O , β‹―, 𝛿 𝑦M for all 𝛿 ∈ 𝐻 βˆ€π‘” ∈ 𝐷, Ξ“ 𝑔 𝑦O ,β‹― , 𝑔 𝑦M = 1 βˆ€π‘” ∈ 𝐷, 𝛿 ∈ 𝐻: 𝑔 ∘ 𝛿 ∈ 𝐷 βˆ€π‘” ∈ 𝐷, 𝛿 ∈ 𝐻 Ξ“ 𝑔 𝛿(𝑦O) , β‹―, 𝑔 𝛿(𝑦M) = 1

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SLIDE 7

Affine invariant codes

  • Kaufman and Sudan in β€˜07
  • 𝔾:any finite field. Let Ξ§ = 𝔾] and let 𝐻 = Aff(n, 𝔾) be the

group of invertible affine maps from 𝔾] β†’ 𝔾]

  • A code 𝐷 βŠ‚ Σ𝔾a which is invariant under the action of

Aff(π‘œ, 𝔾) is called affine invariant i.e.

  • Example
  • Reed-Muller code of degree 𝑒: set of polynomial functions of

degree ≀ 𝑒 from 𝔾] β†’ 𝔾

  • If 𝑔 𝑦 is a degree ≀ 𝑒 polynomial and β„“ 𝑦 = 𝐡𝑦 + 𝑐, then

𝑔 β„“ 𝑦 is also a degree ≀ 𝑒 polynomial

  • Irreducible polynomials, products of two degree 𝑒 polynomials...

Can we construct good LCCs or LTCs using affine invariance? βˆ€π‘” ∈ 𝐷, βˆ€β„“ ∈ Aff(π‘œ, 𝔾), 𝑔 ∘ β„“ ∈ 𝐷

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SLIDE 8

Main Results

  • Achieved by Reed-Muller codes of degree 𝑠 βˆ’ 1
  • Achieved by Lifted Codes of [GKS’13]
  • [Ben-Sasson, Sudan β€˜11] proved the same bounds when Ξ£ is a

subfield of 𝔾 and 𝐷 is a linear code over Ξ£

Locally Correctable Codes

If 𝐷 βŠ‚ Σ𝔾a is an 𝑠-query affine invariant LCC then log 𝐷 ≀ 𝑃M, 𝔾 , i (π‘œMNO) (Note that π‘œ = log|𝔾| 𝑂, where 𝑂 is length of the code)

Locally Testable Codes

If 𝐷 βŠ‚ Σ𝔾a is an 𝑠-query affine invariant LTC then log 𝐷 ≀ 𝑃M, 𝔾 , i (π‘œMNP)

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SLIDE 9

Higher Order Fourier Analysis

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SLIDE 10

Gowers uniformity norms

  • Define multiplicative derivative of 𝑔: 𝔾j

] β†’ β„‚ as

  • Gowers uniformity norm of order 𝑒 + 1 of 𝑔: 𝔾j

] β†’ β„‚

  • If 𝑔 𝑦 = πœ• m - where πœ•: π‘žno root of unity and 𝑕: 𝔾j

] β†’ 𝔾j is

a degree 𝑒 polynomial then

  • Inverse Gowers theorem [Tao, Ziegler ’11]: (π‘ž > 𝑒)

If | 𝑔 |pqrs = Ξ© 1 then 𝑔 is correlated with the phase of a degree 𝑒 polynomial

  • For π‘ž ≀ 𝑒, we get non-classical polynomials

| 𝑔 |pqrs = 𝔽-,vs,β‹―,vqrsβˆˆπ”Ύa Ξ”vs β‹― Ξ”vqrs𝑔 𝑦

O/Pw

Ξ”v𝑔 𝑦 = 𝑔 𝑦 + β„Ž 𝑔 𝑦 | 𝑔 |pqrs = 𝔽-,vs,β‹―,vqrsβˆˆπ”Ύa πœ•yzsβ‹―yzqrsm -

O/Pw

= 1

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SLIDE 11

Von Neumann inequality

  • If | 𝑔 |p{ β‰ͺ 1, then cannot find 𝑔 at β„“ 𝑦} from the values
  • f 𝑕 at β„“ 𝑦O ,β‹― β„“ 𝑦M for a random β„“ ∈~ Aff(𝔾j, π‘œ)
  • Proof: expand Ξ“ in Fourier basis, make linear change of

variables to get expressions like

  • and repeatedly apply Cauchy-Schwarz inequality

𝔽ℓ 𝑔 ∘ β„“ 𝑦} Ξ“ 𝑕 ∘ β„“ 𝑦O , β‹― , 𝑕 ∘ β„“ 𝑦M ≀ 2M| 𝑔 |p{ 𝔽‒s,β‹―,β€’{ 𝑔 βˆ‘π‘¨β€š 𝑕

O Ζ’ βˆ’π‘¨O + βˆ‘π‘¨β€š ⋯𝑕M Ζ’ βˆ’π‘¨M + βˆ‘π‘¨β€š

𝔽‒s,β‹―,β€’{ 𝑔 βˆ‘π‘¨β€š 𝑕

O Ζ’ βˆ’π‘¨O + βˆ‘π‘¨β€š ⋯𝑕M Ζ’ βˆ’π‘¨M + βˆ‘π‘¨β€š

≀ | 𝑔 |p{

π‘•β€š

Ζ’ doesn’t

depend on π‘¨β€š

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SLIDE 12

Proof sketch for LCCs

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SLIDE 13

Some simplifications

  • Assume Ξ£ = βˆ’1,1 , 𝔾 = 𝔾j for some prime π‘ž > 𝑠
  • Assume perfect recovery for codewords

Local Corrector A 𝑦O,𝑦P,β‹―, 𝑦M ∼ β„³- 𝑦 ∈ Ξ§ Ξ“-s,β‹―,-{ 𝑔 𝑦O ,𝑔 𝑦P ,β‹― , 𝑔 𝑦M = 𝑔(𝑦) Codeword word 𝑔: Ξ§ β†’ Ξ£ 𝑦O 𝑦P 𝑦M

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SLIDE 14

Proof Sketch

  • Step 1: Show that any two distinct codewords 𝑔, 𝑕 ∈ 𝐷 must

be 2πœ—-far in 𝑉M-norm i.e. |𝑔 βˆ’ 𝑕| p{ > 2πœ— (von Neumann inequality)

  • Step 2: Construct a small πœ—-net π’ͺ for the set of all functions

in 𝑉M-norm (Inverse Gowers theorem)

  • π’ͺ = {red points}, 𝐷 ={green points},

two green dots cannot fall in the same ball!

  • 𝐷 ≀ π’ͺ

πœ— > 2πœ—

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SLIDE 15

Proof of Step 1

  • Intuitively, if |𝑔 βˆ’ 𝑕| p{ < 2πœ— then the local corrector

cannot distinguish between 𝑔 ∘ β„“,𝑕 ∘ β„“ for a random β„“ ∈ Aff π‘œ, 𝔾j

  • But 𝑔 ∘ β„“, 𝑕 ∘ β„“ are valid codewords by invariance and the

corrector should distinguish them – Contradiction!

Local Corrector A 𝑦O,𝑦P,β‹―, 𝑦M ∼ β„³- 𝑦 ∈ Ξ§ Ξ“-s,β‹―,-{ 𝑔 ∘ β„“ 𝑦O , β‹―, 𝑔 ∘ β„“ 𝑦M Codeword word 𝑔 ∘ β„“: Ξ§ β†’ Ξ£ 𝑦O 𝑦P 𝑦M 𝑔 ∘ β„“(𝑦) 𝑕 ∘ β„“(𝑦) ? ?

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Proof of Step 1

  • Pr
  • [Aβ€˜βˆ˜β„“ outputs 𝑔 ∘ β„“(𝑦)] – Pr
  • [Aβ€˜βˆ˜β„“outputs 𝑕 ∘ β„“(𝑦)]
  • = 1 βˆ’ Pr
  • [𝑔 ∘ β„“ 𝑦 = 𝑕 ∘ β„“(𝑦)]
  • = Ξ” 𝑔,𝑕 β‰₯ 𝑒𝑗𝑑𝑒(𝐷)
  • O

P 𝔽ℓ Pr

  • [Aβ€˜βˆ˜β„“ outputs 𝑔 ∘ β„“ 𝑦 ] βˆ’ Pr
  • [Aβ€˜βˆ˜β„“outputs 𝑕 ∘ β„“ 𝑦 ]
  • 𝔽ℓ 𝔽-𝔽-s,β‹―,-{βˆΌβ„³Λœ 𝑔 ∘ β„“ 𝑦 βˆ’ 𝑕 ∘ β„“ 𝑦

Ξ“

  • s,β‹―,-{ 𝑔 ∘ β„“ 𝑦O ,β‹― ,𝑔 ∘ β„“ 𝑦M
  • 𝔽-𝔽-s,β‹―,-{βˆΌβ„³Λœ 𝔽ℓ

𝑔 ∘ β„“ 𝑦 βˆ’ 𝑕 ∘ β„“ 𝑦 Ξ“

  • s,β‹―,-{ 𝑔 ∘ β„“ 𝑦O ,β‹― ,𝑔 ∘ β„“ 𝑦M
  • Therefore 𝑔 βˆ’ 𝑕

p{ β‰₯ 2 β„’β€šΕ‘β€Ί Ε“ P{

= 2πœ—

≀ 2M 𝑔 βˆ’ 𝑕

p{ (von Neumann inequality)

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SLIDE 17

Proof of Step 2 (small πœ—-net)

  • Decomposition theorem (Green, Tao, Ziegler’11)

βˆ€πœ—, 𝑠 βˆƒπ‘™(πœ—, 𝑠) such that: any β„Ž: 𝔾j

] β†’ βˆ’1,1 can be

πœ—-approximated by a function of 𝑙 degree 𝑠 βˆ’ 1 polynomials in 𝑉M- norm

  • A degree 𝑠 βˆ’ 1 polynomial has π‘œMNO coefficients
  • Gives an epsilon-net of size π’ͺ = exp 𝑃j,M π‘œMNO
  • Thus 𝐷 ≀ π’ͺ = exp 𝑃j,M π‘œMNO

QED!

β„Ž βˆ’ Ξ“ π‘žO, β‹―, π‘ž

p{ < πœ—

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SLIDE 18

Open Questions

  • We show β€œtight” bounds on the size of affine invariant

constant query LCCs and LTCs

  • Improve the dependence on 𝑠, 𝔾 , |Ξ£|
  • Can we prove similar bounds for a more general class of

codes? Codes invariant under some group action and some additional properties?

  • Can we use sparse hypergraph regularity lemmas to

understand the hypergraph structure of local codes?

ARIGATO GOZAIMASU!