On Rigid Matrices andU-Polynomials

Noga Alon, Gil Cohen

( )

( )

Matrix Rigidity

𝑛−𝑘 (𝑛−𝑘 )2

log𝑛

Best explicit construction

𝑛2

𝑘log

𝑛𝑘

11

1

. . .

0

0( ) [Valiant77] A matrix is (k,d)-rigid

if decreasing its rank to k requires changing at least d entries in each row.on average, over the

rows.

Motivation from Algebraic Circuit Complexity

Given an matrix M, how hard is it to compute the linear transformation xMx ?

+

𝑥1𝑥2 𝑥𝑛⋯

+

++

+

Size , depth always suffices.

Size is typically necessary.

Open Problem. Efficiently construct amatrix M that cannot be computed bylinear-size circuits.

Even Easier. and logarithmic-depthsimultaneously.

Motivation from Algebraic Circuit Complexity

Theorem [Valiant77]. If M is -rigid, then any logarithmic-depth circuit for computing M has size .

Fact. There exist -rigid matrices.

Attracted a lot of attention [Friedman93, Lokam95, Shokrollahi- SpielmanStemann97, KashinRazborov98, Lokam06, DeWolf06, AlonPanigrahy- Yekhanin09,

KumarLokamPatankarSarma09, Dvir10, ServedioViola12] and many more related papers.Barrier. Still stuck at .

Set Rigidity[AlonPanigrahyYekhanin09] A set is (k,d)-rigid if for every dimension k subspace U, .

Observation. (k,d)-rigid set .

Pre-[AlonPanigrahyYekhanin09] research focused on matrices, and explored tradeoffs between k,d.

[AlonPanigrahyYekhanin09] fix k=n/2 and try to get the set size m=m(n,d) as small as possible.

Holy Grail. m=O(n)+poly(d).

[AlonPanigrahyYekhanin09] m=nexp(d).

Sounds like a job for a

pseudorandomnist!

Contributions of this Work

Expanders

Codes

Small-Bias Sets

Samplers

Seeded Extractors

Hash Functions

2-Source Extractors

Rigid Sets

Contributions of this Work

Expanders

Codes

Small-Bias Sets

Samplers

Seeded Extractors

Hash Functions

2-Source Extractors

PRG for U-Polynomial

s

Rigid Sets

Contributions of this Work

Expanders

Codes

Small-Bias Sets

Samplers

Seeded Extractors

Hash Functions

2-Source Extractors

PRG for U-Polynomial

s

Rigid Sets

Contributions of this Work

Expanders

Codes

Small-Bias Sets

Samplers

Seeded Extractors

Hash Functions

2-Source Extractors

PRG for U-Polynomial

s

Rigid Sets

U-PolynomialsDefinition. For a subspace U and a constant define the polynomial

𝑝𝑈 (𝑥 )=∑𝑢∈𝑈

𝜌|𝑢|𝑥𝑢

Example.

()

Normalize by the weight enumerator .

Properties. • .• W.h.p. over x, .

Theorem 1

Theorem 1.For every U,x.

Claim. W.h.p, a random set of size O(n) has the following property:

.

Fooling Polynomials• Degree 1 (Small-Bias Sets) [NaorNaor92,

AlonGoldreichHastad- Peralta92, AlonBruckNaorNaorRoth92,

BenAroyaTaShma09].

• Degree d [LubyVelickovicWigderson93, Bogdanov05,

BogdanovViola07, Lovett08, Viola09].

• Sparse Polynomials [LubyVelickovicWigderson93, Viola06,

Agrawal- Bhowmick10].

• U-Polynomials [ThisWork12, YourWork13, … ].

𝑑𝑖𝑠𝑡 (𝑥 ,𝑈 )=min𝑢∈𝑈

|𝑥+𝑢|

Min-distance is hard - shift to energy!

Proof Idea

𝑑𝑖𝑠𝑡 (𝑥 ,𝑈 )=min𝑢∈𝑈

|𝑥+𝑢|

Min-distance is hard - shift to energy!

𝑒𝑛𝑒𝑟𝑔 𝑦𝑈 (𝑥 )=∑𝑢∈𝑈

𝜌|𝑢+𝑥|𝑑𝑖𝑠𝑡 (𝑥 ,𝑈 )=min𝑢∈𝑈

|𝑥+𝑢|

Proof Idea

Step 1) Small implies large (depending on the “density” of U).

Step 2) Observe that is an application of the Fourier noise operator on U’s indicator.

Step 3) Compute the above operator and show it is related to (using MacWilliams Indentity).

𝑒𝑛𝑒𝑟𝑔 𝑦𝑈 (𝑥 )=∑𝑢∈𝑈

𝜌|𝑢+𝑥|

Proof Idea

Rigid Sets from Small-Bias Sets

Expanders

Codes

Small-Bias Sets

Samplers

Seeded Extractors

Hash Functions

2-Source Extractors

PRG for U-Polynomial

s

Rigid Sets

Small-Bias SetsDefinition [NaorNaor92]. A sample space is called -biased if

.

Close-to-optimal explicit constructions (poly in ) [NaorNaor92, AlonGoldreichHastadPeralta92,

AlonBruckNaorNaorRoth92, BenAroyaTaShma09].

Many applications. [NaorNaor92, AlonRoichman94, BenSassonSudan- VadhanWigderson03, Raz05, ViolaWigderson08, Viola09, ArvindSrinivasan10, DeEtesamiTrevisanTulsiani10,

GopalanMekaReingoldTrevisanVadhan12, JafargholiViola13].

(Large..) Small-Bias Sets are Rigid

Idea. Shift from distance to membership.

Theorem 2. Every -biased set is -rigid.

Idea. Shift from distance to membership.

Theorem 2. Every -biased set is -rigid.

(Large..) Small-Bias Sets are Rigid

Lemma [AlonPanigrahyYekhanin09]. The d-neighborhood of any subspace with dimension n/2 can be covered by exp(d) subspaces of dimension 3n/4.

(Large..) Small-Bias Sets are Rigid

||𝑆∩𝑈||𝑆|

−|𝑈|2𝑛 |≤𝜖

A similar lemma was proven (differently) in [ArvindSrinivasan10].

Definition [NaorNaor92]. A sample space is called -biased if

Equivalently. A sample space is called -biased if for any subspace U with co-dim 1

Lemma.

.

Theorem [AlonChung88]. Let G be a d-regular on N vertices with spectral gap . Then, for any subset U of the vertices,

.

Theorem [AlonRoichman94]. Let be an -biased set. Define the graph G on with an edge {u,v} iff . Then, G has spectral gap .

A Proof of the Lemma

𝜖|𝑆||𝑈||𝑆|

Proof. The degree of u in the graph induced by U is

.

Thus, .

Apply [AlonChung88] + [AlonRoichman94].

A Proof of the LemmaProof.

Rigid Sets fromUnbalanced Expanders

Expanders

Codes

Small-Bias Sets

Samplers

Seeded Extractors

Hash Functions

2-Source Extractors

PRG for U-Polynomial

s

Rigid Sets

Unbalanced Expanders

𝑚 𝑛𝑑

𝑠≤ 𝑡 ≥ (2/3 )𝑑𝑠

unique neighbors

Best construction [GuruswamiUmansVadhan06] is close to optimal. Many applications [Upfal-Wigderson87, BenSassonWigderson01, AlekhnovichRazborov01, BuhrmanMiltersenRadhakrishnanVenkatesh02, Alekhnovich- BenSassonRazborovWigderson04, GuruswamiLeeRazborov07, BenAroyaCohen12].

Rigid Sets from Unbalanced Expanders

iff .𝑚 𝑛10

0

v 𝐶={𝑐𝑣 :𝑣∈ [𝑚 ]}Theorem 3. If then C is (k,d/4)-rigid.

Using an optimal unbalanced expander yields a (k,d)-rigid set with size .

Rigid Sets from Unbalanced Expanders

c1

c2

c𝑚c3

u1u2

u𝑚u3

𝑑/ 4

13𝑑⋅2≤|𝑐 𝑖+𝑐 𝑗|≤|𝑐 𝑖+𝑢𝑖|+|𝑐 𝑗+𝑢 𝑗|+|𝑢𝑖+𝑢 𝑗|≤2⋅

𝑑4

+0

has size m.

Rigid Sets from Unbalanced Expanders

c1

c2

c𝑚c3

u1u2

u𝑚u3

𝑑/ 4

13𝑑⋅ 4≤|𝑐 𝑖1

+⋯+𝑐𝑖 4|≤|𝑐 𝑖1+𝑢𝑖 1|+⋯+|𝑐𝑖 4+𝑢𝑖 4|+|𝑢𝑖1

+⋯+𝑢𝑖4|≤4 ⋅𝑑4

+0

has size m.

What about ?

|𝑈 2|=(𝑚2 )

Rigid Sets from Unbalanced Expanders

c1

c2

c𝑚c3

u1u2

u𝑚u3

𝑑/ 4

|𝑈 1∪⋯∪𝑈𝑡 /2|=∑𝑖=0

𝑡 /2

(𝑚𝑖 )>2𝑘=|𝑈|

has size m.

What about ?

|𝑈 2|=(𝑚2 )|𝑈 𝑡 /2|=( 𝑚𝑡 /2)

.

..

𝑈 𝑖∩𝑈 𝑗=∅

The Remote Set Problem [AlonPanigrahyYekhanin09, ArvindSrinivasan10]

Is the rigidity problem easier given a (basis for a) subspace?

Not really! A log(n) barrier.

Related to the Nearest Codeword Problem [BermanKarpinski02, FeigeMicciancio02, Alekhnovich03,

AroraBabaiStern- Sweedyk03, AlonPanigrahyYekhanin09].

Open Problems1) Construct a rigid set with size O(n)+poly(d).

2) Even with size nexp(o(d)).

3)

4) A better algorithm for the Remote Set Problem.

5) Is it at all easier than rigidity?

6) New approach for linear circuit lower bounds.

Thank youfor your attention!