1. Sparse Kindler-Safra Theorem via agreement theorems
Prahladh Harsha
Tata Institute of Fundamental Research
joint work with Irit Dinur and Yuval Filmus

2. Main Contributions:
Result: Structure theorem for low-degree polynomials on biased cube
Kindler-Safra”-type structure theorem for p-biased hypercube
p --- very small, sub-constant, possibly even 𝑝= 𝑂(1) 𝑛
Proof Paradigm:
Application of (high-dimensional) agreement theorems to proving structure theorems for p-biased hypercube
High-dimensional agreement theorem – generalization of direct product testing to larger dimensions

3. Boolean functions in the (standard) hypercube
Can be viewed as a real-valued function
𝑓: 0,1 𝑛 →ℝ
The space of such functions is spanned by 𝜒 𝑆 𝑆⊂ 𝑛 ,
𝑓= 𝑆 𝑓(𝑆) 𝜒 𝑆
f has degree ≤𝑑 iff 𝑓(𝑆) =0 ∀ 𝑆 >𝑑
Basic Junta theorem: If 𝑓: 0,1 𝑛 →{0,1} has degree ≤𝑑
[Nisan-Szegedy ‘94] ⟹ it depends on 𝑂 𝑑 (1) variables (= it is a junta)
Boolean funciton – basic object in CS, and in complexity we want to understand how different measures relate to each other.
Natural approach - look at f as a real valued function

4. Structure theorems, inverse theorems
Structure theorem: if “property” then “structure” often an “inverse” of very easy statement robust = stability version of the theorem General question: when does robust version exist ?
robust
almost

5. Robust versions of junta theorem
What would be a robust version of the basic junta theorem?
[NS]: If 𝑓: 0,1 𝑛 →{0,1} has degree ≤𝑑 ⟹ it depends on 𝑂(1) variables
Even simpler: If 𝑓: 0,1 𝑛 →{0,1} has degree ≤1 ⟹ it is dictator/anti-dictator/constant
Put uniform measure on 0,1 𝑛 and talk about distance of f,g
𝑑𝑖𝑠𝑡 𝑓,𝑔 = 𝔼 𝑥∈ 0,1 𝑛 𝑓 𝑥 −𝑔 𝑥 2
𝜀-close to Boolean or to low degree or both ?
𝑓 Boolean and 𝜀-close to 𝑔 of deg⁡𝑑 ⇔
𝑔 has deg⁡𝑑 and 𝜀-close to 𝑓 Boolean
[Friedgut-Kalai-Naor]: If 𝑓: 0,1 𝑛 →ℝ has degree ≤1, and it is 𝜀-close to Boolean, then it is O(𝜀)-close to dictator/anti-dictator/constant
[Bourgain, Kindler-Safra]: If 𝑓: 0,1 𝑛 →ℝ has degree ≤ 𝑑, and it is 𝜀-close to Boolean, then it is O(𝜀)-close to junta

6. From Boolean to A-valued
The robust junta theorems hold because low degree functions are smooth, not spiky (technically, this is proven via hypercontractivity)
FKN: If 𝑓: 0,1 𝑛 → ℝ has degree ≤1, and it is 𝜀-close to Boolean, then it is O 𝜀 -close to a dictator
What if f attains 3 values and not only 2 ?
Example: 𝑓 𝑥 = 𝑥 1 + 𝑥 2 attains three values 0,1,2 yet is not a dictator
(but it is still a junta)
Theorem “A-valued robust junta theorem”: If 𝑓: 0,1 𝑛 →ℝ has degree ≤𝑑, and it is 𝜀−close to A- valued, then it is O(𝜀)−close to a junta.
(𝐴⊂ℝ a finite set)
Observe that if 𝑔 is a junta, then 𝑔 is A’-valued for some other finite A’
Explain why FKN is true, too many coefs mean that the function is like a Gaussian

7. From Boolean to A-valued
Alternative interpretation: Assume 𝑓: 0,1 𝑛 →ℝ has degree ≤ 𝑑. If 𝑓 is 𝜀-close to A-valued, then 𝑓 is O(𝜀)-close to A’-valued (𝐴′⊂ℝ a finite set)
Parseval’s inequality implies
𝑓 is O(𝜀)-close to A’-valued ⟹ 𝑓 is a junta
Explain why FKN is true, too many coefs mean that the function is like a Gaussian

8. p-biased hypercube
𝜇 𝑝 - product distribution, each bit is 1 independently with probability p
𝜇 𝑝 𝑥 := 𝑝 𝑖 𝑥 𝑖 1−𝑝 𝑛− 𝑖 𝑥 𝑖
Measure concentrates on 𝑥’s with ≈𝑝𝑛 1’s
Studied in various contexts-
Graph properties: sharp threshold phenomena in G(n,p)
Reed Muller decoding from erasures
Hardness of approximation

9. p-biased: Sharp thresholds
A graph on n vertices can be represented as a string 𝑥∈ 0,1 𝑁 where 𝑁= 𝑛 2
A graph property is a function 𝑓: 0,1 𝑛 2 →{0,1}
Example: “the graph is connected”, “the graph has a triangle”
Studying a graph property in G(n,p) is like studying f in the p- biased hypercube
Friedgut-Kalai : all monotone graph properties have a narrow threshold
Friedgut: k-sat has a sharp threshold
Observe: 𝑝 here is very small, e.g. 1/ 𝑁 𝑐 for some constant c
Removed:Friedgut’s Conjecture: monotone functions with low 𝜇 𝑝 -influence must have certain structure

10. p-biased: decoding from erasures
A recent result [KKMPSU’16] showed that Reed-Muller codes with constant rate achieve capacity for decoding from erasures.
Key component = a structure theorem for monotone Boolean functions
A different set of works [ASW’15, SSV’16] showed the same for non- constant rates, using very different ideas.
For all in-between rates – we do not know. To extend further one needs perhaps a better grasp on smaller 𝑝 behavior.

11. p-biased: Hardness of approximation
The Boolean hypercube stars as the long-code gadget in many inapprox reductions
p-biased version is used in hardness of vertex cover, but p=constant
Recent works [KMS,DKKMS] use “the Grassmann graph” and introduce structural conjectures about functions on its vertices. This is also related to the “short code graph” [BKS].
The relevant parameters for the conjectures are analogous to 𝜇 𝑝 for very small p, 𝑝= 𝑂(1) 𝑛 .
This was our motivation. In fact, our result if true for the grassmann would come close to proving the conjecture, but not quite…

12. (Nearly) Boolean low degree functions on the p-biased hypercube
Robust junta theorem applies also to 𝜇 𝑝
Error deteriorates as 𝑝→0 due to use of hypercontractivity
Desire better dependence on 𝜖 even when 𝑝=𝑜(1)
Prob[f=0] = 1- sqrt eps
Prob[f=2] = eps

13. (Nearly) Boolean low degree functions on the p-biased hypercube
Consider 𝑓 𝑥 = 𝑥 1 + 𝑥 2 +…+ 𝑥 𝑠 where 𝑥 𝑖 ∈ 0,1 and 𝑠= 𝜖 𝑝
𝑓 has degree 1, clearly it is not a junta
𝑃𝑟𝑜𝑏 𝑓=0 = 1−𝑝 𝑠 ≈ 𝑒 −√𝜖 ≈ 1− 𝜖
𝑃𝑟𝑜𝑏 𝑓≥2 ≈ 𝑠 2 𝑝 2 ≈𝜖
𝑓 is 𝜖-close to Boolean
𝑓 is √𝜖-close to 0, but we want a more refined approximation
The closest Boolean function is: 𝑔 𝑥 = max⁡(𝑥 1 , 𝑥 2 ,…, 𝑥 𝑠 )
𝑑𝑖𝑠𝑡 (𝑓,𝑔) =𝑂(𝜖)
Filmus’16: If h has degree 1 and 𝜇 𝑝 -close to Boolean, then it looks like 𝑓.
want: If h has degree ≤𝑑 and 𝜇 𝑝 -close to Boolean, then it looks like ???.
Let’s calculate…
Prob[f=0] = (1-p)^s = 1- sqrt eps
Prob[f=2] = s^2 p^2 = eps

14. Looking for structure…
Filmus’16: If h has degree 1 and 𝜇 𝑝 -close to Boolean, then it looks like 𝑓.
want: If h has degree ≤𝑑 and 𝜇 𝑝 -close to Boolean, then it looks like ???.
Naïve guess: maybe there are 𝜖 𝑝 coordinates that control the function?
No: 𝑓 = 𝑥 1 𝑥 2 + 𝑥 3 𝑥 4 + … 𝑥 𝑛−1 𝑥 𝑛 is nearly Boolean for p=O 1 𝑛
Note that a random 𝑥∼𝜇 𝑝 leaves O(1) monomials “alive”

15. The monomial expansion
Consider the multilinear expansion in {0,1} variables i.e.
𝑓 𝑥 = 𝑠 𝑓 𝑠 𝑦 𝑠 where 𝑥 𝑖 ∈ 0,1 and 𝑦 𝑠 = 𝑖∈𝑆 𝑥 𝑖
(do not confuse with the Fourier functions: 𝑥 𝑖 ∈{−1,1} and 𝜒 𝑆 = 𝑖∈𝑆 𝑥 𝑖 )
The monomial-expansion is unique, but the 𝑦 𝑆 functions are not orthogonal
Filmus’16: Let f be a degree 1 function. If f is close to {0,1}-valued, then 𝑓 is close to {-1,0,1}-valued
Definition: 𝑓 is a quantized polynomial if there is a finite set 𝐴 such
that 𝑓 𝑠 ∈𝐴 for every 𝑠⊆[𝑛].
Do not confuse this with

16. Quantized polynomials
Theorem: Let f be a function with degree ≤𝑑. If f is 𝜖-close under 𝜇 𝑝 to an A-valued function, then it is O 𝜖 -close to a quantized polynomial.
For p=1/2 this is the Kindler-Safra robust junta theorem
There’s more: a quantized polynomial q that is nearly Boolean (or A- valued) has further structure.

17. Sparse Juntas If a quantized polynomial is nearly Boolean –
It must be sparse
Even after conditioning on few 𝑥 𝑖 =1, it must still be sparse
Consider the hypergraph H on n vertices whose edges are the non- zero 𝑓 −coefficients
H has branching factor 𝑏 if for all subsets 𝐴⊂[𝑛] and integers 𝑟≥0, there are at most 𝑏 𝑟 hyperedges in H of cardinality |A|+ r containing A .
sparse junta = a quantized polynomial with branching factor 1/p.

18. Main Theorem: sparse Kindler-Safra Theorem
Theorem (main): Let 𝑓 be a function of degree ≤𝑑. If it is 𝜖-close under 𝜇 𝑝 to an A -valued function, then it is O(𝜖)-close to a sparse junta.
So 𝑓 is an “empirical” junta : after selecting x, the number of 𝑓 coefficients that stay “alive” is O(1)
[ compare to Hatami’s pseudo-juntas ]
Thm is tight :
{ nearly- sparse juntas } = { nearly- low degree & A-valued }

19. Proof Given 𝑓 of degree ≤𝑑, 𝜖−close to Boolean.
Earlier structure theorems rely on hyper-contractivity.
As 𝑝→0 hypercontractivity gets weaker and weaker
Instead, we will “divide and conquer” –
Divide: look at random restrictions of 𝑓 to small sub-cubes
Conquer: obtain approximate structure on each subcube
Reunite: recover a global structure

24. ”Divide”: Choose a random subset 𝑆⊂ 𝑛 according to 𝜇 2𝑝 , place zeros outside

25. ”Divide”: Choose a random subset 𝑆⊂ 𝑛 according to 𝜇 2𝑝 , place zeros outside

26. ”Divide”: Choose a random subset 𝑆⊂ 𝑛 according to 𝜇 2𝑝 , place zeros outside
Choose a uniform 𝑥∈ 0,1 𝑆

27. ”Divide”: Choose a random subset 𝑆⊂ 𝑛 according to 𝜇 2𝑝 , place zeros outside
Choose a uniform 𝑥∈ 0,1 𝑆
This describes 𝜇 𝑝 𝑛 as a convex combination of 𝜇 1/2 𝑚 where m is binomially distributed with mean 2pn.

28. The resulting string is distributed according to 𝜇 𝑝 𝑛
”Divide”: Choose a random subset 𝑆⊂ 𝑛 according to 𝜇 2𝑝 , place zeros outside
Choose a uniform 𝑥∈ 0,1 𝑆
This describes 𝜇 𝑝 𝑛 as a convex combination of 𝜇 1/2 𝑚 where m is binomially distributed with mean 2pn.
The resulting string is distributed according to 𝜇 𝑝 𝑛
This describes 𝜇 𝑝 𝑛 as a convex combination of 𝜇 1/2 𝑚 where m is binomially distributed with mean 2pn.

29. Proof outline ”Divide”: “Conquer” : “Reunite”: When can this work?
Let 𝑓 ​ 𝑆 be the function on 0,1 𝑆 obtained by restricting 𝑓 to inputs that are zero outside S
“Conquer” :
For typical 𝑆, 𝑓 ​ 𝑆 is close to Boolean, so we can apply “ 𝜇 junta theorem” of Kindler and Safra and get junta ℎ 𝑆 that approximates 𝑓 ​ 𝑆 .
“Reunite”:
Stitch ℎ 𝑆 together into one global function ℎ: 0,1 𝑛 →ℝ such that typically ℎ ​ 𝑆 = ℎ 𝑆
When can this work?
At the very least require local consistency, i.e, i.e. ℎ 𝑆 1 ​ 𝑆 1 ∩ 𝑆 2 = ℎ 𝑆 2 ​ 𝑆 1 ∩ 𝑆 2
But “Local consistency” ⇒ “Global Consistency” ???

30. Local to Global Agreement
Consider d=1 case
Each ℎ 𝑆 represents a local linear function ℎ 𝑠 :{0,1 } 𝑆 →{0,1}
Local Agreement:
Typically, ℎ 𝑆 1 ​ 𝑆 1 ∩ 𝑆 2 = ℎ 𝑆 2 ​ 𝑆 1 ∩ 𝑆 2
I.e, for most pairs 𝑆 1 and 𝑆 2 , the corresponding two linear functions ℎ 𝑆 1 and ℎ 𝑆 2 agree
Global Agreement:
Does there exist a ”global” linear function ℎ: {0,1 } 𝑛 →{0,1} such that for most 𝑆, we have ℎ ​ 𝑆 = ℎ 𝑆
Direct Product Testing [..., DS]: Local Agreement ⇒ Global Agreement
For larger d, need a high dimensional analogue of this direct product testing

31. High dimensional agreement theorem
General d
Each ℎ 𝑆 represents a degree d function ℎ 𝑠 :{0,1 } 𝑆 →{0,1} (or equivalently a labelled hypergraph with hyperedges of size at most d)
Local Agreement:
Typically, ℎ 𝑆 1 ​ 𝑆 1 ∩ 𝑆 2 = ℎ 𝑆 2 ​ 𝑆 1 ∩ 𝑆 2
I.e, Pr ℎ 𝑆 1 ​ 𝑆 1 ∩ 𝑆 2 = ℎ 𝑆 2 ​ 𝑆 1 ∩ 𝑆 2 ≥1−𝜖
Global Agreement:
Does there exist a ”global” degree d function ℎ: {0,1 } 𝑛 →{0,1} (or equivalently a global hypergraph) such that for most 𝑆, we have
Pr ℎ ​ 𝑆 = ℎ 𝑆 ≥1−𝑂(𝜖)
YES
Furthermore, this global ℎ can be obtained by majority/plurality decoding

32. Proof summary Given 𝑓 of degree ≤𝑑, 𝜖−close to Boolean.
For typical 𝑆, 𝑓 ​ 𝑆 is close to Boolean, so we can apply “ 𝜇 junta theorem” of Kindler and Safra and get a junta ℎ 𝑆 that approximates 𝑓 ​ 𝑆 .
Stitch the local juntas together to get a global function h (using the hypergraph agreement theorem)
Prove that h is close to a sparse junta .

33. Applications Tail bound for sparse juntas
(implies same for nearly low-degree&A-valued)
Sparse juntas must be very biased
(implies that nearly low-degree&A-valued functions must be very biased)
skip

34. Summarizing..
Theorem (main): Let 𝑓 be a function of degree ≤𝑑. If it is 𝜖-close under 𝜇 𝑝 to an A -valued function, then it is O(𝜖)-close to a sparse junta.
Proof via a local-to-global agreement theorem (generalization of direct product testing to larger dimensions)

35. Thank You