1. Joint work with Avishay Tal (IAS) and Jiapeng Zhang (UCSD)
Robust Sensitivity
Shachar Lovett (UCSD)
Joint work with Avishay Tal (IAS) and Jiapeng Zhang (UCSD)

2. Overview Sensitivity conjecture
Robust sensitivity conjecture / theorem
Proof: combinatorics in the hypercube
Conclusions and open problems

3. The sensitivity conjecture

4. Sensitivity conjecture
There has been much work in the last 20 years classifying properties of Boolean functions
It is now known that many properties are equivalent, up to polynomial factors. For example: degree, approximate degree, certificate size, block sensitivity, decision tree complexity…
One property stands out: sensitivity
Sensitivity conjecture: sensitivity is also equivalent to all the rest

5. Sensitivity: definition
Let 𝑓: 0,1 𝑛 →{−1,1} be a Boolean function throughout this talk
(think of it as a 2-coloring of the Boolean hypercube)
Sensitivity at node 𝑥∈ 0,1 𝑛
= # neighbors of x where f assigns a different value
= # edges which touch x crossing the cut defined by f
𝑠 𝑓,𝑥 =|{𝑖∈ 𝑛 :𝑓 𝑥 ≠𝑓 𝑥+ 𝑒 𝑖 }

6. Sensitivity: example
𝑠 𝑓,𝑥 =|{𝑖∈ 𝑛 :𝑓 𝑥 ≠𝑓 𝑥+ 𝑒 𝑖 } 2 3 2 3 1 1

7. Sensitivity conjecture
Functions with low max sensitivity must be “simple”
Max sensitivity of f: 𝑠 𝑚𝑎𝑥 𝑓 = max 𝑥 𝑠(𝑓,𝑥)
Sensitivity conjecture [Nisan’ 91]: if 𝑠 𝑚𝑎𝑥 𝑓 =𝑆 then f can be computed by a polynomial of degree poly(S)
All known upper bounds: deg≤ 2 𝑂 𝑆
[Simon’83, Kenyon-Kutin’04, Ambainis-Bavarian-Gao-Mao-Sun-Zuo ’14]

8. Sensitivity conjecture
Another viewpoint: how can we construct functions with maximal sensitivity S?
Example 1: function depends only on first S inputs
Example 2: same, but adaptively (i.e. a decision tree of depth S)
Sensitivity conjecture: this is it

9. Sensitivity conjecture: Graph theory
Sensitivity conjecture, equivalent version 1 [Gotsman and Linial ’92]
Let 𝐴⊂ 0,1 𝑛 be a set of size 𝐴 ≠ 2 𝑛−1
𝐺[𝐴] = sub-graph of Boolean hypercube induced by A
Conjecture: 𝐺[𝐴] or 𝐺[ 𝐴 𝑐 ] contain a node of degree poly(n)
False if 𝐴 = 2 𝑛−1 . Example: A={nodes with even hamming weight}

10. Sensitivity conjecture: Fourier analysis
Sensitivity conjecture, equivalent version 2
Consider the Fourier decomposition of f:
𝑓 𝑥 = 𝑇∈ 0,1 𝑛 𝑓 𝑇 −1 <𝑇,𝑥>
Conjecture: if 𝑠 𝑚𝑎𝑥 𝑓 =𝑆 then f is computed by a real
polynomial of degree poly(S)
Equivalently: 𝑓 𝑇 =0 whenever 𝑇 ≥𝑝𝑜𝑙𝑦 𝑆

11. Robust sensitivity conjecture

12. Relaxed versions
As it seems that we are stuck on the original sensitivity conjecture, it makes sense to relax it
Recently, a couple of works explored this direction
[Gopalan-Nisan-Servedio-Talwar-Wigderson ’15]:
Functions with bounded max sensitivity are simple in other ways
(for example: max sensitivity S implies circuit size poly(n 𝑆 ) )
[Gopalan-Servedio-Tal-Wigderson ‘16]:
Functions with bounded max sensitivity can be approximated
by low degree polynomials

13. Approximation by polynomials
Assume f has bounded max sensitivity
Sensitivity conjecture: f can be computed by a low degree polynomial
[GSTW]: f can be approximated by a low degree polynomial
Can equivalently be viewed as fast decay of Fourier coefficients of f

14. Approximation by polynomials
Let 𝑠 𝑚𝑎𝑥 𝑓 =𝑆
f is Boolean: Parseval identity gives 𝑇 𝑓 (𝑇) 2 =1
[GSTW]: f approx. within error 𝜖 (in ℓ 2 ) by a polynomial of degree
𝑂 𝑆 log 1 𝜖
Equivalently: Fourier tail decays exponentially fast. For any 𝜖>0:
𝑇 >𝑂 𝑆 log 1 𝜖 𝑓 𝑇 2 ≤𝜖

15. Fourier distribution
Recall that if f is Boolean then by Parseval: 𝑇 𝑓 (𝑇) 2 =1
Can view this as a Fourier distribution induced by f:
Sample T with probability 𝑓 (𝑇) 2 , output |T|
Denote it by 𝐹𝐷=𝐹𝐷 𝑓 . It outputs an integer in 0,…,n
[GSTW]: If 𝑠 𝑚𝑎𝑥 𝑓 =𝑆 then Pr 𝑡∼𝐹𝐷 𝑡>𝑂 𝑆 log 1 𝜖 <𝜖

16. Sensitivity distribution
[GSTW] suggested to also look on distribution of the sensitivities
Sensitivity distribution induced by f:
Sample 𝑥∈ 0,1 𝑛 uniformly, output 𝑠 𝑓,𝑥
Denote is 𝑆𝐷=𝑆𝐷 𝑓 . It also outputs an integer in 0,…,n
Conjecture [GSTW]: If most nodes have low sensitivity, then
most Fourier coefficients are supported on low hamming weight
That is: the sensitivity distribution controls the Fourier distribution

17. Robust sensitivity conjecture
Boolean function 𝑓: 0,1 𝑛 → −1,1
It defines two distributions on 0,…,n:
Fourier distribution FD: sample T with probability 𝑓 𝑇 2 . Output |T|.
Sensitivity distribution SD: sample 𝑥∈ 0,1 𝑛 . Output 𝑠(𝑓,𝑥).
Conjecture [GSTW]: For any 𝑑≥1 there exists a constant 𝑎 𝑑 (independent of n) such that
𝑇 𝑓 𝑇 2 𝑇 𝑑 ≤ 𝑎 𝑑 𝐸 𝑥 𝑠 𝑓,𝑥 𝑑
Equivalently: the d-th moment of SD control the d-th moment of FD
[GSTW]: the ∞-th moment of SD control the d-th moment of FD

18. Robust sensitivity conjecture
Conjecture [GSTW]: Exists constant 𝑎 𝑑 such that
𝑇 𝑓 𝑇 2 𝑇 𝑑 ≤ 𝑎 𝑑 𝐸 𝑥 𝑠 𝑓,𝑥 𝑑
This work: conjecture is true for 𝑎 𝑑 = 𝑑 𝑂(𝑑)
[GSTW] give an example where 𝑎 𝑑 ≥ 𝑑 𝑑(1−𝑜 1 )
To match [GSTW] bounds for max sensitivity we would need 𝑎 𝑑 ≤ 𝑑 𝑑 2 𝑂 𝑑

19. Side note 1: Approximate Gotsman-Linial
Let 𝐴⊂ 0,1 𝑛 be a set of size 𝐴 =(1+𝜖) 2 𝑛−1
𝐺[𝐴] = sub-graph of hypercube induced by A
Corollary of main theorem: In either 𝐺[𝐴] or 𝐺[ 𝐴 𝑐 ], ≥𝑝𝑜𝑙𝑦(𝜖) 2 𝑛 nodes have degree ≥𝑛/𝑝𝑜𝑙𝑦𝑙𝑜𝑔 1/𝜖
Beats the simple averaging argument, which only guarantees a node of degree ≥𝜖𝑛

20. Side note 2: Reverse inequality
Main result: for any 𝑑≥1 𝑇 𝑓 𝑇 2 𝑇 𝑑 ≤ 𝑎 𝑑 𝐸 𝑥 𝑠 𝑓,𝑥 𝑑
[GSTW] asked if the reverse direction holds
Is it true that for any 𝑑≥1
𝑇 𝑓 𝑇 2 𝑇 𝑑 ≥ 𝑏 𝑑 𝐸 𝑥 𝑠 𝑓,𝑥 𝑑 ?

21. Side note 2: Reverse inequality
We can prove a (weak form) of a reverse inequality
For any 𝑘≥1 there exists 𝑘≤𝑑≤ 𝑘 2 such that
𝑇 𝑓 𝑇 2 𝑇 𝑑 ≥ 𝑑 −𝑂 𝑑 𝐸 𝑥 𝑠 𝑓,𝑥 𝑑
This is still work in progress. Hopefully we can prove this for all d.

22. Proof of main theorem

23. Proof: high level 𝑇 𝑓 𝑇 2 |𝑇| 𝑑 ≤ 𝑑 𝑂 𝑑 𝐸 𝑥 𝑠 𝑓,𝑥 𝑑
Main theorem (slightly reformulated):
𝑇 𝑓 𝑇 |𝑇| 𝑑 ≤ 𝑑 𝑂 𝑑 𝐸 𝑥 𝑠 𝑓,𝑥 𝑑
Step 1: Fourier moments (LHS) ≈ number of d-dimensional subcubes on which the restriction of f has maximal degree
Step 2: Find “combinatorial properties” for maximal degree
Step 3: Bound #d-dim max-deg. restrictions by moments of sensitivity (RHS)

24. Step 1: Fourier moments and max-degree restrictions

25. Boolean function restrictions
Let 𝑓: 0,1 𝑛 →{−1,1}
A d-dimensional subcube 𝐶⊂ 0,1 𝑛 is obtained by
fixing all but d variables
The restriction of f to C is a Boolean function on d inputs:
𝑓 ​ 𝐶 : 0,1 𝑑 →{−1,1}
A maximal degree restriction satisfies deg 𝑓 ​ 𝐶 =𝑑
As we will soon see, these play an important role

26. Fourier moments and max-degree restrictions
Main theorem: 𝑇 𝑓 𝑇 |𝑇| 𝑑 ≤ 𝑑 𝑂 𝑑 𝐸 𝑥 𝑠 𝑓,𝑥 𝑑
First step: simplify the LHS
A = 2 n 𝑇 𝑓 𝑇 |𝑇| 𝑑 = 2 𝑛 LHS
B = # d-dimensional maximal degree restrictions of f
Lemma: 2 −d ≤ 𝐴 𝐵 ≤ 2 𝑑
So, we can bound B instead of A, losing at most a 2 2𝑑 factor

27. Fourier moments and max-degree restrictions
Proof of Lemma (sketch):
Define “derivative operator” Δ 𝑖 𝑓 𝑥 =𝑓 𝑥+ 𝑒 𝑖 −𝑓(𝑥)
Apply iteratively d time
Compute D= 𝑥∈ 0,1 𝑛 𝑖 1 ,…, 𝑖 𝑑 ∈ 𝑛 Δ 𝑖 1 … Δ 𝑖 𝑑 𝑓 𝑥 2
Can show: D= 2 2d A and 2 d B≤𝐷≤ 2 3𝑑 𝐵

28. Fourier moments and max-degree restrictions
Conclusion: can forget all about Fourier moments
Instead, bound # of d-dimensional maximal degree restrictions
Goal: identify combinatorial properties of maximal degree functions, then relate them to sensitivity

29. Step 2: Max degree restrictions

30. Max degree functions
We restrict our attention to d-dimensional max degree restrictions of f
We simply denote them by 𝑔: 0,1 𝑑 →{−1,1}
Recall that we assume deg 𝑔 =𝑑
Goal: identify useful combinatorial properties of maximal degree Boolean functions

31. Sensitive paths [GSTW]
P = path in hypercube with vertices 𝑣 1 , 𝑣 2 ,…, 𝑣 𝑚 ∈ 0,1 𝑑
Assume edge directions are 𝑖 1 ,…, 𝑖 𝑚−1 ∈[𝑑] (that is, 𝑣 𝑖+1 = 𝑣 𝑖 + 𝑒 𝑖 )
Definition 1: The path P is sensitive for g if, whenever a direction 𝑖 appears for the first time, say 𝑖=𝑖 𝑡 , then the corresponding edge is sensitive with respect to g, namely 𝑔 𝑣 𝑖 ≠𝑔 𝑣 𝑖+1

32. Sensitive paths: example
Function 𝑔: 0,1 3 →{−1,1}
Path: 𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 , 𝑣 5 , 𝑣 6
Sensitive edges: 𝑣 1 , 𝑣 2 , 𝑣 2 , 𝑣 3 , 𝑣 4 , 𝑣 5
Sensitive nodes: 𝑣 1 , 𝑣 2 , 𝑣 4
Non-sensitive edges: 𝑣 3 , 𝑣 4 , 𝑣 5 , 𝑣 6
𝑣 4
𝑣 3
𝑣 2
𝑣 1
𝑣 5
𝑣 6

33. Sensitive paths
P = path in hypercube with vertices 𝑣 1 , 𝑣 2 ,…, 𝑣 𝑚 ∈ 0,1 𝑑
Assume edge directions are 𝑖 1 ,…, 𝑖 𝑚−1 ∈[𝑑] (that is, 𝑣 𝑖+1 = 𝑣 𝑖 + 𝑒 𝑖 )
Definition 1: The path P is sensitive for g if, whenever a direction 𝑖 appears for the first time, say 𝑖=𝑖 𝑡 , then the corresponding edge is sensitive with respect to g, namely 𝑔 𝑣 𝑖 ≠𝑔 𝑣 𝑖+1
Observation 1: if 𝑔=𝑓 ​ 𝐶 then sensitive for g = sensitive for f

34. Sensitive paths
P = path in hypercube with vertices 𝑣 1 , 𝑣 2 ,…, 𝑣 𝑚 ∈ 0,1 𝑑
Assume edge directions are 𝑖 1 ,…, 𝑖 𝑚−1 ∈[𝑑] (that is, 𝑣 𝑖+1 = 𝑣 𝑖 + 𝑒 𝑖 )
Definition 2: P spans all directions if 𝑖 1 ,…, 𝑖 𝑚−1 ={1,…,𝑑}
(sufficient to just consider sensitive edges)

35. Sensitive paths
P = path in hypercube with vertices 𝑣 1 , 𝑣 2 ,…, 𝑣 𝑚 ∈ 0,1 𝑑
Assume edge directions are 𝑖 1 ,…, 𝑖 𝑚−1 ∈[𝑑] (that is, 𝑣 𝑖+1 = 𝑣 𝑖 + 𝑒 𝑖 )
Definition 2: P spans all directions if 𝑖 1 ,…, 𝑖 𝑚−1 ={1,…,𝑑}
(sufficient to just consider sensitive edges)
Observation 2: if 𝑔=𝑓 ​ 𝐶 has a path P which spans all directions, then P uniquely identifies C

36. Sensitive paths: example
Function 𝑔: 0,1 3 →{−1,1}
Path: 𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 , 𝑣 5 , 𝑣 6
Sensitive edges: 𝑣 1 , 𝑣 2 , 𝑣 2 , 𝑣 3 , 𝑣 4 , 𝑣 5
Sensitive nodes: 𝑣 1 , 𝑣 2 , 𝑣 4
Non-sensitive edges: 𝑣 3 , 𝑣 4 , 𝑣 5 , 𝑣 6
Path identifies supporting 3-dim. cube
𝑣 4
𝑣 3
𝑣 2
𝑣 1
𝑣 5
𝑣 6

37. Paths in max degree functions
Lemma ([GSTW]): If 𝑔: 0,1 𝑑 →{−1,1} has max degree,
then there is a path P in 0,1 𝑑 which is:
Sensitive for g
Spans all directions
Has length O(d)
Corollary: # d-dim max degree restrictions ≤ 2 𝑛 𝑑 𝑂 𝑑 𝑠 𝑚𝑎𝑥 𝑓 𝑑
This is the main result of [GSTW]
with some more refined analysis, 𝑑 𝑂 𝑑 is improved to 2 𝑂 𝑑

38. Paths in max degree functions
Corollary: # d-dim max degree restrictions ≤ 2 𝑛 𝑑 𝑂 𝑑 𝑠 𝑚𝑎𝑥 𝑓 𝑑
Proof idea:
Let C be a d-dim subcube such that 𝑔=𝑓 ​ 𝐶 has maximal degree d
By lemma, there is a path P in C which is:
(i) sensitive for g (= sensitive for f)
(ii) spans all directions
(iii) has length 𝐿=𝑂(𝑑)
P uniquely identifies C; so, bound the # of such P in 0,1 𝑛
Construct 𝑃= 𝑣 1 ,…, 𝑣 𝐿 as follows:
Start vertex 𝑣 1 has 2 𝑛 options
Enumerate indices of sensitive nodes: 𝐿 𝑑 = 2 O d
If 𝑣 𝑖 is a sensitive node, the edge ( 𝑣 𝑖 , 𝑣 𝑖+1 ) has 𝑠 𝑓, 𝑣 𝑖 ≤ 𝑠 𝑚𝑎𝑥 (𝑓) options
Otherwise, the edge ( 𝑣 𝑖 , 𝑣 𝑖+1 ) has ≤𝑑 options (some previously used direction)
#P≤ 2 𝑛 ∗2 O d ∗ 𝑠 𝑚𝑎𝑥 𝑓 𝑑 ∗ 𝑑 𝑂 𝑑 ≤ 2 𝑛 𝑑 O d 𝑠 𝑚𝑎𝑥 𝑓 𝑑

39. Beyond maximal sensitivity
Why did we need to use maximal sensitivity?
Because we needed to bound the sensitivity of each node in a potential path P
New idea: enforce this directly
Let P be a path with sensitive nodes 𝑣 𝑖 1 ,…, 𝑣 𝑖 𝑑 .
Definition 3: P is first maximal if 𝑠 𝑓, 𝑣 𝑖 1 ≥𝑠(𝑓, 𝑣 𝑖 𝑗 ) for all 𝑗=2,…, 𝑑

40. Why first maximal paths help
Assume we had the following lemma: in any d-dim maximal degree restriction 𝑔=𝑓 ​ 𝐶 , there exists a path P which is
Sensitive for g
Spans all directions
First maximal
Has length 𝐿=𝑂(𝑑)
Claim - this allows us to improve the bounds:
# d-dim max degree restrictions ≤ 𝑑 𝑂(𝑑) 𝑥∈ 0,1 𝑛 𝑠 𝑓,𝑥 𝑑
(recall previous bound: 𝑑 𝑂 𝑑 2 𝑛 𝑠 𝑚𝑎𝑥 𝑓 𝑑 )
This is exactly what we want!

41. Improved corollary
New corollary: # d-dim max degree restrictions ≤ 𝑑 𝑂 𝑑 𝑥∈ 0,1 𝑛 𝑠 𝑓,𝑥 𝑑
Proof idea:
Let C be a d-dim subcube such that 𝑔=𝑓 ​ 𝐶 has maximal degree d
There is a path P in C which is:
(i) sensitive for g
(ii) spans all directions
(iii) First maximal
(iv) has length 𝐿=𝑂 𝑑
P uniquely identifies C; so, bound the # of such P.
Construct 𝑃= 𝑣 1 ,…, 𝑣 𝐿 as follows:
Enumerate first vertex 𝑣 1 =𝑥
Indices of sensitive nodes: 𝐿 𝑑 ≤ 2 O d
If 𝑣 𝑖 is a sensitive node, the edge ( 𝑣 𝑖 , 𝑣 𝑖+1 ) has 𝑠 𝑓, 𝑣 𝑖 ≤𝑠 𝑓,𝑥 options
Otherwise, the edge ( 𝑣 𝑖 , 𝑣 𝑖+1 ) has ≤𝑑 options (some previously used direction)
#P≤ 2 𝑂 𝑑 𝑑 𝑂 𝑑 𝑥∈ 0,1 𝑛 𝑠 𝑓,𝑥 𝑑

42. Step 3: Finding first maximal sensitive paths

43. Decoupling f and g Goal: prove the following combinatorial lemma
Given 𝑔=𝑓 ​ 𝐶 of maximal degree d, we want a path P which is
sensitive for g
spans all directions in C
first maximal
has length O(d)
Annoying:
(i),(ii),(iv) can be defined only in terms of g
(iii) defined in terms of f
It would be convenient to get an “f free” definition of first maximal.
This way, can simply study max degree functions g, and forget about f.

44. Decoupling f and g Let 𝑔: 0,1 𝑑 →{−1,1} be our max degree function
Let 𝑤: 0,1 𝑑 →𝑁 be an arbitrary weight function
Let P be a path with sensitive nodes 𝑣 𝑖 1 ,…, 𝑣 𝑖 𝑑 .
Definition: P is first maximal for w if w 𝑣 𝑖 1 ≥𝑤( 𝑣 𝑖 𝑗 ) for all 𝑗=2,…, 𝑑
We only care about 𝑤 𝑥 =𝑠(𝑓,𝑥), but study general weights to decouple f and g

45. First maximal sensitive paths: example
Path 𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 , 𝑣 5 , 𝑣 6 is
Sensitive for 𝑔: 0,1 3 →{−1,1}
First maximal for w 13 11
𝑣 4
𝑣 3
𝑣 2
𝑣 1 4 17 20 19
𝑣 5 4
𝑣 6 30

46. First maximal sensitive paths: example
Path 𝑣 1 , 𝑣 2 , 𝑣 3 , 𝑣 4 , 𝑣 5 , 𝑣 6 is
Sensitive for 𝑔: 0,1 3 →{−1,1}
First maximal for w 13 11
𝑣 4
𝑣 3
𝑣 2
𝑣 1 4 17 20 19
𝑣 5
Sensitive node 4
𝑣 6 30

47. Our main combinatorial lemma
Let 𝑔: 0,1 𝑑 →{−1,1} be any max degree function
Let 𝑤: 0,1 𝑑 →𝑁 be any weight function
Then there exists a path P which is
(i) sensitive for g
(ii) spans all directions
(iii) first maximal for w
(iv) has length O(d)
As we saw, this lemma implies our main theorem

48. Simpler combinatorial lemma
Here we prove a simpler lemma, where the path has length 𝑂( 𝑑 2 ). Proof is by induction on d.
Let 𝑔: 0,1 𝑑 →{−1,1} of maximal degree and 𝑤: 0,1 𝑑 →𝑁
For every edge 𝑒= 𝑢,𝑣 define its weight: 𝑤 𝑒 =max⁡(𝑤 𝑢 ,𝑤 𝑣 )
Let 𝑒 ∗ be a sensitive edge for g of minimal weight. It will be the last edge in the path. Assume wlog that 𝑒 ∗ =(𝑣,𝑣+ 𝑒 𝑑 ) where 𝑣 𝑑 =0.
Assume also wlog that 𝑔 ​ 𝑥 𝑑 =0 has maximal degree, namely 𝑑−1.
By induction, there is a path P’ in 𝐶 ′ ={𝑥: 𝑥 𝑑 =0} which is sensitive for g and first maximal for w.
Extend P’ to end at 𝑣∈ 𝐶 ′ . We can do this as P’ spans all directions in C’, namely {1,…,𝑑−1}.
Add the edge 𝑒 ∗ to P’. This gives us the required path.

49. Simpler combinatorial lemma
Here we prove a simpler lemma, where the path has length 𝑂( 𝑑 2 ). Proof is by induction on d.
Let 𝑔: 0,1 𝑑 →{−1,1} of maximal degree and 𝑤: 0,1 𝑑 →𝑁
For every edge 𝑒= 𝑢,𝑣 define its weight: 𝑤 𝑒 =max⁡(𝑤 𝑢 ,𝑤 𝑣 )
Let 𝑒 ∗ be a sensitive edge for g of minimal weight. It will be the last edge in the path. Assume wlog that 𝑒 ∗ =(𝑣,𝑣+ 𝑒 𝑑 ) where 𝑣 𝑑 =0.
Assume also wlog that 𝑔 ​ 𝑥 𝑑 =0 has maximal degree, namely 𝑑−1.
By induction, there is a path P’ in 𝐶 ′ ={𝑥: 𝑥 𝑑 =0} which is sensitive for g and first maximal for w.
Extend P’ to end at 𝑣∈ 𝐶 ′ . We can do this as P’ spans all directions in C’, namely {1,…,𝑑−1}.
Add the edge 𝑒 ∗ to P’. This gives us the required path.
Extending P’ to v may require d-1 edges
In total we need
(d-1)+(d-2)+…+1=O(d2) edges

50. Reducing the length of the path
Reducing the length of the path to O(d) is unfortunately much more complex (and possibly more interesting)
Is uses various notions of maximal sub-structures in maximal degree functions, some from [GSTW] and some new. Just to through out a few buzzwords:
Sensitive trees
Sensitive orchards
Sensitive chains
Sensitive covers
Sensitive reversible chains …

51. Reducing the length of the path
Reducing the length of the path to O(d) is unfortunately much more complex (and possibly more interesting)
Is uses various notions of maximal sub-structures in maximal degree functions, some from [GSTW] and some new. Just to through out a few buzzwords:
Sensitive trees
Sensitive orchards
Sensitive chains
Sensitive covers
Sensitive reversible chains …

52. A combinatorial conjecture which implies tight bounds
Let 𝑓: 0,1 𝑛 →{−1,1}
Definition [GSTW]: a sensitive tree for f is a tree T in the Boolean hypercube which satisfies two properties:
(i) Each edge 𝑥,𝑥+ 𝑒 𝑖 of T is sensitive for f: 𝑓 𝑥 ≠𝑓 𝑥+ 𝑒 𝑖
(ii) The directions of the edges (e.g. i above) are all distinct
Such a tree immediately gives a first maximal sensitive path, by starting a BFS at the node with largest weight
Conjecture [GSTW]: If f has max degree, then there exists a sensitive tree for f which spans all the directions

53. Related combinatorial conjectures
Let 𝐴⊂ 0,1 𝑛 be a set of size 𝐴 > 2 𝑛−1
𝐺[𝐴] = induced sub-graph of the hypercube on A
Sensitivity conjecture*: Max degree in 𝐺[𝐴] is ≥𝑝𝑜𝑙𝑦 𝑛
Conjecture from last slide*: There is a sub-tree of 𝐺[𝐴], where edges have unique directions, which contains all n directions
An even weaker conjecture: There is a connected component of 𝐺 𝐴 which spans all n directions
Turns out, this is a theorem [Kotlov ’00]
*Actually: true for either 𝐺[𝐴] or 𝐺 𝐴 𝐶 . I suspect that if true, should hold for 𝐺[𝐴].

54. Conclusion
Theorem: If most nodes in a Boolean function have low sensitivity, then most of its Fourier mass is supported on coefficients with low hamming weight
Proof: combinatorics in the Boolean hypercube

55. Thank you! Open problems
1. Can these new techniques shed light on the original sensitivity conjecture?
2. Applications: [GSTW] gives a few applications, but IMO there is no “killer application” yet
3. Bounds: prove theorem with optimal bounds
Conjecture: 𝑇 𝑓 𝑇 |𝑇| 𝑑 ≤ 2 𝑂 𝑑 𝐸 𝑥 𝑠 𝑓,𝑥 𝑑
(we prove this with 𝑑 𝑂 𝑑 instead of 2 𝑂 𝑑 )
Thank you!