fooling’polytopes’theory.stanford.edu/~liyang/polytopes.pdf · fooling’polytopes’...
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Fooling Polytopes
Li-‐Yang Tan (Stanford)
Joint work with Ryan O’Donnell (CMU) and Rocco Servedio (Columbia)
Polytopes over {0,1}n
= IntersecPons of boolean halfspaces
§ Within CS: opPmizaPon ( = {0,1}-‐integer programs Ax ≤ b), complexity theory, learning theory, …
§ Beyond CS: Large body of work in combinatorics, high-‐dimensional geometry, …
F (x) = h1(x) ^ · · · ^ hm(x)
sign(w · x� ✓)Halfspace:
x 2 {0, 1}n
Main complexity measure for this talk: # of facets
m = number of facets of polytope = number of halfspaces
F (x) = h1(x) ^ · · · ^ hm(x)
This talk: can think of m = poly(n), say n10
x 2 {0, 1}n
1 ≤ m ≤ 2n
... then F accepts (Δ + 0.01) fracPon of points
This talk: Discrepancy Sets for Polytopes
For all m-‐facet polytopes F, if F accepts Δ fracPon of inputs in {0,1}n …
{0,1}n
Want small set of points in {0,1}n such that:
Random set of points works great. Want explicit set.
–
Pseudorandom Generators for Polytopes
Truly random input:
Pseudorandom output:
PRG
r bits
n bits
Pseudorandom Generators
DefiniPon: An ε-‐PRG for a class C is an explicit funcPon G : {0,1}r {0,1}n such that: for every funcPon F in C,
!
Goal: minimize seed length r(n,m,ε)
���� Ex⇠{0,1}n
[F (x)]� Es⇠{0,1}r
[F (G(s))]
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G
r bits
n bits
C = { m-‐facet polytopes } This work:
Consider { G(s) : s {0,1}r }
Let G : {0,1}r {0,1}n be an ε-‐PRG for the class m-‐facet polytopes
Image of PRG = Discrepancy Set
!
2 A set of 2r points in {0,1}n
G
r bits
n bits
{0,1}n
Discrepancy set for the class of m-‐facet polytopes
Our main result: PRG for polytopes
§ Previous best seed length had linear dependence on m
An ε-‐PRG for m-‐facet polytopes over {0,1}n with seed length:
poly(log m, 1/ε) log n
Discrepancy set of size n polylog(m)
{0,1}n
Equivalently:
.
Analogous results for other domains
{0,1}n
§ Solid cube [0,1]n
§ Hypergrid {0,1,...,k}n
§ Gaussian space (Rn under N(0,1)n) [HKM10]
§ ...
Standard reducPons
One algorithmic applicaPon: CounPng # of soluPons of {0,1}-‐integer programs
Ax b
x 2 {0, 1}nsubject to
c
Txmaximize
and
Given as input a {0,1}-‐IP with m constraints, there is a determinisPc algorithm that runs in Pme
n poly(log m, 1/ε)
and outputs an esPmate of the fracBon of feasible soluBons, accurate to + ε. −
PRG = input-‐oblivious algorithm
Average w.r.t. fixed discrepancy set works for all possible inputs (all possible {0,1}-‐IPs)
IntersecPons of m general halfspaces (This work)
poly(log m, 1/ε) log n
Class of funcBons: Seed length:
Any funcBon of m general halfspaces [Gopalan, O’Donnell, Wu, Zuckerman 10]
Õ(m log(1/ε)) log n
Comparison with prior results
.
.
IntersecPons of m “regular” halfspaces [Harsha, Klivans, Meka 10]
poly(log m, 1/ε) log n
IntersecPons of m low-‐weight halfspaces [Servedio, T. 17]
poly(log m, 1/ε) polylog n
.
.
PRGs for halfspaces and their generalizaPons
Halfspaces
Polynomial threshold funcPons IntersecBons of halfspaces
§ Diakonikolas, Gopalan, Jaiswal, Servedio, Viola 2009 § Meka, Zuckerman 2009 § Karnin, Rabani, Shpilka 2011 § Kothari, Meka 2015 § Gopalan, Kane, Meka 2015
§ Meka, Zuckerman 2009 § Diakonikolas, Kane, Nelson 2009 § Kane 2011 § Kane 2011 § Kane, Meka 2013 § Kane 2014
§ Harsha, Klivans, Meka 2010 § Gopalan, O’Donnell, Wu, Zuckerman 2010 § Servedio, T. 2017 § This work
Structure of this talk
§ Part I: The connecPon to Central Limit Theorems
§ VersaPle and powerful framework for designing PRGs
[Meka, Zuckerman 09] [Harsha, Klivans, Meka 10]
§ Especially effecPve for analyzing “regular” halfspaces
§ Part II: Our work
§ Challenges in dealing with general halfspaces § New ideas and ingredients in our work
§ New Litlewood–Offord theorem for polytopes
Part I: Background and Context PRGs via Central Limit Theorems
IllustraPve example: Fooling a single “regular” halfspace [MZ10]
Central Limit Theorems
The sum of many independent random variables converges to Gaussian (of same mean and variance).
CDF distance (= Kolmogorov distance): For all θ in R,
CDFs of S vs. Gaussian
S = X1 + · · ·+Xn ⇡ N (µ,�2)
Pr⇥S ✓
⇤⇡ Pr
⇥N ✓
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DefiniPon: An “ε-‐regular” linear form S : Rn R
Regularity and the Berry–Esséen CLT
is one in which no weight is too dominant:
S(x) = w1x1 + · · ·+ wnxn
Berry–Esséen CLT: For x uniform from {−1,1}n,
ObservaPon: Regularity crucial; consider S(x) = x1
!
CDF of x1 ≈ Gaussian
|wi| " · kwk2
S(x) ⇡"
N (µ,�2)
The connecPon between CLTs and pseudorandomness
CLT: S(x) converges to Gaussian in CDF distance
Pseudorandomness: Fool the regular halfspace sign(S(x)−θ)
versus
(for all θ)
Pseudorandom version of Berry–Esséen CLT?
S(x) = w1x1 + · · ·+ wnxnRegular linear form: (x uniform {−1,1}n)
Pruniform x ⇠ {±1}n
⇥S(x) ✓
⇤⇡ Pr
⇥N ✓
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Pruniform x ⇠ {±1}n
⇥S(x) ✓
⇤⇡ Pr
pseudo y ⇠ {±1}n
⇥S(y) ✓
⇤<latexit sha1_base64="h3sVoNExaRBazf9PCVFQ+ziYycI=">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</latexit><latexit sha1_base64="h3sVoNExaRBazf9PCVFQ+ziYycI=">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</latexit><latexit sha1_base64="h3sVoNExaRBazf9PCVFQ+ziYycI=">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</latexit><latexit sha1_base64="h3sVoNExaRBazf9PCVFQ+ziYycI=">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</latexit>
Meka–Zuckerman: PRGs for regular halfspaces via CLTs
Berry–Esséen CLT
Gaussian
[MZ]’s derandomizaBon of BE
S(x) S(y)
Uniform
Pseudorandom
CDF-‐close
CDF-‐close by Δ-‐inequality
x ⇠ {�1, 1}n y ⇠ {�1, 1}n
CDF-‐close
N (µ,�2)<latexit sha1_base64="7Kp9U4wcDqRJrTLBAN5TCDzZUFU=">AAACAXicdVDLSsNAFJ3UV62vqCtxM1iEClKS4qu7ohtXUsFooYllMp20Q2eSMDMRSihu/BU3LlTc+hfu/BsnbYQqeuDC4Zx7ufceP2ZUKsv6NAozs3PzC8XF0tLyyuqaub5xLaNEYOLgiEWi5SNJGA2Jo6hipBULgrjPyI0/OMv8mzsiJI3CKzWMicdRL6QBxUhpqWNuuRypPkYsvRhVXJ7su5L2OLqt7XXMslW1xoBT5NCy60c2tHOlDHI0O+aH241wwkmoMENStm0rVl6KhKKYkVHJTSSJER6gHmlrGiJOpJeOXxjBXa10YRAJXaGCY3V6IkVcyiH3dWd2sPztZeJfXjtRwYmX0jBOFAnxZFGQMKgimOUBu1QQrNhQE4QF1bdC3EcCYaVTK+kQvj+F/xOnVq1XrcuDcuM0T6MItsEOqAAbHIMGOAdN4AAM7sEjeAYvxoPxZLwab5PWgpHPbIIfMN6/AAz/lsI=</latexit><latexit sha1_base64="7Kp9U4wcDqRJrTLBAN5TCDzZUFU=">AAACAXicdVDLSsNAFJ3UV62vqCtxM1iEClKS4qu7ohtXUsFooYllMp20Q2eSMDMRSihu/BU3LlTc+hfu/BsnbYQqeuDC4Zx7ufceP2ZUKsv6NAozs3PzC8XF0tLyyuqaub5xLaNEYOLgiEWi5SNJGA2Jo6hipBULgrjPyI0/OMv8mzsiJI3CKzWMicdRL6QBxUhpqWNuuRypPkYsvRhVXJ7su5L2OLqt7XXMslW1xoBT5NCy60c2tHOlDHI0O+aH241wwkmoMENStm0rVl6KhKKYkVHJTSSJER6gHmlrGiJOpJeOXxjBXa10YRAJXaGCY3V6IkVcyiH3dWd2sPztZeJfXjtRwYmX0jBOFAnxZFGQMKgimOUBu1QQrNhQE4QF1bdC3EcCYaVTK+kQvj+F/xOnVq1XrcuDcuM0T6MItsEOqAAbHIMGOAdN4AAM7sEjeAYvxoPxZLwab5PWgpHPbIIfMN6/AAz/lsI=</latexit><latexit sha1_base64="7Kp9U4wcDqRJrTLBAN5TCDzZUFU=">AAACAXicdVDLSsNAFJ3UV62vqCtxM1iEClKS4qu7ohtXUsFooYllMp20Q2eSMDMRSihu/BU3LlTc+hfu/BsnbYQqeuDC4Zx7ufceP2ZUKsv6NAozs3PzC8XF0tLyyuqaub5xLaNEYOLgiEWi5SNJGA2Jo6hipBULgrjPyI0/OMv8mzsiJI3CKzWMicdRL6QBxUhpqWNuuRypPkYsvRhVXJ7su5L2OLqt7XXMslW1xoBT5NCy60c2tHOlDHI0O+aH241wwkmoMENStm0rVl6KhKKYkVHJTSSJER6gHmlrGiJOpJeOXxjBXa10YRAJXaGCY3V6IkVcyiH3dWd2sPztZeJfXjtRwYmX0jBOFAnxZFGQMKgimOUBu1QQrNhQE4QF1bdC3EcCYaVTK+kQvj+F/xOnVq1XrcuDcuM0T6MItsEOqAAbHIMGOAdN4AAM7sEjeAYvxoPxZLwab5PWgpHPbIIfMN6/AAz/lsI=</latexit>
Meka–Zuckerman’s PRG for regular halfspaces
1. Pseudorandomly hash n variables into 1/ε2 buckets 2. Assign values within each bucket according to O(1)-‐wise independent
distribuPon (independently across buckets)
[Meka–Zuckerman 10]
§ Theorem: Berry–Esséen CLT holds for this distribuPon § Corollary: This is an ε-‐PRG for ε-‐regular halfspaces with seed length
O((log n)/ε2).
. . . y10 y2 y6 y3
y12 y9 y11 y4
y7
This talk: All PRGs = this [MZ] generator (possibly with different parameters)
1/ε2 buckets
Next: Fooling IntersecBons of regular halfspaces
Regular halfspaces IntersecBons of regular halfspaces
Same overall framework, but many cool new ideas and ingredients...
[Meka, Zuckerman 09] [Harsha, Klivans, Meka 10]
1 regular halfspace
IntersecPon of m regular halfspaces
Sum of real-‐valued r.v.’s, none too dominant
Berry-‐Esséen CLT
[HKM10] mulBdimensional CLT nX
i=1
~Xi �! N (~µ,⌃)
Sum of Rm-‐valued r.v.’s, none too dominant
convergence in mulPdimensional CDF distance
nX
i=1
Xi �! N (µ,�2)<latexit sha1_base64="/VLuZsnV5AmsxUnYUED4TjW5frk=">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</latexit><latexit sha1_base64="/VLuZsnV5AmsxUnYUED4TjW5frk=">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</latexit><latexit sha1_base64="/VLuZsnV5AmsxUnYUED4TjW5frk=">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</latexit>
[Meka, Zuckerman 09]
[Harsha, Klivans, Meka 10]
CDF-‐close
HKM’s PRG via their mulPdimensional CLT
Uniform
Pseudorandom
x ⇠ {�1, 1}n y ⇠ {�1, 1}n
Let A be an m x n matrix, where every row of A is regular
Ax
<latexit sha1_base64="UYzBDXfZIuBYgNNhpUnk+LtVyY4=">AAAB+HicdVDLSgMxFL1TX7W+Rl26CRbBVZkRX91V3bis4FihHUomk2lDM5MhyRTL0D9x40LFrZ/izr8xbUeoogdCDufcS05OkHKmtON8WqWFxaXllfJqZW19Y3PL3t65UyKThHpEcCHvA6woZwn1NNOc3qeS4jjgtBUMriZ+a0ilYiK51aOU+jHuJSxiBGsjdW37AnUCwUM1is2VP4y7dtWpOVOgOXLiuPVTF7mFUoUCza790QkFyWKaaMKxUm3XSbWfY6kZ4XRc6WSKppgMcI+2DU1wTJWfT5OP0YFRQhQJaU6i0VSd38hxrCbRzGSMdV/99ibiX14709G5n7MkzTRNyOyhKONICzSpAYVMUqL5yBBMJDNZEeljiYk2ZVVMCd8/Rf8T76hWrzk3x9XGZdFGGfZgHw7BhTNowDU0wQMCQ3iEZ3ixcuvJerXeZqMlq9jZhR+w3r8ANLKTmA==</latexit><latexit sha1_base64="UYzBDXfZIuBYgNNhpUnk+LtVyY4=">AAAB+HicdVDLSgMxFL1TX7W+Rl26CRbBVZkRX91V3bis4FihHUomk2lDM5MhyRTL0D9x40LFrZ/izr8xbUeoogdCDufcS05OkHKmtON8WqWFxaXllfJqZW19Y3PL3t65UyKThHpEcCHvA6woZwn1NNOc3qeS4jjgtBUMriZ+a0ilYiK51aOU+jHuJSxiBGsjdW37AnUCwUM1is2VP4y7dtWpOVOgOXLiuPVTF7mFUoUCza790QkFyWKaaMKxUm3XSbWfY6kZ4XRc6WSKppgMcI+2DU1wTJWfT5OP0YFRQhQJaU6i0VSd38hxrCbRzGSMdV/99ibiX14709G5n7MkzTRNyOyhKONICzSpAYVMUqL5yBBMJDNZEeljiYk2ZVVMCd8/Rf8T76hWrzk3x9XGZdFGGfZgHw7BhTNowDU0wQMCQ3iEZ3ixcuvJerXeZqMlq9jZhR+w3r8ANLKTmA==</latexit><latexit sha1_base64="UYzBDXfZIuBYgNNhpUnk+LtVyY4=">AAAB+HicdVDLSgMxFL1TX7W+Rl26CRbBVZkRX91V3bis4FihHUomk2lDM5MhyRTL0D9x40LFrZ/izr8xbUeoogdCDufcS05OkHKmtON8WqWFxaXllfJqZW19Y3PL3t65UyKThHpEcCHvA6woZwn1NNOc3qeS4jjgtBUMriZ+a0ilYiK51aOU+jHuJSxiBGsjdW37AnUCwUM1is2VP4y7dtWpOVOgOXLiuPVTF7mFUoUCza790QkFyWKaaMKxUm3XSbWfY6kZ4XRc6WSKppgMcI+2DU1wTJWfT5OP0YFRQhQJaU6i0VSd38hxrCbRzGSMdV/99ibiX14709G5n7MkzTRNyOyhKONICzSpAYVMUqL5yBBMJDNZEeljiYk2ZVVMCd8/Rf8T76hWrzk3x9XGZdFGGfZgHw7BhTNowDU0wQMCQ3iEZ3ixcuvJerXeZqMlq9jZhR+w3r8ANLKTmA==</latexit>
Ay<latexit sha1_base64="4Io0uGfokJz1bKhb+lmPLNeO4y4=">AAAB+HicdVDNS8MwHE3n15xfVY9egkPwNFrxa7epF48TrBtsZaRpuoWlSUnSQSn7T7x4UPHqn+LN/8Z0qzBFH4Q83vv9yMsLEkaVdpxPq7K0vLK6Vl2vbWxube/Yu3sPSqQSEw8LJmQ3QIowyomnqWakm0iC4oCRTjC+KfzOhEhFBb/XWUL8GA05jShG2kgD276C/UCwUGWxufJsOrDrTsOZAS6QM8dtnrvQLZU6KNEe2B/9UOA0JlxjhpTquU6i/RxJTTEj01o/VSRBeIyGpGcoRzFRfj5LPoVHRglhJKQ5XMOZuriRo1gV0cxkjPRI/fYK8S+vl+ro0s8pT1JNOJ4/FKUMagGLGmBIJcGaZYYgLKnJCvEISYS1KatmSvj+KfyfeCeNZsO5O623rss2quAAHIJj4IIL0AK3oA08gMEEPIJn8GLl1pP1ar3NRytWubMPfsB6/wI2NpOZ</latexit><latexit sha1_base64="4Io0uGfokJz1bKhb+lmPLNeO4y4=">AAAB+HicdVDNS8MwHE3n15xfVY9egkPwNFrxa7epF48TrBtsZaRpuoWlSUnSQSn7T7x4UPHqn+LN/8Z0qzBFH4Q83vv9yMsLEkaVdpxPq7K0vLK6Vl2vbWxube/Yu3sPSqQSEw8LJmQ3QIowyomnqWakm0iC4oCRTjC+KfzOhEhFBb/XWUL8GA05jShG2kgD276C/UCwUGWxufJsOrDrTsOZAS6QM8dtnrvQLZU6KNEe2B/9UOA0JlxjhpTquU6i/RxJTTEj01o/VSRBeIyGpGcoRzFRfj5LPoVHRglhJKQ5XMOZuriRo1gV0cxkjPRI/fYK8S+vl+ro0s8pT1JNOJ4/FKUMagGLGmBIJcGaZYYgLKnJCvEISYS1KatmSvj+KfyfeCeNZsO5O623rss2quAAHIJj4IIL0AK3oA08gMEEPIJn8GLl1pP1ar3NRytWubMPfsB6/wI2NpOZ</latexit><latexit sha1_base64="4Io0uGfokJz1bKhb+lmPLNeO4y4=">AAAB+HicdVDNS8MwHE3n15xfVY9egkPwNFrxa7epF48TrBtsZaRpuoWlSUnSQSn7T7x4UPHqn+LN/8Z0qzBFH4Q83vv9yMsLEkaVdpxPq7K0vLK6Vl2vbWxube/Yu3sPSqQSEw8LJmQ3QIowyomnqWakm0iC4oCRTjC+KfzOhEhFBb/XWUL8GA05jShG2kgD276C/UCwUGWxufJsOrDrTsOZAS6QM8dtnrvQLZU6KNEe2B/9UOA0JlxjhpTquU6i/RxJTTEj01o/VSRBeIyGpGcoRzFRfj5LPoVHRglhJKQ5XMOZuriRo1gV0cxkjPRI/fYK8S+vl+ro0s8pT1JNOJ4/FKUMagGLGmBIJcGaZYYgLKnJCvEISYS1KatmSvj+KfyfeCeNZsO5O623rss2quAAHIJj4IIL0AK3oA08gMEEPIJn8GLl1pP1ar3NRytWubMPfsB6/wI2NpOZ</latexit>CDF-‐close by Δ-‐inequality
m-‐dimensional Gaussian
(row = weights of a regular halfspace)
y10 y2 y6 y3
y12 y12 y22 y1
CDF-‐close
...
Regular halfspaces IntersecPons of regular halfspaces
Part I:
Part II:
IntersecPons of general halfspaces
[Meka, Zuckerman 09] [Harsha, Klivans, Meka 10]
(Our work)
PRGs via Central Limit Theorems
IntersecPons of m general halfspaces
Regular halfspaces IntersecPons of m regular halfspaces
Other relevant works not discussed in Part I:
§ 1 general halfspace [Meka, Zuckerman 09]
§ Any funcBon of m general halfspaces, but seed length Õ(m) [Gopalan, O’Donnell, Wu, Zuckerman 10]
§ IntersecPon of m low-‐weight halfspaces, seed length polylog(m) [Servedio, T. 17]
Part I: Part II:
Main challenge: CLTs and regularity go hand in hand
Central Limit Theorem false without regularity assumpPon
S(x) = w1x1 + · · ·+ wnxn N (µ,�2)
if there are dominant wi’s
Recall simple example: S(x) = x1
Not close to CDF of any Gaussian
Not CDF-‐close
Not CDF-‐close
Uniform
Pseudorandom
x ⇠ {�1, 1}n y ⇠ {�1, 1}n
Ax
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Ay<latexit sha1_base64="4Io0uGfokJz1bKhb+lmPLNeO4y4=">AAAB+HicdVDNS8MwHE3n15xfVY9egkPwNFrxa7epF48TrBtsZaRpuoWlSUnSQSn7T7x4UPHqn+LN/8Z0qzBFH4Q83vv9yMsLEkaVdpxPq7K0vLK6Vl2vbWxube/Yu3sPSqQSEw8LJmQ3QIowyomnqWakm0iC4oCRTjC+KfzOhEhFBb/XWUL8GA05jShG2kgD276C/UCwUGWxufJsOrDrTsOZAS6QM8dtnrvQLZU6KNEe2B/9UOA0JlxjhpTquU6i/RxJTTEj01o/VSRBeIyGpGcoRzFRfj5LPoVHRglhJKQ5XMOZuriRo1gV0cxkjPRI/fYK8S+vl+ro0s8pT1JNOJ4/FKUMagGLGmBIJcGaZYYgLKnJCvEISYS1KatmSvj+KfyfeCeNZsO5O623rss2quAAHIJj4IIL0AK3oA08gMEEPIJn8GLl1pP1ar3NRytWubMPfsB6/wI2NpOZ</latexit><latexit sha1_base64="4Io0uGfokJz1bKhb+lmPLNeO4y4=">AAAB+HicdVDNS8MwHE3n15xfVY9egkPwNFrxa7epF48TrBtsZaRpuoWlSUnSQSn7T7x4UPHqn+LN/8Z0qzBFH4Q83vv9yMsLEkaVdpxPq7K0vLK6Vl2vbWxube/Yu3sPSqQSEw8LJmQ3QIowyomnqWakm0iC4oCRTjC+KfzOhEhFBb/XWUL8GA05jShG2kgD276C/UCwUGWxufJsOrDrTsOZAS6QM8dtnrvQLZU6KNEe2B/9UOA0JlxjhpTquU6i/RxJTTEj01o/VSRBeIyGpGcoRzFRfj5LPoVHRglhJKQ5XMOZuriRo1gV0cxkjPRI/fYK8S+vl+ro0s8pT1JNOJ4/FKUMagGLGmBIJcGaZYYgLKnJCvEISYS1KatmSvj+KfyfeCeNZsO5O623rss2quAAHIJj4IIL0AK3oA08gMEEPIJn8GLl1pP1ar3NRytWubMPfsB6/wI2NpOZ</latexit><latexit sha1_base64="4Io0uGfokJz1bKhb+lmPLNeO4y4=">AAAB+HicdVDNS8MwHE3n15xfVY9egkPwNFrxa7epF48TrBtsZaRpuoWlSUnSQSn7T7x4UPHqn+LN/8Z0qzBFH4Q83vv9yMsLEkaVdpxPq7K0vLK6Vl2vbWxube/Yu3sPSqQSEw8LJmQ3QIowyomnqWakm0iC4oCRTjC+KfzOhEhFBb/XWUL8GA05jShG2kgD276C/UCwUGWxufJsOrDrTsOZAS6QM8dtnrvQLZU6KNEe2B/9UOA0JlxjhpTquU6i/RxJTTEj01o/VSRBeIyGpGcoRzFRfj5LPoVHRglhJKQ5XMOZuriRo1gV0cxkjPRI/fYK8S+vl+ro0s8pT1JNOJ4/FKUMagGLGmBIJcGaZYYgLKnJCvEISYS1KatmSvj+KfyfeCeNZsO5O623rss2quAAHIJj4IIL0AK3oA08gMEEPIJn8GLl1pP1ar3NRytWubMPfsB6/wI2NpOZ</latexit>
CDF-‐close?
m-‐dimensional Gaussian
This work: bypassing Gaussian middleperson (by necessity) Let A be general m x n matrix
(row = weights of general halfspace, not necessarily regular)
But—will sPll employ CLT proof techniques (even though CLT does not hold!)
§ Lindeberg replacement method
§ Powerful technique for proving CLTs [Lindeberg 22] § [MZ, HKM]’s strategy for the all-‐regular case
§ Non-‐regularity necessitates new ideas and ingredients: § PRGs for CNF formulas [AW85, Nis92, Baz07, Raz09, DETT10, GMR13, ... ] § New Litlewood–Offord theorem for polytopes
Ax
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Uniform x {−1,1}n ~
Ay<latexit sha1_base64="4Io0uGfokJz1bKhb+lmPLNeO4y4=">AAAB+HicdVDNS8MwHE3n15xfVY9egkPwNFrxa7epF48TrBtsZaRpuoWlSUnSQSn7T7x4UPHqn+LN/8Z0qzBFH4Q83vv9yMsLEkaVdpxPq7K0vLK6Vl2vbWxube/Yu3sPSqQSEw8LJmQ3QIowyomnqWakm0iC4oCRTjC+KfzOhEhFBb/XWUL8GA05jShG2kgD276C/UCwUGWxufJsOrDrTsOZAS6QM8dtnrvQLZU6KNEe2B/9UOA0JlxjhpTquU6i/RxJTTEj01o/VSRBeIyGpGcoRzFRfj5LPoVHRglhJKQ5XMOZuriRo1gV0cxkjPRI/fYK8S+vl+ro0s8pT1JNOJ4/FKUMagGLGmBIJcGaZYYgLKnJCvEISYS1KatmSvj+KfyfeCeNZsO5O623rss2quAAHIJj4IIL0AK3oA08gMEEPIJn8GLl1pP1ar3NRytWubMPfsB6/wI2NpOZ</latexit><latexit sha1_base64="4Io0uGfokJz1bKhb+lmPLNeO4y4=">AAAB+HicdVDNS8MwHE3n15xfVY9egkPwNFrxa7epF48TrBtsZaRpuoWlSUnSQSn7T7x4UPHqn+LN/8Z0qzBFH4Q83vv9yMsLEkaVdpxPq7K0vLK6Vl2vbWxube/Yu3sPSqQSEw8LJmQ3QIowyomnqWakm0iC4oCRTjC+KfzOhEhFBb/XWUL8GA05jShG2kgD276C/UCwUGWxufJsOrDrTsOZAS6QM8dtnrvQLZU6KNEe2B/9UOA0JlxjhpTquU6i/RxJTTEj01o/VSRBeIyGpGcoRzFRfj5LPoVHRglhJKQ5XMOZuriRo1gV0cxkjPRI/fYK8S+vl+ro0s8pT1JNOJ4/FKUMagGLGmBIJcGaZYYgLKnJCvEISYS1KatmSvj+KfyfeCeNZsO5O623rss2quAAHIJj4IIL0AK3oA08gMEEPIJn8GLl1pP1ar3NRytWubMPfsB6/wI2NpOZ</latexit><latexit sha1_base64="4Io0uGfokJz1bKhb+lmPLNeO4y4=">AAAB+HicdVDNS8MwHE3n15xfVY9egkPwNFrxa7epF48TrBtsZaRpuoWlSUnSQSn7T7x4UPHqn+LN/8Z0qzBFH4Q83vv9yMsLEkaVdpxPq7K0vLK6Vl2vbWxube/Yu3sPSqQSEw8LJmQ3QIowyomnqWakm0iC4oCRTjC+KfzOhEhFBb/XWUL8GA05jShG2kgD276C/UCwUGWxufJsOrDrTsOZAS6QM8dtnrvQLZU6KNEe2B/9UOA0JlxjhpTquU6i/RxJTTEj01o/VSRBeIyGpGcoRzFRfj5LPoVHRglhJKQ5XMOZuriRo1gV0cxkjPRI/fYK8S+vl+ro0s8pT1JNOJ4/FKUMagGLGmBIJcGaZYYgLKnJCvEISYS1KatmSvj+KfyfeCeNZsO5O623rss2quAAHIJj4IIL0AK3oA08gMEEPIJn8GLl1pP1ar3NRytWubMPfsB6/wI2NpOZ</latexit>
Pseudorandom y {−1,1}n ~
CDF-‐close?
Outline of the rest of the talk (= the structure of our proof)
1. A useful decomposiPon of polytopes
2. “Smooth version” of the problem
3. Proving the smooth version
4. Going from smooth version to actual version
Regularity lemma for a single halfspace
Weights of regular halfspace:
Weights of general halfspace:
(No weight too dominant) Halfspace Regularity Lemma [Servedio 07]
Every halfspace can be made regular* by restricPng a small
number of variables
*or very close to constant
Halfspace Regularity Lemma as a picture
ε-‐regular “tail”
Õ(1/ε2) “head” variables
Weights of a general halfspace sorted by magnitude
Applying the regularity lemma to m halfspaces
1st halfspace 2nd halfspace mth halfspace (m-‐1)st halfspace . . .
§ Each head small, but union of all m heads could cover [n]
§ So the natural strategy of reducing to the all-‐regular case – by “restricPng away” all head variables – does not work
Remark:
A useful mental picture:
m
n
A
1st halfspace
2nd halfspace
mth halfspace
. . .
. . . y10 y2 y6 y3
y12 y9 y11 y4
y7
§ x {−1,1}n uniform § y {−1,1}n pseudorandom: Goal: Ax and Ay are close in
mulPdimensional CDF distance
~ ~
(Each halfspace has few head variables)
Outline of the rest of the talk (= the structure of our proof)
1. A useful decomposiPon of polytopes
2. “Smooth version” of the problem
3. Proving the smooth version
4. Going from smooth version to actual version
A smooth version of CDF distance
b
Ax
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Ob = {0,1}-‐indictor of orthant defined by b
E[ eOb(Ax)] ⇡ E[ eOb(Ay)]<latexit sha1_base64="d2hQoln80YjBgpH9m51pJWkwb3k=">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</latexit><latexit sha1_base64="d2hQoln80YjBgpH9m51pJWkwb3k=">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</latexit><latexit sha1_base64="d2hQoln80YjBgpH9m51pJWkwb3k=">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</latexit>
eOb : Rm ! [0, 1]<latexit sha1_base64="5HN1hMLB4ArVqmMWX4iu+k4/XF0=">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</latexit><latexit sha1_base64="5HN1hMLB4ArVqmMWX4iu+k4/XF0=">AAACG3icbVDLSgMxFM3Ud32NunQTLIILKRnfuhLduPOB1UJnLJnMbRuaeZBklDLMh7jxV9y4UHEluPBvTB+gVg8EDufcS+45fiK40oR8WoWR0bHxicmp4vTM7Ny8vbB4peJUMqiwWMSy6lMFgkdQ0VwLqCYSaOgLuPbbx13/+hak4nF0qTsJeCFtRrzBGdVGqtub7h0PQHMRQOaGVLcYFdlpntd9fIB7gu9nF/lNiF0d4xpZdzxct0ukvE2c/R0HkzLp4Zs4A1JCA5zV7Xc3iFkaQqSZoErVHJJoL6NScyYgL7qpgoSyNm1CzdCIhqC8rBcux6tGCXAjluZFGvfUnxsZDZXqhL6Z7J6rhr2u+J9XS3Vjz8t4lKQaItb/qJEKbHJ2m8IBl8C06BhCmeTmVsxaVFKmTZ9FU4IzHPkvqWyU98vkfKt0eDRoYxItoxW0hhy0iw7RCTpDFcTQPXpEz+jFerCerFfrrT9asAY7S+gXrI8vdHShHA==</latexit><latexit sha1_base64="5HN1hMLB4ArVqmMWX4iu+k4/XF0=">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</latexit>
Ob : Rm ! {0, 1}<latexit sha1_base64="UvAyLDMLDrRfq85GBGvxrfTMxBQ=">AAACEXicdVDLSsNAFJ3UV62vqEs3g0VQkJKIr7oqunFnFWMLTQyT6bQdOnkwMxFKyDe48VfcuFBx686df+MkjVBFDwycOede7r3HixgV0jA+tdLU9MzsXHm+srC4tLyir67diDDmmFg4ZCFve0gQRgNiSSoZaUecIN9jpOUNzzK/dUe4oGFwLUcRcXzUD2iPYiSV5Oo7to/kACOWXKSuB09g/ve85Cq99aEtQ2gnxq5pp9DVq0bNyAEnyIFh1g9NaBZKFRRouvqH3Q1x7JNAYoaE6JhGJJ0EcUkxI2nFjgWJEB6iPukoGiCfCCfJT0rhllK6sBdy9QIJc3WyI0G+ECPfU5XZwuK3l4l/eZ1Y9o6dhAZRLEmAx4N6MYPq0iwf2KWcYMlGiiDMqdoV4gHiCEuVYkWF8H0p/J9Ye7V6zbjcrzZOizTKYANsgm1ggiPQAOegCSyAwT14BM/gRXvQnrRX7W1cWtKKnnXwA9r7FwvSnKE=</latexit><latexit sha1_base64="UvAyLDMLDrRfq85GBGvxrfTMxBQ=">AAACEXicdVDLSsNAFJ3UV62vqEs3g0VQkJKIr7oqunFnFWMLTQyT6bQdOnkwMxFKyDe48VfcuFBx686df+MkjVBFDwycOede7r3HixgV0jA+tdLU9MzsXHm+srC4tLyir67diDDmmFg4ZCFve0gQRgNiSSoZaUecIN9jpOUNzzK/dUe4oGFwLUcRcXzUD2iPYiSV5Oo7to/kACOWXKSuB09g/ve85Cq99aEtQ2gnxq5pp9DVq0bNyAEnyIFh1g9NaBZKFRRouvqH3Q1x7JNAYoaE6JhGJJ0EcUkxI2nFjgWJEB6iPukoGiCfCCfJT0rhllK6sBdy9QIJc3WyI0G+ECPfU5XZwuK3l4l/eZ1Y9o6dhAZRLEmAx4N6MYPq0iwf2KWcYMlGiiDMqdoV4gHiCEuVYkWF8H0p/J9Ye7V6zbjcrzZOizTKYANsgm1ggiPQAOegCSyAwT14BM/gRXvQnrRX7W1cWtKKnnXwA9r7FwvSnKE=</latexit><latexit sha1_base64="UvAyLDMLDrRfq85GBGvxrfTMxBQ=">AAACEXicdVDLSsNAFJ3UV62vqEs3g0VQkJKIr7oqunFnFWMLTQyT6bQdOnkwMxFKyDe48VfcuFBx686df+MkjVBFDwycOede7r3HixgV0jA+tdLU9MzsXHm+srC4tLyir67diDDmmFg4ZCFve0gQRgNiSSoZaUecIN9jpOUNzzK/dUe4oGFwLUcRcXzUD2iPYiSV5Oo7to/kACOWXKSuB09g/ve85Cq99aEtQ2gnxq5pp9DVq0bNyAEnyIFh1g9NaBZKFRRouvqH3Q1x7JNAYoaE6JhGJJ0EcUkxI2nFjgWJEB6iPukoGiCfCCfJT0rhllK6sBdy9QIJc3WyI0G+ECPfU5XZwuK3l4l/eZ1Y9o6dhAZRLEmAx4N6MYPq0iwf2KWcYMlGiiDMqdoV4gHiCEuVYkWF8H0p/J9Ye7V6zbjcrzZOizTKYANsgm1ggiPQAOegCSyAwT14BM/gRXvQnrRX7W1cWtKKnnXwA9r7FwvSnKE=</latexit>
Orthant Ob
Ax ≤ b iff: We will first show:
where
is smooth approximator of
()<latexit sha1_base64="kk6CTAFESn4xIuVViVil2pS3V7g=">AAAB+3icdVDLSgNBEJyNrxhf0Ry9DAbBU9gVX7kFvXjwEME1gWQJs5PZZMjszDLTq4Ql/ooXDype/RFv/o2ThxBFCxqKqm66u8JEcAOu++nkFhaXllfyq4W19Y3NreL2zq1RqabMp0oo3QyJYYJL5gMHwZqJZiQOBWuEg4ux37hj2nAlb2CYsCAmPckjTglYqVMsta+U7AkWgea9PhCt1X2nWHYr7gR4jhy7XvXEw95MKaMZ6p3iR7uraBozCVQQY1qem0CQEQ2cCjYqtFPDEkIHpMdalkoSMxNkk+NHeN8qXRwpbUsCnqjzExmJjRnGoe2MCfTNb28s/uW1UojOgozLJAUm6XRRlAoMCo+TwF2uGQUxtIRQze2tmPaJJhRsXgUbwven+H/iH1aqFff6qFw7n6WRR7toDx0gD52iGrpEdeQjioboET2jF+fBeXJenbdpa86ZzZTQDzjvX/wvlTI=</latexit><latexit sha1_base64="kk6CTAFESn4xIuVViVil2pS3V7g=">AAAB+3icdVDLSgNBEJyNrxhf0Ry9DAbBU9gVX7kFvXjwEME1gWQJs5PZZMjszDLTq4Ql/ooXDype/RFv/o2ThxBFCxqKqm66u8JEcAOu++nkFhaXllfyq4W19Y3NreL2zq1RqabMp0oo3QyJYYJL5gMHwZqJZiQOBWuEg4ux37hj2nAlb2CYsCAmPckjTglYqVMsta+U7AkWgea9PhCt1X2nWHYr7gR4jhy7XvXEw95MKaMZ6p3iR7uraBozCVQQY1qem0CQEQ2cCjYqtFPDEkIHpMdalkoSMxNkk+NHeN8qXRwpbUsCnqjzExmJjRnGoe2MCfTNb28s/uW1UojOgozLJAUm6XRRlAoMCo+TwF2uGQUxtIRQze2tmPaJJhRsXgUbwven+H/iH1aqFff6qFw7n6WRR7toDx0gD52iGrpEdeQjioboET2jF+fBeXJenbdpa86ZzZTQDzjvX/wvlTI=</latexit><latexit sha1_base64="kk6CTAFESn4xIuVViVil2pS3V7g=">AAAB+3icdVDLSgNBEJyNrxhf0Ry9DAbBU9gVX7kFvXjwEME1gWQJs5PZZMjszDLTq4Ql/ooXDype/RFv/o2ThxBFCxqKqm66u8JEcAOu++nkFhaXllfyq4W19Y3NreL2zq1RqabMp0oo3QyJYYJL5gMHwZqJZiQOBWuEg4ux37hj2nAlb2CYsCAmPckjTglYqVMsta+U7AkWgea9PhCt1X2nWHYr7gR4jhy7XvXEw95MKaMZ6p3iR7uraBozCVQQY1qem0CQEQ2cCjYqtFPDEkIHpMdalkoSMxNkk+NHeN8qXRwpbUsCnqjzExmJjRnGoe2MCfTNb28s/uW1UojOgozLJAUm6XRRlAoMCo+TwF2uGQUxtIRQze2tmPaJJhRsXgUbwven+H/iH1aqFff6qFw7n6WRR7toDx0gD52iGrpEdeQjioboET2jF+fBeXJenbdpa86ZzZTQDzjvX/wvlTI=</latexit>
E[Ob(Ax)] ⇡ E[Ob(Ay)] for all b 2 Rm<latexit sha1_base64="oLS5iLfO2JtqZesN4g6QKIF0B+c=">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</latexit><latexit sha1_base64="oLS5iLfO2JtqZesN4g6QKIF0B+c=">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</latexit><latexit sha1_base64="oLS5iLfO2JtqZesN4g6QKIF0B+c=">AAACa3icjVFNTxRBEO0ZQXEFWcUDqIcOCwleNjOGD7mhhMSbaFgg2Rk31b090KGne9JdQ3YzmZP/kBs/wQu/wZ5lJWA0sfrQL6/qVVe9ZoWSDqPoOggfzcw+fjL3tPVsfuH5YvvFy2NnSstFjxtl7CkDJ5TUoocSlTgtrICcKXHCLvab/MmlsE4afYTjQqQ5nGmZSQ7oqUH7R5IDnjNWHdT9CeSgqi/1gG18TJhRQzfO/VWN6ncpTaAorBnR/5OMvSSh/qAYYZUZS0EpusYSqe86fKu/52v1oN2JutEk6D2wFcW72zGNp0yHTONw0L5KhoaXudDIFTjXj6MC0wosSq5E3UpKJwrgF3Am+h5qyIVLq4lZNV33zJA282RGI52w9xUV5K5ZwVc2U7o/cw35t1y/xOxDWkldlCg0v30oKxVFQxvn6VBawVGNPQBupZ+V8nOwwNH/T8ub8HtT+m/Qe9/d7UZfNzt7n6ZuzJE3ZJVskJjskD3ymRySHuHkZ7AYLAcrwU34Knwdvr0tDYOpZok8iHD9F+LHvME=</latexit>
Pr[Ax b] ⇡ Pr[Ay b] for all b 2 Rm<latexit sha1_base64="+/l1gssPw2ZXacEgA3FHCUbSjlM=">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</latexit><latexit sha1_base64="+/l1gssPw2ZXacEgA3FHCUbSjlM=">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</latexit><latexit sha1_base64="+/l1gssPw2ZXacEgA3FHCUbSjlM=">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</latexit>
Ax and Ay are CDF-close
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“mollifier”
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DisconPnuous funcPon!
Smooth approximators of orthants
Two important properPes of Õb [Bentkus 90]:
eOb(v) = EG⇠N (0,1)m
[Ob(v + �G)]<latexit sha1_base64="a17BU7N19FojprmI9WBFHkGnY1s=">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</latexit><latexit sha1_base64="a17BU7N19FojprmI9WBFHkGnY1s=">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</latexit><latexit sha1_base64="a17BU7N19FojprmI9WBFHkGnY1s=">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</latexit>
Standard way of mollifying a funcPon: adding Gaussian noise
1. Good approximaPon of Ob
2. Small derivaPves: for all c > 1,
sup
v2Rm
(X
|↵|=c
|@↵ eOb(v)|)
. (logm)
c/2
�c
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mc many parPal derivaPves
small error region
≈ 1
≈ 0
Orthant Ob
Rm<latexit sha1_base64="zPdAsyhTX1468yfc7dxQYy7M8cE=">AAAB8nicdVDLSgMxFM3UV62vqks3wSK4GjLiq7uiG5dVHFvojCWTZtrQJDMkGaEM/Q03LlTc+jXu/BszbRUVPRA4nHMv9+REKWfaIPTulObmFxaXysuVldW19Y3q5taNTjJFqE8Snqh2hDXlTFLfMMNpO1UUi4jTVjQ8L/zWHVWaJfLajFIaCtyXLGYEGysFgcBmEEX51fhWdKs15B4hr37sQeSiCeCX4s2UGpih2a2+Bb2EZIJKQzjWuuOh1IQ5VoYRTseVINM0xWSI+7RjqcSC6jCfZB7DPav0YJwo+6SBE/X7Ro6F1iMR2ckio/7tFeJfXicz8WmYM5lmhkoyPRRnHJoEFgXAHlOUGD6yBBPFbFZIBlhhYmxNFVvC50/h/8Q/cOsuujysNc5mbZTBDtgF+8ADJ6ABLkAT+ICAFNyDR/DkZM6D8+y8TEdLzmxnG/yA8/oB7KyR0A==</latexit><latexit sha1_base64="zPdAsyhTX1468yfc7dxQYy7M8cE=">AAAB8nicdVDLSgMxFM3UV62vqks3wSK4GjLiq7uiG5dVHFvojCWTZtrQJDMkGaEM/Q03LlTc+jXu/BszbRUVPRA4nHMv9+REKWfaIPTulObmFxaXysuVldW19Y3q5taNTjJFqE8Snqh2hDXlTFLfMMNpO1UUi4jTVjQ8L/zWHVWaJfLajFIaCtyXLGYEGysFgcBmEEX51fhWdKs15B4hr37sQeSiCeCX4s2UGpih2a2+Bb2EZIJKQzjWuuOh1IQ5VoYRTseVINM0xWSI+7RjqcSC6jCfZB7DPav0YJwo+6SBE/X7Ro6F1iMR2ckio/7tFeJfXicz8WmYM5lmhkoyPRRnHJoEFgXAHlOUGD6yBBPFbFZIBlhhYmxNFVvC50/h/8Q/cOsuujysNc5mbZTBDtgF+8ADJ6ABLkAT+ICAFNyDR/DkZM6D8+y8TEdLzmxnG/yA8/oB7KyR0A==</latexit><latexit sha1_base64="zPdAsyhTX1468yfc7dxQYy7M8cE=">AAAB8nicdVDLSgMxFM3UV62vqks3wSK4GjLiq7uiG5dVHFvojCWTZtrQJDMkGaEM/Q03LlTc+jXu/BszbRUVPRA4nHMv9+REKWfaIPTulObmFxaXysuVldW19Y3q5taNTjJFqE8Snqh2hDXlTFLfMMNpO1UUi4jTVjQ8L/zWHVWaJfLajFIaCtyXLGYEGysFgcBmEEX51fhWdKs15B4hr37sQeSiCeCX4s2UGpih2a2+Bb2EZIJKQzjWuuOh1IQ5VoYRTseVINM0xWSI+7RjqcSC6jCfZB7DPav0YJwo+6SBE/X7Ro6F1iMR2ckio/7tFeJfXicz8WmYM5lmhkoyPRRnHJoEFgXAHlOUGD6yBBPFbFZIBlhhYmxNFVvC50/h/8Q/cOsuujysNc5mbZTBDtgF+8ADJ6ABLkAT+ICAFNyDR/DkZM6D8+y8TEdLzmxnG/yA8/oB7KyR0A==</latexit>
Outline of the rest of the talk (= the structure of our proof)
1. A useful decomposiPon of polytopes
2. “Smooth version” of the problem
3. Proving the smooth version
4. Going from smooth version to actual version
Proving the smooth version via a hybrid argument
. . . y10 y2 y6 y3
y12 y9 y11 y4
y7
Ex
[ eOb(Ax)] ⇡ Ey
[ eOb(Ay)]<latexit sha1_base64="DdQ35NqPG8Ts7wvRtHDVXIeVkws=">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</latexit><latexit sha1_base64="DdQ35NqPG8Ts7wvRtHDVXIeVkws=">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</latexit><latexit sha1_base64="DdQ35NqPG8Ts7wvRtHDVXIeVkws=">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</latexit>
Bucket-‐wise hybrid argument [MZ, HKM]:
§ Start will all buckets filled in uniformly (i.e. start with x)
§ Bucket by bucket, “swap out” uniform bits for r-‐wise independent bits
§ Argue that each swap incurs small error
. . . x10 x2 x6 x3
x12 x9 x11 x4
x7
Goal is to “fool” the orthant mollifier:
y pseudorandom: x uniform:
r-‐wise r-‐wise r-‐wise uniform uniform uniform
Single swap in the hybrid argument
Ex
[ eOb(ABx)] ⇡ E
y
[ eOb(ABy)]
<latexit sha1_base64="UEvCo29lKmYnEq5Ge3GLhsGT58U=">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</latexit><latexit sha1_base64="UEvCo29lKmYnEq5Ge3GLhsGT58U=">AAACl3iclVFbS+QwFE676rrjbXZlX/QlOAj6MrTiXmRfxiu+rYKjwrQ7pOmZMZg2IUlXS+hv8r/45r8xnRlBF2XxQDgf3/nOJeckkjNtguDB8z9MTc98nP3UmJtfWFxqfv5yrkWhKHSp4EJdJkQDZzl0DTMcLqUCkiUcLpLr/Tp+8ReUZiI/M6WEOCPDnA0YJcZR/eZdlBFzJaQd+SSxh1XVt1EieKrLzDl7W1W96IalYBhPYayjhNvfTphs7P7ZeynejHFEpFTiFjf+W7t8T+2yrt1vtoJ2MDL8DHwLwp3vIQ4nTAtN7KTfvI9SQYsMckM50boXBtLElijDKIeqERUaJKHXZAg9B3OSgY7taLEVXndMigdCuZcbPGKfZ1iS6Xo6p6xH1//GavK1WK8wg5+xZbksDOR03GhQcGwErq+EU6aAGl46QKhiblZMr4gi1LhbNtwSnn6K3wbdrfZOOzjdbnX2JtuYRatoDW2gEP1AHXSMTlAXUe+r98s78A79Fb/jH/nHY6nvTXKW0QvzTx8BDovQxQ==</latexit><latexit sha1_base64="UEvCo29lKmYnEq5Ge3GLhsGT58U=">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</latexit>
AB m
|B|
B
Ex
[ eOb(Hx+ Tx)] ⇡ Ey
[ eOb(Hy + Ty)]<latexit sha1_base64="YFcG8cc/PjqVil5VCzeWBjntlpQ=">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</latexit><latexit sha1_base64="YFcG8cc/PjqVil5VCzeWBjntlpQ=">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</latexit><latexit sha1_base64="YFcG8cc/PjqVil5VCzeWBjntlpQ=">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</latexit>
Write AB = H + T, where:
§ H contains only the head variables § T contains only the tail variables
()<latexit sha1_base64="kk6CTAFESn4xIuVViVil2pS3V7g=">AAAB+3icdVDLSgNBEJyNrxhf0Ry9DAbBU9gVX7kFvXjwEME1gWQJs5PZZMjszDLTq4Ql/ooXDype/RFv/o2ThxBFCxqKqm66u8JEcAOu++nkFhaXllfyq4W19Y3NreL2zq1RqabMp0oo3QyJYYJL5gMHwZqJZiQOBWuEg4ux37hj2nAlb2CYsCAmPckjTglYqVMsta+U7AkWgea9PhCt1X2nWHYr7gR4jhy7XvXEw95MKaMZ6p3iR7uraBozCVQQY1qem0CQEQ2cCjYqtFPDEkIHpMdalkoSMxNkk+NHeN8qXRwpbUsCnqjzExmJjRnGoe2MCfTNb28s/uW1UojOgozLJAUm6XRRlAoMCo+TwF2uGQUxtIRQze2tmPaJJhRsXgUbwven+H/iH1aqFff6qFw7n6WRR7toDx0gD52iGrpEdeQjioboET2jF+fBeXJenbdpa86ZzZTQDzjvX/wvlTI=</latexit><latexit sha1_base64="kk6CTAFESn4xIuVViVil2pS3V7g=">AAAB+3icdVDLSgNBEJyNrxhf0Ry9DAbBU9gVX7kFvXjwEME1gWQJs5PZZMjszDLTq4Ql/ooXDype/RFv/o2ThxBFCxqKqm66u8JEcAOu++nkFhaXllfyq4W19Y3NreL2zq1RqabMp0oo3QyJYYJL5gMHwZqJZiQOBWuEg4ux37hj2nAlb2CYsCAmPckjTglYqVMsta+U7AkWgea9PhCt1X2nWHYr7gR4jhy7XvXEw95MKaMZ6p3iR7uraBozCVQQY1qem0CQEQ2cCjYqtFPDEkIHpMdalkoSMxNkk+NHeN8qXRwpbUsCnqjzExmJjRnGoe2MCfTNb28s/uW1UojOgozLJAUm6XRRlAoMCo+TwF2uGQUxtIRQze2tmPaJJhRsXgUbwven+H/iH1aqFff6qFw7n6WRR7toDx0gD52iGrpEdeQjioboET2jF+fBeXJenbdpa86ZzZTQDzjvX/wvlTI=</latexit><latexit sha1_base64="kk6CTAFESn4xIuVViVil2pS3V7g=">AAAB+3icdVDLSgNBEJyNrxhf0Ry9DAbBU9gVX7kFvXjwEME1gWQJs5PZZMjszDLTq4Ql/ooXDype/RFv/o2ThxBFCxqKqm66u8JEcAOu++nkFhaXllfyq4W19Y3NreL2zq1RqabMp0oo3QyJYYJL5gMHwZqJZiQOBWuEg4ux37hj2nAlb2CYsCAmPckjTglYqVMsta+U7AkWgea9PhCt1X2nWHYr7gR4jhy7XvXEw95MKaMZ6p3iR7uraBozCVQQY1qem0CQEQ2cCjYqtFPDEkIHpMdalkoSMxNkk+NHeN8qXRwpbUsCnqjzExmJjRnGoe2MCfTNb28s/uW1UojOgozLJAUm6XRRlAoMCo+TwF2uGQUxtIRQze2tmPaJJhRsXgUbwven+H/iH1aqFff6qFw7n6WRR7toDx0gD52iGrpEdeQjioboET2jF+fBeXJenbdpa86ZzZTQDzjvX/wvlTI=</latexit>
Fix bucket B [n] of variables. Let AB = A restricted to columns in B. ✓<latexit sha1_base64="PnZf/mQVWYP7yPcejOC6GMtajH4=">AAAB/nicdVDLSgMxFM3UV62vUcGNm2ARXJUZ8dVd0Y3LCo4tdIaSydxpQzMPk4xQxi78FTcuVNz6He78GzNthSp6IORwzr3k5PgpZ1JZ1qdRmptfWFwqL1dWVtfWN8zNrRuZZIKCQxOeiLZPJHAWg6OY4tBOBZDI59DyBxeF37oDIVkSX6thCl5EejELGSVKS11zx/UTHshhpK/clZkvQcHtqGtWrZo1Bp4hx5ZdP7GxPVWqaIpm1/xwg4RmEcSKciJlx7ZS5eVEKEY5jCpuJiEldEB60NE0JhFILx/nH+F9rQQ4TIQ+scJjdXYjJ5EsIurJiKi+/O0V4l9eJ1PhmZezOM0UxHTyUJhxrBJclIEDJoAqPtSEUMF0Vkz7RBCqdGUVXcL3T/H/xDms1WvW1VG1cT5to4x20R46QDY6RQ10iZrIQRTdo0f0jF6MB+PJeDXeJqMlY7qzjX7AeP8CuE6Www==</latexit><latexit sha1_base64="PnZf/mQVWYP7yPcejOC6GMtajH4=">AAAB/nicdVDLSgMxFM3UV62vUcGNm2ARXJUZ8dVd0Y3LCo4tdIaSydxpQzMPk4xQxi78FTcuVNz6He78GzNthSp6IORwzr3k5PgpZ1JZ1qdRmptfWFwqL1dWVtfWN8zNrRuZZIKCQxOeiLZPJHAWg6OY4tBOBZDI59DyBxeF37oDIVkSX6thCl5EejELGSVKS11zx/UTHshhpK/clZkvQcHtqGtWrZo1Bp4hx5ZdP7GxPVWqaIpm1/xwg4RmEcSKciJlx7ZS5eVEKEY5jCpuJiEldEB60NE0JhFILx/nH+F9rQQ4TIQ+scJjdXYjJ5EsIurJiKi+/O0V4l9eJ1PhmZezOM0UxHTyUJhxrBJclIEDJoAqPtSEUMF0Vkz7RBCqdGUVXcL3T/H/xDms1WvW1VG1cT5to4x20R46QDY6RQ10iZrIQRTdo0f0jF6MB+PJeDXeJqMlY7qzjX7AeP8CuE6Www==</latexit><latexit sha1_base64="PnZf/mQVWYP7yPcejOC6GMtajH4=">AAAB/nicdVDLSgMxFM3UV62vUcGNm2ARXJUZ8dVd0Y3LCo4tdIaSydxpQzMPk4xQxi78FTcuVNz6He78GzNthSp6IORwzr3k5PgpZ1JZ1qdRmptfWFwqL1dWVtfWN8zNrRuZZIKCQxOeiLZPJHAWg6OY4tBOBZDI59DyBxeF37oDIVkSX6thCl5EejELGSVKS11zx/UTHshhpK/clZkvQcHtqGtWrZo1Bp4hx5ZdP7GxPVWqaIpm1/xwg4RmEcSKciJlx7ZS5eVEKEY5jCpuJiEldEB60NE0JhFILx/nH+F9rQQ4TIQ+scJjdXYjJ5EsIurJiKi+/O0V4l9eJ1PhmZezOM0UxHTyUJhxrBJclIEDJoAqPtSEUMF0Vkz7RBCqdGUVXcL3T/H/xDms1WvW1VG1cT5to4x20R46QDY6RQ10iZrIQRTdo0f0jF6MB+PJeDXeJqMlY7qzjX7AeP8CuE6Www==</latexit>
CLTs (e.g. [HKM]) deal with regular linear forms; Presence of H = our main challenge
Want to show:
Part I
Ex
[ eOb(Hx+ Tx)] ⇡ Ey
[ eOb(Hy + Ty)]<latexit sha1_base64="YFcG8cc/PjqVil5VCzeWBjntlpQ=">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</latexit><latexit sha1_base64="YFcG8cc/PjqVil5VCzeWBjntlpQ=">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</latexit><latexit sha1_base64="YFcG8cc/PjqVil5VCzeWBjntlpQ=">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</latexit>
H
T
Entries are small
Rows are sparse
Equivalently, y fools the funcPon z 7! eOb(Hz + Tz)<latexit sha1_base64="0n//+HXi/Bo3CnKHqeLmLSzZu7s=">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</latexit><latexit sha1_base64="0n//+HXi/Bo3CnKHqeLmLSzZu7s=">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</latexit><latexit sha1_base64="0n//+HXi/Bo3CnKHqeLmLSzZu7s=">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</latexit>
MulPdimensional Taylor expansion
Claim:
MulPdimensional Taylor expansion:
Ex
[ eOb(Hx)] ⇡ Ey
[ eOb(Hy)]<latexit sha1_base64="euzEJ3J5IC5AHo6nZnRXp9WDm0c=">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</latexit><latexit sha1_base64="euzEJ3J5IC5AHo6nZnRXp9WDm0c=">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</latexit><latexit sha1_base64="euzEJ3J5IC5AHo6nZnRXp9WDm0c=">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</latexit>
Warmup: Does y fool the zeroth-‐order term?
eOb(Hz) +c�1X
|↵|=1
1
↵!@↵ eOb(Hz)(Tz)↵ ± err
<latexit sha1_base64="Qxt43DEEU0RDEFP5g9dIKBViTYg=">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</latexit><latexit sha1_base64="Qxt43DEEU0RDEFP5g9dIKBViTYg=">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</latexit><latexit sha1_base64="Qxt43DEEU0RDEFP5g9dIKBViTYg=">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</latexit>
(ObservaPon: trivial when H = 0)
Warmup: fooling the zeroth-‐order term
z 7! eOb(Hz)<latexit sha1_base64="V4hk7arfP2G575E6TklSSEaHqVM=">AAACD3icdVC7SgNBFJ2Nrxhfq5Y2g0GMTdgVX+mCNumM4JpANoTZ2ZtkyOyDmVklWfIJNv6KjYWKra2df+PkIUTRAwOHc+5l7jlezJlUlvVpZObmFxaXssu5ldW19Q1zc+tGRomg4NCIR6LuEQmcheAopjjUYwEk8DjUvN7FyK/dgpAsCq9VP4ZmQDohazNKlJZa5v4AuwGJpYqwe8d8UIz7kGpJdSnh6eVw2PIKlcFBy8xbRWsMPEOOLbt0YmN7quTRFNWW+eH6EU0CCBXlRMqGbcWqmRKhGOUwzLmJhJjQHulAQ9OQBCCb6TjQEO9pxcftSOgXKjxWZzdSEkjZDzw9ObpU/vZG4l9eI1Hts2bKwjhRENLJR+2EYx1/1A72mQCqeF8TQgXTt2LaJYJQpTvM6RK+k+L/iXNYLBWtq6N8+XzaRhbtoF1UQDY6RWVUQVXkIIru0SN6Ri/Gg/FkvBpvk9GMMd3ZRj9gvH8BlmCdJA==</latexit><latexit sha1_base64="V4hk7arfP2G575E6TklSSEaHqVM=">AAACD3icdVC7SgNBFJ2Nrxhfq5Y2g0GMTdgVX+mCNumM4JpANoTZ2ZtkyOyDmVklWfIJNv6KjYWKra2df+PkIUTRAwOHc+5l7jlezJlUlvVpZObmFxaXssu5ldW19Q1zc+tGRomg4NCIR6LuEQmcheAopjjUYwEk8DjUvN7FyK/dgpAsCq9VP4ZmQDohazNKlJZa5v4AuwGJpYqwe8d8UIz7kGpJdSnh6eVw2PIKlcFBy8xbRWsMPEOOLbt0YmN7quTRFNWW+eH6EU0CCBXlRMqGbcWqmRKhGOUwzLmJhJjQHulAQ9OQBCCb6TjQEO9pxcftSOgXKjxWZzdSEkjZDzw9ObpU/vZG4l9eI1Hts2bKwjhRENLJR+2EYx1/1A72mQCqeF8TQgXTt2LaJYJQpTvM6RK+k+L/iXNYLBWtq6N8+XzaRhbtoF1UQDY6RWVUQVXkIIru0SN6Ri/Gg/FkvBpvk9GMMd3ZRj9gvH8BlmCdJA==</latexit><latexit sha1_base64="V4hk7arfP2G575E6TklSSEaHqVM=">AAACD3icdVC7SgNBFJ2Nrxhfq5Y2g0GMTdgVX+mCNumM4JpANoTZ2ZtkyOyDmVklWfIJNv6KjYWKra2df+PkIUTRAwOHc+5l7jlezJlUlvVpZObmFxaXssu5ldW19Q1zc+tGRomg4NCIR6LuEQmcheAopjjUYwEk8DjUvN7FyK/dgpAsCq9VP4ZmQDohazNKlJZa5v4AuwGJpYqwe8d8UIz7kGpJdSnh6eVw2PIKlcFBy8xbRWsMPEOOLbt0YmN7quTRFNWW+eH6EU0CCBXlRMqGbcWqmRKhGOUwzLmJhJjQHulAQ9OQBCCb6TjQEO9pxcftSOgXKjxWZzdSEkjZDzw9ObpU/vZG4l9eI1Hts2bKwjhRENLJR+2EYx1/1A72mQCqeF8TQgXTt2LaJYJQpTvM6RK+k+L/iXNYLBWtq6N8+XzaRhbtoF1UQDY6RWVUQVXkIIru0SN6Ri/Gg/FkvBpvk9GMMd3ZRj9gvH8BlmCdJA==</latexit>
y fools the funcPon
H
Rows are sparse eOb(Hz) = EG[Ob(Hz + �G)]
<latexit sha1_base64="IfLLKbahMtFEiozzS9VzjU4sLVk=">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</latexit><latexit sha1_base64="IfLLKbahMtFEiozzS9VzjU4sLVk=">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</latexit><latexit sha1_base64="IfLLKbahMtFEiozzS9VzjU4sLVk=">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</latexit>
Recalling the definiPon of Õb ,
⌘mY
i=1
1[(Hz + �G)i bi]<latexit sha1_base64="19D6xTg8+8M8Kqu0yhWXNIQMdRk=">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</latexit><latexit sha1_base64="19D6xTg8+8M8Kqu0yhWXNIQMdRk=">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</latexit><latexit sha1_base64="19D6xTg8+8M8Kqu0yhWXNIQMdRk=">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</latexit>
Product structure of Õb + Sparsity of H
Suffices for y to fool small-‐width CNFs
Claim:
)<latexit sha1_base64="YpFhaZHBfDESR4GPpU0Zw1k+JAM=">AAAB8XicdVBNSwMxEM3Wr1q/qh69BIvgqeyKX70VvXis4trCdinZNG1Ds8mSzCpl6c/w4kHFq//Gm//GtF2hij4YeLw3w8y8KBHcgOt+OoWFxaXlleJqaW19Y3OrvL1zZ1SqKfOpEkq3ImKY4JL5wEGwVqIZiSPBmtHwcuI375k2XMlbGCUsjElf8h6nBKwUtG94fwBEa/XQKVfcqjsFniMnrlc79bCXKxWUo9Epf7S7iqYxk0AFMSbw3ATCjGjgVLBxqZ0alhA6JH0WWCpJzEyYTU8e4wOrdHFPaVsS8FSdn8hIbMwojmxnTGBgfnsT8S8vSKF3HmZcJikwSWeLeqnAoPDkf9zlmlEQI0sI1dzeiumAaELBplSyIXx/iv8n/lG1VnWvjyv1izyNItpD++gQeegM1dEVaiAfUaTQI3pGLw44T86r8zZrLTj5zC76Aef9CxoQkVY=</latexit><latexit sha1_base64="YpFhaZHBfDESR4GPpU0Zw1k+JAM=">AAAB8XicdVBNSwMxEM3Wr1q/qh69BIvgqeyKX70VvXis4trCdinZNG1Ds8mSzCpl6c/w4kHFq//Gm//GtF2hij4YeLw3w8y8KBHcgOt+OoWFxaXlleJqaW19Y3OrvL1zZ1SqKfOpEkq3ImKY4JL5wEGwVqIZiSPBmtHwcuI375k2XMlbGCUsjElf8h6nBKwUtG94fwBEa/XQKVfcqjsFniMnrlc79bCXKxWUo9Epf7S7iqYxk0AFMSbw3ATCjGjgVLBxqZ0alhA6JH0WWCpJzEyYTU8e4wOrdHFPaVsS8FSdn8hIbMwojmxnTGBgfnsT8S8vSKF3HmZcJikwSWeLeqnAoPDkf9zlmlEQI0sI1dzeiumAaELBplSyIXx/iv8n/lG1VnWvjyv1izyNItpD++gQeegM1dEVaiAfUaTQI3pGLw44T86r8zZrLTj5zC76Aef9CxoQkVY=</latexit><latexit sha1_base64="YpFhaZHBfDESR4GPpU0Zw1k+JAM=">AAAB8XicdVBNSwMxEM3Wr1q/qh69BIvgqeyKX70VvXis4trCdinZNG1Ds8mSzCpl6c/w4kHFq//Gm//GtF2hij4YeLw3w8y8KBHcgOt+OoWFxaXlleJqaW19Y3OrvL1zZ1SqKfOpEkq3ImKY4JL5wEGwVqIZiSPBmtHwcuI375k2XMlbGCUsjElf8h6nBKwUtG94fwBEa/XQKVfcqjsFniMnrlc79bCXKxWUo9Epf7S7iqYxk0AFMSbw3ATCjGjgVLBxqZ0alhA6JH0WWCpJzEyYTU8e4wOrdHFPaVsS8FSdn8hIbMwojmxnTGBgfnsT8S8vSKF3HmZcJikwSWeLeqnAoPDkf9zlmlEQI0sI1dzeiumAaELBplSyIXx/iv8n/lG1VnWvjyv1izyNItpD++gQeegM1dEVaiAfUaTQI3pGLw44T86r8zZrLTj5zC76Aef9CxoQkVY=</latexit>
b
Orthant Ob
Simple but key idea: product of k-‐juntas = width-‐k CNF
Back to the Taylor expansion
More complicated, but same key ideas:
§ Product structure of § Sparsity of H
sup
v2Rm
(X
|↵|=c
|@↵ eOb(v)|)
. (logm)
c/2
�c
<latexit sha1_base64="zcnErT6IswuVo5XUaOgLms06khg=">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</latexit><latexit sha1_base64="zcnErT6IswuVo5XUaOgLms06khg=">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</latexit><latexit sha1_base64="zcnErT6IswuVo5XUaOgLms06khg=">AAACe3icdVHLjtMwFHXCayivAksWWBSkFqGSjBhgFkijYcOOAVFmpLqNbhwntcZ2rNgpqlx/BX/Gjk9hg3DaIg0IjmTp6NyH7z0314IbmyTfo/jS5StXr+1d7924eev2nf7de59N3TaUTWgt6uYsB8MEV2xiuRXsTDcMZC7YaX7+toufLlljeK0+2ZVmMwmV4iWnYIOU9b8S0+rMLTHhChMJdpHn7qOfS0+OeVURh0OCzNyagNALWL+hfk00NJaDyNxW9OQLL5jlomBu04KCcO+9z/LhcrTe9vEYE8GMMVxiUjZA3ZCIusJyNHf0+b73jogwdQFz6rP+IBknG+AL5CBJD1+mON0pA7TDSdb/RoqatpIpSwUYM00TbWeum5IK5nukNUwDPYeKTQNVIJmZuY15Hj8JSoHLuglPWbxRL1Y4kMasZB4yu93M37FO/Fds2try9cxxpVvLFN1+VLYC2xp3l8AFbxi1YhUI0IaHWTFdQLDGhnv1ggm/N8X/J5P98eE4+fBicHS8c2MPPUCP0BCl6BU6Qu/QCZogin5ED6NhNIp+xo/jp/GzbWoc7Wruoz8QH/wChofEbg==</latexit>
Claim: y fools the funcPon z 7! eOb(Hz + Tz)<latexit sha1_base64="0n//+HXi/Bo3CnKHqeLmLSzZu7s=">AAACFHicdZDLSgMxFIYz9VbrrerSTbAIFaXMiLfuim66s0JrC20pmcxpG5q5kGSUduhLuPFV3LhQcevCnW9jph2hiv4Q+PnOOeSc3w44k8o0P43U3PzC4lJ6ObOyura+kd3cupF+KCjUqM990bCJBM48qCmmODQCAcS1OdTtwWVcr9+CkMz3qmoYQNslPY91GSVKo072cIRbLgmk8nHrjjmgGHcg0kj1KeHR1XjcsfPlET7A1dF+J5szC+ZEeMacmFbx1MJWQnIoUaWT/Wg5Pg1d8BTlRMqmZQaqHRGhGOUwzrRCCQGhA9KDprYecUG2o8lVY7yniYO7vtDPU3hCZyci4ko5dG3dGa8rf9di+FetGarueTtiXhAq8Oj0o27Isc4gjgg7TABVfKgNoYLpXTHtE0Go0kFmdAjfl+L/Te2oUCyY18e50kWSRhrtoF2URxY6QyVURhVUQxTdo0f0jF6MB+PJeDXepq0pI5nZRj9kvH8Bd4Wejw==</latexit><latexit sha1_base64="0n//+HXi/Bo3CnKHqeLmLSzZu7s=">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</latexit><latexit sha1_base64="0n//+HXi/Bo3CnKHqeLmLSzZu7s=">AAACFHicdZDLSgMxFIYz9VbrrerSTbAIFaXMiLfuim66s0JrC20pmcxpG5q5kGSUduhLuPFV3LhQcevCnW9jph2hiv4Q+PnOOeSc3w44k8o0P43U3PzC4lJ6ObOyura+kd3cupF+KCjUqM990bCJBM48qCmmODQCAcS1OdTtwWVcr9+CkMz3qmoYQNslPY91GSVKo072cIRbLgmk8nHrjjmgGHcg0kj1KeHR1XjcsfPlET7A1dF+J5szC+ZEeMacmFbx1MJWQnIoUaWT/Wg5Pg1d8BTlRMqmZQaqHRGhGOUwzrRCCQGhA9KDprYecUG2o8lVY7yniYO7vtDPU3hCZyci4ko5dG3dGa8rf9di+FetGarueTtiXhAq8Oj0o27Isc4gjgg7TABVfKgNoYLpXTHtE0Go0kFmdAjfl+L/Te2oUCyY18e50kWSRhrtoF2URxY6QyVURhVUQxTdo0f0jF6MB+PJeDXepq0pI5nZRj9kvH8Bd4Wejw==</latexit>
To bound error term, use fact that Õb has small derivaPves (same as [HKM]):
We consider the mulPdimensional Taylor expansion:
Suffices for y to fool CNFs )<latexit sha1_base64="YpFhaZHBfDESR4GPpU0Zw1k+JAM=">AAAB8XicdVBNSwMxEM3Wr1q/qh69BIvgqeyKX70VvXis4trCdinZNG1Ds8mSzCpl6c/w4kHFq//Gm//GtF2hij4YeLw3w8y8KBHcgOt+OoWFxaXlleJqaW19Y3OrvL1zZ1SqKfOpEkq3ImKY4JL5wEGwVqIZiSPBmtHwcuI375k2XMlbGCUsjElf8h6nBKwUtG94fwBEa/XQKVfcqjsFniMnrlc79bCXKxWUo9Epf7S7iqYxk0AFMSbw3ATCjGjgVLBxqZ0alhA6JH0WWCpJzEyYTU8e4wOrdHFPaVsS8FSdn8hIbMwojmxnTGBgfnsT8S8vSKF3HmZcJikwSWeLeqnAoPDkf9zlmlEQI0sI1dzeiumAaELBplSyIXx/iv8n/lG1VnWvjyv1izyNItpD++gQeegM1dEVaiAfUaTQI3pGLw44T86r8zZrLTj5zC76Aef9CxoQkVY=</latexit><latexit sha1_base64="YpFhaZHBfDESR4GPpU0Zw1k+JAM=">AAAB8XicdVBNSwMxEM3Wr1q/qh69BIvgqeyKX70VvXis4trCdinZNG1Ds8mSzCpl6c/w4kHFq//Gm//GtF2hij4YeLw3w8y8KBHcgOt+OoWFxaXlleJqaW19Y3OrvL1zZ1SqKfOpEkq3ImKY4JL5wEGwVqIZiSPBmtHwcuI375k2XMlbGCUsjElf8h6nBKwUtG94fwBEa/XQKVfcqjsFniMnrlc79bCXKxWUo9Epf7S7iqYxk0AFMSbw3ATCjGjgVLBxqZ0alhA6JH0WWCpJzEyYTU8e4wOrdHFPaVsS8FSdn8hIbMwojmxnTGBgfnsT8S8vSKF3HmZcJikwSWeLeqnAoPDkf9zlmlEQI0sI1dzeiumAaELBplSyIXx/iv8n/lG1VnWvjyv1izyNItpD++gQeegM1dEVaiAfUaTQI3pGLw44T86r8zZrLTj5zC76Aef9CxoQkVY=</latexit><latexit sha1_base64="YpFhaZHBfDESR4GPpU0Zw1k+JAM=">AAAB8XicdVBNSwMxEM3Wr1q/qh69BIvgqeyKX70VvXis4trCdinZNG1Ds8mSzCpl6c/w4kHFq//Gm//GtF2hij4YeLw3w8y8KBHcgOt+OoWFxaXlleJqaW19Y3OrvL1zZ1SqKfOpEkq3ImKY4JL5wEGwVqIZiSPBmtHwcuI375k2XMlbGCUsjElf8h6nBKwUtG94fwBEa/XQKVfcqjsFniMnrlc79bCXKxWUo9Epf7S7iqYxk0AFMSbw3ATCjGjgVLBxqZ0alhA6JH0WWCpJzEyYTU8e4wOrdHFPaVsS8FSdn8hIbMwojmxnTGBgfnsT8S8vSKF3HmZcJikwSWeLeqnAoPDkf9zlmlEQI0sI1dzeiumAaELBplSyIXx/iv8n/lG1VnWvjyv1izyNItpD++gQeegM1dEVaiAfUaTQI3pGLw44T86r8zZrLTj5zC76Aef9CxoQkVY=</latexit>
@↵ eOb<latexit sha1_base64="miSnFXf1UjDzCzi0MX7FA6m9XBQ=">AAACBnicbVBNS8NAEN34WetX1aMgwSJ4Kon41VvRizcrWFtoQphspu3SzQe7G6WE3rz4V7x4UPHqb/Dmv3HTFtTqg4HHezPMzPMTzqSyrE9jZnZufmGxsFRcXlldWy9tbN7IOBUUGzTmsWj5IJGzCBuKKY6tRCCEPsem3z/P/eYtCsni6FoNEnRD6EaswygoLXmlHScBoRhwzwGe9MC5YwEqxgPMLoee75XKVuXIsqvHtmlVrBG+iT0hZTJB3St9OEFM0xAjRTlI2batRLlZvoNyHBadVGICtA9dbGsaQYjSzUZ/DM09rQRmJxa6ImWO1J8TGYRSDkJfd4agenLay8X/vHaqOqduxqIkVRjR8aJOyk0Vm3koZsAEUsUHmgAVTN9q0h4IoEpHV9Qh2NMv/yWNg0q1Yl0dlmtnkzQKZJvskn1ikxNSIxekThqEknvySJ7Ji/FgPBmvxtu4dcaYzGyRXzDevwB9Ipnp</latexit><latexit sha1_base64="miSnFXf1UjDzCzi0MX7FA6m9XBQ=">AAACBnicbVBNS8NAEN34WetX1aMgwSJ4Kon41VvRizcrWFtoQphspu3SzQe7G6WE3rz4V7x4UPHqb/Dmv3HTFtTqg4HHezPMzPMTzqSyrE9jZnZufmGxsFRcXlldWy9tbN7IOBUUGzTmsWj5IJGzCBuKKY6tRCCEPsem3z/P/eYtCsni6FoNEnRD6EaswygoLXmlHScBoRhwzwGe9MC5YwEqxgPMLoee75XKVuXIsqvHtmlVrBG+iT0hZTJB3St9OEFM0xAjRTlI2batRLlZvoNyHBadVGICtA9dbGsaQYjSzUZ/DM09rQRmJxa6ImWO1J8TGYRSDkJfd4agenLay8X/vHaqOqduxqIkVRjR8aJOyk0Vm3koZsAEUsUHmgAVTN9q0h4IoEpHV9Qh2NMv/yWNg0q1Yl0dlmtnkzQKZJvskn1ikxNSIxekThqEknvySJ7Ji/FgPBmvxtu4dcaYzGyRXzDevwB9Ipnp</latexit><latexit sha1_base64="miSnFXf1UjDzCzi0MX7FA6m9XBQ=">AAACBnicbVBNS8NAEN34WetX1aMgwSJ4Kon41VvRizcrWFtoQphspu3SzQe7G6WE3rz4V7x4UPHqb/Dmv3HTFtTqg4HHezPMzPMTzqSyrE9jZnZufmGxsFRcXlldWy9tbN7IOBUUGzTmsWj5IJGzCBuKKY6tRCCEPsem3z/P/eYtCsni6FoNEnRD6EaswygoLXmlHScBoRhwzwGe9MC5YwEqxgPMLoee75XKVuXIsqvHtmlVrBG+iT0hZTJB3St9OEFM0xAjRTlI2batRLlZvoNyHBadVGICtA9dbGsaQYjSzUZ/DM09rQRmJxa6ImWO1J8TGYRSDkJfd4agenLay8X/vHaqOqduxqIkVRjR8aJOyk0Vm3koZsAEUsUHmgAVTN9q0h4IoEpHV9Qh2NMv/yWNg0q1Yl0dlmtnkzQKZJvskn1ikxNSIxekThqEknvySJ7Ji/FgPBmvxtu4dcaYzGyRXzDevwB9Ipnp</latexit>
[Bentkus 90]
eOb(Hz) +X
1|↵|c
1
↵!@↵ eOb(Hz)(Tz)↵ ± err
<latexit sha1_base64="tEWNFxXepDCbGfynkXvGvGMZN/g=">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</latexit><latexit sha1_base64="tEWNFxXepDCbGfynkXvGvGMZN/g=">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</latexit><latexit sha1_base64="tEWNFxXepDCbGfynkXvGvGMZN/g=">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</latexit>
Previous slide
Recap
. . . y10 y2 y6 y3
y12 y9 y11 y4
y7
Ex
[ eOb(Ax)] ⇡ Ey
[ eOb(Ay)]<latexit sha1_base64="DdQ35NqPG8Ts7wvRtHDVXIeVkws=">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</latexit><latexit sha1_base64="DdQ35NqPG8Ts7wvRtHDVXIeVkws=">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</latexit><latexit sha1_base64="DdQ35NqPG8Ts7wvRtHDVXIeVkws=">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</latexit>
What we just sketched Bounding error incurred by a single swap:
. . . x10 x2 x6 x3
x12 x9 x11 x4
x7
Goal is to “fool” the orthant mollifier:
y pseudorandom: x uniform:
r-‐wise r-‐wise r-‐wise uniform uniform uniform
x10 x2 x6
uniform
y10 y2 y6
r-‐wise
Outline of the rest of the talk (= the structure of our proof)
1. A useful decomposiPon of polytopes
2. “Smooth version” of the problem
3. Proving the smooth version
4. Going from smooth version to actual version
small error region
≈ 1
≈ 0
1
0
Ex
[Ob(Ax)] ⇡ Ey
[Ob(Ay)]<latexit sha1_base64="0ybTWB/15GkgfKOkcmcJv90GMEA=">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</latexit><latexit sha1_base64="0ybTWB/15GkgfKOkcmcJv90GMEA=">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</latexit><latexit sha1_base64="0ybTWB/15GkgfKOkcmcJv90GMEA=">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</latexit>
eOb : Rm ! [0, 1]<latexit sha1_base64="1dx6Lx1GMaKrnNeKJiVqle7t6F8=">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</latexit><latexit sha1_base64="1dx6Lx1GMaKrnNeKJiVqle7t6F8=">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</latexit><latexit sha1_base64="1dx6Lx1GMaKrnNeKJiVqle7t6F8=">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</latexit>
Ob : Rm ! {0, 1}<latexit sha1_base64="bwg1OKOo/qSNMNQ/prYW+O2q8MM=">AAACEHicdVDLSsNAFJ3UV62vqEs3g0XoQkoivuqq6MadVYwtNDFMptN26OTBzEQoIb/gxl9x40LFrUt3/o2TNEIVPTBw5px7ufceL2JUSMP41Eozs3PzC+XFytLyyuqavr5xI8KYY2LhkIW84yFBGA2IJalkpBNxgnyPkbY3Osv89h3hgobBtRxHxPHRIKB9ipFUkqvXbB/JIUYsuUhdD57A/O95yVV660NbhtBOjF3TTl29atSNHHCKHBhm49CEZqFUQYGWq3/YvRDHPgkkZkiIrmlE0kkQlxQzklbsWJAI4REakK6iAfKJcJL8ohTuKKUH+yFXL5AwV6c7EuQLMfY9VZntK357mfiX141l/9hJaBDFkgR4MqgfM6gOzeKBPcoJlmysCMKcql0hHiKOsFQhVlQI35fC/4m1V2/Ujcv9avO0SKMMtsA2qAETHIEmOActYAEM7sEjeAYv2oP2pL1qb5PSklb0bIIf0N6/AKfAnHc=</latexit><latexit sha1_base64="bwg1OKOo/qSNMNQ/prYW+O2q8MM=">AAACEHicdVDLSsNAFJ3UV62vqEs3g0XoQkoivuqq6MadVYwtNDFMptN26OTBzEQoIb/gxl9x40LFrUt3/o2TNEIVPTBw5px7ufceL2JUSMP41Eozs3PzC+XFytLyyuqavr5xI8KYY2LhkIW84yFBGA2IJalkpBNxgnyPkbY3Osv89h3hgobBtRxHxPHRIKB9ipFUkqvXbB/JIUYsuUhdD57A/O95yVV660NbhtBOjF3TTl29atSNHHCKHBhm49CEZqFUQYGWq3/YvRDHPgkkZkiIrmlE0kkQlxQzklbsWJAI4REakK6iAfKJcJL8ohTuKKUH+yFXL5AwV6c7EuQLMfY9VZntK357mfiX141l/9hJaBDFkgR4MqgfM6gOzeKBPcoJlmysCMKcql0hHiKOsFQhVlQI35fC/4m1V2/Ujcv9avO0SKMMtsA2qAETHIEmOActYAEM7sEjeAYv2oP2pL1qb5PSklb0bIIf0N6/AKfAnHc=</latexit><latexit sha1_base64="bwg1OKOo/qSNMNQ/prYW+O2q8MM=">AAACEHicdVDLSsNAFJ3UV62vqEs3g0XoQkoivuqq6MadVYwtNDFMptN26OTBzEQoIb/gxl9x40LFrUt3/o2TNEIVPTBw5px7ufceL2JUSMP41Eozs3PzC+XFytLyyuqavr5xI8KYY2LhkIW84yFBGA2IJalkpBNxgnyPkbY3Osv89h3hgobBtRxHxPHRIKB9ipFUkqvXbB/JIUYsuUhdD57A/O95yVV660NbhtBOjF3TTl29atSNHHCKHBhm49CEZqFUQYGWq3/YvRDHPgkkZkiIrmlE0kkQlxQzklbsWJAI4REakK6iAfKJcJL8ohTuKKUH+yFXL5AwV6c7EuQLMfY9VZntK357mfiX141l/9hJaBDFkgR4MqgfM6gOzeKBPcoJlmysCMKcql0hHiKOsFQhVlQI35fC/4m1V2/Ujcv9avO0SKMMtsA2qAETHIEmOActYAEM7sEjeAYv2oP2pL1qb5PSklb0bIIf0N6/AKfAnHc=</latexit>
Ex
[ eOb(Ax)] ⇡ Ey
[ eOb(Ay)]<latexit sha1_base64="DdQ35NqPG8Ts7wvRtHDVXIeVkws=">AAACknicjVFda9swFJW9bsvSfXgre9qLaChkL8EuXbeyh6Urgz4U2kLTBmITZPmmEZEtIclbjPAf6s/Z2/7N5CSFZKxsF8Q9nHvuh+5NJWfahOEvz3+09fjJ09az9vbzFy9fBa/fXGtRKgoDKrhQw5Ro4KyAgWGGw1AqIHnK4SadnTTxm++gNBPFlakkJDm5LdiEUWIcNQ7u4pyYqZB24dPUfqvrsY1TwTNd5c7ZeV2P4h8sA8N4BksdJdyeO2HaPd6Uvk9wTKRUYt7+Z+Hq/wtXTeFx0Al74cLwGvgQRkeHEY5WTAet7GIc/IwzQcscCkM50XoUhdIklijDKIe6HZcaJKEzcgsjBwuSg07sYqc13nNMhidCuVcYvGDXMyzJdTOdUzaD6z9jDfm32Kg0k0+JZYUsDRR02WhScmwEbg6EM6aAGl45QKhiblZMp0QRatwZ224J9z/FD4PBfu+oF14edPpfV9tooXdoF3VRhD6iPjpFF2iAqBd4h94Xr++/9T/7x/7JUup7q5wdtGH+2W8LX88z</latexit><latexit sha1_base64="DdQ35NqPG8Ts7wvRtHDVXIeVkws=">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</latexit><latexit sha1_base64="DdQ35NqPG8Ts7wvRtHDVXIeVkws=">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</latexit>
What we’ve shown: What we’d like to show:
(closeness in CDF distance)
Another conceptual difference/challenge: Boolean vs. Gaussian anPconcentraPon
≈ 1
≈ 0
Proofs of CLTs (e.g. [HKM]): Gaussian anPconcentraPon
Since we are bypassing Gaussians: Have to instead reason about Boolean anPconcentraPon
(Fact: Boolean anPconcentraPon
Gaussian anPconcentraPon)
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AG, G N(0,1)n ~
Litlewood–Offord anPconcentraPon inequality
For all open intervals I ⇢ R of radius 2,
<latexit sha1_base64="gltKzb2mVrW89WLFxAVATWkF5vc=">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</latexit><latexit sha1_base64="gltKzb2mVrW89WLFxAVATWkF5vc=">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</latexit><latexit sha1_base64="gltKzb2mVrW89WLFxAVATWkF5vc=">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</latexit>
Prx⇠{±1}n
[w · x 2 I ] . 1pn.
<latexit sha1_base64="JhAARpKB7Fd3lbl+IEdqrhTLtZI=">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</latexit><latexit sha1_base64="JhAARpKB7Fd3lbl+IEdqrhTLtZI=">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</latexit><latexit sha1_base64="JhAARpKB7Fd3lbl+IEdqrhTLtZI=">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</latexit>
R<latexit sha1_base64="Y4cssu/Y5xLGAWxQ8Y3Sa5FTCBo=">AAAB8HicdVDLSsNAFL2pr1pfVZduBovgqiTiq7uiG5dVjC22oUymk3boZBJmJkIJ+Qs3LlTc+jnu/BsnbYQqemDgcM69zLnHjzlT2rY/rdLC4tLySnm1sra+sblV3d65U1EiCXVJxCPZ8bGinAnqaqY57cSS4tDntO2PL3O//UClYpG41ZOYeiEeChYwgrWR7nsh1iPfT2+yfrVm1+0p0Bw5sZ3GqYOcQqlBgVa/+tEbRCQJqdCEY6W6jh1rL8VSM8JpVuklisaYjPGQdg0VOKTKS6eJM3RglAEKImme0Giqzm+kOFRqEvpmMk+ofnu5+JfXTXRw7qVMxImmgsw+ChKOdITy89GASUo0nxiCiWQmKyIjLDHRpqSKKeH7UvQ/cY/qjbp9fVxrXhRtlGEP9uEQHDiDJlxBC1wgIOARnuHFUtaT9Wq9zUZLVrGzCz9gvX8BRsaQ2w==</latexit><latexit sha1_base64="Y4cssu/Y5xLGAWxQ8Y3Sa5FTCBo=">AAAB8HicdVDLSsNAFL2pr1pfVZduBovgqiTiq7uiG5dVjC22oUymk3boZBJmJkIJ+Qs3LlTc+jnu/BsnbYQqemDgcM69zLnHjzlT2rY/rdLC4tLySnm1sra+sblV3d65U1EiCXVJxCPZ8bGinAnqaqY57cSS4tDntO2PL3O//UClYpG41ZOYeiEeChYwgrWR7nsh1iPfT2+yfrVm1+0p0Bw5sZ3GqYOcQqlBgVa/+tEbRCQJqdCEY6W6jh1rL8VSM8JpVuklisaYjPGQdg0VOKTKS6eJM3RglAEKImme0Giqzm+kOFRqEvpmMk+ofnu5+JfXTXRw7qVMxImmgsw+ChKOdITy89GASUo0nxiCiWQmKyIjLDHRpqSKKeH7UvQ/cY/qjbp9fVxrXhRtlGEP9uEQHDiDJlxBC1wgIOARnuHFUtaT9Wq9zUZLVrGzCz9gvX8BRsaQ2w==</latexit><latexit sha1_base64="Y4cssu/Y5xLGAWxQ8Y3Sa5FTCBo=">AAAB8HicdVDLSsNAFL2pr1pfVZduBovgqiTiq7uiG5dVjC22oUymk3boZBJmJkIJ+Qs3LlTc+jnu/BsnbYQqemDgcM69zLnHjzlT2rY/rdLC4tLySnm1sra+sblV3d65U1EiCXVJxCPZ8bGinAnqaqY57cSS4tDntO2PL3O//UClYpG41ZOYeiEeChYwgrWR7nsh1iPfT2+yfrVm1+0p0Bw5sZ3GqYOcQqlBgVa/+tEbRCQJqdCEY6W6jh1rL8VSM8JpVuklisaYjPGQdg0VOKTKS6eJM3RglAEKImme0Giqzm+kOFRqEvpmMk+ofnu5+JfXTXRw7qVMxImmgsw+ChKOdITy89GASUo0nxiCiWQmKyIjLDHRpqSKKeH7UvQ/cY/qjbp9fVxrXhRtlGEP9uEQHDiDJlxBC1wgIOARnuHFUtaT9Wq9zUZLVrGzCz9gvX8BRsaQ2w==</latexit>
|I|=2
Let w 2 Rnwhere |wi| � 1 for all i.
<latexit sha1_base64="UUvxzzUOfvX8vGRoHAuL38u180Y=">AAACKnicdVBNT9tAEF1ToBC+UjhyGREjcYpsBG25IXrh0ANFBJDiEK0342TFem3tjhsi4//Dhb/CgR4A9coPwQlBAgTv9PTejGbeC1MlLXnegzPxZXJq+uvMbGVufmFxqfpt+dgmmRHYEIlKzGnILSqpsUGSFJ6mBnkcKjwJz38N/ZO/aKxM9BENUmzFvKtlJAWnUmpX9wLCC8p/I4Hbh0BqCGJOvTDMD4sz7UK/hwbBvey3c1lcBl0E34UoMcCVAle69QLa1ZpX90aAV2Tb83e+++CPlRob46BdvQ06ichi1CQUt7bpeym1cm5ICoVFJcgsplyc8y42S6p5jLaVj7IWsF4qndEHUaIJRurrjZzH1g7isJwcBrHvvaH4kdfMKPrZyqVOM0Itng9FmQJKYFgcdKRBQWpQEi6MLH8F0eOGCyrrrZQlvCSFz0ljs75T9/5s1Xb3xm3MsFW2xjaYz36wXbbPDliDCXbFbtgdu3eunX/Og/P/eXTCGe+ssDdwHp8A2vmlwg==</latexit><latexit sha1_base64="UUvxzzUOfvX8vGRoHAuL38u180Y=">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</latexit><latexit sha1_base64="UUvxzzUOfvX8vGRoHAuL38u180Y=">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</latexit> In fact:
[Erdős 45]
✓
n
bn/2c
◆· 2�n
<latexit sha1_base64="GEOQMZCro95ZUGOw04o4hnDwpa8=">AAACFnicdVC7SgNBFJ2NrxhfUUubwSDYGHeDr3RBG8sIxgSyMcxO7iZDZmeWmVkhLPkLG3/FxkLFVuz8GycPIYqe6nDOvdxzTxBzpo3rfjqZufmFxaXscm5ldW19I7+5daNloijUqORSNQKigTMBNcMMh0asgEQBh3rQvxj59TtQmklxbQYxtCLSFSxklBgrtfNFnwNOBfZpT0oN2Ochl1JhcVjy1ZgOfdqRBpdu0wMxbOcLbtEdA8+QY9crn3jYmyoFNEW1nf/wO5ImEQhDOdG66bmxaaVEGUY5DHN+oiEmtE+60LRUkAh0Kx3/NcR7Vung0MYJpTB4rM5upCTSehAFdjIipqd/eyPxL6+ZmPCslTIRJwYEnRwKE46NxKOScIcpoIYPLCFUMZsV0x5RhBpbZc6W8P0p/p/USsVy0b06KlTOp21k0Q7aRfvIQ6eogi5RFdUQRffoET2jF+fBeXJenbfJaMaZ7myjH3DevwC9Z59C</latexit><latexit sha1_base64="GEOQMZCro95ZUGOw04o4hnDwpa8=">AAACFnicdVC7SgNBFJ2NrxhfUUubwSDYGHeDr3RBG8sIxgSyMcxO7iZDZmeWmVkhLPkLG3/FxkLFVuz8GycPIYqe6nDOvdxzTxBzpo3rfjqZufmFxaXscm5ldW19I7+5daNloijUqORSNQKigTMBNcMMh0asgEQBh3rQvxj59TtQmklxbQYxtCLSFSxklBgrtfNFnwNOBfZpT0oN2Ochl1JhcVjy1ZgOfdqRBpdu0wMxbOcLbtEdA8+QY9crn3jYmyoFNEW1nf/wO5ImEQhDOdG66bmxaaVEGUY5DHN+oiEmtE+60LRUkAh0Kx3/NcR7Vung0MYJpTB4rM5upCTSehAFdjIipqd/eyPxL6+ZmPCslTIRJwYEnRwKE46NxKOScIcpoIYPLCFUMZsV0x5RhBpbZc6W8P0p/p/USsVy0b06KlTOp21k0Q7aRfvIQ6eogi5RFdUQRffoET2jF+fBeXJenbfJaMaZ7myjH3DevwC9Z59C</latexit><latexit sha1_base64="GEOQMZCro95ZUGOw04o4hnDwpa8=">AAACFnicdVC7SgNBFJ2NrxhfUUubwSDYGHeDr3RBG8sIxgSyMcxO7iZDZmeWmVkhLPkLG3/FxkLFVuz8GycPIYqe6nDOvdxzTxBzpo3rfjqZufmFxaXscm5ldW19I7+5daNloijUqORSNQKigTMBNcMMh0asgEQBh3rQvxj59TtQmklxbQYxtCLSFSxklBgrtfNFnwNOBfZpT0oN2Ochl1JhcVjy1ZgOfdqRBpdu0wMxbOcLbtEdA8+QY9crn3jYmyoFNEW1nf/wO5ImEQhDOdG66bmxaaVEGUY5DHN+oiEmtE+60LRUkAh0Kx3/NcR7Vung0MYJpTB4rM5upCTSehAFdjIipqd/eyPxL6+ZmPCslTIRJwYEnRwKE46NxKOScIcpoIYPLCFUMZsV0x5RhBpbZc6W8P0p/p/USsVy0b06KlTOp21k0Q7aRfvIQ6eogi5RFdUQRffoET2jF+fBeXJenbfJaMaZ7myjH3DevwC9Z59C</latexit>
PDF of w x .
(Exactly Pght for w = 1n)
A high-‐dimensional Litlewood–Offord inequality (LO: m=1 case)
?
Let A 2 Rm⇥nwhere |Aij | � 1 for all i, j.
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For all orthant boundaries B ⇢ Rmof width 2,
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Prx⇠{±1}n
[Ax 2 B ] .<latexit sha1_base64="6b2WMCRdieGLyKzZTBxVhVqwG/M=">AAACQnicdVDLSgMxFM34tr6qLt0Ei+BCyoz43GnduKxgVWzGkklTDeYxJBmxhPk3N/6AO3/AjQsVty7MtBVU9EK4h3POJfeeJOXM2DB8DIaGR0bHxicmS1PTM7Nz5fmFE6MyTWiDKK70WYIN5UzShmWW07NUUywSTk+T64NCP72h2jAlj203pbHAl5J1GMHWU63yORLYXqnU9boWrq7zvOVQonjbdIVv7jZHhgnkUCpghPILmTfR2j78aYGISVhDazFEnBrjB2CrXAmrYa/gN7AZRrtbEYwGTAUMqt4qP6C2Ipmg0hKOjWlGYWpjh7VlhNO8hDJDU0yu8SVteiixoCZ2vQxyuOKZNuwo7Z+0sMd+n3BYmGJb7ywuNb+1gvxLa2a2sxM7JtPMUkn6H3UyDq2CRaCwzTQllnc9wEQzvyskV1hjYn3sJR/C16Xwf9BYr+5Ww6ONyl5tkMYEWALLYBVEYBvsgUNQBw1AwB14Ai/gNbgPnoO34L1vHQoGM4vgRwUfn3DHsxY=</latexit><latexit sha1_base64="6b2WMCRdieGLyKzZTBxVhVqwG/M=">AAACQnicdVDLSgMxFM34tr6qLt0Ei+BCyoz43GnduKxgVWzGkklTDeYxJBmxhPk3N/6AO3/AjQsVty7MtBVU9EK4h3POJfeeJOXM2DB8DIaGR0bHxicmS1PTM7Nz5fmFE6MyTWiDKK70WYIN5UzShmWW07NUUywSTk+T64NCP72h2jAlj203pbHAl5J1GMHWU63yORLYXqnU9boWrq7zvOVQonjbdIVv7jZHhgnkUCpghPILmTfR2j78aYGISVhDazFEnBrjB2CrXAmrYa/gN7AZRrtbEYwGTAUMqt4qP6C2Ipmg0hKOjWlGYWpjh7VlhNO8hDJDU0yu8SVteiixoCZ2vQxyuOKZNuwo7Z+0sMd+n3BYmGJb7ywuNb+1gvxLa2a2sxM7JtPMUkn6H3UyDq2CRaCwzTQllnc9wEQzvyskV1hjYn3sJR/C16Xwf9BYr+5Ww6ONyl5tkMYEWALLYBVEYBvsgUNQBw1AwB14Ai/gNbgPnoO34L1vHQoGM4vgRwUfn3DHsxY=</latexit><latexit sha1_base64="6b2WMCRdieGLyKzZTBxVhVqwG/M=">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</latexit>
§ 1-‐dimensional LO + union bound:
§ We prove , which we show is Pght
§ Need various technical extensions for our purposes
O(m/pn)
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{ x : Ax ≤ b } \ { x : Ax ≤ b−2 }
O(
plogm/
pn)
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~<latexit sha1_base64="0mgXs7qU24tVY73fsygkWKmFsY8=">AAAB7HicdZDLSgMxFIbPeK31VnXpJlgEV0NGvHVXdOOygtMW2qFk0rSNzWSGJFMoQ9/BjQsVtz6QO9/GtB1FRX8IHL7/HHLOHyaCa4Pxu7OwuLS8slpYK65vbG5tl3Z26zpOFWU+jUWsmiHRTHDJfMONYM1EMRKFgjXC4dXUb4yY0jyWt2acsCAifcl7nBJjUb09YjSbdEpl7J5ir3LmIezimdAX8XJShly1Tumt3Y1pGjFpqCBatzycmCAjynAq2KTYTjVLCB2SPmvZUpKI6SCbbTtBh5Z0US9W9kmDZvT7REYircdRaDsjYgb6tzeFf3mt1PQugozLJDVM0vlHvVQgE6Pp6ajLFaNGjG1BqOJ2V0QHRBFqbEBFG8Lnpej/wj92Ky6+OSlXL/M0CrAPB3AEHpxDFa6hBj5QuIN7eIQnJ3YenGfnZd664OQze/BDzusHf+qPRw==</latexit><latexit sha1_base64="0mgXs7qU24tVY73fsygkWKmFsY8=">AAAB7HicdZDLSgMxFIbPeK31VnXpJlgEV0NGvHVXdOOygtMW2qFk0rSNzWSGJFMoQ9/BjQsVtz6QO9/GtB1FRX8IHL7/HHLOHyaCa4Pxu7OwuLS8slpYK65vbG5tl3Z26zpOFWU+jUWsmiHRTHDJfMONYM1EMRKFgjXC4dXUb4yY0jyWt2acsCAifcl7nBJjUb09YjSbdEpl7J5ir3LmIezimdAX8XJShly1Tumt3Y1pGjFpqCBatzycmCAjynAq2KTYTjVLCB2SPmvZUpKI6SCbbTtBh5Z0US9W9kmDZvT7REYircdRaDsjYgb6tzeFf3mt1PQugozLJDVM0vlHvVQgE6Pp6ajLFaNGjG1BqOJ2V0QHRBFqbEBFG8Lnpej/wj92Ky6+OSlXL/M0CrAPB3AEHpxDFa6hBj5QuIN7eIQnJ3YenGfnZd664OQze/BDzusHf+qPRw==</latexit><latexit sha1_base64="0mgXs7qU24tVY73fsygkWKmFsY8=">AAAB7HicdZDLSgMxFIbPeK31VnXpJlgEV0NGvHVXdOOygtMW2qFk0rSNzWSGJFMoQ9/BjQsVtz6QO9/GtB1FRX8IHL7/HHLOHyaCa4Pxu7OwuLS8slpYK65vbG5tl3Z26zpOFWU+jUWsmiHRTHDJfMONYM1EMRKFgjXC4dXUb4yY0jyWt2acsCAifcl7nBJjUb09YjSbdEpl7J5ir3LmIezimdAX8XJShly1Tumt3Y1pGjFpqCBatzycmCAjynAq2KTYTjVLCB2SPmvZUpKI6SCbbTtBh5Z0US9W9kmDZvT7REYircdRaDsjYgb6tzeFf3mt1PQugozLJDVM0vlHvVQgE6Pp6ajLFaNGjG1BqOJ2V0QHRBFqbEBFG8Lnpej/wj92Ky6+OSlXL/M0CrAPB3AEHpxDFa6hBj5QuIN7eIQnJ3YenGfnZd664OQze/BDzusHf+qPRw==</latexit>
Recap of proof structure
1. A useful decomposiPon of polytopes
2. “Smooth version” of the problem
3. Proving the smooth version
4. Going from smooth version to actual version
≈ 1
≈ 0
§ Previous best seed length had linear dependence on m
§ Many interesPng future direcPons:
§ Other complexity measures of polytopes?
§ ConnecPons to extension complexity?
§ PRGs for other geometric sets?
§ PRG for all convex sets?
Summary
An ε-‐PRG for m-‐facet polytopes over {0,1}n with seed length:
Our main result
of size n polylog(m) Discrepancy set
poly(log m, 1/ε) log n .