The FT-Mollification Method Daniel M. Kane

Outline • • • • •

Introduction Linear Threshold Functions Multiple Linear Threshold Function Degree 2 PTFs Conclusion

K-Independence Def: A collection of random variables x1,…,xn are k-independent if any collection of at most k of the xi are independent. • k-independent families easy to produce • Often behave like fully independent families

Fooling Functions • Want expectations of functions to be correct • Y=(y1,y2,…,yn) Collection of independent random variables • X=(x1,x2,…,xn) k-independent, xi » yi • f:Rn ! R • Want to show that E[f(X)] ¼ E[f(Y)] • Note that if f is a polynomial of total degree at most k, then E[f(X)] = E[f(Y)]

Notation Def: We say that k-independence ²-fools f if for all k-independent X, |E[f(X)]-E[f(Y)]| (2c|W|2)2, log(1/²).

Anti-Concentration • |E[F(Y)] – E[F~(Y)] | · E[|F(Y) – F~(Y)|] • |f – f~| is small except near a • anti-concentrated

Approximation Error • | f – f~ | = O(min{1,|c(x-a)|-2}) ³X

4¡n P r(j < W; Y > ¡aj < 2n c¡1 ) µX ¡n ¶ 2 =O cjW j2 µ ¶ 1 =O : cjW j2

Error = O

´

Anti-Concentration for • To show E[F~(X)] ¼² E[F(X)] we need anticoncentration of Error = O

³X

´ 4¡n P r(j < W; X > ¡aj < 2nc¡1 )

• Have bounds for Y • Need k-indep. forces g(Z) = I[a-b,a+b]() to not have too large expectation. • Same idea only make g < g~ • Obtain g~ by mollifying 2I[a-2b,a+2b]

Fooling LTFs • Errors from approximating f by f~ of size O(1/(c|W|)). – Need c > 1/(² |W|)

• To have small error of f~ need – k > 2(c|W|)2 = O(²-2)

Positive Convolving Function • Often convenient to have ½ ¸ 0 – Then f~ 2 [inf(f), sup(f)]

• Use ½ = |FT(b)|2 for b compactly supported – Normalize by letting s ½ = |b|22 = 1

¯X µ ¶ ¯ ¯ ¯ k (k) (m) (k¡m) j½ j1 = ¯¯ < F T (b) ; F T (b) >¯¯ m Xµk¶ · jxm bj2 jxk¡m bj2 m Xµk¶ · jsupp(b)jk jbj22 = (2jsupp(b)j)k m

Multidimensional Mollification • m –dimensional space 2) for |r|R

³2 2´ m R2

:

• kth derivative (in any direction) is < 2k.

Fooling Intersections of Halfspaces • • • • • •

F a product of m LTFs Depends on bi o.n. basis for the Wi L(X)i:= F(X) = f(L(X)) Multivariate FT-Mollification

Basic Plan • E[F(Y)] ¼² E[F~(Y)] ¼² E[F~(X)] ¼² E[F(X)] • Get f~ by convolving with cm ½(cx) for ½ the multivariate mollification function

Anti-Concentration µ ½ ³ ´ ¾¶ 2 m ~ = O min 1; jf ¡ fj : cd

Where d is distance to the nearest of the m hyperplanes.

~ )j] = O E[jf(Y ) ¡ f(Y

c=-

³

´ 2

m ²

³

´ 2

m c

:

Polynomial Approximation • • • • • • •

Approximate f~ by degree k-1 Taylor poly at 0 Taylor error along line 0 to L(X) Error · |f~(k) |1 E[|L(X)|k ] / k! |f~(k)|1 · |f|1 |½(k)|1 · (2c)k E[|L(X)|k ] · mk/2 kk/2 Error · (4m c2 / k )k/2 k >> m c2 >> m5 ²-2

Quadratic Moment Bound • Thrm: Q a quadratic form, X Gaussian, ³ ´k p E[jQ(X)jk ] = O E[Q(X)] + jQj2 k + jQj1k :

– |Q|2 Frobenius norm – |Q|1 largest eigenvalue

• |L(X)|2 a quadratic form • E[|L(X)|k] = O(m + m k1/2 + k)k/2 • k = O( m4 ²-2 )

Positive Value Approximations • If f~ is bounded, it’s derivatives cannot decay faster than exponentially (look at the FT) • If f~ only bounded on positive numbers can do better (e.g. cos(x1/2)) • In general: – g(x) = f(x2) – f~ = (g * ½c ) (x1/2) – f~(k) = O(c k-1)k – f~ approximates f on R+

Fooling Degree-2 PTFs • F(X) = I[0,1) (p(X)), p degree 2 poly • Cannot use FT-Mollification Naively, f~(p(X)) • Error from polynomial approximation: Error · jf~(k) jE[jp(X)jk ]=k! · ck kk jpjk2 =k! = O (cjpj2 )k :

• Cannot make c big enough.

Decomposition • Suppose |p|2 = 1 • p(X) = Q(X) + L(X) + C – Quadratic, Linear, Constant

• Q(X) = Q+(X) – Q-(X) + Q0(X) – Eigenvalues > ± – Eigenvalues < - ± – |Eigenvalues | · ±

• Q+, Q- positive • |Q0|1 < ±

Decomposition • M(X) = (m1(X),m2(X),m3(X),m4(X)) – m1(X) = Q+(X)1/2 – m2(X) = Q-(X)1/2 – m3(X) = Q0(X) – tr(Q0) – m4(X) = L(X)

• f(w,x,y,z) = I[0,1)(w2 - x2 + y + z + tr(Q0) + C) • F(X) = f(M(X)) • f~ is f convolved with ½c.

Anti-Concentration • |f – f~| small unless M(X) within » c-1 of boundary • |m1(X)|,|m2(X)| = O(±-1/2) w.h.p. • |f – f~| small unless p(X) within » ±-1/2 c-1 of 0 • Gaussian anti-concentration • Error = O( ±-1/4 c-1/2 )

Polynomial Approximation • Error = |f~(k)| E[|M(X)|k] / k! – |f~(k)| = O(c)k – E[|m1(X)|k] , E[|m2(X)|k] = -1/2 k th 1/2 E[k/2 moment of quadratic] = O(k +± ) k k – E[|m3(X)| ] = E[|Q0 –tr(Q0 )| ] = O(k1/2 + k± )k k 1/2 – E[|m4(X)|k] = E[L(X) ] = O(k )k

• Error = O(c

k-1/2

+ c± +

-1/2 -1 k c± k )

Putting it Together • Need: – ck-1/2