Composition theorems in communication complexity

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But note that this holds for any r ∈ G, thus also for the average of them. That is, ∑ χ i (s)χ j (t) = 1 ( ∑ )( ∑ ) χ i (r)χ j (r) χ i (s ′ )χ j (t |G| ′ ) = 0, (s,t)∈T r∈G (s ′ ,t ′ )∈T by the standard orthogonality property of different irreducible characters. ⇐: Since ∑ (s,t)∈T χ i(s)χ j (t) = 0, ∀i ≠ j, we know that T as a function is in span{χ i ⊗χ i : i}. Note that any linear combination of G invariant functions is also G invariant. Thus it remains to check that each basis χ i ⊗ χ i is G invariant, which is easy to see: This finishes the proof. χ i (rs)χ i (rt) = χ i (r)χ i (s)χ i (r)χ i (t) = χ i (s)χ i (t). Another nice property of Abelian groups is that the orthogonality condition condition implies the regularity one. Proposition 4. For an Abelian group G, if either T y,y is G invariant for all y or S x,x is G invariant for all x, then G|{g(x, y) : x ∈ X, y ∈ Y }. Proof. Note that T y,y (s, s) = |{x : g(x, y) = s}|, thus T y,y being G invariant implies that |{x : g(x, y) = s}| = |{x : g(x, y) = t}| for all s, t ∈ G. Thus the column y in matrix [g(x, y)] x,y , when viewed as a multiset, is equal to G repeated |Y |/|G| times. Therefore the whole multiset {g(x, y) : x ∈ X, y ∈ Y } is a multiple of G as well. What we finally get for Abelian groups is the following. Corollary 4. For a sign matrix A = [f(g(x, y))] x,y and an Abelian group G, if d(f, span(Ch Easy )) = Ω(1), and the multisets S x,x′ = {(g(x, y), g(x ′ , y)) : y ∈ Y } and T y,y′ = {(g(x, y), g(x, y ′ )) : x ∈ X} are G invariant for any (x, x ′ ) and any (y, y ′ ), then √ MN Q(A) ≥ log 2 max i∈Hard ‖[χ i (g(x, y))] x,y ‖ − O(1). 5.4 Block composed functions We now consider a special class of functions g: block composed functions. Suppose the group G is a product group G = G 1 × · · · × G t , and g(x, y) = (g 1 (x 1 , y 1 ), · · · , g t (x t , y t )) where x = (x 1 , · · · , x t ) and y = (y 1 , · · · , y t ). That is, both x and y are decomposed into t components and the i-th coordinate of g(x, y) only depends on the i-th components of x and y. The tensor structure makes all the computation easy. Theorem 8 can be generalized to the general product group case for arbitrary groups G i . Definition 7. The ɛ-approximate degree of a class function f on product group G 1 ×· · ·×G t , denoted by d ɛ (f), is the minimum d s.t. ‖f −f ′ ‖ ∞ ≤ ɛ, where f ′ can be represented as a linear combination of irreducible characters with at most d non-identity component characters. Theorem 12. For sign matrix A = [f(g 1 (x 1 , y 1 ), · · · , g t (x t , y t )] x,y where all g i satisfy their orthogonality conditions, we have Q(A) ≥ min {χ i},S ∑ i∈S log 2 √ size(Mgi ) deg(χ i )‖M χi◦g i ‖ − O(1) where the minimum is over all S ⊆ [n] with |S| > deg 1/3 (f), and all non-identity irreducible characters χ i of G i .
Proof. Recall that an irreducible character χ of G is the tensor product of irreducible characters χ i of each component group G i . Let Hard be the set of irreducible characters χ with more than d non-identity component characters. Fix a hard character χ, and denote by S the set of coordinates of its non-identity characters. ‖[χ(g(x, y))]‖ = ‖ ⊗ i∈[t][χ i (g i (x i , y i ))]‖ = ∏ ‖[χ i (g i (x i , y i ))]‖ i∈[t] = ∏ ‖[χ i (g i (x i , y i ))]‖ × ∏ ‖J |Xi|×|Y i|‖ i∈S i /∈S = ∏ ‖M χi◦g i ‖ × ∏ √ size(M gi ) i∈S i /∈S Thus by Theorem 11 and Eq. (5), we have √ ∏ size(Mgi ) Q(A) ≥ log 2 − O(1) (10) deg(χ i ) · ‖M χi◦g i ‖ proving the theorem. i∈S Previous sections as well as [SZ09b] consider the case where all g i ’s are the same and all G i ’s are Z 2 . In this case, the above bound is equal to the one in Theorem 8, and the following proposition says that the group invariance condition degenerates to the strongly balanced property. Proposition 5. For G = Z ×t 2 , the following two conditions for g = (g 1, · · · , g t ) are equivalent: 1. The multisets S x,x′ = {(g(x, y), g(x ′ , y)) : y ∈ Y } and T y,y′ = {(g(x, y), g(x, y ′ )) : x ∈ X} are G invariant for any (x, x ′ ) and any (y, y ′ ), 2. Each matrix [g i (x i , y i )] x i ,yi is strongly balanced. Proof. 1 ⇒ 2: S x,x′ being G invariant implies that for all {z i }, {u i }, {v i }, |{y : z i g i (x i , y i ) = u i , z i g i (x ′i , y i ) = v i , ∀i}| =|{y : g i (x i , y i ) = u i , g i (x ′i , y i ) = v i , ∀i}|. Take x ′ = x and u = v. Now for each i and each row x i , take z i = −1 (where the group Z 2 is represented by {±1}). For all other i ′ ≠ i, take z i ′ = 1. This assignment will show that |{y : g i (x i , y i ) = −u i }| = |{y : g i (x i , y i ) = u i }|. That is, the row x i in matrix [g i (x i , y i )] is balanced. Similarly we can show the balance for each column. 2 ⇒ 1: It is enough to show that for each i and each {z i }, {u i }, {v i }, |{y i : z i g i (x i , y i ) = u i , z i g i (x ′i , y i ) = v i }| =|{y i : g i (x i , y i ) = u i , g i (x ′i , y i ) = v i }|. First consider the case x ′i = x i . If u i ≠ v i then both numbers are 0; if u i = v i then both numbers are |{y i }|/2 by the balance of row x i . Now assume x ′i ≠ x i . Denote a bb ′ = |{y i :
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