An approximation algorithm for approximation rank

An appro ximation algorithm for appro ximation rank T roy Lee ∗ Adi Shraibm an † Marc h 9, 2021 Abstract One of the strongest tec hniques a v a ilable for showing lo w er b ound s on b ounded-err or comm unication complexit y is the logarithm of the appro ximation rank of the comm unication ma trix—the minim um rank of a matrix whic h is close to the comm unication matrix in ℓ ∞ norm. Krause show ed that the logarithm of approximati on rank is a lo w er b ound in the rand omized case, and la ter Buhrman and de W olf sho w ed it could also b e used for quantum comm unication complexit y . As a lo w er b ound tec hnique, appro ximation rank has t w o main drawbac ks: it i s difficult to compute, and it is not kn o wn to lo we r b oun d the mo del of quant um comm unication complexit y with enta nglement . Linial and S hraibman recently introd uced a quantit y , called γ α 2 , to quan tum comm unication complexit y , sh o wing that it can b e us ed to lo w er b oun d comm unication in the mo del with shared en tanglemen t. Here α is a measure of app r o ximation whic h is related to the allo w- able error pr obabilit y of the p roto col. This quanti t y can b e written as a semidefinite pr ogram and giv es b ound s at least as large as man y tec hniques in the l iterature, alt hough it is smaller than the corre- sp ondin g α -appro ximation rank, rk α . W e sho w that in fact log γ α 2 ( A ) and log rk α ( A ) agree u p to small factors. As corollaries we obtain a constan t factor p olynomial time appro ximation algorithm to the loga- rithm of appro ximation rank, and that the loga rithm of appro ximation rank is a low er b oun d for quantum comm unication complexit y with en tanglemen t. ∗ Cent re for Quantum So ft ware a nd Infor mation, University of T e c hnolog y Sydney . Email:troyjlee@gmail.com † The Academic Co lle ge of T el-Aviv-Y affo, T el-Aviv, Isr ael. Email: adish@mta.ac.il 1 1 In tro d uction Often when trying to sho w that a pro blem is computat io nally hard we our- selv es face a computationally hard problem. The minimum cost algorithm for a pro blem is naturally phrased as a n optimization pro blem, and frequen tly tec hniques to low er b ound this cost are also hard com binatorial optimization problems. When ta king suc h a computational view of lo w er bounds, it is natural t o b orrow ideas from approx imation algorithms whic h hav e ha d a go o d deal of success in dealing with NP-har dness. Beginning with the seminal approx i- mation algorithm for MAX CUT of G o emans and Williamson [G W95], a no w common appro ac h to ha r d combinatorial optimization problems is to lo ok at a semidefin ite relaxation of the pro blem with the hop e of sho wing that suc h a relaxation pro vides a go o d appro ximation to the original problem. W e tak e this approac h in dealing with appro ximation rank, an optimiza- tion problem that arises in communic ation complexit y . In communic ation complexit y , in tro duced by Y ao [Y ao79], t w o parties Alice and Bob wish to compute a function f : X × Y → {− 1 , +1 } , where Alice receiv es x ∈ X and Bob receiv es y ∈ Y . The question is how muc h they hav e to comm u- nicate to ev alua te f ( x, y ) for the most difficult pa ir ( x, y ). Ass o ciate to f a | X | -b y- | Y | c ommunic ation matrix M f where M f [ x, y ] = f ( x, y ). A we ll- kno wn low er b ound on the deterministic comm unication complexit y of f due to Mehlhorn a nd Sc hmidt [MS82] is log rk( M f ). This low er b ound has man y nice features—rank is easy to compute, a t least from a theoretical p ersp ec- tiv e, and the famous log rank conjecture of L ov´ asz a nd Saks [LS88] asserts that this b ound is nearly tigh t in the sense that t here is a univ ersal constan t c suc h that (log rk( M f )) c is an upp er b ound on the deterministic comm uni- cation complexit y of f , for ev ery function f . When we lo o k at b ounded-error randomized comm unication complexity , where Alice and Bob ar e allow ed to flip coins and answ er incorrectly with some small probabilit y , the rele v a nt quantit y is no longer rank but appr oxima - tion r a n k . F o r a sign matrix A , the α -approx imation rank, denoted r k α ( A ), is the minim um rank of a matr ix B whic h has the same sign pattern as A and whose en tries ha v e ma g nitude b et w een 1 and α . When used to low er b ound randomized communication complexit y , the appro ximation factor α is related to the allo w able error pro babilit y of the proto col. In the limit as α → ∞ we obtain the sign ra nk, denoted rk ∞ ( A ), the minim um rank of a matrix with the same sign pattern as A . P aturi and Simon [PS86] show ed 2 that the log rk ∞ ( M f ) exactly c haracterizes the un b ounded error complexit y of f , where Alice and Bob only hav e to get the correct answ er on ev ery input with probability strictly larger than 1 / 2. Krause [Kra96] extended this to the b ounded-error case b y show ing that log rk α ( M f ) is a lo w er b ound on the α − 1 2 α -error randomized comm unication complexit y of f . Later, Buhrman and de W olf [BW01] sho w ed that o ne-half this quantit y is also a low er b ound on the bo unded-error quantum comm unication complexit y of f , when the play- ers do not share entanglemen t. Appro ximation ra nk is one of the strong est lo w er b ound t echniq ues a v ailable for either of these b ounded-error mo dels, giving b ounds at least as large as those giv en by the discrepancy metho d, a metho d based on F ourier co efficien t s dev elop ed by Raz [Ra z95], and quan- tum low er metho ds o f Klauc k [K la01] and Ra zb oro v [Ra z03 ]. Notable ex- ceptions include the corruption b ound [Y ao83] and infor ma t io n theory meth- o ds [CSWY01], b oth of which can sho w an Ω( n ) lo w er b ound on the com- m unication complexit y of disjoin tness [Raz92, BYJKS04], whereas the loga- rithm of the appro ximation rank, a nd quantum comm unication complexit y , are Θ( √ n ) for this problem [Raz03, AA05]. In view of the log rank conjec- ture it is natural to conjecture as w ell that a p olynomial in the logarithm of approxim ation rank is an upper b ound on randomized commu nication complexit y . As a low er b ound tec hnique, ho w ev er, approximation rank suffers from t w o deficiencies. The first is that it is quite difficult to compute in practice. Although w e do not kno w if it is NP-hard to compute, the class of prob- lems minimizing rank sub ject to linear constrain ts do es con tain NP-hard instances (see, for example, Section 7.3 in the surv ey of V andenberghe and Bo yd [VB96]). The second drawbac k is that it is not kno wn to low er b ound quan tum comm unication complexit y with en tanglemen t. W e address b oth of t hese problems. W e mak e use of a quan tity γ α 2 whic h w as intro duced in the contex t of comm unication complexit y b y Linia l et al. [LMSS07]. This quantit y can nat urally b e view ed as a semidefinite relaxatio n of rank, a nd it is not hard to sho w that ( 1 α γ α 2 ( A )) 2 ≤ rk α ( A ) fo r a sign matrix A (see Prop o sition 5). W e sho w that this lo w er b ound is in fact fairly tigh t. Theorem 1 L et 1 < α < ∞ . Then for any m -by- n si g n matrix A 1 α 2 γ α 2 ( A ) 2 ≤ rk α ( A ) = 2 16 α 6 ( α − 1) 6 γ α 2 ( A ) 6 ln 3 (8 mn ) . The quantit y γ α 2 ( A ) can b e written as a semidefinite program and so can b e computed up to additiv e error ǫ in time p olynomial in the size of A and 3 log(1 / ǫ ) b y the ellipsoid metho d (see, for example, the textb o ok [GLS88 ]) . Th us Theorem 1 giv es a constan t factor p olynomial time approximation al- gorithm to compute lo g r k α ( A ). Moreo v er, the pro of of this theorem gives a metho d to find a near optimal lo w rank appro ximation to A in r a ndomized p olynomial time. Linial and Shraibman [LS07] hav e sho wn that log γ α 2 ( A ) − log α − 2 is a low er b ound o n the α − 1 2 α -error quan tum comm unication complexit y of the sign matrix A with entanglemen t, thus we also obtain the following corollary . Corollary 2 L et 0 < ǫ < 1 / 2 . L e t Q ∗ ǫ ( A ) b e the q uan tum c omm unic ation c omp l e xity of a m -by- n s i g n matrix A w ith en tanglement. T hen Q ∗ ǫ ( A ) ≥ 1 6 log rk α ǫ ( A ) − 1 2 log log ( mn ) − log α 2 ǫ α ǫ − 1 − O (1) , wher e α ǫ = 1 1 − 2 ǫ . The log log f a ctor is necessary as the n -bit equality function with comm u- nication matrix of size 2 n -b y-2 n has approxim ation ra nk Ω( n ) [Alo09], but can b e solv ed by a b ounded-error quan tum pro t o col with entangleme nt—or randomized proto col with public coins—with O (1) bits of comm unication. This corollary means that appro ximation rank cannot b e used to sho w a large gap b etw een the mo dels of quantum comm unication complexit y with and without en tanglemen t, if indeed suc h a gap exists. Our pro of w orks roughly as follows. Note that the rank of a m -b y- n matrix A is the smallest k suc h that A can b e factored as A = X Y T where X is a m -b y- k matrix and Y is a n -by- k matrix. The factorization norm γ 2 ( A ) can b e defined as min X,Y : X Y T = A r ( X ) r ( Y ) where r ( X ) is the largest ℓ 2 norm of a ro w of X . Let X 0 , Y 0 b e an optimal solution to this program so that all row s of X 0 , Y 0 ha v e squared ℓ 2 norm at most γ 2 ( A ). The problem is that, although the row s o f X 0 , Y 0 ha v e small ℓ 2 norm, they might still ha v e large dimension . Intuitiv ely , how eve r, if the rows of X 0 ha v e small ℓ 2 norm but X 0 has man y columns, then one w ould think that man y of the columns are rather sparse and one could someho w compres s the matrix without causing to o muc h damage. The Johnson-Lindenstrauss dimension reduction lemma [JL84] can b e used to mak e this in tuition precise. W e randomly pro ject X 0 and Y 0 to matrices X 1 , Y 1 with column space of dimension roughly ln( mn ) γ α 2 ( A ) 2 . One can argue that with high probabilit y after suc h a pro jection X 1 Y T 1 still pro vides a decen t approx imation to A . In the second step of the pro o f, w e do 4 an error reduc tion step to sho w that one can then impro v e this approx imation without increasing the rank of X 1 Y T 1 b y to o m uc h. Ben-Da vid, Eiron, and Simon [BES02] hav e previously used this dimen- sion reduction techniq ue to sho w that rk ∞ ( A ) = O (ln( mn ) γ ∞ 2 ( A ) 2 ) fo r a sign matrix A . In this limiting case, how eve r, γ ∞ 2 ( A ) fails to b e a low er bo und on rk ∞ ( A ). Buhrman, V ereshc ha gin, and de W olf [BVW07], and independently Shersto v [She08], hav e giv en an example of a sign matrix A where γ ∞ 2 ( A ) is exp o nen tially larger than rk ∞ ( A ). 2 Preliminaries W e mak e use of the Johnson-Lindenstrauss lemma [JL84]. W e state it here in a form from [BES02] whic h is most con v enien t for our use. Lemma 3 (Cor ollary 19, [BES02]) L et x, y ∈ R r . L et R b e a r andom k -by- r matrix with entries ind e p endent an d identic al ly distribute d ac c or di n g to the normal distribution with me an 0 and varianc e 1 . Then for every δ > 0 Pr R  |h Rx, Ry i − h x, y i| ≥ δ 2  k x k 2 2 + k y k 2 2   ≤ 4 exp( − δ 2 k / 8) . 2.1 Matrix notation W e will w ork with real matrices and v ectors throughout this pap er. F o r a v ector u , we use k u k for the ℓ 2 norm of u , and k u k ∞ for t he ℓ ∞ norm of u . F or a matrix A , let A T denote the transp ose of A . W e let A ◦ B denote the en trywise pro duct of A and B . W e use S n + to denote the set of n -b y- n symmetric p ositive semidefinite matrices. F or a symmetric p ositive semidefinite matrix M let λ 1 ( M ) ≥ · · · ≥ λ n ( M ) ≥ 0 b e the eigenv a lues of M . W e define the i th singular v alue of A , denoted σ i ( A ), as σ i ( A ) = p λ i ( AA T ). The ra nk o f A , denoted rk( A ) is the n um b er o f no nzero singular v alues of A . W e will use sev eral matrix norms. • Sp ectral or op erator norm: k A k = σ 1 ( A ). • T race norm: k A k tr = P i σ i ( A ). • F rob enius norm: k A k F = p P i σ i ( A ) 2 . 5 One can alternativ ely see that k A k 2 F = T r( AA T ) = P i,j A [ i, j ] 2 . Our main to ol will b e the factorizatio n norm γ 2 [TJ89], in tro duced in the con text of complexity measures o f matrices by Linial et al. [LMSS07]. This norm can naturally b e view ed a s a semidefinite programming relaxation of rank as we now explain. W e tak e the follo wing as our primary definition o f γ 2 : Definition 4 ( [TJ89, LMSS07]) L et A b e a m -by- n matrix. Then γ 2 ( A ) = min X,Y : X Y T = A r ( X ) r ( Y ) , wher e r ( X ) is the lar gest ℓ 2 norm of a r ow of X . W e can write γ 2 ( A ) as the optim um v alue of a semidefinite program as follo ws. γ 2 ( A ) = min P ∈ S m + n + c P [ i, i ] ≤ c for all i = 1 , . . . , m + n P [ i, j + m ] = A [ i, j ] for i = 1 , . . . , m, j = 1 , . . . , n This is b ecause giv en a factor ization X Y T = A , we can create a p o sitive semidefinite matrix P =  X X T X Y T Y X T Y Y T  satisfying the constraints of this semidefinite program. Conv ersely , giv en a p ositiv e semidefinite matrix P satisfying the constrain ts of the program, w e can write P = Z Z T and let X b e the first m rows of Z and Y the last n ro ws of Z to obtain a factorization of A = X Y T . The quantit y γ 2 can equiv alently b e written a s the optimum of a max- imization problem kno wn a s the Sc h ur pro duct op erator norm: γ 2 ( A ) = max X : k X k =1 k A ◦ X k . The b o ok of Bhatia (Thm. 3.4.3 [Bha07]) con tains a nice discussion of this equiv a lence and attributes it to an unpublished man uscript of Haag erup. An alternat iv e pro of can b e obtained b y dualizing the ab o v e semidefinite progra m [LS ˇ S08]. More con v enien t for our purp oses will b e a form ulation of γ 2 in terms of the trace nor m. One can see that this next formulation is equiv alen t to the Sc h ur pro duct op erator norm formulation using the fact that k A k tr = max B : k B k≤ 1 T r ( AB T ). 6 Prop osition 5 (cf. [LS ˇ S08]) L et A b e a matrix. Then γ 2 ( A ) = max u,v k u k = k v k =1 k A ◦ v u T k tr F ro m this fo r mulation w e can easily see the connection of γ 2 to matrix rank. This connection is w ell kno wn in Banac h spaces theory , where it is pro v ed in a more general setting, but the follo wing pro of is more elemen tary . Prop osition 6 ( [TJ89, LS ˇ S08]) L et A b e a matrix. Then rk( A ) ≥ γ 2 ( A ) 2 k A k 2 ∞ . Pro of: Let u, v b e unit vec tors suc h that γ 2 ( A ) = k A ◦ v u T k tr . As the rank of A is equal t o the n um b er of no nzero singular v alues of A , w e see b y the Cauc h y-Sc h w arz inequalit y that rk( A ) ≥ k A k 2 tr k A k 2 F . As rk( A ◦ v u T ) ≤ rk( A ) w e obta in rk( A ) ≥ k A ◦ v u T k 2 tr k A ◦ v u T k 2 F ≥ γ 2 ( A ) 2 k A k 2 ∞ ✷ Finally , we define the approximate v ersion of the γ 2 norm. Definition 7 ( [LS07]) L et A b e a sign matrix, and let α ≥ 1 . γ α 2 ( A ) = min B :1 ≤ A [ i,j ] B [ i,j ] ≤ α γ 2 ( B ) γ ∞ 2 ( A ) = min B :1 ≤ A [ i,j ] B [ i,j ] γ 2 ( B ) W e define approximation rank similarly . 7 Definition 8 (approxim ation r ank) L et A b e a sign matrix, and α ≥ 1 . rk α ( A ) = min B :1 ≤ A [ i,j ] B [ i,j ] ≤ α rk( B ) rk ∞ ( A ) = min B :1 ≤ A [ i,j ] B [ i,j ] rk( B ) As corollary of Prop osition 6 w e get Corollary 9 L et A b e a sign matrix and α ≥ 1 . rk α ( A ) ≥ 1 α 2 γ α 2 ( A ) 2 3 Main Result In t his section we presen t our main r esult relating γ α 2 ( A ) and rk α ( A ). W e sho w this in t w o steps: first using dimension reduction w e upp er b ound rk α ′ ( A ) in terms of γ α 2 ( A ) where α ′ is sligh tly larger than α . In the second step o f error reduction w e show ho w to decrease the error bac k to α without increasing the rank to o m uc h. 3.1 Dimension reduction Theorem 10 L et A b e a m -by- n sign matrix a nd α ≥ 1 . Then for any 0 < t < 1 rk α + t 1 − t ( A ) ≤ 8 γ α 2 ( A ) 2 ln(8 mn ) t 2 Pro of: Supp ose that γ α 2 ( A ) = γ . By t he formulation in Definition 4, this means there is a set of v ectors x i ∈ R r for i = 1 , . . . , m and y j ∈ R r for j = 1 , . . . , n suc h that • 1 ≤ h x i , y j i A [ i, j ] ≤ α for all i, j • k x i k 2 , k y j k 2 ≤ γ for all i, j . Applying the Johnson- L indenstrauss lemma (Lemma 3) with δ = t/γ w e ha v e Pr R [ |h Rx i , R y j i − h x i , y j i| ≥ t ] ≤ 4 exp  − t 2 k 8 γ 2  , 8 where the probability is tak en o v er k -by- r matrices with eac h en try c hosen indep enden tly and iden tically distributed according to t he standard no r - mal distribution. By taking k = 8 γ 2 ln(8 mn ) /t 2 w e can make the fail- ure probability at most 1 / (2 mn ). Th us by a union b ound w e ha v e that |h Rx i , R y j i − h x i , y j i| ≤ t for all i, j with probabilit y at least 1 / 2 ov er the c hoice of R . Th us there exists a matrix R 0 where this holds and by defining the m -by- n mat r ix B where B ( i, j ) = 1 1 − t h R 0 x i , R 0 y j i , we see that B has rank at most k and giv es an α + t 1 − t -approx imation to A . ✷ 3.2 Error-reduction In this section, we will see ho w to improv e the approxim ation factor a matrix A ′ giv es to a sign matrix A without increasing its rank by to o m uc h. W e do this b y applying a low-degree p o lynomial appro ximation of the sign function to t he entries of A ′ . This tec hnique has b een used sev eral times b efore. The tric k of controlling the v a lue of inner pro ducts h x, y i by taking (sums of ) tensor pro ducts of x and y can b e found in Kr ivine’s pro of of Grothendiec k’s inequalit y [Kri7 9]; results more sp ecifically related to our context can b e found, for example, in [Alo03, KS07 ]. W e first need a lemma of Alon [Alo03] a b out ho w applying a degree d p olynomial to a matrix entryw ise can increase its rank. F or completeness we giv e the pro of of a w eak er vers ion of this lemma here. Let p ( x ) = a 0 + a 1 x + . . . + a d x d b e a degree d p olynomial. F o r a mat r ix A , w e define p ( A ) to b e the matrix a 0 J + a 1 A + . . . + a d A ◦ d where A ◦ s is the matrix whose ( i, j ) en try is A [ i, j ] s , and J is the all ones matrix. Lemma 11 L et A b e a matrix and p b e a de gr e e d p olynomial . Then rk( p ( A )) ≤ ( d + 1)rk( A ) d Pro of: The result follo ws using subadditivit y of rank and that rank is mul- tiplicativ e under tensor pro duct. W e ha v e rk( A ◦ s ) ≤ rk( A ⊗ s ) = rk( A ) s since A ◦ s is a submatrix of A ⊗ s . ✷ In general for any constan ts 1 < β ≤ α < ∞ one can show that there is a constan t c suc h that rk β ( A ) ≤ rk α ( A ) c b y lo oking a t lo w degree approxim a- tions of the sign function (see Corollary 1 of [K S0 7] fo r suc h a statemen t). 9 As we are in terested in the sp ecial case where α, β are quite close, we giv e an explicit construction in a n attempt to k eep the exp onen t as small as p ossible. Prop osition 12 Fix ǫ > 0 . L et a 3 = 1 / (2 + 6 ǫ + 4 ǫ 2 ) , and a 1 = 1 + a 3 . Then the p olynomial p ( x ) = a 1 x − a 3 x 3 maps [1 , 1 + 2 ǫ ] into [1 , 1 + ǫ ] and [ − 1 − 2 ǫ, − 1] into [ − 1 − ǫ, − 1] . Pro of: As p is an o dd p olynomial, w e only need to che c k that it maps [1 , 1 + 2 ǫ ] in to [1 , 1 + ǫ ]. With our c hoice of a 1 , a 3 , w e see that p (1) = p (1 + 2 ǫ ) = 1. F urthermore, p ( x ) ≥ 1 for all x ∈ [1 , 1 + 2 ǫ ], th us w e just need to che c k t ha t the maxim um v alue of p ( x ) in this interv al do es not exceed 1 + ǫ . Calculus sho ws that the maximum v alue of p ( x ) is attained at x = ( 1+ a 3 3 a 3 ) 1 / 2 . Plugging this in to the expression for p ( x ), w e see that the maxi- m um v alue is max x ∈ [1 , 1+2 ǫ ] p ( x ) = 2 3 √ 3 (1 + a 3 ) 3 / 2 √ a 3 . W e w ant to show that this is at most 1 + ǫ , or equiv alently that 2 3 √ 3 √ 2 + 6 ǫ + 4 ǫ 2 1 + ǫ  3 + 6 ǫ + 4 ǫ 2 2 + 6 ǫ + 4 ǫ 2  3 / 2 ≤ 1 . One can v erify that this inequalit y is true for all ǫ ≥ 0. ✷ 3.3 Putting ev erything together No w w e are ready to put ev erything together. Theorem 1 L et 1 < α < ∞ . Then for any m -by- n si g n matrix A 1 α 2 γ α 2 ( A ) 2 ≤ rk α ( A ) = 2 16 α 6 ( α − 1) 6 γ α 2 ( A ) 6 ln 3 (8 mn ) . Pro of: W e first a pply Theorem 10 with t = α − 1 2 α . With this choice, α + t 1 − t = 2 α − 1, th us w e obtain rk 2 α − 1 ( A ) ≤ 32 α 2 ( α − 1) 2 γ α 2 ( A ) 2 ln(8 mn ) . 10 No w we can use the p olynomial constructed in Prop osition 12 and Lemma 11 to obtain rk α ( A ) ≤ 2 r k 2 α − 1 ( A ) 3 ≤ 2 16 α 6 ( α − 1) 6 γ α 2 ( A ) 6 ln 3 (8 mn ) . ✷ 4 Discuss ion and op en problems One o f the f undamental questions of quan tum inf o rmation is the p ow er of en tanglemen t. If we believ e that there can b e a large gap b etw een the com- m unication complexit y of a function with and without en tanglemen t then w e m ust dev elop techniq ues to low er b o und quan tum comm unication complexit y without en tanglemen t t hat do not also w ork for communic ation complexit y with en tanglemen t. W e ha v e eliminated one of these p ossibilities in appro x- imation rank. As can b e seen in Theorem 1, the relationship b etw een γ α 2 ( A ) and rk α ( A ) w eak ens as α → ∞ b ecause the lower b ound b ecomes worse . Indeed, Buhrman, V ereshc hagin, and de W olf [BVW07], a nd indep enden tly Shersto v [She08], ha v e give n examples where γ ∞ 2 ( A ) is exp onentially larg er tha n rk ∞ ( A ). It is an in teresting op en problem to find a p olynomial time approxim ation al- gorithm for the sign ra nk rk ∞ ( A ). It is know n that the sign rank itself is NP-hard to compute [BF G + 09, BK 1 5]. Ac kno wledgmen ts W e would lik e to tha nk Ronald de W olf for helpful commen ts o n an earlier v ersion of this manusc ript and Gideon Sch ec h tman for helpful conv ersations. W e also thank Shalev Ben-Da vid for pointing out an error in a previous v ersion o f the pro of of Theorem 1 0. This w ork conducted while TL w as at Rutgers Univ ersit y , supp orted by a NSF mathematical sciences p ostdo ctoral fello wship. 11 References [AA05] S. Aaronson and A. Am bainis. Quan tum searc h of spatial re- gions. The ory of Computing , 1:47–79, 2005 . [Alo03] N. Alon. Problems and results in extremal com binatorics, part i. Discr ete Mathematics , 273 :3 1–53, 200 3. [Alo09] N. Alon. P erturb ed iden tit y matrices ha v e high r a nk: pro of and applications. Combinatorics, Pr ob ability, an d Co mputing , 18:3– 15, 2009. [BES02] S. Ben-Da vid, N. Eiron, and H. Simon. Limitations of learning via embeddings in Euclide an half spaces. Journal of Machine L e arning R ese ar ch , 3:441–461 , 2002. [BF G + 09] Ronen Basri, Pedro F. F elzenszw alb, Ro ss B. Girshic k, Dav id W. Jacobs, and Caro line J. Kliv ans. Visibility constrain ts on fea- tures of 3d ob jects. In 2009 IEEE Comp uter So ciety Confer enc e on Com p uter Vision and Pattern R e c o gnition (CVPR 2009 ), 2 0 - 25 June 2009, Miam i, Florida, USA , pages 1231–12 38. IEEE Computer So ciet y , 2009. [Bha07] R. Bhatia. Positive definite m a tric es . Princeton Univ ersit y Press, 2007. [BK15] Amey Bhang ale and Sw astik Ko ppart y . The complexit y of com- puting the minim um rank of a sign pattern matrix. CoRR , abs/1503.0448 6, 20 1 5. [BVW07] H. Buhrman, N. V ereshc hagin, and R. de W olf . O n computation and comm unication with small bias. In Pr o c e e din g s of the 22nd IEEE Confer en c e on Computational Complexi ty , pages 24– 32. IEEE, 2007. [BW01] H. Buhrman and R. de W olf. Communic ation complexit y lo w er b ounds by p olynomials. In Pr o c e e dings o f the 16th IEEE Con- fer enc e on Computational Complexity , pages 120–130, 2001. 12 [BYJKS04] Z. Bar-Y ossef, T. Ja yram, R. Kumar, and D. Siv akumar. In- formation statistics approac h to data stream and comm unica- tion complexit y . Journal of Com puter and System Scien c es , 68(4):702– 732, 2004 . [CSWY01] A. Chakrabarti, Y. Shi, A. Wirth, and A. Y ao. Informational complexit y and the direct sum problem for sim ultaneous message complexit y . In Pr o c e e dings of the 42nd IEEE Symp osium on F oundations of Computer Scienc e , pages 270– 278. IEE E, 2 001. [GLS88] M . Gr¨ otsc hel, L . Lov´ asz, and A. Sc hrijv er. Ge ometric A lgorithms and Co mbinatorial Optimization . Springer-V erlag, 1988. [GW95] M. Go emans and D . Williamson. Impro v ed approxim ation al- gorithms for maxim um cut and satisfiabilit y problems using semidefinite programming. Journal o f the ACM , 42:1 1 15–1145, 1995. [JL84] W. Johnson and J. Lindenstrauss. Extensions of Lipsc hitz map- ping in to Hilb ert space. Co ntemp or ary Mathematics , 26:189–2 0 6, 1984. [Kla01] H. Klauck . Low er b ounds f or quan tum communication complex- it y . In Pr o c e e din gs of the 42nd IEEE Symp osium on F oundations of Com puter Scie nc e . IEEE, 2001. [Kra96] M. Krause. Geometric argumen ts yield b etter b ounds fo r thresh- old circuits and distributed computing. The or etic al Computer Scienc e , 156:9 9–117, 1996. [Kri79] J. Krivine. Constan tes de Grothendiec k et fonctions de type p ositif sur les sph ` eres. A dv. Math. , 31:16 – 30, 1 9 79. [KS07] A. Kliv ans and A. Shersto v. A lo w er b o und fo r agnostically learning disjunctions. In Pr o c e e dings of the 20th Confer e nc e on L e arning Th e ory , 2007. [LMSS07] N. Linial, S. Mendelson, G. Sc hec h tman, and A. Shraib- man. Complexit y measures of sign matrices. Combi n atoric a , 27(4):439– 463, 2007 . 13 [LS88] L. Lov´ asz and M. Saks. M¨ obius f unctions and communic ation complexit y . In Pr o c e e dings of the 29 th IEEE Symp os i um on F oundations of Computer Scienc e , pages 81–9 0. IEEE, 198 8. [LS07] N. Linial and A. Shra ibman. Low er b ounds in comm unication complexit y based on factorization norms. In Pr o c e e dings of the 39th ACM Symp osium on the Th e ory of C omputing , pag es 69 9 – 708. A CM, 2007. [LS ˇ S08] T. Lee, A. Shraibman, and R. ˇ Spalek. A direct pro duct theorem for discrepancy . In Pr o c e e dings of the 23r d IEEE C onfer en c e on Computational C omplexity , pages 71–80. IEEE , 2008. [MS82] K. Mehlhorn and E. Schmidt. Las V ega s is b etter than deter- minism in VLSI a nd distributed computing. In Pr o c e e dings of the 14th ACM Symp osi um on the T h e ory of Computing , pages 330–337. A CM, 1982. [PS86] R. P aturi and J. Simon. Probabilistic comm unication complex- it y . Journal o f Computer and System Sc i e nc es , 33 (1):106–12 3 , 1986. [Raz92] A. Razb orov. On the distributional complexit y of disjoin tness. The o r etic al Computer Scienc e , 106:3 85–390, 1 992. [Raz95] R. Raz. F ourier analysis for probabilistic comm unication com- plexit y . Computational Complexity , 5(3/4):205–2 21, 1995. [Raz03] A. R azb oro v. Quan tum comm unication complexit y of symmetric predicates. Izv e stiya: Mathematics , 67(1):14 5–159, 20 03. [She08] A. Shersto v. Halfspace matrices. Computational Com plexity , 17(2):149– 178, 2008 . [TJ89] N. T o mczak-Jaegermann. B anach-Mazur d istanc es and finite- dimensional op e r ator ide als , v olume 38 of Pitman Mono gr aphs and Surveys in Pur e and Applie d Mathematics . Longman Scien- tific & T ec hnical, Ha r low, 1989. [VB96] L. V andenberghe and S. Boyd. Semidefinite programming. SI AM R eview , 38:49 –95, 1996. 14 [Y ao7 9] A. Y ao. So me complexit y questions related to distributive com- puting. In Pr o c e e dings of the 11th A CM Symp osium on the The- ory o f Computing , pages 209–2 13. ACM , 1979. [Y ao8 3] A. Y ao. Low er b ounds b y proba bilistic argumen ts. In Pr o c e e d- ings of the 24th IEEE Symp o sium on F oundations of Computer Scienc e , pages 420–4 28, 1 983. 15