Shachar Lovett
email: shachar.lovett@weizmann.ac.il

LOWER BOUNDS FOR ADAPTIVE LINEARITY TESTS

Keywords: 1998 ACM Subject Classification: F.2.2, G.2.1 Property testing, Linearity testing, Adaptive tests, Lower bounds

Linearity tests are randomized algorithms which have oracle access to the truth table of some function f, and are supposed to distinguish between linear functions and functions which are far from linear. Linearity tests were first introduced by Blum, Luby and Rubenfeld in [BLR93], and were later used in the PCP theorem, among other applications. The quality of a linearity test is described by its correctness c -the probability it accepts linear functions, its soundness s -the probability it accepts functions far from linear, and its query complexity q -the number of queries it makes.

Linearity tests were studied in order to decrease the soundness of linearity tests, while keeping the query complexity small (for one reason, to improve PCP constructions). Samorodnitsky and Trevisan constructed in [ST00] the Complete Graph Test, and prove that no Hyper Graph Test can perform better than the Complete Graph Test. Later in [ST06] they prove, among other results, that no non-adaptive linearity test can perform better than the Complete Graph Test. Their proof uses the algebraic machinery of the Gowers Norm. A result by Harsha and Raskhodnikova [BHR05] allows to generalize this lower bound also to adaptive linearity tests.

. We also study, like [ST06], the behavior of linearity tests on quadratic functions. However, instead of analyzing the Gowers Norm of certain functions, we provide a more direct combinatorial proof, studying the behavior of linearity tests on random quadratic functions. This proof technique also lets us prove directly the lower bound also for adaptive linearity tests.

Introduction

We study the relation between the number of queries and soundness of adaptive linearity tests. A linearity test (over the field F 2 for example) is a randomized algorithm which has oracle access to the truth table of a function f : {0, 1} n → {0, 1}, and needs to distinguish between the following two extreme cases:

(1) f is linear (2) f is far from linear functions A function f is called linear if it can be written as f (x 1 , ..., x n ) = a 1 x 1 + ... + a n x n , with a 1 , ..., a n ∈ F 2 . The agreement of two functions f, g : {0, 1} n → {0, 1} is defined as d(f, g) = |P x [f (x) = g(x)] -P x [f (x) = g(x)]|. f is far from linear functions if it has small agreement with all linear functions (we make this definition precise in Section 2).

Linearity tests were first introduced by Blum, Luby and Rubenfeld in [START_REF] Blum | Self testing/correcting with applications to numerical problems[END_REF]. They presented the following test (coined the BLR test), which makes only 3 queries to f :

(1) Choose x, y ∈ {0, 1} n at random (2) Verify that f (x + y) = f (x) + f (y). Bellare et al. [START_REF] Bellare | Linearity testing over characteristic two[END_REF] gave a tight analysis of the BLR test. It is obvious that the BLR test always accepts a linear function. They have shown that if the test accepts a function f with probability 1/2 + ǫ, then f has agreement at least 2ǫ with some linear function.

For a linearity test, we define that it has completeness c if it accepts any linear function with probability of at least c. A test has perfect completeness if c = 1. A linearity test has soundness s if it accepts any function f with agreement at most ǫ with all linear functions, with probability of at most s + ǫ ′ , where ǫ ′ → 0 when ǫ → 0. We define the query complexity q of a test as the maximal number of queries it performs. In the case of the BLR test, it has perfect completeness, soundness s = 1/2 (with ǫ ′ = 2ǫ) and query complexity q = 3.

If one repeats a linearity test with query complexity q and soundness s independently t times, the query complexity grows to q ′ = qt while the soundness reduces to s ′ = s t . So, it makes sense to define the amortized query complexity q of a test as q = q/ log 2 (1/s). Independent repetition of a test doesn't change it's amortized query complexity. Notice that the BLR test has amortized query complexity q = 3.

Linearity tests are a key ingredient in the PCP theorem, started in the works of Arora and Safra [START_REF] Arora | Probabilistic checking of proofs: A new characterization of NP[END_REF] and Arora, Lund, Motwani, Sudan and Szegedy [START_REF] Arora | Proof verification and hardness of approximation problems[END_REF]. In order to improve PCP constructions, linearity tests were studied in order to improve their amortized query complexity.

Samorodnitsky and Trevisan [START_REF] Samorodnitsky | A PCP characterization of NP with optimal amortized query complexity[END_REF] have generalized the basic BLR linearity test. They introduced the Complete Graph Test. The Complete Graph Test (on k vertices) is:

(1) Choose x 1 , ...,

x k ∈ {0, 1} n independently (2) Verify f (x i + x j ) = f (x i ) + f (x j ) for all i, j
This test has perfect completeness and query complexity q = k 2 + k. They show that all the k 2 tests that the Complete Graph Test performs are essentially independent, i.e. that the test has soundness s = 2 -( k 2 ) . This makes this test have amortized query complexity q = 1 + θ(1/ √ q). They show that this test is optimal among the family of Hyper-Graph Tests (see [START_REF] Samorodnitsky | A PCP characterization of NP with optimal amortized query complexity[END_REF] for definition of this family of linearity tests), and raise the question of whether the Complete Graph Test is optimal among all linearity tests, i.e. does a test with the same query complexity but with better soundness exist? They partially answer this question in [START_REF] Samorodnitsky | Gowers uniformity, influence of variables, and PCPs[END_REF], where (among many other results) they show that no non-adaptive linearity test can perform better than the Complete Graph Test. A test is called non-adaptive if it first chooses q locations in the truth table of f , then queries them, and based on the results accept or rejects f . Otherwise, a test is called adaptive. An adaptive test may decide on its query locations based on the values of f in previous queries.

The proof technique of [START_REF] Samorodnitsky | Gowers uniformity, influence of variables, and PCPs[END_REF] uses the algebraic analysis of the Gowers Norm of certain functions. The Gowers Norm is a measure of local closeness of a function to a low degree polynomial. For more details regarding the definition and properties of the Gowers Norm, see [START_REF] Green | An inverse theorem for the Gowers U3 norm[END_REF] and [START_REF] Samorodnitsky | Low-degree tests at large distances[END_REF].

Ben-Sasson, Harsha and Raskhodnikova prove in [START_REF] Ben-Sasson | Some 3CNF properties are hard to test[END_REF] that any adaptive linearity test with completeness c, soundness s and query complexity q can be transformed into a non-adaptive linearity test with the same query complexity, perfect completeness and soundness s ′ = s + 1 -c. Combining their result with the result of [START_REF] Samorodnitsky | Gowers uniformity, influence of variables, and PCPs[END_REF] proves the lower bound also for adaptive linearity tests.

We also prove the same optimal lower bound for adaptive linearity test, but our proof technique is arguably simpler and more direct than the one used in [START_REF] Samorodnitsky | Gowers uniformity, influence of variables, and PCPs[END_REF]. We also study, like [START_REF] Samorodnitsky | Gowers uniformity, influence of variables, and PCPs[END_REF], the behavior of linearity tests on quadratic functions. However, instead of employing algebraic analysis of the Gowers Norm of certain functions, we provide a more direct combinatorial proof, studying the behavior of linearity tests on random quadratic functions. This proof technique also lets us prove directly the lower bound also for adaptive linearity tests.

Our techniques

We model adaptive tests using test trees. A test tree T is a binary tree, where in each inner vertex v there is some label x(v) ∈ {0, 1} n , and the leaves are labeled with either accept or reject. Running a test tree on a function f is done by querying at each stage f on the label of the current vertex (starting at the root), and following one of the two edges leaving the vertex, depending on the query response. When reaching a leaf, its label (accept or reject) is the value of that f gets in T . An adaptive test T can always be modeled as first randomly choosing a test tree from some set {T i }, according to some distribution on the test trees, then running the test tree on f . It turns out that in order to prove a lower bound which matches the upper bound of the Complete Graph Test, it is enough to consider functions f which are quadratic. Actually, it's enough to consider f which is a random quadratic function.

A function f is quadratic if it can be presented as

f (x 1 , ..., x n ) = i,j a i,j x i x j + i b i x i +c
for some values a i,j , b i , c ∈ F 2 . We study the behavior of running test trees on a random linear function, and on a random quadratic function.

The main idea is as follows. Let v be some inner vertex in a test tree T , with the path from the root of T to v being v 0 , ..., v k-1 , v. If x(v) is linearly dependent on x(v 0 ), ..., x(v k-1 ), then when running T on any linear function, the value of f (x(v)) can be deduced from the already known values of f (x(v 0 )), ..., f (x(v k-1 )). Therefore, if the vertex v is reached, then the same edge leaving v will always be taken by any linear function. Additionally, if x(v) is linearly independent of x(v 0 ), ..., x(v k-1 ), then either v is never reached running T on linear functions, or the two edges leaving v are taken with equal probability when running T on a random linear function. A similar analysis can be made when running T on quadratic functions, replacing linear dependence with a corresponding notion of quadratic dependence.

Using this observation, we can define the linear rank of a leaf v, marked l(v), as the linear rank of labels on the path from the root to v. We prove that running the test tree T on a random linear function reaches v with probability 2 -l(v) . Similarly, we define the quadratic rank of a leaf v, marked q(v), as the quadratic rank of those labels, and we proving that running T on a random quadratic function reaches v with probability 2 -q(v) . We prove that the quadratic rank of any set cannot be much larger than its linear rank, and in particular that q(v) ≤ l(v) 2 + l(v) for all leaves v. We use this inequality to prove that a test which has completeness c and query complexity q accepts a random quadratic function with a probability of at least c -1 + 2 -q+φ(q) , where φ(q) is defined as the unique non-negative solution to φ(q) 2 + φ(q) = q. We use this to show that any linearity test with completeness c and query complexity q must have s ≥ 2 -q+φ(q) . In particular, the Complete Graph Test on k vertices has perfect completeness, soundness s = 2 -( k 2 ) and query complexity q = k 2 + k. Since φ(q) = k the Complete Graph Test is optimal among all adaptive tests with the same query complexity.

In fact, we prove a stronger claim. We say that a test T has average query complexity q if for any function f , the average number of queries performed is at most q. In particular any test with query complexity q also has average query complexity q. We prove that for any test with completeness c and average query complexity q, the soundness is at least s ≥ 2 -q+φ(q) . We present and analyze linearity tests over F 2 . Linearity tests can also be considered over larger fields or groups. Our lower bound actually generalizes easily to any finite field, but for ease of presentation, and since the techniques are exactly the same, we present everything over F 2 . We comment further on the modifications required for general finite fields in Section 2.

Preliminaries

Linearity tests

We call a function f : {0, 1} n → {0, 1} linear if it can be written as f (x 1 , ..., x n ) = a 1 x 1 + ... + a n x n for some a 1 , ..., a n ∈ {0, 1} where addition and multiplication are in F 2 .

A linearity test is a randomized algorithm with oracle access to the truth table of f , which is supposed to distinguish the following two extreme cases:

(1) f is linear (accept)

(2) f is ǫ-far from linear functions (reject) where the agreement of two functions f, g : {0, 1} → {0, 1} is defined as

d(f, g) = |P r x [f (x) = g(x)] -P r x [f (x) = g(x)
]|, and f is ǫ-far from linear functions if the agreement it has with any linear function is at most ǫ.

We now follow with some standard definition regarding linearity tests (or more generally, property tests). We say a test has completeness c if for any linear function f the test accepts with probability at least c. A test has perfect completeness if c = 1. We say a test has soundness s if for any f which is ǫ-far from linear the test accepts with probability at most s + ǫ ′ , where ǫ ′ → 0 when ǫ → 0 (in fact, we talk about a family of linearity tests, for n → ∞, but we ignore this subtle point).

A test is said to have query complexity q if it accesses the truth-table of f at most q times (for any choice of it's internal randomness). A test is said to have average query complexity q if for any function f , the average number of accesses (over the internal randomness of the test) done to the truth table of f is at most q. Obviously, any test with query complexity q is also a test with average query complexity q.

We say a test is non-adaptive if it chooses all the locations it's going to query in the truth table of f before reading any of their values. Otherwise, we call the test adaptive.

We now turn to model adaptive tests in a way that will be more convenient for our analysis. We first define a test tree and running a test tree on a function.

Definition 2.1. A test tree on functions {0, 1} n → {0, 1} is a rooted binary tree T . On each inner vertex of the tree v there is a label x(v) ∈ {0, 1} n . On each leaf there is a label of either accept or reject.

Definition 2.2. Running a test tree T on a function f is done as follows. We start at the root of the tree v 0 , read the value of f (x(v 0 )), and according to the value take the left or the right edge leaving v 0 . We continue in this fashion on inner vertices of T until we reach a leaf of T . The value of f in T is the value of the end leaf (i.e. accept or reject), and the depth of f in T is the depth of the end vertex of f in T .

Using these definitions, we can now model adaptive tests. We identify an adaptive test T on functions {0, 1} n → {0, 1} with a distribution of binary trees {T i } (also on functions {0, 1} n → {0, 1}). Running the test T on a function f is done by randomly choosing one of the trees T i (according to their distribution), and then running the test tree T i on f . The result of the function f in the test tree T i is the result the test T returns on f .

Notice that a test has query complexity q iff all trees T i has depth at most q, and has average query complexity q iff for any function f , the average depth reached in a random tree from {T i } is at most q.

In order to define our main theorem, we will define the following function. For x > 0 define φ(x) as the unique real positive solution to φ(x) 2 /2 + φ(x)/2 = x. Notice that for positive integer φ(x), this is the same as φ(x) 2 + φ(x) = x. The following is the main theorem of this paper: Theorem 2.3. (main theorem) Let T be an adaptive test with completeness c, soundness s and average query complexity q ≥ 1. Then s + 1 -c ≥ 2 -q+φ(q) . Notice that for large q, φ(q) ≈ √ 2q, also √ q ≤ φ(q) ≤ √ 2q, so we get that in particular, s + 1 -c ≥ 2 -q+θ( √ q ). The Complete Graph Test was presented in [START_REF] Samorodnitsky | A PCP characterization of NP with optimal amortized query complexity[END_REF]. The test (on a graph with k vertices) can be described as choosing x 1 , ..., x k at random, and querying f at x i (for i = 1..k) and on x i + x j (for 1 ≤ i < j ≤ k). The test accepts f if for any i, j

f (x i ) + f (x j ) + f (x i + x j ) = 0
In [START_REF] Samorodnitsky | A PCP characterization of NP with optimal amortized query complexity[END_REF] it is proven that the Complete Graph Test has perfect completeness and soundness s = 2 -( k 2 ) . The total number of queries performed is q = k + k 2 , so by our definitions, k = φ(q) and s = 2 -q+φ(q) . We have the following corollary:

Corollary 2.4. The Complete Graph Test is optimal among all adaptive linearity tests.

Remark 2.5. We state and prove all results for functions f : {0, 1} n → {0, 1}. In fact, the lower bound result on adaptive linearity tests holds for functions f : F n → F for any finite field F, and not just F 2 , with only minor adjustments to the definitions and proofs. We need to make the following modifications:

(1) Define "ǫ-far from linear functions" for general fields (2) Test trees should have |F | edges leaving each edge instead of 2 (3) The proof that random quadratic functions are far from linear, proved in Section 5, should be slightly modified Since the results follow simply for any finite field, we chose to present the results over F 2 to make the presentation simpler and clearer.

Quadratic functions

We will see that in order to prove Theorem 2.3, it will be enough to limit the functions f to be quadratic. We say a function f is quadratic if it can be written as:

f (x 1 , ..., x n ) = i,j a i,j x i x j + i b i x i + c for some a i,j , b i , c ∈ F 2 .
In fact, for our usage, we will force our quadratic functions f to have f (0) = 0 (equivalently, c = 0 in the above description). So, throughout this paper, when speaking of quadratic functions, we actually speak of quadratic functions f with the added condition f (0) = 0.

We will study the dynamics of a test tree T in a linearity test T, in two cases -when applied to a uniformly random linear function, and when applied to a uniformly random quadratic function.

The following technical lemma is the key ingredient to the proof of the Theorem 2.3.

Lemma 3.1. Let T be an adaptive linearity test with completeness c and average query complexity q. Then running T on a random quadratic function returns accept with probability at least c -1 + 2 -q+φ(q) .

In order to prove Theorem 2.3, we will also need the following simple lemma:

Lemma 3.2. Let f be a random quadratic function. Then the probability that f is not 2 -Ω(n) -far from linear functions is 2 -Ω(n) .

Theorem 2.3 now follows directly from Lemmas 3.1 and 3.2. We sketch now it's proof following the two lemmas.

Proof. (of the main theorem) The average probability that T returns accept on a random quadratic function which is 2 -Ω(n) -far from linear functions is at least c-1+2 -q+φ(q) -2 -Ω(n) . So, there exists some quadratic function f which is 2 -Ω(n) -far from linear and on which T returns accept with probability at least c -1 + 2 -q+φ(q) -2 -Ω(n) . Taking n → ∞ shows that s + 1 -c ≥ 2 -1+φ(q) . The remainder of the paper is organized as follows. Lemma 3.1 is proved in Section 4, and Lemma 3.2 is proved in Section 5.

Linearity test applied to a random quadratic function

We study tests and test trees applied to linear and quadratic functions, in order to prove Lemma 3.1. Let T be an adaptive test with completeness c and average query complexity q. Let T be a some test tree which is a part of the test T.

We start by studying the dynamics of applying T to linear functions. Assume we know that f is a linear function, and we are at some vertex v ∈ T , where the path from the root to v is v 0 , .., v k-1 , v. Assume x(v) is linearly dependant on x(v 0 ), ..., x(v k-1 ). Since we know f is linear, we can deduce the value of x(v) from x(v 0 ), ..., x(v k-1 ), and so we will always follow the same edge leaving v when we apply T to any linear function. On the other hand, if x(v) is linearly independent of x(v 0 ), ..., x(v k-1 ), we know that when we apply T to a random linear function, either we never reach v, or we have equal chances of taking any of the two edges leaving v.

This gives rise to the following formal definition:

Definition 4.1. Let v be a leaf in T , where the path from the root to v is v 0 , v 1 , ..., v k-1 , v.

We define the linear degree of v, marked l(v), to be the linear rank of x(v 0 ), ..., x(v k-1 ).

We define L T to be the set of leaves of T to which linear functions can arrive. i.e, v ∈ L if the path from the root to v, v 0 , ..., v k-1 , v always takes the "correct" edge leaving any vertex v i with x(v i ) linearly dependent on x(v 0 ), ..., x(v i-1 ).

The following lemma formalizes the discussion above:

Lemma 4.2. For any test tree T :

(1) For any v ∈ L T , the probability that a random linear function will arrive to v is

2 -l(v) (2) v∈L T 2 -l(v) = 1
For v ∈ L T , we define c(v) to be 1 if the value of v is accept, and c(v) = 0 otherwise. Since the completeness of T is c, we have that the probability that T returns accept on a random linear function is at least c. On the other hand, for any test tree T in T, the probability that a random linear function will return accept is exactly v) . So, the following lemma follows:

v∈L T c(v)2 -l(
Lemma 4.3. E T v∈L T c(v)2 -l(v) ≥ c
where by E T here and throughout the paper we mean the average value of a random test tree T in T.

We now generalize the concept of linear dependence to quadratic functions.

Definition 4.4. Let x 1 , ..., x k ∈ {0, 1} n .

(1) We say x 1 , ..., x k are quadratically dependent if there are constants a 1 , ..., a k ∈ F 2 , not all zero, s.t. for any quadratic function f we have:

a 1 f (x 1 ) + ... + a k f (x k ) = 0.
otherwise will call x 1 , ..., x k quadratically independent.

(2) We say x k is quadratically dependent on x 1 , ..., x k-1 if there are constants a 1 , ..., a k-1 ∈ F 2 s.t. for any quadratic function f we have:

f (x k ) = a 1 f (x 1 ) + ... + a k-1 f (x k-1 ).
Otherwise we say x k is quadratically independent of x 1 , ..., x k-1 . (3) We define the quadratic dimension of x 1 , ..., x k to be the size of the largest subset of {x 1 , ..., x k } which is quadratically independent.

This definition may seem obfuscated, but the following alternative yet equivalent definition will clarify it. The space of quadratic functions over {0, 1} n is a linear space over F 2 . Let M be it's generating matrix, i.e. the rows of M are a base for the linear space (in particular, the dimensions of M are ( n 2 + n)× 2 n ). A column of M corresponds to an input x ∈ {0, 1} n . Now, x 1 , ..., x k are quadratically dependent iff the columns corresponding to them are linearly dependent, and similarly for the other definitions.

Notice that the usual definition of linear dependence is equivalent to this more complex definition, when applied to the linear space of all linear functions.

We now can repeat the informal discussion at the start of this section, except this time for quadratic functions, with all the reasoning left intact. Let v ∈ T be a vertex, with path from the root being v 0 , ..., v k-1 , v. Assume x(v) is quadratically dependent on x(v 0 ), ..., x(v k-1 ), and f is any quadratic function. The value of f (x(v)) can be deduced from the already known values of f (x(v 0 )), ..., f (x(v k-1 )), and so only one edge leaving v will be taken on all quadratic functions. Alternatively, if x(v) is quadratically independent on x(v 0 ), ..., x(v k-1 ), then a random quadratic function either never reaches v, or has equal chances of taking each of the two edges leaving v.

This leads to the following definition and lemma for quadratic degree of a vertex v ∈ T , similar to the ones for linear degree. Definition 4.5. Let v be a leaf in T , where the path from the root to v is v 0 , v 1 , ..., v k-1 , v. We define the quadratic degree of v, marked q(v), to be the quadratic rank of x(v 0 ), ..., x(v k-1 ).

We define Q T to be the set of leaves of T to which quadratic functions can arrive. Naturally L T ⊆ Q T . The following lemma on quadratic degree follows from the discussion above:

Lemma 4.6. For any test tree T :

(1) For any v ∈ Q T , the probability that a random quadratic function will arrive to v is

2 -q(v) (2) v∈Q 2 -q(v) = 1 (3) For any v ∈ L T we have q(v) ≥ l(v)
Last, we mark the depth of a vertex v ∈ T by d(v). Since T has average query complexity q, we know that for any function f , the average depth of running a random tree T of T on f is at most q. So, this also holds for a random linear function. However, the average depth a random linear function arrives on a tree T is exactly d(v)2 -l(v) , so the following lemma follows.

Lemma 4.7.

E T v∈L T d(v)2 -l(v) ≤ q
We now wish to make a connection between q(v) and l(v) for vertices v ∈ L T . First, we prove that following lemma: Lemma 4.8. For any x 1 , ..., x k ∈ {0, 1} n there are coefficients a i,j , b i ∈ F 2 s.t. for any quadratic function f we have:

f (x 1 + ... + x k ) = i,j a i,j f (x i + x j ) + i b i f (x i )
Proof. Let f (x) by some polynomial of degree d. It's derivative in the y direction is defined to be f y (x) = f (x + y) -f (x). It's easy to see that the degree of f y as a function of x is at most d -1. So, taking 3 derivatives from a quadratic function makes it the zero function, and so in particular for any quadratic function f, we we take it's derivatives in directions x, y and z, and evaluate the result at 0, we get that (((f x ) y ) z (0) = 0 Opening this expression yields:

f (x + y + z) -f (x + y) -f (x + z) -f (y + z) + f (x) + f (y) + f (z) -f (0) = 0
Since f (0) = 0, we can express f (x + y + z) as a sum of application of f on an element, or sum of two elements in {x, y, z}. This proves the lemma for k = 3. For k > 3 we use simple induction. Now we can bound l(v) in term of q(v). We first prove a result bounding in general the linear rank of a set by it's quadratic rank.

Lemma 4.9. Let {x 1 , ..., x k } be elements in {0, 1} n . Let l be the their linear rank, and q their quadratic rank. Then

q ≤ l 2 + l
Proof. Let S ⊂ {x 1 , ..., x k } be a maximal quadratic independent set. |S| = q. The linear rank of S is also l. Let S ′ ⊂ S be a maximal set of linearly independent elements of S.

|S ′ | = l. Assume w.l.o.g that S ′ = {x 1 , ..., x l }. Since every x ∈ S is linearly dependent on S ′ , it can be written as a sum of some of the elements of S ′ . Using Lemma 4.8, we get that for any x ∈ S there exists coefficients a

∈ F 2 s.t for any quadratic function f :

f (x) = 1≤i<j≤l a (x) i,j f (x i + x j ) + 1≤i≤l b (x) i f (x i )
We have assumed that all the elements of S are quadratically independent. For this to hold, the above equations in the symbolic variables f (x i + x j ) and f (x i ) must be linearly independent. So the number of equations q must be at most the number of variables, which is l 2 + l. So, we get that:

q = |S| ≤ l 2 + l Lemma 4.10. For any leaf v ∈ L T , l(v) ≥ φ(q(v))
Proof. Let v 0 , ..., v k-1 , v be the path in T from the root to v. Let x i = x(v i ) for i = 0..k -1. Apply lemma 4.9 on {x 0 , ..., x k-1 } to get that q(v) ≤ l(v) 2 + l(v). Reversing this formula, since φ(x) is monotone, we get that l(v) ≥ φ(q(v)).

We can now prove our main technical lemma (Lemma 3.1). We start with some technical lemmas. We define ψ(x) to be x -φ(x) for x ≥ 1, and 0 for x < 1. Notice that ψ is continuous, and ψ(x) = x -φ(x) for any non-negative integer x. Hence, using Lemma 4.10 we get that: Lemma 4.11. For any vertex v in a tree T , q(v) -l(v) ≤ ψ(q(v)).

Lemma 4.12. ψ is increasing and convex.

Proof. Since ψ is continuous and constant for x ≤ 1, it's enough to prove the claim for x > 1 (for increasing it's clear, and once we've proved ψ is increasing, it shows it's enough to prove convexity for x > 1). We first show ψ is increasing.

For x > 1, define y = φ(x), so x = y 2 /2 + y/2 and ψ(y) = y 2 /2 -y/2. If x > 1 then y = φ(x) > 1, hence dψ dx > 0 for x > 1, and so ψ is increasing. To show that ψ in convex,

d 2 ψ dx 2 = d y-1/2 y+1/2 dy dy dx = 1 (y + 1/2) 3 > 0
We are now finally ready to prove Lemma 3.1.

Proof. (of Lemma 3.1) We need to prove that any test T with completeness c and average query complexity q ≥ 1 accepts a random quadratic function with probability at least c -1 + 2 -ψ(q) . Let us mark the probability the test accepts a random quadratic function by p. Let p T mark the probability that a tree T accepts a random quadratic function. p T is at least the probability that a random quadratic function reaches a leaf in L T which is labeled accept. So:

p T ≥ v∈L T c(v)2 -q(v) We now follow to analyze p = E T [p T ]. p ≥ E T [ v∈L T c(v)2 -q(v) ] = E T [ v∈L T 2 -l(v) c(v)2 -q(v)+l(v) ]
.

We divide the sum in the right side into two parts, p 0 -p 1 , with p 0 , p 1 ≥ 0, where:

p 0 = E T [ v∈L T 2 -l(v) 2 -q(v)+l(v) ]
. and

p 1 = E T [ v∈L T 2 -l(v) (1 -c(v))2 -q(v)+l(v) ]
.

We start by analyzing p 1 . Since for any v always q(v) ≥ l(v) we have:

p 1 ≤ E T [ v∈L T 2 -l(v) (1 -c(v))]
Recall that by Lemma 4.6 for any tree T we have

v∈L T 2 -l(v) = 1
and by Lemma 4.3 we have

E T [ v∈L T 2 -l(v) c(v)] ≥ c
so we conclude that:

p 1 ≤ 1 -c
We move to analyze p 0 . Since E T [ v∈L T 2 -l(v) ] = 1 and since the function X → 2 X is concave, we have by Jensen's inequality that:

p 0 ≥ 2 E T [ v∈L T 2 -l(v) (-q(v) + l(v))]
Now, we have that q(v) -l(v) ≤ ψ(q(v)) by Lemma 4.12, and also by the same lemma, since q(v) ≤ d(v), we get ψ(q(v)) ≤ ψ(d(v)). So we get:

E T [ v∈L T 2 -l(v) (q(v) -l(v))] ≤ E T [ v∈L T 2 -l(v) ψ(d(v))]
.

Since by Lemma 4.12 ψ is convex, we get that again by Jensen's inequality we get that this is at most ψ(E T [ v∈L T 2 -l(v) d(v)]). By Lemma 4.7

E T [ v∈L T 2 -l(v) d(v)] ≤ q
where q is the average query complexity of T. So, we conclude that p 0 ≥ 2 -ψ(q) , and in total p ≥ p 0 -p 1 ≥ 2 -ψ(q) + c -1

Random quadratic function is far from linear

In this section we prove Lemma 3.2, i.e. that a random quadratic function is far from linear. We will use commonly known facts about quadratic functions.

Any quadratic function can be written as:

f (x) = x t Ax+ < x, b >
The correlation of f with some linear function g is the g-th Fourier coefficient of f . The Fourier coefficients of quadratic functions are well studied. In particular, it is known that all the Fourier coefficients of f have the same absolute value, and that the number of non-zero Fourier coefficients is 2 rank(A+A t ) . So, in order to show that f has no large correlation with some linear function, it's enough to show that B = A + A t has high rank. In particular, in order to show that f is 2 -Ω(n) -far from linear functions, we need to show that B has rank Ω(n). We will show that the probability that a random quadratic function has rank less than n/4 is 2 -Ω(n) . We will use the following lemma:

Lemma 5.1. The number of matrices of rank at most k is at most n k 2 nk . Using Lemma 5.1, it's easy to prove Lemma 3.2. The number of matrices of rank at most n/4 is at most 2 n 2 /4(1+o(1)) . For a random quadratic function, B is a random symmetric matrix with zero diagonal, and so the probability that B has rank less than n/4 is 2 -n 2 /4(1+o(1)) = 2 -Ω(n) . Now we finish by proving Lemma 5.1.

Proof. Let B be a matrix of rank at most k. There are n k options to choose k rows which span the row span of the matrix, each other row have at most 2 k options since it must be in the row span of k specific rows. So, the number of possibilities for rank k matrices is at most:

n k (2 k ) n-k ≤ n k 2 nk

This work is licensed under the Creative Commons Attribution-NoDerivs License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nd/3.0/.

Acknowledgement

I thank my supervisor, Omer Reingold, for useful comments and for his constant support and interest in the work. I thank Alex Samorodnitsky and Prahladh Harsha for helpful discussions.

Research supported by grant 1300/05 from the Israel Science Foundation. c S. Lovett