Kevin Matulef

Property Testing Lower Bounds via Communication Complexity

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a Boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010a). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.

Testing k-wise and almost k-wise independence

In this work, we consider the problems of testing whether adistribution over (0,1ⁿ) is <i>k</i>-wise (resp. (ε,k)-wise) independentusing samples drawn from that distribution. For the problem of distinguishing <i>k</i>-wise independent distributions from those that are δ-far from <i>k</i>-wise independence in statistical distance, we upper bound the number ofrequired samples by Õ(n^k/δ²) and lower bound it by Ω(n^k-1/2/δ) (these bounds hold for constant<i>k</i>, and essentially the same bounds hold for general <i>k</i>). Toachieve these bounds, we use Fourier analysis to relate adistributions distance from <i>k</i>-wise independence to its biases, a measure of the parity imbalance it induces on a setof variables. The relationships we derive are tighter than previouslyknown, and may be of independent interest. To distinguish (ε,k)-wise independent distributions from thosethat are δ-far from (ε,k)-wise independence in statistical distance, we upper bound thenumber of required samples by O(k log n / δ²ε²) and lower bound it by Ω(√ k log n / 2^k(ε+δ)√ log 1/2^k(ε+δ)). Although these bounds are anexponential improvement (in terms of <i>n</i> and <i>k</i>) over thecorresponding bounds for testing <i>k</i>-wise independence, we give evidence thatthe time complexity of testing (ε,k)-wise independence isunlikely to be poly(n,1/ε,1/δ) for k=Θ(log n),since this would disprove a plausible conjecture concerning the hardness offinding hidden cliques in random graphs. Under the conjecture, ourresult implies that for, say, k = log n and ε = 1 / n^0.99,there is a set of (ε,k)-wise independent distributions, and a set of distributions at distance δ=1/n^0.51 from (ε,k)-wiseindependence, which are indistinguishable by polynomial time algorithms.

Testing ±1-weight halfspace

We consider the problem of testing whether a Boolean function f :{ - 1,1} n -->{ - 1,1} is a ±1-weight halfspace , i.e. a function of the form f (x ) = sgn(w 1 x 1 + w 2 x 2 + ... + w n x n ) where the weights w i take values in { - 1,1}. We show that the complexity of this problem is markedly different from the problem of testing whether f is a general halfspace with arbitrary weights. While the latter can be done with a number of queries that is independent of n [7], to distinguish whether f is a ±-weight halfspace versus e -far from all such halfspaces we prove that nonadaptive algorithms must make Ω(logn ) queries. We complement this lower bound with a sublinear upper bound showing that

Lower bounds for testing computability by small width OBDDs

O(\sqrt{n}\cdot

Testing (subclasses of) halfspaces

poly

Testing Halfspaces

(\frac{1}{\epsilon}))

Testing halfspaces

queries suffice.

Testing +/- 1-Weight Halfspaces

We consider the problem of testing whether a function f : {0, 1}n → {0, 1} is computable by a read-once, width-2 ordered binary decision diagram (OBDD), also known as a branching program. This problem has two variants: one where the variables must occur in a fixed, known order, and one where the variables are allowed to occur in an arbitrary order. We show that for both variants, any nonadaptive testing algorithm must make Ω(n) queries, and thus any adaptive testing algorithm must make Ω(log n) queries. We also consider the more general problem of testing computability by width-w OBDDs where the variables occur in a fixed order. We show that for any constant w ≥ 4, Ω(n) queries are required, resolving a conjecture of Goldreich [15]. We prove all of our lower bounds using a new technique of Blais, Brody, and Matulef [6], giving simple reductions from known hard problems in communication complexity to the testing problems at hand. Our result for width-2 OBDDs provides the first example of the power of this technique for proving strong nonadaptive bounds.

Efficiently Testing Sparse GF(2) Polynomials

We address the problem of testing whether a Boolean-valued function f is a halfspace, i.e. a function of the form f(x) = sgn(wċx-θ). We consider halfspaces over the continuous domain Rn (endowed with the standard multivariate Gaussian distribution) as well as halfspaces over the Boolean cube {-1, 1}n (endowed with the uniform distribution). In both cases we give an algorithm that distinguishes halfspaces from functions that are e-far from any halfspace using only poly(1/e) queries, independent of the dimension n. In contrast to the case of general halfspaces, we show that testing natural subclasses of halfspaces can be markedly harder; for the class of {-1, 1}-weight halfspaces, we show that a tester must make at least O(log n) queries. We complement this lower bound with an upper bound showing that O(√n queries suffice.