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We construct a small set of explicit linear transformations mapping $\mathbb{R}^n$ to $\mathbb{R}^t$, where $t=O(\log (\gamma^{-1}) \epsilon^{-2})$, such that the $L_2$ norm of any vector in $\mathbb{R}^n$ is distorted by at most $1\pm \epsilon$ in at least a fraction of $1 - \gamma$ of the transformations in the set. Albeit the tradeoff between the size of the set and the success probability is suboptimal compared with probabilistic arguments, we nevertheless are able to apply our construction to a number of problems. In particular, we use it to construct an $\epsilon$-sample (or pseudorandom generator) for linear threshold functions on $\mathbb{S}^{n-1}$ for $\epsilon = o(1)$. We also use it to construct an $\epsilon$-sample for spherical digons in $\mathbb{S}^{n-1}$ for $\epsilon = o(1)$. This construction leads to an efficient oblivious derandomization of the Goemans-Williamson Max-Cut algorithm and similar approximation algorithms (i.e., we construct a small set of hyperplanes such that for any instance we can choose one of them to generate a good solution). Our technique for constructing an $\epsilon$-sample for linear threshold functions on the sphere is considerably different than previous techniques that rely on $k$-wise independent sample spaces.

For any 00, we give an efficient deterministic construction of a linear subspace V ⊆ Rn, of dimension (1-e)n in which the lp and lr norms are the same up to a multiplicative factor of poly(e-1) (after proper normalization). As a corollary we get a deterministic compressed sensing algorithm (Basis Pursuit) for a new range of parameters. In particular, for any constant e>0 and p Re n with the l1/lp guarantee for (n ⋅ poly(e))-sparse vectors. Namely, let x be a vector in Rn whose l1 distance from a k-sparse vector (for some k=n ⋅ poly(e)) is δ. The algorithm, given Ax as input, outputs an n dimensional vector y such that ||x-y||p ≤ δ k1/p-1. In particular this gives a weak form of the l2/l1 guarantee.Our construction has the additional benefit that when viewed as a matrix, A has at most O(1) non-zero entries in each row. As a result, both the encoding (computing Ax) and decoding (retrieving x from Ax) can be computed efficiently.

We give the first sub-exponential time deterministic polynomial identity testing algorithm for depth-4 multilinear circuits with a small top fan-in. More accurately, our algorithm works for depth-4 circuits with a plus gate at the top (also known as ΣΠΣΠ circuits) and has a running time of exp(poly(log(n),log(s),k)) where n is the number of variables, s is the size of the circuit and k is the fan-in of the top gate. In particular, when the circuit is of polynomial (or quasi-polynomial) size, our algorithm runs in quasi-polynomial time. In [AV08], it was shown that derandomizing polynomial identity testing for general ΣΠΣΠ circuits implies a derandomization of polynomial identity testing in general arithmetic circuits. Prior to this work sub-exponential time deterministic algorithms were known for depth-$3$ circuits with small top fan-in and for very restricted versions of depth-4 circuits.The main ingredient in our proof is a new structural theorem for multilinear ΣΠΣΠ(k) circuits. Roughly, this theorem shows that any nonzero multilinear ΣΠΣΠ(k) circuit contains an `embedded' nonzero multilinear ΣΠΣ(k) circuit. Using ideas from previous works on identity testing of sums of read-once formulas and of depth-3 multilinear circuits, we are able to exploit this structure and obtain an identity testing algorithm for multilinear ΣΠΣΠ(k) circuits.

In this paper we give reconstruction algorithms for depth-3 arithmetic circuits with $k$ multiplication gates (also known as $\Sigma\Pi\Sigma(k)$ circuits), where $k=O(1)$. Namely, we give an algorithm that when given a black box holding a $\Sigma\Pi\Sigma(k)$ circuit $C$ over a field $\F$ as input, makes queries to the black box (possibly over a polynomial sized extension field of $\F$) and outputs a circuit $C'$ computing the same polynomial as $C$. In particular we obtain the following results. 1) When $C$ is a multilinear $\Sigma\Pi\Sigma(k)$ circuit (i.e. each of its multiplication gates computes a multilinear polynomial) then our algorithm runs in polynomial time (when $k$ is a constant) and outputs a multilinear $\Sigma\Pi\Sigma(k)$ circuits computing the same polynomial. 2) In the general case, our algorithm runs in quasi-polynomial time and outputs a generalized depth-3 circuit (as defined in \cite{KarninShpilka08}) with $k$ multiplication gates. For example, the polynomials computed by generalized depth-3 circuits can be computed by quasi-polynomial sized depth-3 circuits. In fact, our algorithm works in the slightly more general case where the black box holds a generalized depth-3 circuits. Prior to this work there were reconstruction algorithms for several different models of bounded depth circuits: the well studied class of depth-2 arithmetic circuits (that compute sparse polynomials) and its close by model of depth-3 set-multilinear circuits. For the class of depth-3 circuits only the case of $k=2$ (i.e. $\Sigma\Pi\Sigma(2)$ circuits) was known. Our proof technique combines ideas from [Shpilka09] and [KarninShpilka08] with some new ideas. Our most notable new ideas are: We prove the existence of a unique canonical representation of depth-3 circuits. This enables us to work with a specific representation in mind. Another technical contribution is an isolation lemma for depth-3 circuits that enables us to reconstruct a single multiplication gate of the circuit.

In this paper we consider the problem of determining whether an unknown arithmetic circuit, for which we have oracle access, computes the identically zero polynomial. This problem is known as the black-box polynomial identity testing (PIT) problem. Our focus is on polynomials that can be written in the form $f(\bar x) = \sum\nolimits_{i = 1}^k {h_i (\bar x) \cdot g_i (\bar x)} $, where each h i is a polynomial that depends on only ρ linear functions, and each g i is a product of linear functions (when h i = 1, for each i, then we get the class of depth-3 circuits with k multiplication gates, also known as ΣΠΣ(k) circuits, but the general case is much richer). When maxi (deg(h i · g i )) = d we say that f is computable by a ΣΠΣ(k; d;ρ) circuit. We obtain the following results. A deterministic black-box identity testing algorithm for ΣΠΣ(k; d;ρ) circuits that runs in quasi-polynomial time (for ρ=polylog(n+d)). In particular this gives the first black-box quasi-polynomial time PIT algorithm for depth-3 circuits with k multiplication gates. A deterministic black-box identity testing algorithm for read-k ΣΠΣ circuits (depth-3 circuits where each variable appears at most k times) that runs in time $n^{2^{O(k^2 )} } $. In particular this gives a polynomial time algorithm for k=O(1). Our results give the first sub-exponential black-box PIT algorithm for circuits of depth higher than 2. Another way of stating our results is in terms of test sets for the underlying circuit model. A test set is a set of points such that if two circuits get the same values on every point of the set then they compute the same polynomial. Thus, our first result gives an explicit test set, of quasi-polynomial size, for ΣΠΣ(k; d;ρ) circuits (when ρ=polylog(n+d)). Our second result gives an explicit polynomial size test set for read-k depth-3 circuits. The proof technique involves a construction of a family of affine subspaces that have a rank-preserving property that is inspired by the construction of linear seeded extractors for affine sources of Gabizon and Raz [9], and a generalization of a theorem of [8] regarding the structure of identically zero depth-3 circuits with bounded top fan-in.