A number of recent works have studied algorithms for entrywise p-low rank approximation, namely algorithms which given an n×d matrix A (with n ≥ d), output a rank-k matrix B minimizing A − B p p = i,j |Ai,j − Bi,j| p when p > 0; and A − B 0 = i,j [Ai,j = Bi,j] for p = 0, where [·] is the Iverson bracket, that is, A − B 0 denotes the number of entries (i, j) for which Ai,j = Bi,j. For p = 1, this is often considered more robust than the SVD, while for p = 0 this corresponds to minimizing the number of disagreements, or robust PCA. This problem is known to be NP-hard for p ∈ {0, 1}, already for k = 1, and while there are polynomial time approximation algorithms, their approximation factor is at best poly(k). It was left open if there was a polynomial-time approximation scheme (PTAS) for p-approximation for any p ≥ 0. We show the following:1. On the algorithmic side, for p ∈ (0, 2), we give the first n poly(k/ε) time (1 + ε)approximation algorithm. For p = 0, there are various problem formulations, a common one being the binary setting in which A ∈ {0, 1} n×d and B = U · V , where U ∈ {0, 1} n×k and V ∈ {0, 1} k×d . There are also various notions of multiplication U · V , such as a matrix product over the reals, over a finite field, or over a Boolean semiring. We give the first almost-linear time approximation scheme for what we call the Generalized Binary 0-Rank-k problem, for which these variants are special cases. Our algorithm computes (1 + ε)-approximation in time (1/ε) 2 O(k) /ε 2 · nd 1+o(1) , where o(1) hides a factor (log log d) 1.1 / log d. In addition, for the case of finite fields of constant size, we obtain an alternate PTAS running in time n · d poly(k/ε) . Definition 2. (Generalized Binary 0 -Rank-k) Given a matrix A ∈ {0, 1} n×d with n ≥ d, an integer k, and an inner product function ., . :Our first result for p = 0 is as follows.Theorem 2 (PTAS for p = 0). For any ε ∈ (0, 1 2 ), there is a (1+ε)-approximation algorithm for the Generalized Binary 0 -Rank-k problem running in time (1/ε) 2 O(k) /ε 2 · nd 1+o(1) and succeeds with constant probability 1 , where o(1) hides a factor (log log d)Hence, we obtain the first almost-linear time approximation scheme for the Generalized Binary 0 -Rank-k problem, for any constant k. In particular, this yields the first polynomial time (1+ε)-approximation for constant k for 0 -low rank approximation of binary matrices when the underlying field is F 2 or the Boolean semiring. Even for k = 1, no PTAS was known before.Theorem 2 is doubly-exponential in k, and we show below that this is necessary for any approximation algorithm for Generalized Binary 0 -Rank-k. However, in the special case when the base field is F 2 , or more generally F q and A, U, and V have entries belonging to F q , it is possible to obtain an algorithm running in time n·d poly(k/ε) , which is an improvement for certain super-constant values of k and ε. We formally define the problem and state our result next. Definition 3. (Entrywise 0 -Rank-k Approximation over F q ) Given an n × d matrix A with e...

For an n-variate order-d tensor A, defineA, x ⊗d to be the maximum value taken by the tensor on the unit sphere. It is known that for a random tensor with i.We study the problem of efficiently certifying upper bounds on A max via the natural relaxation from the Sum of Squares (SoS) hierarchy. Our results include: * Supported by NSF grants CCF-1422045 and CCF-1526092.

Abstract.We prove an analog of Parikh's theorem for weighted context-free grammars over commutative, idempotent semirings, and exhibit a stochastic context-free grammar with behavior that cannot be realized by any stochastic right-linear context-free grammar. Finally, we show that every unary stochastic context-free grammar with polynomially-bounded ambiguity has an equivalent stochastic right-linear context-free grammar.

We study the problem of computing the p → q operator norm of a matrix A in R m×n , defined as ||A|| p→q := sup x∈R n \{0} ||Ax|| q /||x|| p . This problem generalizes the spectral norm of a matrix (p = q = 2) and the Grothendieck problem (p = ∞, q = 1), and has been widely studied in various regimes.When p ≥ q, the problem exhibits a dichotomy: constant factor approximation algorithms are known if 2 is in [q, p], and the problem is hard to approximate within almost polynomial factors when 2 is not in [q,p]. For the case when 2 is in [q, p] we prove almost matching approximation and NP-hardness results.The regime when p < q, known as hypercontractive norms, is particularly significant for various applications but much less well understood. The case with p = 2 and q > 2 was studied by [Barak et. al., STOC'12] who gave sub-exponential algorithms for a promise version of the problem (which captures small-set expansion) and also proved hardness of approximation results based on the Exponential Time Hypothesis. However, no NPhardness of approximation is known for these problems for any p < q.We prove the first NP-hardness result for approximating hypercontractive norms. We show that for any 1 < p < q < ∞ with 2 not in [p, q], ||A|| p→q is hard to approximate within 2 O(log 1−ε n) assuming NP is not contained in BPTIME(2 log O(1) n ).

We consider the following basic problem: given an n-variate degree-d homogeneous polynomial f with real coefficients, compute a unit vector x ∈ R n that maximizes | f (x)|. Besides its fundamental nature, this problem arises in diverse contexts ranging from tensor and operator norms to graph expansion to quantum information theory. The homogeneous degree 2 case is efficiently solvable as it corresponds to computing the spectral norm of an associated matrix, but the higher degree case is NP-hard.We give approximation algorithms for this problem that offer a trade-off between the approximation ratio and running time: in n O(q) time, we get an approximation within factor O d ((n/q) d/2−1 ) for arbitrary polynomials, O d ((n/q) d/4−1/2 ) for polynomials with non-negative coefficients, and O d ( m/q) for sparse polynomials with m monomials. The approximation guarantees are with respect to the optimum of the level-q sum-of-squares (SoS) SDP relaxation of the problem (though our algorithms do not rely on actually solving the SDP). Known polynomial time algorithms for this problem rely on "decoupling lemmas." Such tools are not capable of offering a tradeoff like our results as they blow up the number of variables by a factor equal to the degree. We develop new decoupling tools that are more efficient in the number of variables at the expense of less structure in the output polynomials. This enables us to harness the benefits of higher level SoS relaxations. Our decoupling methods also work with "folded polynomials," which are polynomials with polynomials as coefficients. This allows us to exploit easy substructures (such as quadratics) by considering them as coefficients in our algorithms.We complement our algorithmic results with some polynomially large integrality gaps for d-levels of the SoS relaxation. For general polynomials this follows from known results for random polynomials, which yield a gap of Ω d (n d/4−1/2 ). For polynomials with non-negative coefficients, we prove anΩ(n 1/6 ) gap for the degree 4 case, based on a novel distribution of 4-uniform hypergraphs. We establish an n Ω(d) gap for general degree d, albeit for a slightly weaker (but still very natural) relaxation. Toward this, we give a method to lift a level-4 solution matrix M to a higher level solution, under a mild technical condition on M.From a structural perspective, our work yields worst-case convergence results on the performance of the sum-of-squares hierarchy for polynomial optimization. Despite the popularity of SoS in this context, such results were previously only known for the case of q = Ω(n). Open problems 53A Oracle Lower Bound 57To remedy this, one can instead consider Λ(h 2 ) which is a relaxation of h 2 , since (h 2 ) max = h 2 2 . More generally, for a degree-d homogeneous polynomial f and an integer q divisible by 2d, we have the upper estimateThe following result shows that Λ f q/d d/q approximates f 2 within polynomial factors, and also gives an algorithm to approximate f 2 with respect to the upper bound Λ f q/d d/q . In th...

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We consider the ( p , r )-Grothendieck problem, which seeks to maximize the bilinear form y T Ax for an input matrix A ∈ R m×n over vectors x, y with x p = y r = 1. The problem is equivalent to computing the p → r * operator norm of A, where r * is the dual norm to r . The case p = r = ∞ corresponds to the classical Grothendieck problem. Our main result is an algorithm for arbitrary p, r ≥ 2 with approximation ratio (1 + ε 0 )/(sinh −1 (1) • γ p * γ r * ) for some fixed ε 0 ≤ 0.00863. Here γ t denotes the t'th norm of the standard Gaussian. Comparing this with Krivine's approximation ratio (π/2)/ sinh −1 (1) for the original Grothendieck problem, our guarantee is off from the best known hardness factor of (γ p * γ r * ) −1 for the problem by a factor similar to Krivine's defect (up to the constant (1 + ε 0 )).Our approximation follows by bounding the value of the natural vector relaxation for the problem which is convex when p, r ≥ 2. We give a generalization of random hyperplane rounding using Hölder-duals of Gaussian projections rather than taking the sign. We relate the performance of this rounding to certain hypergeometric functions, which prescribe necessary transformations to the vector solution before the rounding is applied. Unlike Krivine's Rounding where the relevant hypergeometric function was arcsin, we have to study a family of hypergeometric functions. The bulk of our technical work then involves methods from complex analysis to gain detailed information about the Taylor series coefficients of the inverses of these hypergeometric functions, which then dictate our approximation factor.Our result also implies improved bounds for "factorization through n 2 " of operators from n p to m q (when p ≥ 2 ≥ q), and our work provides modest supplementary evidence for an intriguing parallel between factorizability, and constant-factor approximability.