Subspaces of tensors with high analytic rank

It is shown that for any subspace V⊆Fn×⋯×np of d-tensors, if dim(V)≥tnd−1, then there is subspace W⊆V of dimension at least t/(dr)−1 whose nonzero elements all have analytic rank Ωd,p(r). As an application, we generalize a result of Altman on Szemerédi's theorem with random differences

Gaussian width bounds with applications to arithmetic progressions in random settings

Motivated by two problems on arithmetic progressions (APs)—concerning large deviations for AP counts in random sets and random differences in Szemer´edi’s theorem— we prove upper bounds on the Gaussian width of the image of the n-dimensional Boolean hypercube under a mapping ψ : Rn → Rk, where each coordinate is a constant-degree multilinear polynomial with 0/1 coefficients. We show the following applications of our bounds. Let [Z/NZ]p be the random subset of Z/NZ containing each element independently with probability p. • Let Xk be the number of k-term APs in [Z/NZ]p. We show that a precise estimate on the large deviation rate log Pr[Xk ≥ (1 + δ)EXk] due to Bhattacharya, Ganguly, Shao and Zhao is valid if

Document Similarity from Vector Space Densities

We propose a computationally light method for estimating similarities between text documents, which we call the density similarity (DS) method. The method is based on a word embedding in a high-dimensional Euclidean space and on kernel regression, and takes into account semantic relations among words. We find that the accuracy of this method is virtually the same as that of a state-of-the-art method, while the gain in speed is very substantial. Additionally, we introduce generalized versions of the top-k accuracy metric and of the Jaccard metric of agreement between similarity models.Comment: 12 pages, 3 figure

Quantum query algorithms are completely bounded forms

We prove a characterization of quantum query algorithms in terms of polynomials satisfying a certain (completely bounded) norm constraint. Based on this, we obtain a refined notion of approximate polynomial degree that equals the quantum query complexity, answering a question of Aaronson et al. (CCC’16). Using this characterization, we show that many polynomials of degree at least 4 are far from those coming from quantum query algorithms. Our proof is based on a fundamental result of Christensen and Sinclair (J. Funct. Anal., 1987) that generalizes the well-known Stinespring representation for quantum channels to multilinear forms. We also give a simple and short proof of one of the results of Aaronson et al. showing an equivalence between one-query quantum algorithms and bounded quadratic polynomials

Subspaces of tensors with high analytic rank

It is shown that if V ⊆ F p n ×⋯×np is a subspace of d-tensors with dimension at least tnd-1, then there is a subspace W ⊆ V of dimension at least t/(dr)−1 p is a subspace of d-tensors with dimension whose nonzero elements all have analytic rank Ωd,p(r). As an application, we generalize a result of Altman on Szemerédi's theorem with random differences

On the existence of 0/1 polytopes with high semidefinite extension complexity

In Rothvoß (Math Program 142(1–2):255–268, 2013) it was shown that there exists a 0/1 polytope (a polytope whose vertices are in {0, 1}n) such that any higher-dimensional polytope projecting to it must have 2Ω(n) facets, i.e., its linear extension complexity is exponential. The question whether there exists a 0/1 polytope with high positive semidefinite extension complexity was left open. We answer this question in the affirmative by showing that there is a 0/1 polytope such that any spectrahedron projecting to it must be the intersection of a semidefinite cone of dimension 2Ω(n) and an affine space. Our proof relies on a new technique to rescale semidefinite factorizations

Malaria intervention scale-up in Africa : effectiveness predictions for health programme planning tools, based on dynamic transmission modelling

Scale-up of malaria prevention and treatment needs to continue to further important gains made in the past decade, but national strategies and budget allocations are not always evidence-based. Statistical models were developed summarizing dynamically simulated relations between increases in coverage and intervention impact, to inform a malaria module in the Spectrum health programme planning tool.; The dynamic Plasmodium falciparum transmission model OpenMalaria was used to simulate health effects of scale-up of insecticide-treated net (ITN) usage, indoor residual spraying (IRS), management of uncomplicated malaria cases (CM) and seasonal malaria chemoprophylaxis (SMC) over a 10-year horizon, over a range of settings with stable endemic malaria. Generalized linear regression models (GLMs) were used to summarize determinants of impact across a range of sub-Sahara African settings.; Selected (best) GLMs explained 94-97 % of variation in simulated post-intervention parasite infection prevalence, 86-97 % of variation in case incidence (three age groups, three 3-year horizons), and 74-95 % of variation in malaria mortality. For any given effective population coverage, CM and ITNs were predicted to avert most prevalent infections, cases and deaths, with lower impacts for IRS, and impacts of SMC limited to young children reached. Proportional impacts were larger at lower endemicity, and (except for SMC) largest in low-endemic settings with little seasonality. Incremental health impacts for a given coverage increase started to diminish noticeably at above ~40 % coverage, while in high-endemic settings, CM and ITNs acted in synergy by lowering endemicity. Vector control and CM, by reducing endemicity and acquired immunity, entail a partial rebound in malaria mortality among people above 5 years of age from around 5-7 years following scale-up. SMC does not reduce endemicity, but slightly shifts malaria to older ages by reducing immunity in child cohorts reached.; Health improvements following malaria intervention scale-up vary with endemicity, seasonality, age and time. Statistical models can emulate epidemiological dynamics and inform strategic planning and target setting for malaria control

Outlaw distributions and locally decodable codes

Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. Despite decades of study, the optimal trade-off between query complexity and codeword length is far from understood. In this work, we give a new characterization of LDCs using distributions over Boolean functions whose expectation is hard to approximate (in L∞ norm) with a small number of samples. We coin the term “outlaw distributions” for such distributions since they “defy” the Law of Large Numbers. We show that the existence of outlaw distributions over sufficiently “smooth” functions implies the existence of constant query LDCs and vice versa. We give several candidates for outlaw distributions over smooth functions coming from finite field incidence geometry, additive combinatorics and hypergraph (non)expanders. We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters