A Quantum Lovasz Local Lemma

The Lovasz Local Lemma (LLL) is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of "weakly dependent" criteria. We show that the LLL extends to a much more general geometric setting, where events are replaced with subspaces and probability is replaced with relative dimension, which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions. Our result immediately applies to the k-QSAT problem: For instance we show that any collection of rank 1 projectors with the property that each qubit appears in at most $2^k/(e \cdot k)$ of them, has a joint satisfiable state. We then apply our results to the recently studied model of random k-QSAT. Recent works have shown that the satisfiable region extends up to a density of 1 in the large k limit, where the density is the ratio of projectors to qubits. Using a hybrid approach building on work by Laumann et al. we greatly extend the known satisfiable region for random k-QSAT to a density of $\Omega(2^k/k^2)$. Since our tool allows us to show the existence of joint satisfying states without the need to construct them, we are able to penetrate into regions where the satisfying states are conjectured to be entangled, avoiding the need to construct them, which has limited previous approaches to product states.Comment: 19 page

On the cost-complexity of multi-context systems

Multi-context systems provide a powerful framework for modelling information-aggregation systems featuring heterogeneous reasoning components. Their execution can, however, incur non-negligible cost. Here, we focus on cost-complexity of such systems. To that end, we introduce cost-aware multi-context systems, an extension of non-monotonic multi-context systems framework taking into account costs incurred by execution of semantic operators of the individual contexts. We formulate the notion of cost-complexity for consistency and reasoning problems in MCSs. Subsequently, we provide a series of results related to gradually more and more constrained classes of MCSs and finally introduce an incremental cost-reducing algorithm solving the reasoning problem for definite MCSs

Consistency and convergence rate of phylogenetic inference via regularization

It is common in phylogenetics to have some, perhaps partial, information about the overall evolutionary tree of a group of organisms and wish to find an evolutionary tree of a specific gene for those organisms. There may not be enough information in the gene sequences alone to accurately reconstruct the correct "gene tree." Although the gene tree may deviate from the "species tree" due to a variety of genetic processes, in the absence of evidence to the contrary it is parsimonious to assume that they agree. A common statistical approach in these situations is to develop a likelihood penalty to incorporate such additional information. Recent studies using simulation and empirical data suggest that a likelihood penalty quantifying concordance with a species tree can significantly improve the accuracy of gene tree reconstruction compared to using sequence data alone. However, the consistency of such an approach has not yet been established, nor have convergence rates been bounded. Because phylogenetics is a non-standard inference problem, the standard theory does not apply. In this paper, we propose a penalized maximum likelihood estimator for gene tree reconstruction, where the penalty is the square of the Billera-Holmes-Vogtmann geodesic distance from the gene tree to the species tree. We prove that this method is consistent, and derive its convergence rate for estimating the discrete gene tree structure and continuous edge lengths (representing the amount of evolution that has occurred on that branch) simultaneously. We find that the regularized estimator is "adaptive fast converging," meaning that it can reconstruct all edges of length greater than any given threshold from gene sequences of polynomial length. Our method does not require the species tree to be known exactly; in fact, our asymptotic theory holds for any such guide tree.Comment: 34 pages, 5 figures. To appear on The Annals of Statistic

Comment on "Support Vector Machines with Applications"

Comment on "Support Vector Machines with Applications" [math.ST/0612817]Comment: Published at http://dx.doi.org/10.1214/088342306000000475 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

On Multi-Relational Link Prediction with Bilinear Models

We study bilinear embedding models for the task of multi-relational link prediction and knowledge graph completion. Bilinear models belong to the most basic models for this task, they are comparably efficient to train and use, and they can provide good prediction performance. The main goal of this paper is to explore the expressiveness of and the connections between various bilinear models proposed in the literature. In particular, a substantial number of models can be represented as bilinear models with certain additional constraints enforced on the embeddings. We explore whether or not these constraints lead to universal models, which can in principle represent every set of relations, and whether or not there are subsumption relationships between various models. We report results of an independent experimental study that evaluates recent bilinear models in a common experimental setup. Finally, we provide evidence that relation-level ensembles of multiple bilinear models can achieve state-of-the art prediction performance

Fast Quantum Algorithm for Solving Multivariate Quadratic Equations

In August 2015 the cryptographic world was shaken by a sudden and surprising announcement by the US National Security Agency NSA concerning plans to transition to post-quantum algorithms. Since this announcement post-quantum cryptography has become a topic of primary interest for several standardization bodies. The transition from the currently deployed public-key algorithms to post-quantum algorithms has been found to be challenging in many aspects. In particular the problem of evaluating the quantum-bit security of such post-quantum cryptosystems remains vastly open. Of course this question is of primarily concern in the process of standardizing the post-quantum cryptosystems. In this paper we consider the quantum security of the problem of solving a system of {\it $m$ Boolean multivariate quadratic equations in $n$ variables} (\MQb); a central problem in post-quantum cryptography. When $n=m$, under a natural algebraic assumption, we present a Las-Vegas quantum algorithm solving \MQb{} that requires the evaluation of, on average, $O(2^{0.462n})$ quantum gates. To our knowledge this is the fastest algorithm for solving \MQb{}

Exponential Lower Bounds and Separation for Query Rewriting

We establish connections between the size of circuits and formulas computing monotone Boolean functions and the size of first-order and nonrecursive Datalog rewritings for conjunctive queries over OWL 2 QL ontologies. We use known lower bounds and separation results from circuit complexity to prove similar results for the size of rewritings that do not use non-signature constants. For example, we show that, in the worst case, positive existential and nonrecursive Datalog rewritings are exponentially longer than the original queries; nonrecursive Datalog rewritings are in general exponentially more succinct than positive existential rewritings; while first-order rewritings can be superpolynomially more succinct than positive existential rewritings