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We consider the problem of finding a compromise between the opinions of a group of individuals on a number of mutually independent, binary topics. In this paper, we quantify the loss in representativeness that results from requiring the outcome to have majority support, in other words, the "price of majority support". Each individual is assumed to support an outcome if they agree with the outcome on at least as many topics as they disagree on. Our results can also be seen as quantifying Anscombes paradox which states that topic-wise majority outcome may not be supported by a majority. To measure the representativeness of an outcome, we consider two metrics. First, we look for an outcome that agrees with a majority on as many topics as possible. We prove that the maximum number such that there is guaranteed to exist an outcome that agrees with a majority on this number of topics and has majority support, equals $\ceil{(t+1)/2}$ where $t$ is the total number of topics. Second, we count the number of times a voter opinion on a topic matches the outcome on that topic. The goal is to find the outcome with majority support with the largest number of matches. We consider the ratio between this number and the number of matches of the overall best outcome which may not have majority support. We try to find the maximum ratio such that an outcome with majority support and this ratio of matches compared to the overall best is guaranteed to exist. For 3 topics, we show this ratio to be $5/6\approx 0.83$. In general, we prove an upper bound that comes arbitrarily close to $2\sqrt{6}-4\approx 0.90$ as $t$ tends to infinity. Furthermore, we numerically compute a better upper and a non-matching lower bound in the relevant range for $t$.

A homomorphism $f$ from a guest graph $G$ to a host graph $H$ is locally bijective, injective or surjective if for every $u\in V(G)$, the restriction of $f$ to the neighbourhood of $u$ is bijective, injective or surjective, respectively. The corresponding decision problems, LBHOM, LIHOM and LSHOM, are well studied both on general graphs and on special graph classes. Apart from complexity results when the problems are parameterized by the treewidth and maximum degree of the guest graph, the three problems still lack a thorough study of their parameterized complexity. This paper fills this gap: we prove a number of new FPT, W[1]-hard and para-NP-complete results by considering a hierarchy of parameters of the guest graph $G$. For our FPT results, we do this through the development of a new algorithmic framework that involves a general ILP model. To illustrate the applicability of the new framework, we also use it to prove FPT results for the Role Assignment problem, which originates from social network theory and is closely related to locally surjective homomorphisms.

Grover's algorithm, a well-know quantum search algorithm, allows one to find the correct item in a database, with quadratic speedup. In this paper we adapt Grover's algorithm to the problem of finding a correct answer to a natural language question in English, thus contributing to the growing field of Quantum Natural Language Processing. Using a grammar that can be interpreted as tensor contractions, each word is represented as a quantum state that serves as input to the quantum circuit. We here introduce a quantum measurement to contract the representations of words, resulting in the representation of larger text fragments. Using this framework, a representation for the question is found that contains all the possible answers in equal quantum superposition, and allows for the building of an oracle that can detect a correct answer, being agnostic to the specific question. Furthermore, we show that our construction can deal with certain types of ambiguous phrases by keeping the various different meanings in quantum superposition.

Crystal Structure Prediction (csp) is one of the central and most challenging problems in materials science and computational chemistry. In csp, the goal is to find a configuration of ions in 3D space that yields the lowest potential energy. Finding an efficient procedure to solve this complex optimisation question is a well known open problem. Due to the exponentially large search space, the problem has been referred in several materials-science papers as "NP-Hard and very challenging" without a formal proof. This paper fills a gap in the literature providing the first set of formally proven NP-Hardness results for a variant of csp with various realistic constraints. In particular, we focus on the problem of removal: the goal is to find a substructure with minimal potential energy, by removing a subset of the ions. Our main contributions are NP-Hardness results for the csp removal problem, new embeddings of combinatorial graph problems into geometrical settings, and a more systematic exploration of the energy function to reveal the complexity of csp. In a wider context, our results contribute to the analysis of computational problems for weighted graphs embedded into the three-dimensional Euclidean space.

Building a deep learning model for a Question-Answering (QA) task requires a lot of human effort, it may need several months to carefully tune various model architectures and find a best one. It's even harder to find different excellent models for multiple datasets. Recent works show that the best model structure is related to the dataset used, and one single model cannot adapt to all tasks. In this paper, we propose an automated Question-Answering framework, which could automatically adjust network architecture for multiple datasets. Our framework is based on an innovative evolution algorithm, which is stable and suitable for multiple dataset scenario. The evolution algorithm for search combine prior knowledge into initial population and use a performance estimator to avoid inefficient mutation by predicting the performance of candidate model architecture. The prior knowledge used in initial population could improve the final result of the evolution algorithm. The performance estimator could quickly filter out models with bad performance in population as the number of trials increases, to speed up the convergence. Our framework achieves 78.9 EM and 86.1 F1 on SQuAD 1.1, 69.9 EM and 72.5 F1 on SQuAD 2.0. On NewsQA dataset, the found model achieves 47.0 EM and 62.9 F1.

Visual question answering (VQA) demands simultaneous comprehension of both the image visual content and natural language questions. In some cases, the reasoning needs the help of common sense or general knowledge which usually appear in the form of text. Current methods jointly embed both the visual information and the textual feature into the same space. However, how to model the complex interactions between the two different modalities is not an easy task. In contrast to struggling on multimodal feature fusion, in this paper, we propose to unify all the input information by natural language so as to convert VQA into a machine reading comprehension problem. With this transformation, our method not only can tackle VQA datasets that focus on observation based questions, but can also be naturally extended to handle knowledge-based VQA which requires to explore large-scale external knowledge base. It is a step towards being able to exploit large volumes of text and natural language processing techniques to address VQA problem. Two types of models are proposed to deal with open-ended VQA and multiple-choice VQA respectively. We evaluate our models on three VQA benchmarks. The comparable performance with the state-of-the-art demonstrates the effectiveness of the proposed method.

We propose a new method of estimation in topic models, that is not a variation on the existing simplex finding algorithms, and that estimates the number of topics K from the observed data. We derive new finite sample minimax lower bounds for the estimation of A, as well as new upper bounds for our proposed estimator. We describe the scenarios where our estimator is minimax adaptive. Our finite sample analysis is valid for any number of documents (n), individual document length (N_i), dictionary size (p) and number of topics (K), and both p and K are allowed to increase with n, a situation not handled well by previous analyses. We complement our theoretical results with a detailed simulation study. We illustrate that the new algorithm is faster and more accurate than the current ones, although we start out with a computational and theoretical disadvantage of not knowing the correct number of topics K, while we provide the competing methods with the correct value in our simulations.

Question Answering for complex questions is often modeled as a graph construction or traversal task, where a solver must build or traverse a graph of facts that answer and explain a given question. This "multi-hop" inference has been shown to be extremely challenging, with few models able to aggregate more than two facts before being overwhelmed by "semantic drift", or the tendency for long chains of facts to quickly drift off topic. This is a major barrier to current inference models, as even elementary science questions require an average of 4 to 6 facts to answer and explain. In this work we empirically characterize the difficulty of building or traversing a graph of sentences connected by lexical overlap, by evaluating chance sentence aggregation quality through 9,784 manually-annotated judgments across knowledge graphs built from three free-text corpora (including study guides and Simple Wikipedia). We demonstrate semantic drift tends to be high and aggregation quality low, at between 0.04% and 3%, and highlight scenarios that maximize the likelihood of meaningfully combining information.

We propose the inverse problem of Visual question answering (iVQA), and explore its suitability as a benchmark for visuo-linguistic understanding. The iVQA task is to generate a question that corresponds to a given image and answer pair. Since the answers are less informative than the questions, and the questions have less learnable bias, an iVQA model needs to better understand the image to be successful than a VQA model. We pose question generation as a multi-modal dynamic inference process and propose an iVQA model that can gradually adjust its focus of attention guided by both a partially generated question and the answer. For evaluation, apart from existing linguistic metrics, we propose a new ranking metric. This metric compares the ground truth question's rank among a list of distractors, which allows the drawbacks of different algorithms and sources of error to be studied. Experimental results show that our model can generate diverse, grammatically correct and content correlated questions that match the given answer.

Questions that require counting a variety of objects in images remain a major challenge in visual question answering (VQA). The most common approaches to VQA involve either classifying answers based on fixed length representations of both the image and question or summing fractional counts estimated from each section of the image. In contrast, we treat counting as a sequential decision process and force our model to make discrete choices of what to count. Specifically, the model sequentially selects from detected objects and learns interactions between objects that influence subsequent selections. A distinction of our approach is its intuitive and interpretable output, as discrete counts are automatically grounded in the image. Furthermore, our method outperforms the state of the art architecture for VQA on multiple metrics that evaluate counting.