43 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Machine Learning Algorithm for the Scansion of Old Saxon Poetry

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    Several scholars designed tools to perform the automatic scansion of poetry in many languages, but none of these tools deal with Old Saxon or Old English. This project aims to be a first attempt to create a tool for these languages. We implemented a Bidirectional Long Short-Term Memory (BiLSTM) model to perform the automatic scansion of Old Saxon and Old English poems. Since this model uses supervised learning, we manually annotated the Heliand manuscript, and we used the resulting corpus as labeled dataset to train the model. The evaluation of the performance of the algorithm reached a 97% for the accuracy and a 99% of weighted average for precision, recall and F1 Score. In addition, we tested the model with some verses from the Old Saxon Genesis and some from The Battle of Brunanburh, and we observed that the model predicted almost all Old Saxon metrical patterns correctly misclassified the majority of the Old English input verses

    Discovering causal relations and equations from data

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    Physics is a field of science that has traditionally used the scientific method to answer questions about why natural phenomena occur and to make testable models that explain the phenomena. Discovering equations, laws, and principles that are invariant, robust, and causal has been fundamental in physical sciences throughout the centuries. Discoveries emerge from observing the world and, when possible, performing interventions on the system under study. With the advent of big data and data-driven methods, the fields of causal and equation discovery have developed and accelerated progress in computer science, physics, statistics, philosophy, and many applied fields. This paper reviews the concepts, methods, and relevant works on causal and equation discovery in the broad field of physics and outlines the most important challenges and promising future lines of research. We also provide a taxonomy for data-driven causal and equation discovery, point out connections, and showcase comprehensive case studies in Earth and climate sciences, fluid dynamics and mechanics, and the neurosciences. This review demonstrates that discovering fundamental laws and causal relations by observing natural phenomena is revolutionised with the efficient exploitation of observational data and simulations, modern machine learning algorithms and the combination with domain knowledge. Exciting times are ahead with many challenges and opportunities to improve our understanding of complex systems

    QSETH strikes again: finer quantum lower bounds for lattice problem, strong simulation, hitting set problem, and more

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    While seemingly undesirable, it is not a surprising fact that there are certain problems for which quantum computers offer no computational advantage over their respective classical counterparts. Moreover, there are problems for which there is no `useful' computational advantage possible with the current quantum hardware. This situation however can be beneficial if we don't want quantum computers to solve certain problems fast - say problems relevant to post-quantum cryptography. In such a situation, we would like to have evidence that it is difficult to solve those problems on quantum computers; but what is their exact complexity? To do so one has to prove lower bounds, but proving unconditional time lower bounds has never been easy. As a result, resorting to conditional lower bounds has been quite popular in the classical community and is gaining momentum in the quantum community. In this paper, by the use of the QSETH framework [Buhrman-Patro-Speelman 2021], we are able to understand the quantum complexity of a few natural variants of CNFSAT, such as parity-CNFSAT or counting-CNFSAT, and also are able to comment on the non-trivial complexity of approximate-#CNFSAT; both of these have interesting implications about the complexity of (variations of) lattice problems, strong simulation and hitting set problem, and more. In the process, we explore the QSETH framework in greater detail than was (required and) discussed in the original paper, thus also serving as a useful guide on how to effectively use the QSETH framework.Comment: 34 pages, 2 tables, 2 figure

    Traversing the FFT Computation Tree for Dimension-Independent Sparse Fourier Transforms

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    We consider the well-studied Sparse Fourier transform problem, where one aims to quickly recover an approximately Fourier kk-sparse vector x^Cnd\widehat{x} \in \mathbb{C}^{n^d} from observing its time domain representation xx. In the exact kk-sparse case the best known dimension-independent algorithm runs in near cubic time in kk and it is unclear whether a faster algorithm like in low dimensions is possible. Beyond that, all known approaches either suffer from an exponential dependence on the dimension dd or can only tolerate a trivial amount of noise. This is in sharp contrast with the classical FFT of Cooley and Tukey, which is stable and completely insensitive to the dimension of the input vector: its runtime is O(NlogN)O(N\log N) in any dimension dd for N=ndN=n^d. Our work aims to address the above issues. First, we provide a translation/reduction of the exactly kk-sparse FT problem to a concrete tree exploration task which asks to recover kk leaves in a full binary tree under certain exploration rules. Subsequently, we provide (a) an almost quadratic in kk time algorithm for this task, and (b) evidence that a strongly subquadratic time for Sparse FT via this approach is likely impossible. We achieve the latter by proving a conditional quadratic time lower bound on sparse polynomial multipoint evaluation (the classical non-equispaced sparse FT) which is a core routine in the aforementioned translation. Thus, our results combined can be viewed as an almost complete understanding of this approach, which is the only known approach that yields sublinear time dimension-independent Sparse FT algorithms. Subsequently, we provide a robustification of our algorithm, yielding a robust cubic time algorithm under bounded 2\ell_2 noise. This requires proving new structural properties of the recently introduced adaptive aliasing filters combined with a variety of new techniques and ideas

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Matrix Multiplication Verification Using Coding Theory

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    We study the Matrix Multiplication Verification Problem (MMV) where the goal is, given three n×nn \times n matrices AA, BB, and CC as input, to decide whether AB=CAB = C. A classic randomized algorithm by Freivalds (MFCS, 1979) solves MMV in O~(n2)\widetilde{O}(n^2) time, and a longstanding challenge is to (partially) derandomize it while still running in faster than matrix multiplication time (i.e., in o(nω)o(n^{\omega}) time). To that end, we give two algorithms for MMV in the case where ABCAB - C is sparse. Specifically, when ABCAB - C has at most O(nδ)O(n^{\delta}) non-zero entries for a constant 0δ<20 \leq \delta < 2, we give (1) a deterministic O(nωε)O(n^{\omega - \varepsilon})-time algorithm for constant ε=ε(δ)>0\varepsilon = \varepsilon(\delta) > 0, and (2) a randomized O~(n2)\widetilde{O}(n^2)-time algorithm using δ/2log2n+O(1)\delta/2 \cdot \log_2 n + O(1) random bits. The former algorithm is faster than the deterministic algorithm of K\"{u}nnemann (ESA, 2018) when δ1.056\delta \geq 1.056, and the latter algorithm uses fewer random bits than the algorithm of Kimbrel and Sinha (IPL, 1993), which runs in the same time and uses log2n+O(1)\log_2 n + O(1) random bits (in turn fewer than Freivalds's algorithm). We additionally study the complexity of MMV. We first show that all algorithms in a natural class of deterministic linear algebraic algorithms for MMV (including ours) require Ω(nω)\Omega(n^{\omega}) time. We also show a barrier to proving a super-quadratic running time lower bound for matrix multiplication (and hence MMV) under the Strong Exponential Time Hypothesis (SETH). Finally, we study relationships between natural variants and special cases of MMV (with respect to deterministic O~(n2)\widetilde{O}(n^2)-time reductions)

    Dynamic Dynamic Time Warping

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    The Dynamic Time Warping (DTW) distance is a popular similarity measure for polygonal curves (i.e., sequences of points). It finds many theoretical and practical applications, especially for temporal data, and is known to be a robust, outlier-insensitive alternative to the \frechet distance. For static curves of at most nn points, the DTW distance can be computed in O(n2)O(n^2) time in constant dimension. This tightly matches a SETH-based lower bound, even for curves in R1\mathbb{R}^1. In this work, we study \emph{dynamic} algorithms for the DTW distance. Here, the goal is to design a data structure that can be efficiently updated to accommodate local changes to one or both curves, such as inserting or deleting vertices and, after each operation, reports the updated DTW distance. We give such a data structure with update and query time O(n1.5logn)O(n^{1.5} \log n), where nn is the maximum length of the curves. As our main result, we prove that our data structure is conditionally \emph{optimal}, up to subpolynomial factors. More precisely, we prove that, already for curves in R1\mathbb{R}^1, there is no dynamic algorithm to maintain the DTW distance with update and query time~\makebox{O(n1.5δ)O(n^{1.5 - \delta})} for any constant δ>0\delta > 0, unless the Negative-kk-Clique Hypothesis fails. In fact, we give matching upper and lower bounds for various trade-offs between update and query time, even in cases where the lengths of the curves differ.Comment: To appear at SODA2

    Automated Reasoning

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    This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book

    Sparse {Fourier Transform} by Traversing {Cooley-Tukey FFT} Computation Graphs

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    Computing the dominant Fourier coefficients of a vector is a common task in many fields, such as signal processing, learning theory, and computational complexity. In the Sparse Fast Fourier Transform (Sparse FFT) problem, one is given oracle access to a dd-dimensional vector xx of size NN, and is asked to compute the best kk-term approximation of its Discrete Fourier Transform, quickly and using few samples of the input vector xx. While the sample complexity of this problem is quite well understood, all previous approaches either suffer from an exponential dependence of runtime on the dimension dd or can only tolerate a trivial amount of noise. This is in sharp contrast with the classical FFT algorithm of Cooley and Tukey, which is stable and completely insensitive to the dimension of the input vector: its runtime is O(NlogN)O(N\log N) in any dimension dd. In this work, we introduce a new high-dimensional Sparse FFT toolkit and use it to obtain new algorithms, both on the exact, as well as in the case of bounded 2\ell_2 noise. This toolkit includes i) a new strategy for exploring a pruned FFT computation tree that reduces the cost of filtering, ii) new structural properties of adaptive aliasing filters recently introduced by Kapralov, Velingker and Zandieh'SODA'19, and iii) a novel lazy estimation argument, suited to reducing the cost of estimation in FFT tree-traversal approaches. Our robust algorithm can be viewed as a highly optimized sparse, stable extension of the Cooley-Tukey FFT algorithm. Finally, we explain the barriers we have faced by proving a conditional quadratic lower bound on the running time of the well-studied non-equispaced Fourier transform problem. This resolves a natural and frequently asked question in computational Fourier transforms. Lastly, we provide a preliminary experimental evaluation comparing the runtime of our algorithm to FFTW and SFFT 2.0
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