313 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Low Complexity Subshifts have Discrete Spectrum

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    We prove results about subshifts with linear (word) complexity, meaning that lim supp(n)n<\limsup \frac{p(n)}{n} < \infty, where for every nn, p(n)p(n) is the number of nn-letter words appearing in sequences in the subshift. Denoting this limsup by CC, we show that when C<43C < \frac{4}{3}, the subshift has discrete spectrum, i.e. is measurably isomorphic to a rotation of a compact abelian group with Haar measure. We also give an example with C=32C = \frac{3}{2} which has a weak mixing measure. This partially answers an open question of Ferenczi, who asked whether C=53C = \frac{5}{3} was the minimum possible among such subshifts; our results show that the infimum in fact lies in [43,32][\frac{4}{3}, \frac{3}{2}]. All results are consequences of a general S-adic/substitutive structure proved when C<43C < \frac{4}{3}

    Learning Neural Graph Representations in Non-Euclidean Geometries

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    The success of Deep Learning methods is heavily dependent on the choice of the data representation. For that reason, much of the actual effort goes into Representation Learning, which seeks to design preprocessing pipelines and data transformations that can support effective learning algorithms. The aim of Representation Learning is to facilitate the task of extracting useful information for classifiers and other predictor models. In this regard, graphs arise as a convenient data structure that serves as an intermediary representation in a wide range of problems. The predominant approach to work with graphs has been to embed them in an Euclidean space, due to the power and simplicity of this geometry. Nevertheless, data in many domains exhibit non-Euclidean features, making embeddings into Riemannian manifolds with a richer structure necessary. The choice of a metric space where to embed the data imposes a geometric inductive bias, with a direct impact on the performance of the models. This thesis is about learning neural graph representations in non-Euclidean geometries and showcasing their applicability in different downstream tasks. We introduce a toolkit formed by different graph metrics with the goal of characterizing the topology of the data. In that way, we can choose a suitable target embedding space aligned to the shape of the dataset. By virtue of the geometric inductive bias provided by the structure of the non-Euclidean manifolds, neural models can achieve higher performances with a reduced parameter footprint. As a first step, we study graphs with hierarchical structures. We develop different techniques to derive hierarchical graphs from large label inventories. Noticing the capacity of hyperbolic spaces to represent tree-like arrangements, we incorporate this information into an NLP model through hyperbolic graph embeddings and showcase the higher performance that they enable. Second, we tackle the question of how to learn hierarchical representations suited for different downstream tasks. We introduce a model that jointly learns task-specific graph embeddings from a label inventory and performs classification in hyperbolic space. The model achieves state-of-the-art results on very fine-grained labels, with a remarkable reduction of the parameter size. Next, we move to matrix manifolds to work on graphs with diverse structures and properties. We propose a general framework to implement the mathematical tools required to learn graph embeddings on symmetric spaces. These spaces are of particular interest given that they have a compound geometry that simultaneously contains Euclidean as well as hyperbolic subspaces, allowing them to automatically adapt to dissimilar features in the graph. We demonstrate a concrete implementation of the framework on Siegel spaces, showcasing their versatility on different tasks. Finally, we focus on multi-relational graphs. We devise the means to translate Euclidean and hyperbolic multi-relational graph embedding models into the space of symmetric positive definite (SPD) matrices. To do so we develop gyrocalculus in this geometry and integrate it with the aforementioned framework

    Transformers Learn Shortcuts to Automata

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    Algorithmic reasoning requires capabilities which are most naturally understood through recurrent models of computation, like the Turing machine. However, Transformer models, while lacking recurrence, are able to perform such reasoning using far fewer layers than the number of reasoning steps. This raises the question: what solutions are learned by these shallow and non-recurrent models? We find that a low-depth Transformer can represent the computations of any finite-state automaton (thus, any bounded-memory algorithm), by hierarchically reparameterizing its recurrent dynamics. Our theoretical results characterize shortcut solutions, whereby a Transformer with o(T)o(T) layers can exactly replicate the computation of an automaton on an input sequence of length TT. We find that polynomial-sized O(logT)O(\log T)-depth solutions always exist; furthermore, O(1)O(1)-depth simulators are surprisingly common, and can be understood using tools from Krohn-Rhodes theory and circuit complexity. Empirically, we perform synthetic experiments by training Transformers to simulate a wide variety of automata, and show that shortcut solutions can be learned via standard training. We further investigate the brittleness of these solutions and propose potential mitigations

    On the interplay between vortices and harmonic flows: Hodge decomposition of Euler's equations in 2d

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    Let Σ\Sigma be a compact manifold without boundary whose first homology is nontrivial. Hodge decomposition of the incompressible Euler's equation in terms of 1-forms yields a coupled PDE-ODE system. The L2L^2-orthogonal components are a `pure' vorticity flow and a potential flow (harmonic, with the dimension of the homology). In this paper we focus on NN point vortices on a compact Riemann surface without boundary of genus gg, with a metric chosen in the conformal class. The phase space has finite dimension 2N+2g2N+ 2g. We compute a surface of section for the motion of a single vortex (N=1N=1) on a torus (g=1g=1) with a non-flat metric, that shows typical features of non-integrable 2-dof Hamiltonians. In contradistinction, for flat tori the harmonic part is constant. Next, we turn to hyperbolic surfaces (g2 g \geq 2), having constant curvature -1, with discrete symmetries. Fixed points of involutions yield vortex crystals in the Poincar\'e disk. Finally we consider multiply connected planar domains. The image method due to Green and Thomson is viewed in the Schottky double. The Kirchhoff-Routh hamiltonian given in C.C. Lin's celebrated theorem is recovered by Marsden-Weinstein reduction from 2N+2g2N+2g to 2N2N. The relation between the electrostatic Green function and the hydrodynamical Green function is clarified. A number of questions are suggested

    Codes and Pseudo-Geometric Designs from the Ternary mm-Sequences with Welch-type decimation d=23(n1)/2+1d=2\cdot 3^{(n-1)/2}+1

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    Pseudo-geometric designs are combinatorial designs which share the same parameters as a finite geometry design, but which are not isomorphic to that design. As far as we know, many pseudo-geometric designs have been constructed by the methods of finite geometries and combinatorics. However, none of pseudo-geometric designs with the parameters S(2,q+1,(qn1)/(q1))S\left (2, q+1,(q^n-1)/(q-1)\right ) is constructed by the approach of coding theory. In this paper, we use cyclic codes to construct pseudo-geometric designs. We firstly present a family of ternary cyclic codes from the mm-sequences with Welch-type decimation d=23(n1)/2+1d=2\cdot 3^{(n-1)/2}+1, and obtain some infinite family of 2-designs and a family of Steiner systems S(2,4,(3n1)/2)S\left (2, 4, (3^n-1)/2\right ) using these cyclic codes and their duals. Moreover, the parameters of these cyclic codes and their shortened codes are also determined. Some of those ternary codes are optimal or almost optimal. Finally, we show that one of these obtained Steiner systems is inequivalent to the point-line design of the projective space PG(n1,3)\mathrm{PG}(n-1,3) and thus is a pseudo-geometric design.Comment: 15 pages. arXiv admin note: text overlap with arXiv:2206.15153, arXiv:2110.0388

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Distance-Biregular Graphs and Orthogonal Polynomials

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    This thesis is about distance-biregular graphs– when they exist, what algebraic and structural properties they have, and how they arise in extremal problems. We develop a set of necessary conditions for a distance-biregular graph to exist. Using these conditions and a computer, we develop tables of possible parameter sets for distancebiregular graphs. We extend results of Fiol, Garriga, and Yebra characterizing distance-regular graphs to characterizations of distance-biregular graphs, and highlight some new results using these characterizations. We also extend the spectral Moore bounds of Cioaba et al. to semiregular bipartite graphs, and show that distance-biregular graphs arise as extremal examples of graphs meeting the spectral Moore bound

    A visual representation of the Steiner triple systems of order 13

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    Steiner triple systems (STSs) are a basic topic in combinatorics. In an STS the elements can be collected in threes in such a way that any pair of elements is contained in a unique triple. The two smallest nontrivial STSs, with 7 and 9 elements, arise in the context of finite geometry and nonsingular cubic curves, and have well-known pictorial representations. On the contrary, an STS with 13 elements does not have an intrinsic geometric nature, nor a natural pictorial illustration. In this paper we present a visual representation of the two non-isomorphic Steiner triple systems of order 13 by means of a regular hexagram. The thirteen points of each system are the vertices of the twelve equilateral triangles inscribed in the hexagram. In the case of the non-cyclic system, our representation allows one to visualize in a simple, elegant and highly symmetric way the twenty-six triples, the six automorphisms and their orbits, the eight quadrilaterals, the ten mitres, the thirteen grids, the four 3-colouring patterns, the block-colouring and some distinguished ovals. Our construction is based on a very simple idea (seeing the blocks as much as possible as equilateral triangles), which can be further extended to get new representations of the STSs of order 7 and 9, and of one of the STSs of order 15
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