1,565 research outputs found

    Classical and quantum algorithms for scaling problems

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    This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Ultrahomogeneous tensor spaces

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    A cubic space is a vector space equipped with a symmetric trilinear form. Using categorical Fra\"iss\'e theory, we show that there is a universal ultrahomogeneous cubic space VV of countable infinite dimension, which is unique up to isomorphism. The automorphism group GG of VV is quite large and, in some respects, similar to the infinite orthogonal group. We show that GG is a linear-oligomorphic group (a class of groups we introduce), and we determine the algebraic representation theory of GG. We also establish some model-theoretic results about VV: it is ω\omega-categorical (in a modified sense), and has quantifier elimination (for vectors). Our results are not specific to cubic spaces, and hold for a very general class of tensor spaces; we view these spaces as linear analogs of the relational structures studied in model theory.Comment: 33 page

    Towards a General Complex Systems Model of Economic Sanctions with Some Results Outlining Consequences of Sanctions on the Russian Economy and the World

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    The main purpose of this paper is to present a complex nonlinear modelling approach to analyzing mixed capitalist economic systems. An application of a more elaborate version of this model is to explore the consequences of sanctions on the Russian economy and evaluate the model’s predictive successes or failures. Furthermore, the formal expanded nonlinear model presented in the appendix may be seen as an initial step to put the analysis of economic sanctions within a formal complex socio-economic systems framework. The results obtained from this structural complex multisectoral model so far seem fairly accurate in terms of agreement with measured values of observable economic variables. The political consequences are uncertain and are to be explored separately in a companion paper and ultimately in a book length treatment. Methodologically, the paper also presents the case for using Social Accounting Matrix (SAM)-based models for understanding problems of analyzing sanctions in an economywide context. Linear as well as Nonlinear models are presented in the appendix. The nonlinear modelling approach might prove to be especially relevant for studying the properties of multiple equilibria and complex dynamics

    Impossibility Results for Lattice-Based Functional Encryption Schemes

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    Functional Encryption denotes a form of encryption where a master secret key-holder can control which functions a user can evaluate on encrypted data. Learning With Errors (LWE) (Regev, STOC\u2705) is known to be a useful cryptographic hardness assumption which implies strong primitives such as, for example, fully homomorphic encryption (Brakerski-Vaikuntanathan, FOCS\u2711) and lockable obfuscation (Goyal et al., Wichs et al., FOCS\u2717). Despite its strength, however, there is just a limited number of functional encryption schemes which can be based on LWE. In fact, there are functional encryption schemes which can be achieved by using pairings but for which no secure instantiations from lattice-based assumptions are known: function-hiding inner product encryption (Lin, Baltico et al., CRYPTO\u2717) and compact quadratic functional encryption (Abdalla et al., CRYPTO\u2718). This raises the question whether there are some mathematical barriers which hinder us from realizing function-hiding and compact functional encryption schemes from lattice-based assumptions as LWE. To study this problem, we prove an impossibility result for function-hiding functional encryption schemes which meet some algebraic restrictions at ciphertext encryption and decryption. Those restrictions are met by a lot of attribute-based, identity-based and functional encryption schemes whose security stems from LWE. Therefore, we see our results as important indications why it is hard to construct new functional encryption schemes from LWE and which mathematical restrictions have to be overcome to construct secure lattice-based functional encryption schemes for new functionalities

    Stabilisation for varieties in polynomial functors

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    Implicit Loss of Surjectivity and Facial Reduction: Theory and Applications

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    Facial reduction, pioneered by Borwein and Wolkowicz, is a preprocessing method that is commonly used to obtain strict feasibility in the reformulated, reduced constraint system. The importance of strict feasibility is often addressed in the context of the convergence results for interior point methods. Beyond the theoretical properties that the facial reduction conveys, we show that facial reduction, not only limited to interior point methods, leads to strong numerical performances in different classes of algorithms. In this thesis we study various consequences and the broad applicability of facial reduction. The thesis is organized in two parts. In the first part, we show the instabilities accompanied by the absence of strict feasibility through the lens of facially reduced systems. In particular, we exploit the implicit redundancies, revealed by each nontrivial facial reduction step, resulting in the implicit loss of surjectivity. This leads to the two-step facial reduction and two novel related notions of singularity. For the area of semidefinite programming, we use these singularities to strengthen a known bound on the solution rank, the Barvinok-Pataki bound. For the area of linear programming, we reveal degeneracies caused by the implicit redundancies. Furthermore, we propose a preprocessing tool that uses the simplex method. In the second part of this thesis, we continue with the semidefinite programs that do not have strictly feasible points. We focus on the doubly-nonnegative relaxation of the binary quadratic program and a semidefinite program with a nonlinear objective function. We closely work with two classes of algorithms, the splitting method and the Gauss-Newton interior point method. We elaborate on the advantages in building models from facial reduction. Moreover, we develop algorithms for real-world problems including the quadratic assignment problem, the protein side-chain positioning problem, and the key rate computation for quantum key distribution. Facial reduction continues to play an important role for providing robust reformulated models in both the theoretical and the practical aspects, resulting in successful numerical performances

    Convex valuations, from Whitney to Nash

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    We consider the Whitney problem for valuations: does a smooth jj-homogeneous translation-invariant valuation on Rn\mathbb R^n exist that has given restrictions to a fixed family SS of linear subspaces? A necessary condition is compatibility: the given valuations must coincide on intersections. We show that for S=Grr(Rn)S=\mathrm{Gr}_r(\mathbb R^n), the grassmannian of rr-planes, this condition becomes sufficient once r≥j+2r\geq j+2. This complements the Klain and Schneider uniqueness theorems with an existence statement, and provides a recursive description of the image of the cosine transform. Informally speaking, we show that the transition from densities to valuations is localized to codimension 22. We then look for conditions on SS when compatibility is also sufficient for extensibility, in two distinct regimes: finite arrangements of subspaces, and compact submanifolds of the grassmannian. In both regimes we find unexpected flexibility. As a consequence of the submanifold regime, we prove a Nash-type theorem for valuations on compact manifolds, from which in turn we deduce the existence of Crofton formulas for all smooth valuations on manifolds. As an intermediate step of independent interest, we construct Crofton formulas for all odd translation-invariant valuations.Comment: 53 page

    On Elliott's conjecture and applications

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    Let f:N→Df:\mathbb{N}\to \mathbb{D} be a multiplicative function. Under the merely necessary assumption that ff is non-pretentious (in the sense of Granville and Soundararajan), we show that for any pair of distinct integer shifts h1,h2h_1,h_2 the two-point correlation 1x∑n≤xf(n+h1)f‾(n+h2)\frac{1}{x}\sum_{n\leq x}{f(n+h_1)\overline{f}(n+h_2)} tends to 00 along a set of x∈Nx\in\mathbb{N} of full upper logarithmic density. We also show that the same result holds for the kk-point correlations 1x∑n≤xf(n+h1)⋯f(n+hk)\frac{1}{x}\sum_{n\leq x}{f(n+h_1)\cdots f(n+h_k)} if kk is odd and ff is a real-valued non-pretentious function. Previously, the vanishing of correlations was known only under stronger non-pretentiousness hypotheses on ff by the works of Tao, and Tao and the third author. We derive several applications, including: (i) A classification of ±1\pm 1-valued completely multiplicative functions that omit a length four sign pattern, solving a 1974 conjecture of R.H. Hudson. (ii) A proof that a class of "Liouville-like" functions satisfies the unweighted Elliott conjecture of all orders, solving a problem of de la Rue. (iii) Constructing examples of multiplicative f:N→{−1,0,1}f:\mathbb{N}\to \{-1,0,1\} with a given (unique) Furstenberg system, answering a question of Lema\'nczyk. (iv) A density version of the Erd\H{o}s discrepancy theorem of Tao.Comment: 54 page

    Introduction to Riemannian Geometry and Geometric Statistics: from basic theory to implementation with Geomstats

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    International audienceAs data is a predominant resource in applications, Riemannian geometry is a natural framework to model and unify complex nonlinear sources of data.However, the development of computational tools from the basic theory of Riemannian geometry is laborious.The work presented here forms one of the main contributions to the open-source project geomstats, that consists in a Python package providing efficient implementations of the concepts of Riemannian geometry and geometric statistics, both for mathematicians and for applied scientists for whom most of the difficulties are hidden under high-level functions. The goal of this monograph is two-fold. First, we aim at giving a self-contained exposition of the basic concepts of Riemannian geometry, providing illustrations and examples at each step and adopting a computational point of view. The second goal is to demonstrate how these concepts are implemented in Geomstats, explaining the choices that were made and the conventions chosen. The general concepts are exposed and specific examples are detailed along the text.The culmination of this implementation is to be able to perform statistics and machine learning on manifolds, with as few lines of codes as in the wide-spread machine learning tool scikit-learn. We exemplify this with an introduction to geometric statistics
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