945 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

    To have done with theory? Baudrillard, or the literal confrontation with reality

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    Baudrillard, Eluding the temptation to reinterpret Jean Baudrillard once more, this work started from the ambition to consider his thought in its irreducibility, that is, in a radically literal way. Literalness is a recurring though overlooked term in Baudrillard’s oeuvre, and it is drawn from the direct concatenation of words in poetry or puns and other language games. It does not indicate a realist positivism but a principle that considers the metamorphoses and mutual alteration of things in their singularity without reducing them to a general equivalent (i.e. the meaning of words in a poem, which destroys its appearances). Reapplying the idea to Baudrillard and finding other singular routes through his “passwords” is a way to short-circuit its reductio ad realitatem and reaffirm its challenge to the hegemony of global integration. Even in the literature dedicated to it, this exercise has been rarer than the ‘hermeneutical’ one, where Baudrillard’s oeuvre was taken as a discourse to be interpreted and explained (finding an equivalent for its singularity). In plain polemic with any ideal of conformity between theory and reality (from which our present conformisms arguably derive, too), Baudrillard conceived thought not as something to be verified but as a series of hypotheses to be repeatedly radicalised – he often described it as a “spiral”, a form which challenges the codification of things, including its own. Coherent with this, the thesis does not consider Baudrillard’s work either a reflection or a prediction of reality but, instead, an out-and-out act, a precious singular object which, interrogated, ‘thinks’ us and our current events ‘back’. In the second part, Baudrillard’s hypotheses are taken further and measured in their capacity to challenge the reality of current events and phenomena. The thesis confronts the ‘hypocritical’ position of critical thinking, which accepts the present principle of reality. It questions the interminability of our condition, where death seems thinkable only as a senseless interruption of the apparatus. It also confronts the solidarity between orthodox and alternative realities of the COVID pandemic and the Ukrainian invasion, searching for what is irreducible to the perfect osmosis of “virtual and factual”. Drawing equally from the convulsions of globalisation and the psychopathologies of academics, from DeLillo’s fiction and Baudrillard’s lesser-studied influences, this study evaluates the irreversibility of our system against the increasingly silent challenges of radical thought. It looks for what an increasingly pessimistic late Baudrillard called ‘rogue singularities’: forms which, often outside the conventional realms one would expect to find them, constitute potential sources of the fragility of global power. ‘To have done with theory’ does not mean abandoning radical thought and, together with it, the singularity of humanity. It means, as the thesis concludes, the courage to leave conventional ideas of theory and listen to less audible voices which, at the heart of this “enormous conspiracy”, whisper — as a mysterious lady in Mariupol did to Putin — “It’s all not true! It’s all for show!”

    Learning algebraic structures with the help of Borel equivalence relations

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    We study algorithmic learning of algebraic structures. In our framework, a learner receives larger and larger pieces of an arbitrary copy of a computable structure and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. The learning is successful if the conjectures eventually stabilize to a correct guess. We prove that a family of structures is learnable if and only if its learning domain is continuously reducible to the relation E0 of eventual agreement on reals. This motivates a novel research program, that is, using descriptive set theoretic tools to calibrate the (learning) complexity of nonlearnable families. Here, we focus on the learning power of well-known benchmark Borel equivalence relations (i.e., E1, E2, E3, Z0, and Eset)

    SPDH-Sign: towards Efficient, Post-quantum Group-based Signatures

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    In this paper, we present a new diverse class of post-quantum group-based Digital Signature Schemes (DSS). The approach is significantly different from previous examples of group-based digital signatures and adopts the framework of group action-based cryptography: we show that each finite group defines a group action relative to the semidirect product of the group by its automorphism group, and give security bounds on the resulting signature scheme in terms of the group-theoretic computational problem known as the Semidirect Discrete Logarithm Problem (SDLP). Crucially, we make progress towards being able to efficiently compute the novel group action, and give an example of a parameterised family of groups for which the group action can be computed for any parameters, thereby negating the need for expensive offline computation or inclusion of redundancy required in other schemes of this type

    Degree Spectra, and Relative Acceptability of Notations

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    On linear, fractional, and submodular optimization

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    In this thesis, we study four fundamental problems in the theory of optimization. 1. In fractional optimization, we are interested in minimizing a ratio of two functions over some domain. A well-known technique for solving this problem is the Newton– Dinkelbach method. We propose an accelerated version of this classical method and give a new analysis using the Bregman divergence. We show how it leads to improved or simplified results in three application areas. 2. The diameter of a polyhedron is the maximum length of a shortest path between any two vertices. The circuit diameter is a relaxation of this notion, whereby shortest paths are not restricted to edges of the polyhedron. For a polyhedron in standard equality form with constraint matrix A, we prove an upper bound on the circuit diameter that is quadratic in the rank of A and logarithmic in the circuit imbalance measure of A. We also give circuit augmentation algorithms for linear programming with similar iteration complexity. 3. The correlation gap of a set function is the ratio between its multilinear and concave extensions. We present improved lower bounds on the correlation gap of a matroid rank function, parametrized by the rank and girth of the matroid. We also prove that for a weighted matroid rank function, the worst correlation gap is achieved with uniform weights. Such improved lower bounds have direct applications in submodular maximization and mechanism design. 4. The last part of this thesis concerns parity games, a problem intimately related to linear programming. A parity game is an infinite-duration game between two players on a graph. The problem of deciding the winner lies in NP and co-NP, with no known polynomial algorithm to date. Many of the fastest (quasi-polynomial) algorithms have been unified via the concept of a universal tree. We propose a strategy iteration framework which can be applied on any universal tree

    (b2023 to 2014) The UNBELIEVABLE similarities between the ideas of some people (2006-2016) and my ideas (2002-2008) in physics (quantum mechanics, cosmology), cognitive neuroscience, philosophy of mind, and philosophy (this manuscript would require a REVOLUTION in international academy environment!)

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    (b2023 to 2014) The UNBELIEVABLE similarities between the ideas of some people (2006-2016) and my ideas (2002-2008) in physics (quantum mechanics, cosmology), cognitive neuroscience, philosophy of mind, and philosophy (this manuscript would require a REVOLUTION in international academy environment!

    From the Hardness of Detecting Superpositions to Cryptography: Quantum Public Key Encryption and Commitments

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    Recently, Aaronson et al. (arXiv:2009.07450) showed that detecting interference between two orthogonal states is as hard as swapping these states. While their original motivation was from quantum gravity, we show its applications in quantum cryptography. 1. We construct the first public key encryption scheme from cryptographic \emph{non-abelian} group actions. Interestingly, the ciphertexts of our scheme are quantum even if messages are classical. This resolves an open question posed by Ji et al. (TCC '19). We construct the scheme through a new abstraction called swap-trapdoor function pairs, which may be of independent interest. 2. We give a simple and efficient compiler that converts the flavor of quantum bit commitments. More precisely, for any prefix X,Y ∈\in {computationally,statistically,perfectly}, if the base scheme is X-hiding and Y-binding, then the resulting scheme is Y-hiding and X-binding. Our compiler calls the base scheme only once. Previously, all known compilers call the base schemes polynomially many times (Cr\'epeau et al., Eurocrypt '01 and Yan, Asiacrypt '22). For the security proof of the conversion, we generalize the result of Aaronson et al. by considering quantum auxiliary inputs.Comment: 51 page
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