58,703 research outputs found

    Efficient inference in the transverse field Ising model

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    In this paper we introduce an approximate method to solve the quantum cavity equations for transverse field Ising models. The method relies on a projective approximation of the exact cavity distributions of imaginary time trajectories (paths). A key feature, novel in the context of similar algorithms, is the explicit separation of the classical and quantum parts of the distributions. Numerical simulations show accurate results in comparison with the sampled solution of the cavity equations, the exact diagonalization of the Hamiltonian (when possible) and other approximate inference methods in the literature. The computational complexity of this new algorithm scales linearly with the connectivity of the underlying lattice, enabling the study of highly connected networks, as the ones often encountered in quantum machine learning problems

    The Topology of Negatively Associated Distributions

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    We consider the sets of negatively associated (NA) and negatively correlated (NC) distributions as subsets of the space M\mathcal{M} of all probability distributions on Rn\mathbb{R}^n, in terms of their relative topological structures within the topological space of all measures on a given measurable space. We prove that the class of NA distributions has a non-empty interior with respect to the topology of the total variation metric on M\mathcal{M}. We show however that this is not the case in the weak topology (i.e. the topology of convergence in distribution), unless the underlying probability space is finite. We consider both the convexity and the connectedness of these classes of probability measures, and also consider the two classes on their (widely studied) restrictions to the Boolean cube in Rn\mathbb{R}^n

    Nonlinear realisation approach to topologically massive supergravity

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    We develop a nonlinear realisation approach to topologically massive supergravity in three dimensions, with and without a cosmological term. It is a natural generalisation of a similar construction for N=1{\cal N}=1 supergravity in four dimensions, which was recently proposed by one of us. At the heart of both formulations is the nonlinear realisation approach to gravity which was given by Volkov and Soroka fifty years ago in the context of spontaneously broken local supersymmetry. In our setting, the action for cosmological topologically massive supergravity is invariant under two different local supersymmetries. One of them acts on the Goldstino, while the other supersymmetry leaves the Goldstino invariant. The former can be used to gauge away the Goldstino, and then the resulting action coincides with that given in the literature.Comment: 29 page

    Rank-based linkage I: triplet comparisons and oriented simplicial complexes

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    Rank-based linkage is a new tool for summarizing a collection SS of objects according to their relationships. These objects are not mapped to vectors, and ``similarity'' between objects need be neither numerical nor symmetrical. All an object needs to do is rank nearby objects by similarity to itself, using a Comparator which is transitive, but need not be consistent with any metric on the whole set. Call this a ranking system on SS. Rank-based linkage is applied to the KK-nearest neighbor digraph derived from a ranking system. Computations occur on a 2-dimensional abstract oriented simplicial complex whose faces are among the points, edges, and triangles of the line graph of the undirected KK-nearest neighbor graph on SS. In SK2|S| K^2 steps it builds an edge-weighted linkage graph (S,L,σ)(S, \mathcal{L}, \sigma) where σ({x,y})\sigma(\{x, y\}) is called the in-sway between objects xx and yy. Take Lt\mathcal{L}_t to be the links whose in-sway is at least tt, and partition SS into components of the graph (S,Lt)(S, \mathcal{L}_t), for varying tt. Rank-based linkage is a functor from a category of out-ordered digraphs to a category of partitioned sets, with the practical consequence that augmenting the set of objects in a rank-respectful way gives a fresh clustering which does not ``rip apart`` the previous one. The same holds for single linkage clustering in the metric space context, but not for typical optimization-based methods. Open combinatorial problems are presented in the last section.Comment: 37 pages, 12 figure

    Graph Convex Hull Bounds as generalized Jensen Inequalities

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    Jensen's inequality is ubiquitous in measure and probability theory, statistics, machine learning, information theory and many other areas of mathematics and data science. It states that, for any convex function f ⁣:KRf\colon K \to \mathbb{R} on a convex domain KRdK \subseteq \mathbb{R}^{d} and any random variable XX taking values in KK, E[f(X)]f(E[X])\mathbb{E}[f(X)] \geq f(\mathbb{E}[X]). In this paper, sharp upper and lower bounds on E[f(X)]\mathbb{E}[f(X)], termed "graph convex hull bounds", are derived for arbitrary functions ff on arbitrary domains KK, thereby strongly generalizing Jensen's inequality. Establishing these bounds requires the investigation of the convex hull of the graph of ff, which can be difficult for complicated ff. On the other hand, once these inequalities are established, they hold, just like Jensen's inequality, for any random variable XX. Hence, these bounds are of particular interest in cases where ff is fairly simple and XX is complicated or unknown. Both finite- and infinite-dimensional domains and codomains of ff are covered, as well as analogous bounds for conditional expectations and Markov operators.Comment: 12 pages, 1 figur

    Entanglement entropy for spherically symmetric regular black holes

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    The Bardeen and Hayward spacetimes are here considered as standard configurations of spherically symmetric regular black holes. Assuming the thermodynamics of such objects to be analogous to standard black holes, we compute the island formula in the regime of small topological charge and vacuum energy, respectively for Bardeen and Hayward spacetimes. Late and early-time domains are separately discussed, with particular emphasis on the island formations. We single out conditions under which it is not possible to find out islands at early-times and how our findings depart from the standard Schwarzschild case. Motivated by th fact that those configurations extend Reissner-Nordstr\"{o}m and Schwarzschild-de Sitter metrics through the inclusion of regularity behavior at r=0r=0, we show how the effects of regularity induces modifications on the overall entanglement entropy. Finally, the Page time is also computed and we thus show which asymptotic values are expected for it, for all the configurations under exam. The Page time shows slight departures than the Schwarzschild case, especially for the Hayward case, while the Bardeen regular black hole turns out to be quite indistinguishable from the Schwarzschild case.Comment: 11 pages, 4 table

    Can you hear your location on a manifold?

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    We introduce a variation on Kac's question, "Can one hear the shape of a drum?" Instead of trying to identify a compact manifold and its metric via its Laplace--Beltrami spectrum, we ask if it is possible to uniquely identify a point xx on the manifold, up to symmetry, from its pointwise counting function Nx(λ)=λjλej(x)2, N_x(\lambda) = \sum_{\lambda_j \leq \lambda} |e_j(x)|^2, where here Δgej=λj2ej\Delta_g e_j = -\lambda_j^2 e_j and eje_j form an orthonormal basis for L2(M)L^2(M). This problem has a physical interpretation. You are placed at an arbitrary location in a familiar room with your eyes closed. Can you identify your location in the room by clapping your hands once and listening to the resulting echos and reverberations? The main result of this paper provides an affirmative answer to this question for a generic class of metrics. We also probe the problem with a variety of simple examples, highlighting along the way helpful geometric invariants that can be pulled out of the pointwise counting function NxN_x.Comment: 26 pages, 1 figur

    Barren plateaus in quantum tensor network optimization

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    We analyze the barren plateau phenomenon in the variational optimization of quantum circuits inspired by matrix product states (qMPS), tree tensor networks (qTTN), and the multiscale entanglement renormalization ansatz (qMERA). We consider as the cost function the expectation value of a Hamiltonian that is a sum of local terms. For randomly chosen variational parameters we show that the variance of the cost function gradient decreases exponentially with the distance of a Hamiltonian term from the canonical centre in the quantum tensor network. Therefore, as a function of qubit count, for qMPS most gradient variances decrease exponentially and for qTTN as well as qMERA they decrease polynomially. We also show that the calculation of these gradients is exponentially more efficient on a classical computer than on a quantum computer

    Less sculptural more intellectual: conceptualizing landscape in the architecture of 1990s and 2000s

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    The aim of this paper is to discuss the radical shift which emerges in the 1990s and enhances architecture in the 2000s by turning it into a less sculptural more intellectual field of design. Hence, architects rather focus on ground than figure in design projects. This leads them to interrogate the conventional relationships between figure and ground enabling figure to dominate the ground in architecture for decades. They discover the mutual relationships between figure and ground, and design grounded structures instead of ungrounded sculptures. These artificial structures seem like the extensions of the natural landscape, as such the conceptual and categorical distinction between artificial and natural blurs in architecture. Another conceptual blurring emerges between the concepts of landscape, ground, and field. These are generally used as interchangeable concepts, but landscape encompasses ground and field, making it a more comprehensive concept for architects. It is revealed in the paper that landscape is a re-emerging concept which refers to the conceptual shift from form and function to flow and force in architecture. Landscape, therefore, awaits to be explored as a field of flows and forces by even more architects in this century in which cities are characterized by sculptural forms and objects

    Changes in PRC1 activity during interphase modulate lineage transition in pluripotent cells

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    The potential of pluripotent cells to respond to developmental cues and trigger cell differentiation is enhanced during the G1 phase of the cell cycle, but the molecular mechanisms involved are poorly understood. Variations in polycomb activity during interphase progression have been hypothesized to regulate the cell-cycle-phase-dependent transcriptional activation of differentiation genes during lineage transition in pluripotent cells. Here, we show that recruitment of Polycomb Repressive Complex 1 (PRC1) and associated molecular functions, ubiquitination of H2AK119 and three-dimensional chromatin interactions, are enhanced during S and G2 phases compared to the G1 phase. In agreement with the accumulation of PRC1 at target promoters upon G1 phase exit, cells in S and G2 phases show firmer transcriptional repression of developmental regulator genes that is drastically perturbed upon genetic ablation of the PRC1 catalytic subunit RING1B. Importantly, depletion of RING1B during retinoic acid stimulation interferes with the preference of mouse embryonic stem cells (mESCs) to induce the transcriptional activation of differentiation genes in G1 phase. We propose that incremental enrolment of polycomb repressive activity during interphase progression reduces the tendency of cells to respond to developmental cues during S and G2 phases, facilitating activation of cell differentiation in the G1 phase of the pluripotent cell cycle
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