IA Scholar Query: Two-Restricted One Context Unification is in Polynomial Time.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgMon, 08 Aug 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Online Matching Frameworks under Stochastic Rewards, Product Ranking, and Unknown Patience
https://scholar.archive.org/work/5oxp63w7jfe4zgc2h5gs6noxre
We study generalizations of online bipartite matching in which each arriving vertex (customer) views a ranked list of offline vertices (products) and matches to (purchases) the first one they deem acceptable. The number of products that the customer has patience to view can be stochastic and dependent on the products seen. We develop a framework that views the interaction with each customer as an abstract resource consumption process, and derive new results for these online matching problems under the adversarial, non-stationary, and IID arrival models, assuming we can (approximately) solve the product ranking problem for each single customer. To that end, we show new results for product ranking under two cascade-click models: an optimal algorithm when each item has its own hazard rate for making the customer depart, and a 1/2-approximate algorithm when the customer has a general item-independent patience distribution. We also present a constant-factor 0.027-approximate algorithm in a new model where items are not initially available and arrive over time. Finally, we present three negative results of interest: one formalizing the notion of a stochasticity gap exhibited by existing approaches to this problem, an example showing the analysis of SimpleGreedy in existing work to be tight, and another one for the single-customer problem in which any constant-factor approximation is impossible when compared to a benchmark that knows the realization of the patience in advance.Brian Brubach, Nathaniel Grammel, Will Ma, Aravind Srinivasanwork_5oxp63w7jfe4zgc2h5gs6noxreMon, 08 Aug 2022 00:00:00 GMTClassical groups and Haar integration
https://scholar.archive.org/work/ctqxr7ssrfgmbpf3oz6e4wyf4y
This is an introduction to the closed subgroups G⊂ U_N, with all needed preliminaries included. We discuss the basic theory, and we perform some probability computations, in the finite case. In the general case, we explain how the Weingarten integration formula works, and we present some basic N→∞ applications.Teo Banicawork_ctqxr7ssrfgmbpf3oz6e4wyf4ySun, 07 Aug 2022 00:00:00 GMTContextual Search in the Presence of Adversarial Corruptions
https://scholar.archive.org/work/ngtrjneaq5bxzgp4b75y6gopbu
We study contextual search, a generalization of binary search in higher dimensions, which captures settings such as feature-based dynamic pricing. Standard formulations of this problem assume that agents act in accordance with a specific homogeneous response model. In practice, however, some responses may be adversarially corrupted. Existing algorithms heavily depend on the assumed response model being (approximately) accurate for all agents and have poor performance in the presence of even a few such arbitrary misspecifications. We initiate the study of contextual search when some of the agents can behave in ways inconsistent with the underlying response model. In particular, we provide two algorithms, one based on multidimensional binary search methods and one based on gradient descent. We show that these algorithms attain near-optimal regret in the absence of adversarial corruptions and their performance degrades gracefully with the number of such agents, providing the first results for contextual search in any adversarial noise model. Our techniques draw inspiration from learning theory, game theory, high-dimensional geometry, and convex analysis.Akshay Krishnamurthy, Thodoris Lykouris, Chara Podimata, Robert Schapirework_ngtrjneaq5bxzgp4b75y6gopbuSat, 06 Aug 2022 00:00:00 GMTRobust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning
https://scholar.archive.org/work/bq2pa2fpq5aylmptavp3h3wj4u
The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations based on associated stochastic differential equations (SDEs), which allow the minimization of corresponding losses using gradient-based optimization methods. In respective numerical implementations it is therefore crucial to rely on adequate gradient estimators that exhibit low variance in order to reach convergence accurately and swiftly. In this article, we rigorously investigate corresponding numerical aspects that appear in the context of linear Kolmogorov PDEs. In particular, we systematically compare existing deep learning approaches and provide theoretical explanations for their performances. Subsequently, we suggest novel methods that can be shown to be more robust both theoretically and numerically, leading to substantial performance improvements.Lorenz Richter, Julius Bernerwork_bq2pa2fpq5aylmptavp3h3wj4uFri, 05 Aug 2022 00:00:00 GMTBasic introduction to higher-spin theories
https://scholar.archive.org/work/fjehkyh6rrfazhgurghloqfzga
This is a collection of my lecture notes on the higher-spin theory course given for students at the Institute for Theoretical and Mathematical Physics, Lomonosov Moscow State University. The goal of these lectures is to give an introduction to higher-spin theories accessible to master level students which would enable them to read the higher-spin literature. I start by introducing basic relevant notions of representation theory and the associated field-theoretic descriptions. Focusing on massless symmetric fields I review different approaches to interactions as well as the no-go results. I end the lectures by reviewing some of the currently available positive results on interactions of massless higher-spin fields, namely, holographic, Chern-Simons and chiral higher-spin theories.Dmitry Ponomarevwork_fjehkyh6rrfazhgurghloqfzgaFri, 05 Aug 2022 00:00:00 GMTSpectral Universality of Regularized Linear Regression with Nearly Deterministic Sensing Matrices
https://scholar.archive.org/work/guhch37pqrbexo6wkvuirna5pm
It has been observed that the performances of many high-dimensional estimation problems are universal with respect to underlying sensing (or design) matrices. Specifically, matrices with markedly different constructions seem to achieve identical performance if they share the same spectral distribution and have "generic" singular vectors. We prove this universality phenomenon for the case of convex regularized least squares (RLS) estimators under a linear regression model with additive Gaussian noise. Our main contributions are two-fold: (1) We introduce a notion of universality classes for sensing matrices, defined through a set of deterministic conditions that fix the spectrum of the sensing matrix and precisely capture the previously heuristic notion of generic singular vectors; (2) We show that for all sensing matrices that lie in the same universality class, the dynamics of the proximal gradient descent algorithm for solving the regression problem, as well as the performance of RLS estimators themselves (under additional strong convexity conditions) are asymptotically identical. In addition to including i.i.d. Gaussian and rotational invariant matrices as special cases, our universality class also contains highly structured, strongly correlated, or even (nearly) deterministic matrices. Examples of the latter include randomly signed versions of incoherent tight frames and randomly subsampled Hadamard transforms. As a consequence of this universality principle, the asymptotic performance of regularized linear regression on many structured matrices constructed with limited randomness can be characterized by using the rotationally invariant ensemble as an equivalent yet mathematically more tractable surrogate.Rishabh Dudeja, Subhabrata Sen, Yue M. Luwork_guhch37pqrbexo6wkvuirna5pmThu, 04 Aug 2022 00:00:00 GMTIndependent Policy Gradient for Large-Scale Markov Potential Games: Sharper Rates, Function Approximation, and Game-Agnostic Convergence
https://scholar.archive.org/work/2lfulnvsmvelji5bdgoa4zstti
We examine global non-asymptotic convergence properties of policy gradient methods for multi-agent reinforcement learning (RL) problems in Markov potential games (MPG). To learn a Nash equilibrium of an MPG in which the size of state space and/or the number of players can be very large, we propose new independent policy gradient algorithms that are run by all players in tandem. When there is no uncertainty in the gradient evaluation, we show that our algorithm finds an ϵ-Nash equilibrium with O(1/ϵ^2) iteration complexity which does not explicitly depend on the state space size. When the exact gradient is not available, we establish O(1/ϵ^5) sample complexity bound in a potentially infinitely large state space for a sample-based algorithm that utilizes function approximation. Moreover, we identify a class of independent policy gradient algorithms that enjoys convergence for both zero-sum Markov games and Markov cooperative games with the players that are oblivious to the types of games being played. Finally, we provide computational experiments to corroborate the merits and the effectiveness of our theoretical developments.Dongsheng Ding and Chen-Yu Wei and Kaiqing Zhang and Mihailo R. Jovanovićwork_2lfulnvsmvelji5bdgoa4zsttiThu, 04 Aug 2022 00:00:00 GMTSparse Continuous Distributions and Fenchel-Young Losses
https://scholar.archive.org/work/2yqsquvnjzhirjdhvhwim4eawe
Exponential families are widely used in machine learning, including many distributions in continuous and discrete domains (e.g., Gaussian, Dirichlet, Poisson, and categorical distributions via the softmax transformation). Distributions in each of these families have fixed support. In contrast, for finite domains, recent work on sparse alternatives to softmax (e.g., sparsemax, α-entmax, and fusedmax), has led to distributions with varying support. This paper develops sparse alternatives to continuous distributions, based on several technical contributions: First, we define Ω-regularized prediction maps and Fenchel-Young losses for arbitrary domains (possibly countably infinite or continuous). For linearly parametrized families, we show that minimization of Fenchel-Young losses is equivalent to moment matching of the statistics, generalizing a fundamental property of exponential families. When Ω is a Tsallis negentropy with parameter α, we obtain "deformed exponential families," which include α-entmax and sparsemax (α=2) as particular cases. For quadratic energy functions, the resulting densities are β-Gaussians, an instance of elliptical distributions that contain as particular cases the Gaussian, biweight, triweight, and Epanechnikov densities, and for which we derive closed-form expressions for the variance, Tsallis entropy, and Fenchel-Young loss. When Ω is a total variation or Sobolev regularizer, we obtain a continuous version of the fusedmax. Finally, we introduce continuous-domain attention mechanisms, deriving efficient gradient backpropagation algorithms for α∈{1, 4/3, 3/2, 2}. Using these algorithms, we demonstrate our sparse continuous distributions for attention-based audio classification and visual question answering, showing that they allow attending to time intervals and compact regions.André F. T. Martins, Marcos Treviso, António Farinhas, Pedro M. Q. Aguiar, Mário A. T. Figueiredo, Mathieu Blondel, Vlad Niculaework_2yqsquvnjzhirjdhvhwim4eaweThu, 04 Aug 2022 00:00:00 GMTLIPIcs, Volume 239, TYPES 2021, Complete Volume
https://scholar.archive.org/work/uxb5tcwg6bflpdk4rslta5pn6a
LIPIcs, Volume 239, TYPES 2021, Complete VolumeHenning Basold, Jesper Cockx, Silvia Ghilezanwork_uxb5tcwg6bflpdk4rslta5pn6aThu, 04 Aug 2022 00:00:00 GMTFormalized functional analysis with semilinear maps
https://scholar.archive.org/work/tjlxwg5cb5hr5iz7kgsiclj5ga
Semilinear maps are a generalization of linear maps between vector spaces where we allow the scalar action to be twisted by a ring homomorphism such as complex conjugation. In particular, this generalization unifies the concepts of linear and conjugate-linear maps. We implement this generalization in Lean's mathlib library, along with a number of important results in functional analysis which previously were impossible to formalize properly. Specifically, we prove the Fréchet-Riesz representation theorem and the spectral theorem for compact self-adjoint operators generically over real and complex Hilbert spaces. We also show that semilinear maps have applications beyond functional analysis by formalizing the one-dimensional case of a theorem of Dieudonné and Manin that classifies the isocrystals over an algebraically closed field with positive characteristic.Frédéric Dupuis, Robert Y. Lewis, Heather Macbeth, June Andronick, Leonardo de Mourawork_tjlxwg5cb5hr5iz7kgsiclj5gaWed, 03 Aug 2022 00:00:00 GMTA violation of the Tsirelson bound in the pre-quantum theory of trace dynamics
https://scholar.archive.org/work/cilkwmuxejdo5iykfwdo3ojzp4
The term Bell's theorem refers to a set of closely related results which imply that quantum mechanics is incompatible with local hidden variable theories. Bell's inequality is the statement that if measurements are performed independently on two space-like separated particles of an entangled pair, the assumption that outcomes depend on hidden variables implies an upper bound on the correlations between the outcomes. Quantum mechanics predicts correlations which violate this upper bound. The CHSH inequality is a specific Bell inequality in which classical correlation (i.e. if local hidden variables exist) can take the maximum value of 2. Quantum mechanics violates this bound, allowing for a higher bound on the correlation, which can take the maximum value 2√(2), known as the Tsirelson bound. Popescu and Rohrlich showed that the assumption of relativistic causality allows for an even higher bound on the CHSH correlation, this value being 4. Why is the bound coming from causality higher than the Tsirelson bound? Are there relativistic causal dynamical theories which violate the Tsirelson bound? In the present paper we answer this question in the affirmative. We show that the pre-quantum theory of trace dynamics, from which quantum theory is emergent as a thermodynamic approximation, permits the CHSH correlation to take values higher than 2√(2). We interpret our findings to suggest that quantum theory is approximate, and emergent from the more general theory of trace dynamics.Rabsan G. Ahmed, Tejinder P. Singhwork_cilkwmuxejdo5iykfwdo3ojzp4Wed, 03 Aug 2022 00:00:00 GMTUse and Abuse of Instance Parameters in the Lean Mathematical Library
https://scholar.archive.org/work/rof3xdk27rhihk7srigfkjzgo4
The Lean mathematical library mathlib features extensive use of the typeclass pattern for organising mathematical structures, based on Lean's mechanism of instance parameters. Related mechanisms for typeclasses are available in other provers including Agda, Coq and Isabelle with varying degrees of adoption. This paper analyses representative examples of design patterns involving instance parameters in the current Lean 3 version of mathlib, focussing on complications arising at scale and how the mathlib community deals with them.Anne Baanen, June Andronick, Leonardo de Mourawork_rof3xdk27rhihk7srigfkjzgo4Wed, 03 Aug 2022 00:00:00 GMTDynamical quantum phase transitions from random matrix theory
https://scholar.archive.org/work/7b3dtjoepnaehkuxpvsw5kzxm4
We uncover a novel dynamical quantum phase transition, using random matrix theory and its associated notion of planar limit. We study it for the isotropic XY Heisenberg spin chain. For this, we probe its real-time dynamics through the Loschmidt echo. This leads to the study of a random matrix ensemble with a complex weight, whose analysis requires novel technical considerations, that we develop. We obtain three main results: 1) There is a third order phase transition at a rescaled critical time, that we determine. 2) The third order phase transitions persists away from the thermodynamic limit. 3) For times below the critical value, the difference between the thermodynamic limit and a finite chain decreases exponentially with the system size. All these results depend in a rich manner on the parity of the number of flipped spins of the quantum state conforming the fidelity.David Pérez-García, Leonardo Santilli, Miguel Tierzwork_7b3dtjoepnaehkuxpvsw5kzxm4Tue, 02 Aug 2022 00:00:00 GMTControlled Discovery and Localization of Signals via Bayesian Linear Programming
https://scholar.archive.org/work/siwekynyw5gsbpfxdv2tej2oty
Scientists often must simultaneously discover signals and localize them as precisely as possible. For instance, in genetic fine-mapping, high correlations between nearby genetic variants make it hard to identify the exact locations of causal variants. So the statistical task is to output as many disjoint regions containing a signal as possible, each as small as possible, while controlling false positives. Similar problems arise in any application where signals cannot be perfectly localized, such as locating stars in astronomical surveys and change point detection in time series data. With this motivation, we introduce Bayesian Linear Programming (BLiP), a Bayesian method for jointly detecting and localizing signals. BLiP overcomes an extremely high-dimensional and non-convex problem to verifiably nearly maximize expected power while provably controlling false positives. BLiP is very computationally efficient and can wrap around nearly any Bayesian model and algorithm. Applying BLiP to existing state-of-the-art analyses of UK Biobank data (for genetic fine-mapping) and the Sloan Digital Sky Survey (for astronomical point source detection) increased power by 30-120% in just a few minutes of computation. BLiP is implemented in the new packages pyblip (Python) and blipr (R).Asher Spector, Lucas Jansonwork_siwekynyw5gsbpfxdv2tej2otyMon, 01 Aug 2022 00:00:00 GMTPointwise Weyl law for graphs from quantized interval maps
https://scholar.archive.org/work/qpf2gefic5aulkfqxsfzm4xvza
We prove an analogue of the pointwise Weyl law for families of unitary matrices obtained from quantization of one-dimensional interval maps. This quantization for interval maps was introduced by Pako\'nski et al. [J. Phys. A: Math. Gen. 34 9303-9317 (2001)] as a model for quantum chaos on graphs. Since we allow shrinking spectral windows in the pointwise Weyl law, as a consequence we obtain for these models a strengthening of the quantum ergodic theorem from Berkolaiko et al. [Commun. Math. Phys. 273 137-159 (2007)], and show in the semiclassical limit that a family of randomly perturbed quantizations has approximately Gaussian eigenvectors. We also examine further the specific case where the interval map is the doubling map.Laura Shouwork_qpf2gefic5aulkfqxsfzm4xvzaMon, 01 Aug 2022 00:00:00 GMTLaplacian-level meta-generalized gradient approximation for solid and liquid metals
https://scholar.archive.org/work/givyaz7zlvb2hktlt444ytj4se
We derive and motivate a Laplacian-level, orbital-free meta-generalized-gradient approximation (LL-MGGA) for the exchange-correlation energy, targeting accurate ground-state properties of sp and sd metallic condensed matter, in which the density functional for the exchange-correlation energy is only weakly nonlocal due to perfect long-range screening. Our model for the orbital-free kinetic energy density restores the fourth-order gradient expansion for exchange to the r^2SCAN meta-GGA [Furness et al., J. Phys. Chem. Lett. 11, 8208 (2020)], yielding a LL-MGGA we call OFR2. OFR2 matches the accuracy of SCAN for prediction of common lattice constants and improves the equilibrium properties of alkali metals, transition metals, and intermetallics that were degraded relative to the PBE GGA values by both SCAN and r^2SCAN. We compare OFR2 to the r^2SCAN-L LL-MGGA [D. Mejia-Rodriguez and S.B. Trickey, Phys. Rev. B 102, 121109 (2020)] and show that OFR2 tends to outperform r^2SCAN-L for the equilibrium properties of solids, but r^2SCAN-L much better describes the atomization energies of molecules than OFR2 does. For best accuracy in molecules and non-metallic condensed matter, we continue to recommend SCAN and r^2SCAN. Numerical performance is discussed in detail, and our work provides an outlook to machine learning.Aaron D. Kaplan, John P. Perdewwork_givyaz7zlvb2hktlt444ytj4seMon, 01 Aug 2022 00:00:00 GMTLower bounds for learning quantum states with single-copy measurements
https://scholar.archive.org/work/iyqd3g4xkvhftbvkot32uct2we
We study the problems of quantum tomography and shadow tomography using measurements performed on individual, identical copies of an unknown d-dimensional state. We first revisit a known lower bound due to Haah et al. (2017) on quantum tomography with accuracy ϵ in trace distance, when the measurements choices are independent of previously observed outcomes (i.e., they are nonadaptive). We give a succinct proof of this result. This leads to stronger lower bounds when the learner uses measurements with a constant number of outcomes. In particular, this rigorously establishes the optimality of the folklore "Pauli tomography" algorithm in terms of its sample complexity. We also derive novel bounds of Ω(r^2 d/ϵ^2) and Ω(r^2 d^2/ϵ^2) for learning rank r states using arbitrary and constant-outcome measurements, respectively, in the nonadaptive case. In addition to the sample complexity, a resource of practical significance for learning quantum states is the number of different measurements used by an algorithm. We extend our lower bounds to the case where the learner performs possibly adaptive measurements from a fixed set of exp(O(d)) measurements. This implies in particular that adaptivity does not give us any advantage using single-copy measurements that are efficiently implementable. We also obtain a similar bound in the case where the goal is to predict the expectation values of a given sequence of observables, a task known as shadow tomography. Finally, in the case of adaptive, single-copy measurements implementable with polynomial-size circuits, we prove that a straightforward strategy based on computing sample means of the given observables is optimal.Angus Lowe, Ashwin Nayakwork_iyqd3g4xkvhftbvkot32uct2weMon, 01 Aug 2022 00:00:00 GMTDark energy: EFTs and supergravity
https://scholar.archive.org/work/wdyew7mperdkrb2zc7csvuouqe
The subject of this thesis is cosmological implications of string compactifications understood in a broad sense. In the first half of the thesis, we will begin by reviewing the four-dimensional description of the tree-level perturbative type IIB action. We will then review a number of open questions in cosmology and their relevance with regards to the remainder of the thesis. We will first explore some of these cosmological questions from the perspective of effective field theories motivated by supergravity. From the naturalness of dark energy and how to obtain a naturally light dark energy field in terms of the clockwork mechanism and the Dvali-Kaloper-Sorbo four-form mixing. We also discuss the coincidence problem for dynamical models of dark energy consistent with a quintessence field slowly rolling down a potential slope, of the type one would expect from the asymptotics of moduli space. In the second half of the thesis, we introduce the effects of perturbative and non-perturbative corrections to the tree-level type IIB action. We then focus on obtaining a viable model of quintessence from the type IIB effective field theory. However, we are able to show that such a model must have a non-supersymmetric Minkowski vacuum at leading order. When we consider the effects of quantum fluctuations during the early Universe, we see that such models must have extremely fine-tuned initial conditions to describe a slow-rolling scalar field at present times. We conclude that quintessence faces more challenges than a true cosmological constant, to the point that quintessence is very unattractive for model building modulo a ruling out of the cosmological constant by observations. Following this line of reasoning, we consider whether other perturbative corrections can generate de Sitter solutions in an appropriate setting.Francesc Cunillerawork_wdyew7mperdkrb2zc7csvuouqeSat, 30 Jul 2022 00:00:00 GMTFast quantum circuit cutting with randomized measurements
https://scholar.archive.org/work/4a45cal35bdbfkqkbvglkp4auu
We propose a new method to extend the size of a quantum computation beyond the number of physical qubits available on a single device. This is accomplished by randomly inserting measure-and-prepare channels to express the output state of a large circuit as a separable state across distinct devices. Our method employs randomized measurements, resulting in a sample overhead that is O(4^k / ε ^2), where ε is the accuracy of the computation and k the number of parallel wires that are "cut" to obtain smaller sub-circuits. We also show an information-theoretic lower bound of Ω(2^k / ε ^2) for any comparable procedure. We use our techniques to show that circuits in the Quantum Approximate Optimization Algorithm (QAOA) with p entangling layers can be simulated by circuits on a fraction of the original number of qubits with an overhead that is roughly 2^O(pκ), where κ is the size of a known balanced vertex separator of the graph which encodes the optimization problem. We obtain numerical evidence of practical speedups using our method applied to the QAOA, compared to prior work. Finally, we investigate the practical feasibility of applying the circuit cutting procedure to large-scale QAOA problems on clustered graphs by using a 30-qubit simulator to evaluate the variational energy of a 129-qubit problem as well as carry out a 62-qubit optimization.Angus Lowe, Matija Medvidović, Anthony Hayes, Lee J. O'Riordan, Thomas R. Bromley, Juan Miguel Arrazola, Nathan Killoranwork_4a45cal35bdbfkqkbvglkp4auuFri, 29 Jul 2022 00:00:00 GMTTowards an unified theory of the fundamental physical interactions based on the underlying geometric structure of the tangent bundle
https://scholar.archive.org/work/kx6vorihpjd5ngby7jcthvpdgu
This paper pursues the hypothesis that the tangent bundle (TB) with the central extended little groups of the SO(3,1) group as gauge group is the underlying geometric structure for an unified theory of the fundamental physical interactions. Based on this hypothesis as a first step recently I presented a generalized theory of electroweak interaction (including hypothetical Dark Matter particles) [1]. The vertical Laplacian of the tangent bundle possesses the same form as the Hamiltonian of a 2D semiconductor Quantum Hall system. This explains fractional charge quantization of quarks and the existence of lepton and quark families. As will be shown the SU(3) colour symmetry for strong interaction arises in the TB as an emergent symmetry similar as Chern-Simon gauge symmetries in Quantum Hall systems. This predicts a signature of quark confinement as an universal large-scale property of the Chern-Simon fields and induces a new understanding of the vacuum as the ground state occupied with a condensate of quark-antiquark pairs. The gap for quark-antiquark pairing is calculated in the mean-field approximation which allows a numerical estimation of the characteristic parameters of the vacuum such as its chemical potential, the quark condensation parameter and the vacuum energy. Note that previously a gauge theoretical understanding of gravity has been achieved by considering the translation group T(3,1) in the TB as gauge group. Therefore the theory presented here can be considered as a new type of unified theory for all known fundamental interactions linked with the geometrization program of physics.Joachim Herrmannwork_kx6vorihpjd5ngby7jcthvpdguFri, 29 Jul 2022 00:00:00 GMT