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Author: Grafiati

Published: 1 June 2024

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1

Jiang, Yu Hang, Tong Liu, Zhiya Lou, Jeffrey S. Rosenthal, Shanshan Shangguan, Fei Wang, and Zixuan Wu. "Markov Chain Confidence Intervals and Biases." International Journal of Statistics and Probability 11, no. 1 (December 21, 2021): 29. http://dx.doi.org/10.5539/ijsp.v11n1p29.

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We derive explicit asymptotic confidence intervals for any Markov chain Monte Carlo (MCMC) algorithm with finite asymptotic variance, started at any initial state, without requiring a Central Limit Theorem nor reversibility nor geometric ergodicity nor any bias bound. We also derive explicit non-asymptotic confidence intervals assuming bounds on the bias or first moment, or alternatively that the chain starts in stationarity. We relate those non-asymptotic bounds to properties of MCMC bias, and show that polynomially ergodicity implies certain bias bounds. We also apply our results to several numerical examples. It is our hope that these results will provide simple and useful tools for estimating errors of MCMC algorithms when CLTs are not available.

2

Zhou, Lin, and Mehul Motani. "Non-Asymptotic Converse Bounds and Refined Asymptotics for Two Source Coding Problems." IEEE Transactions on Information Theory 65, no. 10 (October 2019): 6414–40. http://dx.doi.org/10.1109/tit.2019.2920893.

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3

Xia, Dong. "Non-asymptotic bounds for percentiles of independent non-identical random variables." Statistics & Probability Letters 152 (September 2019): 111–20. http://dx.doi.org/10.1016/j.spl.2019.04.018.

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4

Menozzi, Stéphane, and Vincent Lemaire. "On Some non Asymptotic Bounds for the Euler Scheme." Electronic Journal of Probability 15 (2010): 1645–81. http://dx.doi.org/10.1214/ejp.v15-814.

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Yang, Xiaowei, Lu Pan, Kun Cheng, and Chao Liu. "Optimal Non-Asymptotic Bounds for the Sparse β Model." Mathematics 11, no. 22 (November 17, 2023): 4685. http://dx.doi.org/10.3390/math11224685.

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This paper investigates the sparse β model with 𝓁1 penalty in the field of network data models, which is a hot topic in both statistical and social network research. We present a refined algorithm designed for parameter estimation in the proposed model. Its effectiveness is highlighted through its alignment with the proximal gradient descent method, stemming from the convexity of the loss function. We study the estimation consistency and establish an optimal bound for the proposed estimator. Empirical validations facilitated through meticulously designed simulation studies corroborate the efficacy of our methodology. These assessments highlight the prospective contributions of our methodology to the advanced field of network data analysis.

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Raj Jhunjhunwala, Prakirt, Daniela Hurtado-Lange, and Siva Theja Maguluri. "Exponential Tail Bounds on Queues: A Confluence of Non- Asymptotic Heavy Traffic and Large Deviations." ACM SIGMETRICS Performance Evaluation Review 51, no. 4 (February 22, 2024): 18–19. http://dx.doi.org/10.1145/3649477.3649488.

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In general, obtaining the exact steady-state distribution of queue lengths is not feasible. Therefore, we focus on establishing bounds for the tail probabilities of queue lengths. We examine queueing systems under Heavy Traffic (HT) conditions and provide exponentially decaying bounds for the probability P(∈q > x), where ∈ is the HT parameter denoting how far the load is from the maximum allowed load. Our bounds are not limited to asymptotic cases and are applicable even for finite values of ∈, and they get sharper as ∈ - 0. Consequently, we derive non-asymptotic convergence rates for the tail probabilities. Furthermore, our results offer bounds on the exponential rate of decay of the tail, given by -1/2 log P(∈q > x) for any finite value of x. These can be interpreted as non-asymptotic versions of Large Deviation (LD) results. To obtain our results, we use an exponential Lyapunov function to bind the moment-generating function of queue lengths and apply Markov's inequality. We demonstrate our approach by presenting tail bounds for a continuous time Join-the-shortest queue (JSQ) system.

7

Függer, Matthias, Thomas Nowak, and Manfred Schwarz. "Tight Bounds for Asymptotic and Approximate Consensus." Journal of the ACM 68, no. 6 (December 31, 2021): 1–35. http://dx.doi.org/10.1145/3485242.

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Agreeing on a common value among a set of agents is a fundamental problem in distributed computing, which occurs in several variants: In contrast to exact consensus, approximate variants are studied in systems where exact agreement is not possible or required, e.g., in human-made distributed control systems and in the analysis of natural distributed systems, such as bird flocking and opinion dynamics. We study the time complexity of two classical agreement problems: non-terminating asymptotic consensus and terminating approximate consensus. Asymptotic consensus, requires agents to repeatedly set their outputs such that the outputs converge to a common value within the convex hull of initial values; approximate consensus requires agents to eventually stop setting their outputs, which must then lie within a predefined distance of each other. We prove tight lower bounds on the contraction ratios of asymptotic consensus algorithms subject to oblivious message adversaries, from which we deduce bounds on the time complexity of approximate consensus algorithms. In particular, the obtained bounds show optimality of asymptotic and approximate consensus algorithms presented by Charron-Bost et al. (ICALP’16) for certain systems, including the strongest oblivious message adversary in which asymptotic and approximate consensus are solvable. As a corollary we also obtain asymptotically tight bounds for asymptotic consensus in the classical asynchronous model with crashes. Central to the lower-bound proofs is an extended notion of valency, the set of reachable limits of an asymptotic consensus algorithm starting from a given configuration. We further relate topological properties of valencies to the solvability of exact consensus, shedding some light on the relation of these three fundamental problems in dynamic networks.

8

Gu, Yujie, and Ofer Shayevitz. "On the Non-Adaptive Zero-Error Capacity of the Discrete Memoryless Two-Way Channel." Entropy 23, no. 11 (November 15, 2021): 1518. http://dx.doi.org/10.3390/e23111518.

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We study the problem of communicating over a discrete memoryless two-way channel using non-adaptive schemes, under a zero probability of error criterion. We derive single-letter inner and outer bounds for the zero-error capacity region, based on random coding, linear programming, linear codes, and the asymptotic spectrum of graphs. Among others, we provide a single-letter outer bound based on a combination of Shannon’s vanishing-error capacity region and a two-way analogue of the linear programming bound for point-to-point channels, which, in contrast to the one-way case, is generally better than both. Moreover, we establish an outer bound for the zero-error capacity region of a two-way channel via the asymptotic spectrum of graphs, and show that this bound can be achieved in certain cases.

9

Lim, Fabian, and Vladimir Stojanovic. "On U-Statistics and Compressed Sensing II: Non-Asymptotic Worst-Case Analysis." Signal Processing, IEEE Transactions on 61, no. 10 (April 2013): 2486–97. http://dx.doi.org/10.1109/tsp.2013.2255041.

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In another related work, U-statistics were used for non-asymptotic average-case analysis of random compressed sensing matrices. In this companion paper the same analytical tool is adopted differently-here we perform non-asymptotic worst-case analysis. Simple union bounds are a natural choice for worst-case analyses, however their tightness is an issue (and questioned in previous works). Here we focus on a theoretical U-statistical result, which potentially allows us to prove that these union bounds are tight. To our knowledge, this kind of (powerful) result is completely new in the context of CS. This general result applies to a wide variety of parameters, and is related to (Stein-Chen) Poisson approximation. In this paper, we consider i) restricted isometries, and ii) mutual coherence. For the bounded case, we show that -th order restricted isometry constants have tight union bounds, when the measurements m = O (k(1.5(+ log(n/k))). Here, we require the restricted isometries to grow linearly in , however we conjecture that this result can be improved to allow them to be fixed. Also, we show that mutual coherence (with the standard estimate √(4 log n)/m) have very tight union bounds. For coherence, the normalization complicates general discussion, and we consider only Gaussian and Bernoulli cases here.

10

Cheng, Xu, Zhipeng Liao, and Ruoyao Shi. "On uniform asymptotic risk of averaging GMM estimators." Quantitative Economics 10, no. 3 (2019): 931–79. http://dx.doi.org/10.3982/qe711.

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This paper studies the averaging GMM estimator that combines a conservative GMM estimator based on valid moment conditions and an aggressive GMM estimator based on both valid and possibly misspecified moment conditions, where the weight is the sample analog of an infeasible optimal weight. We establish asymptotic theory on uniform approximation of the upper and lower bounds of the finite‐sample truncated risk difference between any two estimators, which is used to compare the averaging GMM estimator and the conservative GMM estimator. Under some sufficient conditions, we show that the asymptotic lower bound of the truncated risk difference between the averaging estimator and the conservative estimator is strictly less than zero, while the asymptotic upper bound is zero uniformly over any degree of misspecification. The results apply to quadratic loss functions. This uniform asymptotic dominance is established in non‐Gaussian semiparametric nonlinear models.

11

Afser, Huseyin. "Statistical Classification via Robust Hypothesis Testing: Non-Asymptotic and Simple Bounds." IEEE Signal Processing Letters 28 (2021): 2112–16. http://dx.doi.org/10.1109/lsp.2021.3119230.

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Fidler, Markus, Brenton Walker, and Yuming Jiang. "Non-Asymptotic Delay Bounds for Multi-Server Systems with Synchronization Constraints." IEEE Transactions on Parallel and Distributed Systems 29, no. 7 (July 1, 2018): 1545–59. http://dx.doi.org/10.1109/tpds.2017.2779872.

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Toumpis, Stavros. "Asymptotic Capacity Bounds for Wireless Networks with Non-Uniform Traffic Patterns." IEEE Transactions on Wireless Communications 7, no. 6 (June 2008): 2231–42. http://dx.doi.org/10.1109/twc.2008.061010.

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Maggioni, Mauro, Stanislav Minsker, and Nate Strawn. "Dictionary Learning and Non-Asymptotic Bounds for Geometric Multi-Resolution Analysis." PAMM 14, no. 1 (December 2014): 1013–16. http://dx.doi.org/10.1002/pamm.201410486.

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15

Cork, Josh, Derek Harland, and Thomas Winyard. "A model for gauged skyrmions with low binding energies." Journal of Physics A: Mathematical and Theoretical 55, no. 1 (December 13, 2021): 015204. http://dx.doi.org/10.1088/1751-8121/ac3c81.

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Abstract We consider gauged skyrmions with boundary conditions which break the gauge from SU(2) to U(1) in models derived from Yang–Mills theory. After deriving general topological energy bounds, we approximate charge 1 energy minimisers using KvBLL calorons with non-trivial asymptotic holonomy, use them to calibrate the model to optimise the ratio of energy to lower bound, and compare them with solutions to full numerical simulation. Skyrmions from calorons with non-trivial asymptotic holonomy exhibit a non-zero magnetic dipole moment, which we calculate explicitly, and compare with experimental values for the proton and the neutron. We thus propose a way to develop a physically realistic Skyrme–Maxwell theory, with the potential for exhibiting low binding energies.

16

Grove, Adam J., Joseph Y. Halpern, and Daphne Koller. "Asymptotic conditional probabilities: The non-unary case." Journal of Symbolic Logic 61, no. 1 (March 1996): 250–76. http://dx.doi.org/10.2307/2275609.

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AbstractMotivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences φ and θ, we consider the structures with domain {1, …, N} that satisfy θ, and compute the fraction of them in which φ is true. We then consider what happens to this fraction as N gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kiĭ [24], if there is a non-unary predicate symbol in the vocabulary, asymptotic conditional probabilities do not always exist. We extend this result to show that asymptotic conditional probabilities do not always exist for any reasonable notion of limit. Liogon'kiĭ also showed that the problem of deciding whether the limit exists is undecidable. We analyze the complexity of three problems with respect to this limit: deciding whether it is well-defined, whether it exists, and whether it lies in some nontrivial interval. Matching upper and lower bounds are given for all three problems, showing them to be highly undecidable.

17

Fujikoshi, Y. "Non-uniform error bounds for asymptotic expansions of scale mixtures of distributions." Journal of Multivariate Analysis 27, no. 1 (October 1988): 194–205. http://dx.doi.org/10.1016/0047-259x(88)90125-x.

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18

Guigues, Vincent, Anatoli Juditsky, and Arkadi Nemirovski. "Non-asymptotic confidence bounds for the optimal value of a stochastic program." Optimization Methods and Software 32, no. 5 (July 28, 2017): 1033–58. http://dx.doi.org/10.1080/10556788.2017.1350177.

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19

Wang, Fei, and Yuhao Deng. "Non-Asymptotic Bounds of AIPW Estimators for Means with Missingness at Random." Mathematics 11, no. 4 (February 6, 2023): 818. http://dx.doi.org/10.3390/math11040818.

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The augmented inverse probability weighting is well known for its double robustness in missing data and causal inference. If either the propensity score model or the outcome regression model is correctly specified, the estimator is guaranteed to be consistent. Another important property of the augmented inverse probability weighting is that it can achieve first-order equivalence to the oracle estimator in which all nuisance parameters are known, even if the fitted models do not converge at the parametric root-n rate. We explore the non-asymptotic properties of the augmented inverse probability weighting estimator to infer the population mean with missingness at random. We also consider inferences of the mean outcomes on the observed group and on the unobserved group.

20

Pflug, Georg Ch. "Non-asymptotic confidence bounds for stochastic approximation algorithms with constant step size." Monatshefte f�r Mathematik 110, no. 3-4 (September 1990): 297–314. http://dx.doi.org/10.1007/bf01301683.

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21

Du, Yubo, and Keyou You. "Distributed adaptive greedy quasi-Newton methods with explicit non-asymptotic convergence bounds." Automatica 165 (July 2024): 111629. http://dx.doi.org/10.1016/j.automatica.2024.111629.

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22

GERLACH, MORITZ, and JOCHEN GLÜCK. "Lower bounds and the asymptotic behaviour of positive operator semigroups." Ergodic Theory and Dynamical Systems 38, no. 8 (May 2, 2017): 3012–41. http://dx.doi.org/10.1017/etds.2017.9.

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If $(T_{t})$ is a semigroup of Markov operators on an $L^{1}$-space that admits a non-trivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as $t\rightarrow \infty$. In this article we generalize and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result, we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalize a theorem of Ding on semigroups of Frobenius–Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results. Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups.

23

Sason, Igal. "Tight Bounds on the Rényi Entropy via Majorization with Applications to Guessing and Compression." Entropy 20, no. 12 (November 22, 2018): 896. http://dx.doi.org/10.3390/e20120896.

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This paper provides tight bounds on the Rényi entropy of a function of a discrete random variable with a finite number of possible values, where the considered function is not one to one. To that end, a tight lower bound on the Rényi entropy of a discrete random variable with a finite support is derived as a function of the size of the support, and the ratio of the maximal to minimal probability masses. This work was inspired by the recently published paper by Cicalese et al., which is focused on the Shannon entropy, and it strengthens and generalizes the results of that paper to Rényi entropies of arbitrary positive orders. In view of these generalized bounds and the works by Arikan and Campbell, non-asymptotic bounds are derived for guessing moments and lossless data compression of discrete memoryless sources.

24

Gortzen, Simon, and Anke Schmeink. "Non-Asymptotic Bounds on the Performance of Dual Methods for Resource Allocation Problems." IEEE Transactions on Wireless Communications 13, no. 6 (June 2014): 3430–41. http://dx.doi.org/10.1109/twc.051414.131480.

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25

Kumar, Bhumesh, Vivek Borkar, and Akhil Shetty. "Non-asymptotic error bounds for constant stepsize stochastic approximation for tracking mobile agents." Mathematics of Control, Signals, and Systems 31, no. 4 (October 25, 2019): 589–614. http://dx.doi.org/10.1007/s00498-019-00249-4.

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26

Derumigny, Alexis, and Jean-David Fermanian. "On kernel-based estimation of conditional Kendall’s tau: finite-distance bounds and asymptotic behavior." Dependence Modeling 7, no. 1 (September 30, 2019): 292–321. http://dx.doi.org/10.1515/demo-2019-0016.

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AbstractWe study nonparametric estimators of conditional Kendall’s tau, a measure of concordance between two random variables given some covariates. We prove non-asymptotic pointwise and uniform bounds, that hold with high probabilities. We provide “direct proofs” of the consistency and the asymptotic law of conditional Kendall’s tau. A simulation study evaluates the numerical performance of such nonparametric estimators. An application to the dependence between energy consumption and temperature conditionally to calendar days is finally provided.

27

NGAMPITIPAN, TRITOS, and PETARPA BOONSERM. "BOUNDING THE GREYBODY FACTORS FOR NON-ROTATING BLACK HOLES." International Journal of Modern Physics D 22, no. 09 (June 26, 2013): 1350058. http://dx.doi.org/10.1142/s0218271813500582.

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Semiclassical black holes emit radiation called Hawking radiation. Such radiation, as seen by an asymptotic observer far outside the black hole, differs from the original radiation near the horizon of the black hole by a redshift factor and the so-called "greybody factor." In this paper, we concentrate on the greybody factor; various bounds for the greybody factors of non-rotaging black holes are obtained, concentrating primarily on charged Reissner–Nordström (RN) and RN–de Sitter black holes. These bounds can be derived using a 2 × 2 transfer matrix formalism. It is found that the charges of black holes act as efficient barriers. Furthermore, adding extra dimensions to spacetime can shield Hawking radiation. Finally, it is also found that the cosmological constant can increase the emission rate of Hawking radiation.

28

Kim, Yoon Young. "Uncoupled Wave Systems and Dispersion in an Infinite Solid Cylinder." Journal of Applied Mechanics 56, no. 2 (June 1, 1989): 347–55. http://dx.doi.org/10.1115/1.3176089.

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In this study, it is shown that there exist uncoupled wave systems for general non-axisymmetric wave propagation in an infinite isotropic cylinder. Two cylindrical surface conditions corresponding to the uncoupled wave systems are discussed. The solutions of the uncoupled wave systems are shown to provide proper bounds of Pochhammer’s equation for a free cylindrical surface. The bounds, which are easy to construct for any Fourier number in the circumferential direction, can be used to trace the branches of Pochhammer’s equation. They also give insight into the modal composition of the branches of Pochhammer’s equation at and between the intersections of the bounds. More refined dispersion relations of Pochhammer’s equation are possible through an asymptotic analysis of the itersections of the branches of Pochhammer’s equation with one family of the bounds. The asymptotic nature of wave motion corresponding to large wave numbers, imaginary or complex, for Pochhammer’s equation is studied. The wave motion is asymptotically equivoluminal for large imaginary wave numbers, and is characterized by coupled dilatation and shear for large complex wave numbers.

29

Hadj, Taieb, M. A. Hammami, and M. Hammi. "On the global uniform stability analysis of non-autonomous dynamical systems: A survey." Mathematica Moravica 26, no. 2 (2022): 1–48. http://dx.doi.org/10.5937/matmor2202001t.

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In this survey, we introduce the notion of stability of time varying nonlinear systems. In particular we investigate the notion of global practical exponential stability for non-autonomous systems. The proposed approach for stability analysis is based on the determination of the bounds of perturbations that characterize the asymptotic convergence of the solutions to a closed ball centered at the origin.

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Lindholm, Mathias, and Felix Wahl. "On the variance parameter estimator in general linear models." Metrika 83, no. 2 (November 6, 2019): 243–54. http://dx.doi.org/10.1007/s00184-019-00751-4.

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Abstract In the present note we consider general linear models where the covariates may be both random and non-random, and where the only restrictions on the error terms are that they are independent and have finite fourth moments. For this class of models we analyse the variance parameter estimator. In particular we obtain finite sample size bounds for the variance of the variance parameter estimator which are independent of covariate information regardless of whether the covariates are random or not. For the case with random covariates this immediately yields bounds on the unconditional variance of the variance estimator—a situation which in general is analytically intractable. The situation with random covariates is illustrated in an example where a certain vector autoregressive model which appears naturally within the area of insurance mathematics is analysed. Further, the obtained bounds are sharp in the sense that both the lower and upper bound will converge to the same asymptotic limit when scaled with the sample size. By using the derived bounds it is simple to show convergence in mean square of the variance parameter estimator for both random and non-random covariates. Moreover, the derivation of the bounds for the above general linear model is based on a lemma which applies in greater generality. This is illustrated by applying the used techniques to a class of mixed effects models.

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Lei, Antonio, David Loeffler, and Zerbes Sarah Livia. "On the Asymptotic Growth of Bloch-Kato-Shafarevich-Tate Groups of Modular Forms Over Cyclotomic Extensions." Canadian Journal of Mathematics 69, no. 4 (August 1, 2017): 826–50. http://dx.doi.org/10.4153/cjm-2016-034-x.

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AbstractWe study the asymptotic behaviour of the Bloch-Kato-Shafarevich-Tate group of a modular form f over the cyclotomic ℤp-extension of ℚ under the assumption that f is non-ordinary at p. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using p-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara, and Sprung for supersingular elliptic curves.

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Vuuren, Jan H. van, and John Norbury. "Permanence and asymptotic stability in diagonally convex reaction–diffusion systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 128, no. 1 (1998): 147–72. http://dx.doi.org/10.1017/s0308210500027219.

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Reaction–diffusion systems are widely used to model competition in, for example, the scientific fields of biology, chemistry, medicine and industry. It is often not too difficult to establish positive uniform upper bounds on solution components of such systems, but the task of establishing strictly positive uniform lower bounds (when they exist) can be quite troublesome. Conditions for the existence of such lower bounds in Lotka–Volterra competitve models are already well known. A general context for the understanding of these conditions is provided in this paper, by establishing more general permanence criteria (for the nonexplosion and noncollapse to zero of solutions) in the class of diagonally convex competitive reaction–diffusion systems with zero flux Neumann boundary conditions. This class admits most famous competition models as special cases. The asymptotic (large time) behaviour of positive solutions within the bounds of permanence is also considered and it is shown that the methods of proof for asymptotic stability (and hence resilience) normally associated with order-preserving systems (such as comparison arguments) are also applicable, in a slightly generalised form, to competitive systems as long as the competitive interactions are not too strong. The general criteria for permanence obtained here provide a natural method for developing new and easily verifiable permanence conditions for a host of non Lotka–Volterra competition models, as is illustrated by considering three famous special cases. In one of these cases known results are recovered, while in the other two cases new conditions for solution permanence are established.

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MATSUTA, Tetsunao, and Tomohiko UYEMATSU. "New Non-Asymptotic Bounds on Numbers of Codewords for the Fixed-Length Lossy Compression." IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E99.A, no. 12 (2016): 2116–29. http://dx.doi.org/10.1587/transfun.e99.a.2116.

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Alekseychuk, A. N. "Non-Asymptotic Lower Bounds for the Data Complexity of Statistical Attacks on Symmetric Cryptosystems." Cybernetics and Systems Analysis 54, no. 1 (January 27, 2018): 83–93. http://dx.doi.org/10.1007/s10559-018-0009-0.

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Wu, Shi-Liang, Guang-Sheng Chen, and Cheng-Hsiung Hsu. "Exact asymptotic behavior of pulsating traveling waves for a periodic monostable lattice dynamical system." Proceedings of the American Mathematical Society 149, no. 4 (February 12, 2021): 1697–710. http://dx.doi.org/10.1090/proc/15369.

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This paper is concerned with the pulsating traveling waves of a periodic lattice dynamical system with monostable nonlinearity. We first establish the exponential upper and lower bounds of the pulsating wave profiles at minus infinity. Then we prove the uniqueness result and derive the asymptotic behavior of all non-critical monostable pulsating traveling waves. This might be the first time to obtain the exact asymptotic behavior of the pulsating traveling waves for periodic discrete systems towards the unstable steady state.

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Dunster, T. M. "Uniform asymptotic expansions for the Whittaker functions M κ , μ ( z ) and W κ , μ ( z ) with μ large." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2252 (August 2021): 20210360. http://dx.doi.org/10.1098/rspa.2021.0360.

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Uniform asymptotic expansions are derived for Whittaker’s confluent hypergeometric functions M κ , μ ( z ) and W κ , μ ( z ) , as well as the numerically satisfactory companion function W − κ , μ ( z e − π i ) . The expansions are uniformly valid for μ → ∞ , 0 ≤ κ / μ ≤ 1 − δ < 1 and 0 ≤ arg ⁡ ( z ) ≤ π . By using appropriate connection and analytic continuation formulae, these expansions can be extended to all unbounded non-zero complex z . The approximations come from recent asymptotic expansions involving elementary functions and Airy functions, and explicit error bounds are either provided or available.

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Dostert, Maria, and Alexander Kolpakov. "Kissing number in non-Euclidean spaces of constant sectional curvature." Mathematics of Computation 90, no. 331 (April 13, 2021): 2507–25. http://dx.doi.org/10.1090/mcom/3622.

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This paper provides upper and lower bounds on the kissing number of congruent radius r > 0 r > 0 spheres in hyperbolic H n \mathbb {H}^n and spherical S n \mathbb {S}^n spaces, for n ≥ 2 n\geq 2 . For that purpose, the kissing number is replaced by the kissing function κ H ( n , r ) \kappa _H(n, r) , resp. κ S ( n , r ) \kappa _S(n, r) , which depends on the dimension n n and the radius r r . After we obtain some theoretical upper and lower bounds for κ H ( n , r ) \kappa _H(n, r) , we study their asymptotic behaviour and show, in particular, that κ H ( n , r ) ∼ ( n − 1 ) ⋅ d n − 1 ⋅ B ( n − 1 2 , 1 2 ) ⋅ e ( n − 1 ) r \kappa _H(n,r) \sim (n-1) \cdot d_{n-1} \cdot B(\frac {n-1}{2}, \frac {1}{2}) \cdot e^{(n-1) r} , where d n d_n is the sphere packing density in R n \mathbb {R}^n , and B B is the beta-function. Then we produce numeric upper bounds by solving a suitable semidefinite program, as well as lower bounds coming from concrete spherical codes. A similar approach allows us to locate the values of κ S ( n , r ) \kappa _S(n, r) , for n = 3 , 4 n= 3,\, 4 , over subintervals in [ 0 , π ] [0, \pi ] with relatively high accuracy.

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Vieira, Lucas B., Simon Milz, Giuseppe Vitagliano, and Costantino Budroni. "Witnessing environment dimension through temporal correlations." Quantum 8 (January 10, 2024): 1224. http://dx.doi.org/10.22331/q-2024-01-10-1224.

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We introduce a framework to compute upper bounds for temporal correlations achievable in open quantum system dynamics, obtained by repeated measurements on the system. As these correlations arise by virtue of the environment acting as a memory resource, such bounds are witnesses for the minimal dimension of an effective environment compatible with the observed statistics. These witnesses are derived from a hierarchy of semidefinite programs with guaranteed asymptotic convergence. We compute non-trivial bounds for various sequences involving a qubit system and a qubit environment, and compare the results to the best known quantum strategies producing the same outcome sequences. Our results provide a numerically tractable method to determine bounds on multi-time probability distributions in open quantum system dynamics and allow for the witnessing of effective environment dimensions through probing of the system alone.

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Cheraghchi, Mahdi, and Joao Ribeiro. "Non-Asymptotic Capacity Upper Bounds for the Discrete-Time Poisson Channel With Positive Dark Current." IEEE Communications Letters 25, no. 12 (December 2021): 3829–32. http://dx.doi.org/10.1109/lcomm.2021.3120706.

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Flores, Salvador. "Sharp non-asymptotic performance bounds for $$\ell _1$$ ℓ 1 and Huber robust regression estimators." TEST 24, no. 4 (March 14, 2015): 796–812. http://dx.doi.org/10.1007/s11749-015-0435-5.

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Vaskouski, Maksim M. "Random walks on Cayley graphs of complex reflection groups." Journal of the Belarusian State University. Mathematics and Informatics, no. 3 (November 19, 2021): 51–56. http://dx.doi.org/10.33581/2520-6508-2021-3-51-56.

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Asymptotic properties of random walks on minimal Cayley graphs of complex reflection groups are investigated. The main result of the paper is theorem on fast mixing for random walks on Cayley graphs of complex reflection groups. Particularly, bounds of diameters and isoperimetric constants, a known result on fast fixing property for expander graphs play a crucial role to obtain the main result. A constructive way to prove a special case of Babai’s conjecture on logarithmic order of diameters for complex reflection groups is proposed. Basing on estimates of diameters and Cheeger inequality, there is obtained a non-trivial lower bound for spectral gaps of minimal Cayley graphs on complex reflection groups.

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Barreira, Luis M. "A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems." Ergodic Theory and Dynamical Systems 16, no. 5 (October 1996): 871–927. http://dx.doi.org/10.1017/s0143385700010117.

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AbstractA non-additive version of the thermodynamic formalism is developed. This allows us to obtain lower and upper bounds for the dimension of a broad class of Cantor-like sets. These are constructed with a decreasing sequence of closed sets that may satisfy no asymptotic behavior. Moreover, they can be coded by arbitrary symbolic dynamics, and the geometry of the construction may depend on all the symbolic past. Applications include estimates of dimension for hyperbolic sets of maps that need not be differentiable.

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Huang, Wen, and Xintao Wu. "Robustly Improving Bandit Algorithms with Confounded and Selection Biased Offline Data: A Causal Approach." Proceedings of the AAAI Conference on Artificial Intelligence 38, no. 18 (March 24, 2024): 20438–46. http://dx.doi.org/10.1609/aaai.v38i18.30027.

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This paper studies bandit problems where an agent has access to offline data that might be utilized to potentially improve the estimation of each arm’s reward distribution. A major obstacle in this setting is the existence of compound biases from the observational data. Ignoring these biases and blindly fitting a model with the biased data could even negatively affect the online learning phase. In this work, we formulate this problem from a causal perspective. First, we categorize the biases into confounding bias and selection bias based on the causal structure they imply. Next, we extract the causal bound for each arm that is robust towards compound biases from biased observational data. The derived bounds contain the ground truth mean reward and can effectively guide the bandit agent to learn a nearly-optimal decision policy. We also conduct regret analysis in both contextual and non-contextual bandit settings and show that prior causal bounds could help consistently reduce the asymptotic regret.

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CLARK, PETE L., BRIAN COOK, and JAMES STANKEWICZ. "TORSION POINTS ON ELLIPTIC CURVES WITH COMPLEX MULTIPLICATION (WITH AN APPENDIX BY ALEX RICE)." International Journal of Number Theory 09, no. 02 (December 5, 2012): 447–79. http://dx.doi.org/10.1142/s1793042112501436.

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We present seven theorems on the structure of prime order torsion points on CM elliptic curves defined over number fields. The first three results refine bounds of Silverberg and Prasad–Yogananda by taking into account the class number of the CM order and the splitting of the prime in the CM field. In many cases we can show that our refined bounds are optimal or asymptotically optimal. We also derive asymptotic upper and lower bounds on the least degree of a CM-point on X1(N). Upon comparison to bounds for the least degree for which there exist infinitely many rational points on X1(N), we deduce that, for sufficiently large N, X1(N) will have a rational CM point of degree smaller than the degrees of at least all but finitely many non-CM points.

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Lindzey, Nathan, and Ansis Rosmanis. "A tight lower bound for non-coherent index erasure." Quantum Information and Computation 22, no. 7&8 (May 2022): 594–626. http://dx.doi.org/10.26421/qic22.7-8-3.

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The index erasure problem is a quantum state generation problem that asks a quantum computer to prepare a uniform superposition over the image of an injective function given by an oracle. We prove a tight $\Omega(\sqrt{n})$ lower bound on the quantum query complexity of the non-coherent case of the problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state. This resolves an open question of Ambainis et al., who gave a tight bound for the coherent case, the case where the ancillary memory must return to its initial state. To prove our main result, we first extend the automorphism principle}of Hoyer et al. to the general adversary method of Lee et al. for state generation problems, which allows one to exploit the symmetries of these problems to lower bound their quantum query complexity. Using this method, we establish a strong connection between the quantum query complexity of non-coherent symmetric state generation problems and the Krein parameters of an association scheme defined on injective functions. In particular, we use the spherical harmonics a finite symmetric Gelfand pair associated with the space of injective functions to obtain asymptotic bounds on certain Krein parameters, from which the main result follows.

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Ishimwe, Didier, KimHao Nguyen, and ThanhVu Nguyen. "Dynaplex: analyzing program complexity using dynamically inferred recurrence relations." Proceedings of the ACM on Programming Languages 5, OOPSLA (October 20, 2021): 1–23. http://dx.doi.org/10.1145/3485515.

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Being able to detect program runtime complexity is useful in many tasks (e.g., checking expected performance and identifying potential security vulnerabilities). In this work, we introduce a new dynamic approach for inferring the asymptotic complexity bounds of recursive programs. From program execution traces, we learn recurrence relations and solve them using pattern matching to obtain closed-form solutions representing the complexity bounds of the program. This approach allows us to efficiently infer simple recurrence relations that represent nontrivial, potentially nonlinear polynomial and non-polynomial, complexity bounds. We present Dynaplex, a tool that implements these ideas to automatically generate recurrence relations from execution traces. Our preliminary results on popular and challenging recursive programs show that Dynaplex can learn precise relations capturing worst-case complexity bounds (e.g., O ( n log n ) for mergesort, O (2 n ) for Tower of Hanoi and O ( n 1.58 ) for Karatsuba’s multiplication algorithm).

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Zhu, Yuanzheng. "Convergence Rates for the Constrained Sampling via Langevin Monte Carlo." Entropy 25, no. 8 (August 18, 2023): 1234. http://dx.doi.org/10.3390/e25081234.

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Sampling from constrained distributions has posed significant challenges in terms of algorithmic design and non-asymptotic analysis, which are frequently encountered in statistical and machine-learning models. In this study, we propose three sampling algorithms based on Langevin Monte Carlo with the Metropolis–Hastings steps to handle the distribution constrained within some convex body. We present a rigorous analysis of the corresponding Markov chains and derive non-asymptotic upper bounds on the convergence rates of these algorithms in total variation distance. Our results demonstrate that the sampling algorithm, enhanced with the Metropolis–Hastings steps, offers an effective solution for tackling some constrained sampling problems. The numerical experiments are conducted to compare our methods with several competing algorithms without the Metropolis–Hastings steps, and the results further support our theoretical findings.

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Bogatyrev, S. A., F. Götze, and V. V. Ulyanov. "Non-uniform bounds for short asymptotic expansions in the CLT for balls in a Hilbert space." Journal of Multivariate Analysis 97, no. 9 (October 2006): 2041–56. http://dx.doi.org/10.1016/j.jmva.2006.01.010.

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Rizk, Amr, and Markus Fidler. "Non-asymptotic end-to-end performance bounds for networks with long range dependent fBm cross traffic." Computer Networks 56, no. 1 (January 2012): 127–41. http://dx.doi.org/10.1016/j.comnet.2011.07.027.

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Kloeckner, Benoit R. "Effective perturbation theory for simple isolated eigenvalues of linear operators." Journal of Operator Theory 81, no. 1 (December 15, 2018): 175–94. http://dx.doi.org/10.7900/jot.2017dec22.2179.

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We propose a new approach to the spectral theory of perturbed linear operators in the case of a simple isolated eigenvalue. We obtain two kinds of results: ``radius bounds'' which ensure perturbation theory applies for perturbations up to an explicit size, and ``regularity bounds'' which control the variations of eigendata to any order. Our method is based on the implicit function theorem and proceeds by establishing differential inequalities on two natural quantities: the norm of the projection to the eigendirection, and the norm of the reduced resolvent. We obtain completely explicit results without any assumption on the underlying Banach space. In companion articles, on the one hand we apply the regularity bounds to Markov chains, obtaining non-asymptotic concentration and Berry-Esseen inequalities with explicit constants, and on the other hand we apply the radius bounds to transfer operators of intermittent maps, obtaining explicit high-temperature regimes where a spectral gap occurs.