Optimal Lower Bound Research Articles

Minimization of the lowest positive Neumann-Dirichlet eigenvalue for general indefinite Sturm-Liouville problems

The aim of this paper is to obtain the sharp estimate for the lowest positive eigenvalue λ0ND+ for the general Sturm–Liouville problemy″=q(t)y+λm(t)y, with the Neumann-Dirichlet boundary conditions, where q is a nonnegative potential and another potential m admits to change sign. First, we will study the optimal lower bound for the smallest positive eigenvalue in the measure differential equations to make our results more applicable. Second, based on the relationship between the minimization problem of the smallest positive eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest positive eigenvalue for the general Sturm–Liouville equation.

The irregularity strength of dense graphs — on asymptotically optimal solutions of problems of Faudree, Jacobson, Kinch and Lehel

The irregularity strength of a graph G, s(G), is the least k such that there exists a {1,2,…,k}-weighting of the edges of G attributing distinct weighted degrees to all vertices, or equivalently the least k allowing the creation of a multigraph with nonrecurring degrees by blowing each edge e of G to at most k copies of e. In 1991 Faudree, Jacobson, Kinch and Lehel asked for the optimal lower bound for the minimum degree of a graph G of order n which implies that s(G)≤3. More generally, they also posed a similar question regarding the upper bound s(G)≤K for any given constant K. We provide asymptotically tight solutions of these problems by proving that such optimal lower bound is of order 1K−1n for every fixed integer K≥3.

Sharp estimates of lowest positive Neumann eigenvalue for general indefinite Sturm-Liouville problems

Given two measures μ,ν and their total variations, we study the minimization of Neumann eigenvalues for measure differential equationdy•=y(t)dμ(t)+λydν(t). By solving the infinitely dimensional minimization problem of the lowest positive Neumann eigenvalue for the measure differential equation, we obtain the optimal lower bound of the lowest positive Neumann eigenvalue for the Sturm-Liouville problemy″=q(t)y+λm(t)y, where q(t) is a nonnegative potential function and the other potential function m(t) admits to change sign.

Multipartite entanglement detection based on the generalized state-dependent entropic uncertainty relation for multiple measurements

We present the generalized state-dependent entropic uncertainty relations in multiple measurements setting, and the optimal lower bound is obtained by considering different measurement sequences. We then apply this uncertainty relation to witness entanglement, and give the experimentally accessible lower bounds on both bipartite and tripartite entanglements. This method of detecting entanglement is applied to a physical system of two particles on a one-dimensional lattice, and GHZ-Werner state. It is shown that, for measurements that are not in mutually unbiased bases, this new entropic uncertainty relation is superior to the previous state-independent one in entanglement detection. Furthermore, we conduct an online experiment to detect multipartite entanglement of GHZ states up to 10 qubits on Quafu cloud quantum computation platform. Our results might play important roles in detecting multipartite entanglement experimentally.

Linearized symmetric multi-block ADMM with indefinite proximal regularization and optimal proximal parameter

The proximal term plays a significant role in the literature of proximal Alternating Direction Method of Multipliers (ADMM), since (positive-definite or indefinite) proximal terms can promote convergence of ADMM and further simplify the involved subproblems. However, an overlarge proximal parameter decelerates the convergence numerically, though the convergence can be established with it. In this paper, we thus focus on a Linearized Symmetric ADMM (LSADMM) with proper proximal terms for solving a family of multi-block separable convex minimization models, and we determine an optimal (smallest) value of the proximal parameter while convergence of this LSADMM can be still ensured. Our LSADMM partitions the data into two group variables and updates the Lagrange multiplier twice in different forms with suitable step sizes. The region of the proximal parameter, involved in the second group subproblems, is partitioned into the union of three different sets. We show the global convergence and sublinear ergodic convergence rate of LSADMM for the two cases, while a counter-example is given to illustrate that convergence of LSADMM can not be guaranteed for the remaining case. Theoretically, we obtain the optimal lower bound of the proximal parameter. Numerical experiments on the so-called latent variable Gaussian graphical model selection problems are presented to demonstrate performance of the proposed algorithm and the significant advantage of the optimal lower bound of the proximal parameter.

Circuit Lower Bounds for MCSP from Local Pseudorandom Generators

The Minimum Circuit Size Problem (MCSP) asks if a given truth table of a Boolean function f can be computed by a Boolean circuit of size at most θ, for a given parameter θ. We improve several circuit lower bounds for MCSP, using pseudorandom generators (PRGs) that are local; a PRG is called local if its output bit strings, when viewed as the truth table of a Boolean function, can be computed by a Boolean circuit of small size. We get new and improved lower bounds for MCSP that almost match the best-known lower bounds against several circuit models. Specifically, we show that computing MCSP, on functions with a truth table of length N , requires • N 3− o (1) -size de Morgan formulas, improving the recent N 2− o (1) lower bound by Hirahara and Santhanam (CCC, 2017), • N 2− o (1) -size formulas over an arbitrary basis or general branching programs (no non-trivial lower bound was known for MCSP against these models), and • 2 Ω( N 1/( d +1.01)) -size depth- d AC 0 circuits, improving the (implicit, in their work) exponential size lower bound by Allender et al. (SICOMP, 2006). The AC 0 lower bound stated above matches the best-known AC 0 lower bound (for PARITY) up to a small additive constant in the depth. Also, for the special case of depth-2 circuits (i.e., CNFs or DNFs), we get an optimal lower bound of 2 Ω( N ) for MCSP.

Optimal lower bounds for Donaldson's J-functional

In this paper we provide an explicit formula for the optimal lower bound of Donaldson's J-functional, in the sense of finding explicitly the optimal constant in the definition of coercivity, which always exists and takes negative values in general. This constant is positive precisely if the J-equation admits a solution, and the explicit formula has a number of applications. First, this leads to new existence criteria for constant scalar curvature Kähler (cscK) metrics in terms of Tian's alpha invariant. Moreover, we use the above formula to discuss Calabi dream manifolds and an analogous notion for the J-equation, and show that for surfaces the optimal bound is an explicitly computable rational function which typically tends to minus infinity as the underlying class approaches the boundary of the Kähler cone, even when the underlying Kähler classes admit cscK metrics. As a final application we show that if the Lejmi-Székelyhidi conjecture holds, then the optimal bound coincides with its algebraic counterpart, the set of J-semistable classes equals the closure of the set of uniformly J-stable classes in the Kähler cone, and there exists an optimal degeneration for uniform J-stability.

Minimization of eigenvalues for the Camassa–Holm equation

A key basis for seeking solutions of the Camassa–Holm equation is to understand the associated spectral problem [Formula: see text] We will study in this paper the optimal lower bound of the smallest eigenvalue for the Camassa–Holm equation with the Neumann boundary condition when the [Formula: see text] norm of potentials is given. First, we will study the optimal lower bound for the smallest eigenvalue in the measure differential equations to make our results more applicable. Second, Based on the relationship between the minimization problem of the smallest eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest eigenvalue for the Camassa–Holm equation.

Generalized ADMM with optimal indefinite proximal term for linearly constrained convex optimization

We consider the generalized alternating direction method of multipliers (ADMM) for linearly constrained convex optimization. Many problems derived from practical applications have showed that usually one of the subproblems in the generalized ADMM is hard to solve, thus a special proximal term is added. In the literature, the proximal term can be indefinite which plays a vital role in accelerating numerical performance. In this paper, we are devoted to deriving the optimal lower bound of the proximal parameter and result in the generalized ADMM with optimal indefinite proximal term. The global convergence and the \begin{document}$ O(1/t) $\end{document} convergence rate measured by the iteration complexity of the proposed method are proved. Moreover, some preliminary numerical experiments on LASSO and total variation-based denoising problems are presented to demonstrate the efficiency of the proposed method and the advantage of the optimal lower bound.

Optimal Online Two-Way Trading with Bounded Number of Transactions

We consider a two-way trading problem, where investors buy and sell a stock whose price moves within a certain range. Naturally they want to maximize their profit. Investors can perform up to k trades, where each trade must involve the full amount. We give optimal algorithms for three different models which differ in the knowledge of how the price fluctuates. In the first model, there are global minimum and maximum bounds m and M. We first show an optimal lower bound of varphi (where varphi =M/m) on the competitive ratio for one trade, which is the bound achieved by trivial algorithms. Perhaps surprisingly, when we consider more than one trade, we can give a better algorithm that loses only a factor of varphi ^{2/3} (rather than varphi ) per additional trade. Specifically, for k trades the algorithm has competitive ratio varphi ^{(2k+1)/3}. Furthermore we show that this ratio is the best possible by giving a matching lower bound. In the second model, m and M are not known in advance, and just varphi is known. We show that this only costs us an extra factor of varphi ^{1/3}, i.e., both upper and lower bounds become varphi ^{(2k+2)/3}. Finally, we consider the bounded daily return model where instead of a global limit, the fluctuation from one day to the next is bounded, and again we give optimal algorithms, and interestingly one of them resembles common trading strategies that involve stop loss limits.

Continuity and minimization of spectrum related with the periodic Camassa–Holm equation

An important point in looking for period solutions of the Camassa–Holm equation is to understand the associated spectral problemy″=14y+λm(t)y. The first aim of this paper is to study the dependence of eigenvalues for the periodic Camassa–Holm Equation on potentials as an infinitely dimensional parameter. To be precise, we prove that as nonlinear functionals of potentials, eigenvalues for the periodic Camassa–Holm Equation are continuous in potentials with respect to the weak topologies in the Lp Lebesgue spaces. The second aim of this paper is to find the optimal lower bound of the lowest eigenvalue for the periodic Camassa–Holm Equation when the L1 norm of potentials are given. In order to make our results more applicable, we will find the optimal lower bound for the lowest eigenvalue in the more general setting of measure differential equations.

Noisy Low-Tubal-Rank Tensor Completion Through Iterative Singular Tube Thresholding

In many applications, data organized in tensor form contains noise and missing entries. In this paper, the goal is to complete a tensor from its partial noisy observations. Specifically, we consider low-tubal-rank tensors sampled by element-wise Bernoulli sampling with additive sub-exponential noise. Algorithmically, a soft-impute-like algorithm, namely iterative singular tube thresholding (ISTT), is proposed. Statistically, bound on the estimation error of ISTT is explored. First, the estimation error is upper bounded non-asymptotically. Then, the minimax optimal lower bound of the estimation error for tensors with limited tubal-rank and bounded $l_\infty $ -norm is established, indicating that the proposed upper bound is order optimal up to a logarithm factor. Numerical simulations show that the proposed upper bound can precisely predict the scaling behavior of the estimation error of ISTT. Compared with several state-of-the-art convex algorithms, the effectiveness of ISTT is demonstrated through experiments on real-world data sets.

A lower bound on the release of differentially private integer partitions

We consider the problem of privately releasing integer partitions. This problem is of high practical interest, being related to the publication of password frequency lists or the degree distribution of social networks. In this work, we show that any ε-differentially private mechanism releasing a partition of a sufficiently large non-negative integer N must incur a minimax risk of order Ω(N/ε). Moreover, for small values of N, we provide an optimal lower bound of order Ω(N).

Up to n: Pragmatic inference about an optimal lower bound

This paper focuses on English directional modified numerals up to n, which triggers opposite inference patterns in speaker-uncertainty and authoritative-permission contexts. I propose that these opposite inference patterns are due to pragmatic inference about an unspecified semantic lower bound of up to n, based on its similarities to gradable adjectives and vague characteristics. The value of the semantic lower bound in different contexts is predicted by a general pragmatic principle of interaction between informativity and applicability independently motivated in previous probabilistic models on gradable adjectives.

Lower bound of quantum uncertainty from extractable classical information

The sum of entropic uncertainties for the measurement of two non-commuting observables is not always reduced by the amount of entanglement (quantum memory) between two parties, and in certain cases may be impacted by quantum correlations beyond entanglement (discord). An optimal lower bound of entropic uncertainty in the presence of any correlations may be determined by fine-graining. Here we express the uncertainty relation in a new form where the maximum possible reduction of uncertainty is shown to be given by the extractable classical information. We show that the lower bound of uncertainty matches with that using fine-graining for several examples of two-qubit pure and mixed entangled states, and also separable states with non-vanishing discord. Using our uncertainty relation we further show that even in the absence of any quantum correlations between the two parties, the sum of uncertainties may be reduced with the help of classical correlations.

Optimal Lower Bound of the Resonance Widths for the Helmholtz Resonator

Under a geometric assumption on the region near the end of its neck, we prove an optimal exponential lower bound on the widths of resonances for a general two-dimensional Helmholtz resonator. An extension of the result to the n-dimensional case, \({n \leq 12}\), is also obtained.

An Optimal Threshold Strategy in the Two-Envelope Problem with Partial Information

In this paper we propose a strategy that gives an optimal lower bound of the average gain for the two-envelope problem within the McDonnell and Abbott (2009) and McDonnell et al. (2011) framework. We obtain this result with partial information about the probability distribution of the envelope's contents.

An Optimal Threshold Strategy in the Two-Envelope Problem with Partial Information

In this paper we propose a strategy that gives an optimal lower bound of the average gain for the two-envelope problem within the McDonnell and Abbott (2009) and McDonnell et al. (2011) framework. We obtain this result with partial information about the probability distribution of the envelope's contents.

Optimal Lower Bounds for Locality-Sensitive Hashing (Except When q is Tiny)

We study lower bounds for Locality-Sensitive Hashing (LSH) in the strongest setting: point sets in {0,1} d under the Hamming distance. Recall that H is said to be an ( r , cr , p , q )-sensitive hash family if all pairs x , y ∈ {0,1} d with dist( x , y ) ≤ r have probability at least p of collision under a randomly chosen h ∈ H, whereas all pairs x , y ∈ {0, 1} d with dist( x , y ) ≥ cr have probability at most q of collision. Typically, one considers d → ∞, with c > 1 fixed and q bounded away from 0. For its applications to approximate nearest-neighbor search in high dimensions, the quality of an LSH family H is governed by how small its ρ parameter ρ = ln(1/ p )/ln(1/ q ) is as a function of the parameter c . The seminal paper of Indyk and Motwani [1998] showed that for each c ≥ 1, the extremely simple family H = { x ↦ x i : i ∈ [ d ]} achieves ρ ≤ 1/ c . The only known lower bound, due to Motwani et al. [2007], is that ρ must be at least ( e 1/c - 1)/( e 1/c + 1) ≥ .46/ c (minus o d (1)). The contribution of this article is twofold. (1) We show the “optimal” lower bound for ρ : it must be at least 1/ c (minus o d (1)). Our proof is very simple, following almost immediately from the observation that the noise stability of a boolean function at time t is a log-convex function of t . (2) We raise and discuss the following issue: neither the application of LSH to nearest-neighbor search nor the known LSH lower bounds hold as stated if the q parameter is tiny. Here, “tiny” means q = 2 -Θ(d) , a parameter range we believe is natural.

Almost Optimal Lower Bounds for Problems Parameterized by Clique-Width

We obtain asymptotically tight algorithmic bounds for Max-Cut and Edge Dominating Set problems on graphs of bounded clique-width. We show that on an $n$-vertex graph of clique-width $t$ both problems (1) cannot be solved in time $f(t)n^{o(t)}$ for any function $f$ of $t$ unless exponential time hypothesis fails, and (2) can be solved in time $n^{O(t)}$.