Showing papers on "Upper and lower bounds published in 1994"
TL;DR: An effective technique in locating a source based on intersections of hyperbolic curves defined by the time differences of arrival of a signal received at a number of sensors is proposed and is shown to attain the Cramer-Rao lower bound near the small error region.
Abstract: An effective technique in locating a source based on intersections of hyperbolic curves defined by the time differences of arrival of a signal received at a number of sensors is proposed. The approach is noniterative and gives an explicit solution. It is an approximate realization of the maximum-likelihood estimator and is shown to attain the Cramer-Rao lower bound near the small error region. Comparisons of performance with existing techniques of beamformer, spherical-interpolation, divide and conquer, and iterative Taylor-series methods are made. The proposed technique performs significantly better than spherical-interpolation, and has a higher noise threshold than divide and conquer before performance breaks away from the Cramer-Rao lower bound. It provides an explicit solution form that is not available in the beamforming and Taylor-series methods. Computational complexity is comparable to spherical-interpolation but substantially less than the Taylor-series method. >
2,202 citations
TL;DR: This paper establishes that the interior-reflective Newton approach is globally and quadratically convergent, and develops a specific example of interior- reflective Newton methods which can be used for large-scale and sparse problems.
Abstract: We consider a new algorithm, an interior-reflective Newton approach, for the problem of minimizing a smooth nonlinear function of many variables, subject to upper and/or lower bounds on some of the variables. This approach generatesstrictly feasible iterates by using a new affine scaling transformation and following piecewise linear paths (reflection paths). The interior-reflective approach does not require identification of an "activity set". In this paper we establish that the interior-reflective Newton approach is globally and quadratically convergent. Moreover, we develop a specific example of interior-reflective Newton methods which can be used for large-scale and sparse problems.
1,101 citations
TL;DR: A formula for the capacity of arbitrary single-user channels without feedback is proved and capacity is shown to equal the supremum, over all input processes, of the input-output inf-information rate defined as the liminf in probability of the normalized information density.
Abstract: A formula for the capacity of arbitrary single-user channels without feedback (not necessarily information stable, stationary, etc.) is proved. Capacity is shown to equal the supremum, over all input processes, of the input-output inf-information rate defined as the liminf in probability of the normalized information density. The key to this result is a new converse approach based on a simple new lower bound on the error probability of m-ary hypothesis tests among equiprobable hypotheses. A necessary and sufficient condition for the validity of the strong converse is given, as well as general expressions for /spl epsiv/-capacity. >
907 citations
02 Jan 1994
TL;DR: Differentially uniform mappings as discussed by the authors have also desirable cryptographic properties: large distance from affine functions, high nonlinear order and efficient computability, and have also been used in DES-like ciphers.
Abstract: This work is motivated by the observation that in DES-like ciphers it is possible to choose the round functions in such a way that every non-trivial one-round characteristic has small probability. This gives rise to the following definition. A mapping is called differentially uniform if for every non-zero input difference and any output difference the number of possible inputs has a uniform upper bound. The examples of differentially uniform mappings provided in this paper have also other desirable cryptographic properties: large distance from affine functions, high nonlinear order and efficient computability.
859 citations
TL;DR: The modified Cramer-Rao bound (CRB) is introduced which, like the true CRB, is a lower bound to the error variance of any parameter estimator.
Abstract: We introduce the modified Cramer-Rao bound (CRB) which, like the true CRB, is a lower bound to the error variance of any parameter estimator. The modified CRB proves useful when, in addition to the parameter to be estimated, the observed data also depend on other unwanted parameters. The relationship between the modified and true CRB is established and applications are discussed regarding the estimation of carrier-frequency offset, carrier phase, and timing epoch in linearly modulated signals. Modified CRBs for phase and timing estimation have been already discussed in previous works where it is shown that several practical carrier-phase and clock recovery circuits do attain such bounds. Frequency discrimination, instead, is not so well-represented in the literature and a significant contribution of this paper is the calculation of the modified CRB for frequency estimation. This bound is compared with the performance of some frequency detectors and it is concluded that further work is needed in search of more efficient frequency discrimination methods. >
540 citations
TL;DR: It is shown that the sum capacity of the symbol-synchronous code-division multiple-access channel with equal average-input-energy constraints is maximized precisely by those spreading sequence multisets that meet Welch's lower bound on total squared correlation.
Abstract: It is shown that the sum capacity of the symbol-synchronous code-division multiple-access channel with equal average-input-energy constraints is maximized precisely by those spreading sequence multisets that meet Welch's lower bound on total squared correlation. It is further shown that the symmetric capacity of the channel determined by these same sequence multisets is equal to the sum capacity. >
387 citations
TL;DR: In this paper, the capacity of a communication channel is the maximum rate at which information can be transmitted without error from the channel's input to its output, i.e., the time it takes to decode the information from the input to the output.
Abstract: The capacity C of a communication channel is the maximum rate at which information can be transmitted without error from the channel's input to its output. The authors review quantum limits on the capacity that can be achieved with linear bosonic communication channels that have input power P. The limits arise ultimately from the Einstein relation that a field quantum at frequency f has energy E=hf. A single linear bosonic channel corresponds to a single transverse mode of the bosonic field i.e., to a particular spatial dependence in the plane orthogonal to the propagation direction and to a particular spin state or polarization. For a single channel the maximum communication rate is CWB=( ln 2)2P3h bits/s. This maximum rate can be achieved by a "number-state channel," in which information is encoded in the number of quanta in the bosonic field and in which this information is recovered at the output by counting quanta. Derivations of the optimum capacity CWB are reviewed. Until quite recently all derivations assumed, explicitly or implicitly, a number-state channel. They thus left open the possibility that other techniques for encoding information on the bosonic field, together with other ways of detecting the field at the output, might lead to a greater communication rate. The authors present their own general derivation of the single-channel capacity upper bound, which applies to any physically realizable technique for encoding information on the bosonic field and to any physically realizable detection scheme at the output. They also review the capacities of coherent communication channels that encode information in coherent states and in quadrature-squeezed states. A three-dimensional bosonic channel can employ many transverse modes as parallel single channels. An upper bound on the information flux that can be transferred down parallel bosonic channels is derived.
378 citations
TL;DR: It is noted that these networks are not likely to solve polynomially NP-hard problems, as the equality “ p = np ” in the model implies the almost complete collapse of the standard polynomial hierarchy.
365 citations
TL;DR: Lower bounds for the time complexity of a class of deterministic algorithms for the dictionary problem are proved and this class encompasses realistic hashing-based schemes that use linear space.
Abstract: The dynamic dictionary problem is considered: provide an algorithm for storing a dynamic set, allowing the operations insert, delete, and lookup. A dynamic perfect hashing strategy is given: a randomized algorithm for the dynamic dictionary problem that takes $O(1)$ worst-case time for lookups and $O(1)$ amortized expected time for insertions and deletions; it uses space proportional to the size of the set stored. Furthermore, lower bounds for the time complexity of a class of deterministic algorithms for the dictionary problem are proved. This class encompasses realistic hashing-based schemes that use linear space. Such algorithms have amortized worst-case time complexity $\Omega(\log n)$ for a sequence of $n$ insertions and lookups; if the worst-case lookup time is restricted to $k$, then the lower bound becomes $\Omega(k\cdot n^{1/k})$.
297 citations
TL;DR: It is proved the existence of an efficient “simulation” of randomized on-line algorithms by deterministic ones, which is best possible in general.
Abstract: Against in adaptive adversary, we show that the power of randomization in on-line algorithms is severely limited! We prove the existence of an efficient "simulation" of randomized on-line algorithms by deterministic ones, which is best possible in general. The proof of the upper bound is existential. We deal with the issue of computing the efficient deterministic algorithm, and show that this is possible in very general cases.
291 citations
TL;DR: A new adaptive algorithm is introduced which ensures that the plant can be locally stabilized and an upper bound on the plant control parameter is required to be known.
Abstract: This paper deals with the problem of adaptively controlling a linear time-invariant plant in the presence of constraints on the input amplitude. We introduce a new adaptive algorithm which ensures that the plant can be locally stabilized. In addition to the standard assumptions which are required for adaptive control in the ideal case, an upper bound on the plant control parameter is required to be known. The results are evaluated by simulation studies. >
TL;DR: In this article, lower bounds on the minimum cost of managing certain production-distribution networks with setup costs at all stages and stochastic demands were established through novel cost-allocation schemes.
Abstract: We establish lower bounds on the minimum cost of managing certain production-distribution networks with setup costs at all stages and stochastic demands. These networks include serial, assembly, and one-warehouse multi-retailer systems. We obtain the bounds through novel cost-allocation schemes. We evaluate the bounds' performance for one-warehouse multi-retailer systems by comparing them with simple, heuristic policies. The bounds are quite tight for systems with a small number of retailers. We also present simplified proof of known optimality results for serial and assembly systems.
TL;DR: In this paper, a Riccati equation approach is proposed to solve the problem of Kalman filter design for uncertain systems and a suboptimal covariance upper bound can be computed by a convex optimization.
TL;DR: The algorithm for computing the BKK bound for the number of isolated solutions of a polynomial system with a sparse monomial structure is described and the algorithmic construction of the cheater’s homotopy or the coefficienthomotopy is obtained.
Abstract: This paper is concerned with the problem of finding all isolated solutions of a polynomial system The BKK bound, defined as the mixed volume of the Newton polytopes of the polynomials in the system, is a sharp upper bound for the number of isolated solutions in $\mathbb{C}_0^n ,\mathbb{C}_0 = \mathbb{C} \backslash \{ 0\} $, of a polynomial system with a sparse monomial structure First an algorithm is described for computing the BKK bound Following the lines of Bernshtei˘n’s proof, the algorithmic construction of the cheater’s homotopy or the coefficient homotopy is obtained The mixed homotopy methods can be combined with the random product start systems based on a generalized Bezout number Applications illustrate the effectiveness of the new approach
TL;DR: A general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems is given, which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution.
Abstract: We give a general framework for deriving a posteriori error estimates for approximate solutions of nonlinear problems. In a first step it is proven that the error of the approximate solution can be bounded from above and from below by an appropriate norm of its residual. In a second step this norm of the residual is bounded from above and from below by a similar norm of a suitable finite-dimensional approximation of the residual. This quantity can easily be evaluated, and for many practical applications sharp explicit upper and lower bounds are readily obtained. The general results are then applied to finite element discretizations of scalar quasi-linear elliptic partial differential equations of 2nd order, the eigenvalue problem for scalar linear elliptic operators of 2nd order, and the stationary incompressible Navier-Stokes equations. They immediately yield a posteriori error estimates, which can easily be computed from the given data of the problem and the computed numerical solution and which give global upper and local lower bounds on the error of the numerical solution.
TL;DR: In this paper, it was shown that for n sufficiently large log 2 T(r,n) n ⩽8 log 2 r r r 2 holds for any ε > 0, where r is the maximum number of subsets of an n-set satisfying a given condition.
TL;DR: In this paper, a statistical test has been constructed to test, for some x ∈ [a, b], against the hypothesis, for all x ∆ [a and b], where a and b are any two real numbers, and has an upper bound α on the asymptotic size.
Abstract: A distribution function F is said to stochastically dominate another distribution function G in the second-order sense if , for all x. Second-order stochastic dominance plays an important role in economics, finance, and accounting. Here a statistical test has been constructed to test , for some x ∈ [a, b], against the hypothesis , for all x ∈ [a, b], where a and b are any two real numbers. The test has been shown to be consistent and has an upper bound α on the asymptotic size. The test is expected to have usefulness for comparison of random prospects for risk averters.
TL;DR: A new technique for obtaining upper and lower bounds on the performance of Markovian queueing networks and scheduling policies is introduced, and analytic bounds which improve upon Kingman's bound (1970) for E/sub 2//M/1 queues are obtained.
Abstract: Except for the class of queueing networks and scheduling policies admitting a product form solution for the steady-state distribution, little is known about the performance of such systems. For example, if the priority of a part depends on its class (e.g., the buffer that the part is located in), then there are no existing results on performance, or even stability. In most applications such as manufacturing systems, however, one has to choose a control or scheduling policy, i.e., a priority discipline, that optimizes a performance objective. In this paper the authors introduce a new technique for obtaining upper and lower bounds on the performance of Markovian queueing networks and scheduling policies. Assuming stability, and examining the consequence of a steady state for general quadratic forms, the authors obtain a set of linear equality constraints on the mean values of certain random variables that determine the performance of the system. Further, the conservation of time and material gives an augmenting set of linear equality and inequality constraints. Together, these allow the authors to bound the performance, either above or below, by solving a linear program. The authors illustrate this technique on several typical problems of interest in manufacturing systems. For an open re-entrant line modeling a semiconductor plant, the authors plot a bound on the mean delay (called cycle-time) as a function of line loading. It is shown that the last buffer first serve policy is almost optimal in light traffic. For another such line, it is shown that it dominates the first buffer first serve policy. For a set of open queueing networks, the authors compare their lower bounds with those obtained by another method of Ou and Wein (1992). For a closed queueing network, the authors bracket the performance of all buffer priority policies, including the suggested priority policy of Harrison and Wein (1990). The authors also study the asymptotic heavy traffic limits of the lower and upper bounds. For a manufacturing system with machine failures, it is shown how the performance changes with failure and repair rates. For systems with finite buffers, the authors show how to bound the throughput. Finally, the authors illustrate the application of their method to GI/GI/1 queues. The authors obtain analytic bounds which improve upon Kingman's bound (1970) for E/sub 2//M/1 queues. >
TL;DR: In this paper, a new formulation for the multiple-depot vehicle scheduling problem with side constraints is given, whose continuous relaxation is amenable to be solved by column generation, and it is shown that the continuous relaxation of the set partitioning formulation provides a much tighter lower bound than the additive bound procedure previously applied to this problem.
Abstract: We give a new formulation to the multiple-depot vehicle scheduling problem as a set partitioning problem with side constraints, whose continuous relaxation is amenable to be solved by column generation. We show that the continuous relaxation of the set partitioning formulation provides a much tighter lower bound than the additive bound procedure previously applied to this problem. We also establish that the additive bound technique cannot provide tighter bounds than those obtained by Lagrangian decomposition, in the framework in which it has been used so far. Computational results that illustrate the robustness of the combined set partitioning-column generation approach are reported for problems four to five times larger than the largest problems that have been exactly solved in the literature. Finally, we show that the gap associated with the additive bound based on the assignment and shortest path relaxations can be arbitrarily bad in the general case, and as bad as 50% in the symmetric case.
TL;DR: This follow up article introduces and analyzes a new TGFSR variant having better k-distribution property, and provides an efficient algorithm to obtain the order of equidistribution, together with a tight upper bound on the order.
Abstract: The twisted GFSR generators proposed in a previous article have a defect in k-distribution for k larger than the order of recurrence. In this follow up article, we introduce and analyze a new TGFSR variant having better k-distribution property. We provide an efficient algorithm to obtain the order of equidistribution, together with a tight upper bound on the order. We discuss a method to search for generators attaining this bound, and we list some of these such generators. The upper bound turns out to be (sometimes far) less than the maximum order of equidistribution for a generator of that period length, but far more than that for a GFSR with a working are of the same size.
27 Jun 1994
TL;DR: The lower bound on d-capacity, the capacity of a deterministic arbitrarily varying channel defined by a bipartite graph, is shown to be not tight in general, but C/sub d/(W)>0 iff this bound is positive.
Abstract: For discrete memoryless channels {W: X/spl rarr/Y} we consider decoders, possibly suboptimal, which minimize a metric defined additively by a given function d(x, y)/spl ges/0. The largest rate achievable by codes with such a decoder is called the d-capacity C/sub d/(W). The choice d(x, y)=0 if and only if (iff) W(y|x)>0 makes C/sub d/(W) equal to the "zero undetected error" or "erasures-only" capacity C/sub eo/(W). The graph-theoretic concepts of Shannon capacity (1956, 1974) and Sperner capacity are also special cases of d-capacity, viz. for a noiseless channel with a suitable {0, 1}-valued function d. We show that the lower bound on d-capacity given previously by Csiszar and Korner (1980), and Hui (1983), is not tight in general, but C/sub d/(W)>0 iff this bound is positive. The "product space" improvement of the lower bound is considered,and a "product space characterization" of C/sub eo/(W) is obtained. We also determine the erasures-only (e.o.) capacity of a deterministic arbitrarily varying channel defined by a bipartite graph, and show that it equals capacity. We conclude with a list of challenging open problems. >
TL;DR: The first correct proof that, for a random oracleA, PPA is properly contained in PSPACEA is given, which allows the new result that an AC0 circuit with one majority gate cannot approximate parity to be proved.
Abstract: We consider the problem of approximating a Boolean functionf∶{0,1}
n
→{0,1} by the sign of an integer polynomialp of degreek. For us, a polynomialp(x) predicts the value off(x) if, wheneverp(x)≥0,f(x)=1, and wheneverp(x)<0,f(x)=0. A low-degree polynomialp is a good approximator forf if it predictsf at almost all points. Given a positive integerk, and a Boolean functionf, we ask, “how good is the best degreek approximation tof?” We introduce a new lower bound technique which applies to any Boolean function. We show that the lower bound technique yields tight bounds in the casef is parity. Minsky and Papert [10] proved that a perceptron cannot compute parity; our bounds indicate exactly how well a perceptron canapproximate it. As a consequence, we are able to give the first correct proof that, for a random oracleA, PP
A
is properly contained in PSPACE
A
. We are also able to prove the old AC0 exponential-size lower bounds in a new way. This allows us to prove the new result that an AC0 circuit with one majority gate cannot approximate parity. Our proof depends only on basic properties of integer polynomials.
TL;DR: A general principle is derived demonstrating that by partitioning the feasible set, the duality gap, existing between a nonconvex program and its lagrangian dual, can be reduced, and in important special cases, even eliminated.
Abstract: We derive a general principle demonstrating that by partitioning the feasible set, the duality gap, existing between a nonconvex program and its lagrangian dual, can be reduced, and in important special cases, even eliminated. The principle can be implemented in a Branch and Bound algorithm which computes an approximate global solution and a corresponding lower bound on the global optimal value. The algorithm involves decomposition and a nonsmooth local search. Numerical results for applying the algorithm to the pooling problem in oil refineries are given.
TL;DR: A new probabilistic clock synchronization algorithm that can guarantee a much smaller bound on the clock skew than most existing algorithms is presented and it is shown that an upper bound onThe probability of invalidity decreases exponentially with the number of synchronization messages transmitted.
Abstract: Presents and analyzes a new probabilistic clock synchronization algorithm that can guarantee a much smaller bound on the clock skew than most existing algorithms. The algorithm is probabilistic in the sense that the bound on the clock skew that it guarantees has a probability of invalidity associated with it. However, the probability of invalidity may be made extremely small by transmitting a sufficient number of synchronization messages. It is shown that an upper bound on the probability of invalidity decreases exponentially with the number of synchronization messages transmitted. A closed-form expression that relates the probability of invalidity to the clock skew and the number of synchronization messages is also derived. >
TL;DR: The first nontrivial general upper bound for this problem is shown, and it almost establishes a long-standing conjecture that the complexity of the envelope isO(nd-2λq(n) for some constantq depending on the shape and degree of the surfaces.
Abstract: We consider the problem of bounding the combinatorial complexity of the lower envelope ofn surfaces or surface patches ind-space (d?3), all algebraic of constant degree, and bounded by algebraic surfaces of constant degree. We show that the complexity of the lower envelope ofn such surface patches isO(nd?1+?), for any ?>0; the constant of proportionality depends on ?, ond, ons, the maximum number of intersections among anyd-tuple of the given surfaces, and on the shape and degree of the surface patches and of their boundaries. This is the first nontrivial general upper bound for this problem, and it almost establishes a long-standing conjecture that the complexity of the envelope isO(nd-2?q(n)) for some constantq depending on the shape and degree of the surfaces (where ?q(n) is the maximum length of (n, q) Davenport-Schinzel sequences). We also present a randomized algorithm for computing the envelope in three dimensions, with expected running timeO(n2+?), and give several applications of the new bounds.
TL;DR: This paper reformulated the rate-distortion problem in terms of the optimal mapping from the unit interval with Lebesgue measure that would induce the desired reproduction probability density and shows how the number of "symbols" grows as the system undergoes phase transitions.
Abstract: In rate-distortion theory, results are often derived and stated in terms of the optimizing density over the reproduction space. In this paper, the problem is reformulated in terms of the optimal mapping from the unit interval with Lebesgue measure that would induce the desired reproduction probability density. This results in optimality conditions that are "random relatives" of the known Lloyd (1982) optimality conditions for deterministic quantizers. The validity of the mapping approach is assured by fundamental isomorphism theorems for measure spaces. We show that for the squared error distortion, the optimal reproduction random variable is purely discrete at supercritical distortion (where the Shannon (1948) lower bound is not tight). The Gaussian source is thus the only source that produces continuous reproduction variables for the entire range of positive rate. To analyze the evolution of the optimal reproduction distribution, we use the mapping formulation and establish an analogy to statistical mechanics. The solutions are given by the distribution at isothermal statistical equilibrium, and are parameterized by the temperature in direct correspondence to the parametric solution of the variational equations in rate-distortion theory. The analysis of an annealing process shows how the number of "symbols" grows as the system undergoes phase transitions. Thus, an algorithm based on the mapping approach often needs but a few variables to find the exact solution, while the Blahut (1972) algorithm would only approach it at the limit of infinite resolution. Finally, a quick "deterministic annealing" algorithm to generate the rate-distortion curve is suggested. The resulting curve is exact as long as continuous phase transitions in the process are accurately followed. >
Journal Article•
TL;DR: The examples of differentially uniform mappings provided in this paper have also other desirable cryptographic properties: large distance from affine functions, high nonlinear order and efficient computability.
Abstract: This work is motivated by the observation that in DES-like ciphers it is possible to choose the round functions in such a way that every non-trivial one-round characteristic has small probability. This gives rise to the following definition. A mapping is called differentially uniform if for every non-zero input difference and any output difference the number of possible inputs has a uniform upper bound. The examples of differentially uniform mappings provided in this paper have also other desirable cryptographic properties: large distance from affine functions, high nonlinear order and efficient computability.
TL;DR: An upper bound on performance loss is derived that is slightly tighter than that in Bertsekas (1987), and the extension of the bound to Q-learning is shown to provide a partial theoretical rationale for the approximation of value functions.
Abstract: Many reinforcement learning approaches can be formulated using the theory of Markov decision processes and the associated method of dynamic programming (DP) The value of this theoretical understanding, however, is tempered by many practical concerns One important question is whether DP-based approaches that use function approximation rather than lookup tables can avoid catastrophic effects on performance This note presents a result of Bertsekas (1987) which guarantees that small errors in the approximation of a task's optimal value function cannot produce arbitrarily bad performance when actions are selected by a greedy policy We derive an upper bound on performance loss that is slightly tighter than that in Bertsekas (1987), and we show the extension of the bound to Q-learning (Watkins, 1989) These results provide a partial theoretical rationale for the approximation of value functions, an issue of great practical importance in reinforcement learning
27 Jun 1994
TL;DR: The bound has immediate application to the design of large families of phase-shift-keying sequences having low correlation and an alphabet of size p/sup e/.
Abstract: We present an analog of the well-known Weil-Carlitz-Uchiyama (1948, 1957) upper bound for exponential sums over finite fields for exponential sums over Galois rings. Some examples are given where the bound is tight. The bound has immediate application to the design of large families of phase-shift-keying sequences having low correlation and an alphabet of size p/sup e/. p, prime, e/spl ges/2. Some new constructions of eight-phase sequences are provided. >
Journal Article•
TL;DR: This paper proves lower bounds of the form exp(ne d), ed > 0, on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d, for any constant d, is the largest lower bound for the strongest proof system, for which any superpolynomial lower bounds are known.
Abstract: We prove lower bounds of the form exp(ne d), ed > 0, on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d, for any constant d. This is the largest lower bound for the strongest proof system, for which any superpolynomial lower bounds are known.