Showing papers on "Upper and lower bounds published in 1991"

Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics

Abstract: Continuing the development of the structured singular value approach to robust control design, the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty. The problem is shown to be equivalent to a smooth constrained finite-dimensional optimization problem. In view of the fact that the functional to be maximized may have several local extrema, an upper bound on mu whose computation is numerically tractable is established; this leads to a sufficient condition of robust stability and performance. A historical perspective on the development of the mu theory is included. >

Control oriented system identification: a worst-case/deterministic approach in H/sub infinity /

Abstract: The authors formulate and solve two related control-oriented system identification problems for stable linear shift-invariant distributed parameter plants In each of these problems the assumed a priori information is minimal, consisting only of a lower bound on the relative stability of the plant, an upper bound on a certain gain associated with the plant, and an upper bound on the noise level The first of these problems involves identification of a point sample of the plant frequency response from a noisy, finite, output time series obtained in response to an applied sinusoidal input with frequency corresponding to the frequency point of interest This problem leads naturally to the second problem, which involves identification of the plant transfer function in H/sub infinity / from a finite number of noisy point samples of the plant frequency response Concrete plans for identification algorithms are provided for each of these two problems >

Eigenvalues, diameter, and mean distance in graphs

Abstract: It is well-known that the second smallest eigenvalue? 2 of the difference Laplacian matrix of a graphG is related to the expansion properties ofG. A more detailed analysis of this relation is given. Upper and lower bounds on the diameter and the mean distance inG in terms of? 2 are derived.

Bounds on the Number of Hidden Neurons in

Abstract: Absrract-This paper investigates some fundamental issues concerning the capability of multilayer perceptrons with one hidden layer. The studies are focused on realizations of functions which map from a finite subset of E” into E“. Both real-valued and binary-valued functions are considered. In particular, a least upper bound is derived for the number of hidden neurons needed to realize an arbitrary function which maps from a finite subset of E“ into E“. A nontrivial lower bound is also obtained for realizations of injective functions. This result will be useful in studying pattern recognition and database retrieval. In addition, an upper hound is given for realizing binary-valued functions that are related to pattern classification problems.

Finding Optimal (s, S) Policies Is About As Simple As Evaluating a Single Policy

Abstract: In this paper, a new algorithm for computing optimal (s, S) policies is derived based upon a number of new properties of the infinite horizon cost function c(s, S) as well as a new upper bound for optimal order-up-to levels S* and a new lower bound for optimal reorder levels s*. The algorithm is simple and easy to understand. Its computational complexity is only 2.4 times that required to evaluate a (specific) single (s, S) policy. The algorithm applies to both periodic review and continuous review inventory systems.

Bounds on the number of hidden neurons in multilayer perceptrons

Abstract: Fundamental issues concerning the capability of multilayer perceptrons with one hidden layer are investigated. The studies are focused on realizations of functions which map from a finite subset of E/sup n/ into E/sup d/. Real-valued and binary-valued functions are considered. In particular, a least upper bound is derived for the number of hidden neurons needed to realize an arbitrary function which maps from a finite subset of E/sup n/ into E/sup d/. A nontrivial lower bound is also obtained for realizations of injective functions. This result can be applied in studies of pattern recognition and database retrieval. An upper bound is given for realizing binary-valued functions that are related to pattern-classification problems. >

An adaptive controller which provides an arbitrarily good transient and steady-state response

Abstract: A model reference adaptive control problem is posed In the problem, the objective is not the usual one of forcing the error between the plant output and the reference model output asymptotically to zero, but instead, it is that of forcing this error to be less than a (arbitrarily small) prespecified constant after a (arbitrarily short) prespecified period of time, with a (arbitrarily small) prespecified upper bound on the amount of overshoot It is shown that to achieve this goal for a stabilizable and detectable, single-input single-output linear time-invariant (LTI) plant, it is necessary and sufficient that the plant be minimum phase Knowledge of an upper bound on the plant order, of the relative degree, and of the sign of the high-frequency gain is not required The controller proposed consists of an LTI compensator together with a switching mechanism to adjust the compensator parameters If an upper bound on the relative degree is available, the compensator has dynamics of order equal to this upper bound less one; otherwise, the order of the compensator is adjusted as well as its parameters >

The Cramer-Rao lower bound for signals with constant amplitude and polynomial phase

Abstract: The authors derive the Cramer-Rao lower bound (CRLB) for complex signals with constant amplitude and polynomial phase, measured in additive Gaussian white noise. The exact bound requires numerical inversion of an ill-conditioned matrix, while its O(N/sup -1/) approximation is free of matrix inversion. The approximation is tested for several typical parameter values and is found to be excellent in most cases. The formulas derived are of practical value in several radar applications, such as electronic intelligence systems (ELINT) for special pulse-compression radars, and motion estimation from Doppler measurements. Consequently, it is of interest to analyze the best possible performance of potential estimators of the phase coefficients, as a function of signal parameters, the signal-to-noise ratio, the sampling rate, and the number of measurements. This analysis is carried out. >

Lower bounds for non-commutative computation

Abstract: JVe consider algebraic computations which are not allowed to rely on the commut,at,ivity of multiplication. We obtain various lower bounds for algebraic formula size in this model: (1) Computing the determinant is as hard as computing the permanent and tight exponential upper and lower bounds are given. (2) Computation cannot be parallelized, as opposed to in the commutative case – this solves in the negative an open problem of hliller et al [8]. (3) The question of the power of negation in this model is shown to be closely related to a well known open problem relating communication complexity and rank. We then take modest steps towards extending our results to general, commutative algebraic computation, and prove exponential lower bounds for monotone algebraic circuit size, as well as for the size of certain types of constant depth algebraic circuits.

On the exponent of the all pairs shortest path problem

Abstract: The upper bound on the exponent,?, of matrix multiplication over a ring that was three in 1968 has decreased several times and since 1986 it has been 2.376. On the other hand, the exponent of the algorithms known for the all pairs shortest path problem has stayed at three all these years even for the very special case of directed graphs with uniform edge lengths. In this paper we give an algorithm of timeO(n?log3n),?=(3+?)/2, for the case of edge lengths in {?1, 0, 1}. Thus, for the current known bound on?, we get a bound on the exponent,?<2.688. In case of integer edge lengths with absolute value bounded above byM, the time bound isO((Mn)?log3n) and the exponent is less than 3 forM=O(n?), for?<0.116 and the current bound on?.

The Pointwise Stationary Approximation for Queues with Nonstationary Arrivals

Abstract: We empirically explore the accuracy of an easily computed approximation for long run, average performance measures such as expected delay and probability of delay in multiserver queueing systems with exponential service times and periodic sinusoidal Poisson arrival processes. The pointwise stationary approximation is computed by integrating over time that is taking the expectation of the formula for the stationary performance measure with the arrival rate that applies at each point in time. This approximation, which has been empirically confirmed as a tight upper bound of the true value, is shown to be very accurate for a range of parameter values corresponding to a reasonably broad spectrum of real systems.

Linear FM signal parameter estimation from discrete-time observations

Abstract: The authors consider the problem of estimating the parameters of a complex linear FM signal from a finite number of noisy discrete-time observations An estimation algorithm is proposed, and its asymptotic (large sample) performance is analyzed The algorithm is computationally simple, consisting of two fast Fourier transforms (FFTs) accompanied by one-dimensional searches for maxima The variance of the estimates is shown to be close to the Cramer-Rao lower bound when the signal-to-noise ratio is 0 dB and above The authors applied the algorithm to the problem of estimating the kinematic parameters of an accelerating target by pulse-Doppler radar A representative test case was used to exhibit the usefulness of the algorithm for this problem, and to verify the analytical results by Monte Carlo simulations >

A collection of robust control problems leading to LMIs

Abstract: The authors consider four control-related problems, all of which involve reformulation into linear matrix inequalities (LMIs). The problems are: structured singular value ( mu ) upper bound synthesis for constant matrix problems; robust-state-feedback problem with quadratic stability criteria for uncertain systems; optimal, constant, block diagonal, similarity scaling for full information and state feedback H/sub infinity / problem; and a partial theory for optimal performance in systems which depend on several independent variables. >

Local expansion of vertex-transitive graphs and random generation in finite groups

Abstract: Heuristic algorithms manipulating finite groups often work under the assumption that certain operations lead to “random” elements of the group. While polynomial time methods to construct uniform random elements of permutation groups have been known for over two decades, no such methods have previously been known for more general cases such as matrix groups over finite fields. We present a Monte Carlo algorithm which constructs an efficient nearly uniform random generator for finite groups G in a very general setting. The algorithm presumes a priori knowledge of an upper bound n on log |G|. The random generator is constructed and works in time, polynomial in this upper bound n. The process admits high degree of parallelization: after a preprocessing of length O(n logn) with O(n) processors, the construction of each random element costs O(logn) time with O(n) processors. We use the computational model of “black box groups”: group elements are encoded as (0, 1)-strings of uniform length; and an oracle performs group operations at unit cost. The group G is given by a list of generators. The random generator will produce each group element with probability (1/|G|)(1 ± ) where can be prescribed to be an arbitrary exponentially small function of n. ∗Research supported in part by NSF Grant CCR-8710078.

Lower bounds of rates of decay for solutions to the Navier-Stokes equations

Abstract: where a(p) = 3(2/p 1) and the constant C depends on the L2 and Lnorms of the initial data u0. This paper deals with the more subtle problem of deriving lower bounds on the energy decay rates. We show that for a certain class of initial data the solutions u(x, t) to the 2D and 3D Navier-Stokes equations admit an algebraic lower bound on the energy decay. Specifically, there are two cases to consider. In the first case, the average of the initial data f u0 dx is nonzero. This case was treated in the earlier paper [5] where it was established that

Branch and bound algorithm for computing the minimum stability degree of parameter‐dependent linear systems

Abstract: We consider linear systems with unspecified parameters that lie between given upper and lower bounds. Except for a few special cases, the computation of many quantities of interest for such systems can be performed only through an exhaustive search in parameter space. We present a general branch and bound algorithm that implements this search in a systematic manner and apply it to computing the minimum stability degree.

Edge-isoperimetric inequalities in the grid

Abstract: The grid graph is the graph on [k] n ={0,...,k−1} n in whichx=(x i ) 1 is joined toy=(y i ) 1 if for somei we have |x i −y i |=1 andx j =y j for allj≠i. In this paper we give a lower bound for the number of edges between a subset of [k] n of given cardinality and its complement. The bound we obtain is essentially best possible. In particular, we show that ifA⊂[k] n satisfiesk n /4≤|A|≤3k n /4 then there are at leastk n−1 edges betweenA and its complement. Our result is apparently the first example of an isoperimetric inequality for which the extremal sets do not form a nested family. We also give a best possible upper bound for the number of edges spanned by a subset of [k] n of given cardinality. In particular, forr=1,...,k we show that ifA⊂[k] n satisfies |A|≤r n then the subgraph of [k] n induced byA has average degree at most 2n(1−1/r).

Parametric characterization of general partially coherent beams propagating through ABCD optical systems

Abstract: Within the formalism of the Wigner distribution function, a new parameter is proposed, which characterizes arbitrary tridimensional partially coherent beams and is invariant through ABCD optical systems. The relationship between such a parameter and the bidimensional concept of beam quality is analyzed. An absolute lower bound that the new parameter can reach is also shown.

Information rates for a discrete-time Gaussian channel with intersymbol interference and stationary inputs

Abstract: Bounds are presented on I/sub i.i.d./-the achievable information rate for a discrete Gaussian Channel with intersymbol interference (ISI) present and i.i.d. channel input symbols governed by an arbitrary predetermined distribution p/sub x/(x). The lower and upper bounds on I/sub i.i.d./ and I are formulated. The bounds on I/sub i.i.d./ are calculated for independent equiprobably binary channel symbols and for causal channels with ISI memory of degree one and two. The bounds on I/sub i.i.d./ are compared to the approximated (by Monte Carlo methods) known value of I/sub i.i.d./ and their tightness is considered. An application of the new lower bound on I/sub i.i.d./ yields an improvement on previously reported lower bounds for the capacity of the continuous-time strictly bandlimited (or bandpass) Gaussian channel with either peak or simultaneously peak power and bandlimiting constraints imposed on the channel's input waveform. >

Bounds on least dilatations

Abstract: We consider the collection of all pseudo-Anosov homeomorphisms on a surface of fixed topological type. To each such homeomorphism is associated a real-valued invariant, called its dilatation (which is greater than one), and we define the spectrum of the surface to be the collection of logarithms of dilatations of pseudo-Anosov maps supported on the surface. The spectrum is a natural object of study from the topological, geometric, and dynamical points of view. We are concerned in this paper with the least element of the spectrum, and explicit upper and lower bounds on this least element are derived in terms of the topological type of the surface; train tracks are the main tool used in establishing our estimates.

Space and time bounds on indexing 3D models from 2D images

Abstract: Model-based visual recognition systems often match groups of image features to groups of model features to form initial hypotheses, which are then verified. In order to accelerate recognition considerably, the model groups can be arranged in an index space (hashed) offline such that feasible matches are found by indexing into this space. For the case of 2D images and 3D models consisting of point features, bounds on the space required for indexing and on the speedup that such indexing can achieve are demonstrated. It is proved that, even in the absence of image error, each model must be represented by a 2D surface in the index space. This places an unexpected lower bound on the space required to implement indexing and proves that no quantity is invariant for all projections of a model into the image. Theoretical bounds on the speedup achieved by indexing in the presence of image error are also determined, and an implementation of indexing for measuring this speedup empirically is presented. It is found that indexing can produce only a minimal speedup on its own. However, when accompanied by a grouping operation, indexing can provide significant speedups that grow exponentially with the number of features in the groups. >

Lower Bounds for Deterministic and Nondeterministic Branching Programs

Abstract: We survey lower bounds established for the complexity of computing explicitly given Boolean functions by switching-and-rectifier networks, branching programs and switching networks. We first consider the unrestricted case and then proceed to various restricted models. Among these are monotone networks, bounded-width devices, oblivious devices and read-k times only devices.

A Lower Bound on Probabilistic Algorithms for Distributive Ring Coloring

Abstract: Suppose that n processors are arranged in a ring and can communicate only with their immediate neighbors. It is shown that any probabilistic algorithm for 3 coloring the ring must take at least $\frac{1}{2}\log^* n - 2$ rounds, otherwise the probability that all processors are colored legally is less than $\frac{1}{2}$. A similar time bound holds for selecting a maximal independent set. The bound is tight (up to a constant factor) in light of the deterministic algorithms of Cole and Vishkin [Inform, and Control, 70 (1986), pp. 32–53] and extends the lower bound for deterministic algorithms of Linial [Proc. 28th IEEE Foundations of Computer Science Symposium, 1987, pp. 331–335].

An integral representation and bounds on the effective diffusivity in passive advection by laminar and turbulent flows

Abstract: Precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection-diffusion by a steady incompressible velocity field are developed here. Under these conditions, a rigorous Stieltjes integral representation for effective diffusivity in turbulent transport is derived. This representation is valid for all Peclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the Peclet number. One consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all Peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series. Explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all Peclet numbers. A nonlocal variational principle for effective diffusivity is developed along with applications to advection-diffusion by random arrays of vortices. A new class of rigorous examples is introduced. These examples have an explicit Stieltjes measure for the effective diffusivity; furthermore, the effective diffusivity behaves likek0(Pe)1/2 in the limit of large Peclet numbers wherek0 is the molecular diffusivity. Formal analogies with the theory of composite materials are exploited systematically.

A Lagrangian based approach for the asymmetric Generalized Traveling Salesman Problem

Abstract: This paper presents an optimal approach for the asymmetric Generalized Traveling Salesman Problem (GTSP). The GTSP is defined on a directed graph in which the nodes are grouped into m predefined, mutually exclusive and exhaustive sets with the arc set containing no intraset arcs. The problem is to find a minimum cost m-arc directed cycle which includes exactly one node from each set. Our approach employs a Lagrangian relaxation to compute a lower bound on the total cost of an optimal solution. The lower bound and a heuristically determined upper bound are used to identify and remove arcs and nodes which are guaranteed not to be in an optimal solution. Finally, we use an efficient branch-and-bound procedure which exploits the multiple choice structure of the node sets. We present computational results for the optimal approach tested on a series of randomly generated problems. The results show success on a range of problems with up to 104 nodes.

New upper bounds in Klee's measure problem

Abstract: New upper bounds for the measure problem of Klee are given which significantly improve the previous bounds for dimensions greater than two. An $O(n^{d / 2} \log n,n)$ time-space upper bound is obtained and used to compute the measure of a set of n boxes in Euclidean d-space. The solution is based on a new data structure, which is called an orthogonal partition tree. This structure has other applications as well.

The expressive power of voting polynomials

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.

On the conductance of order Markov chains

Abstract: Let Q be a convex solid in ℝn, partitioned into two volumes u and v by an area s. We show that s>min(u,v)/diam Q, and use this inequality to obtain the lower bound n-5/2 on the conductance of order Markov chains, which describe nearly uniform generators of linear extensions for posets of size n. We also discuss an application of the above results to the problem of sorting of posets.