Unbounded Weights Research Articles

Markov decision processes with iterated coherent risk measures

This paper considers a Markov decision process in Borel state and action spaces with the aggregated (or say iterated) coherent risk measure to be minimised. For this problem, we establish the Bellman optimality equation as well as the value and policy iteration algorithms, and show the existence of a deterministic stationary optimal policy. The cost function, while being allowed to be unbounded from below (in the sense that its negative part needs be bounded by some nonnegative real-valued possibly unbounded weight function), can be arbitrarily unbounded from above and possibly infinitely valued.

Numerical Exact Controllability of the 1D Heat Equation: Duality and Carleman Weights

This article is devoted to the numerical computation of distributed null controls for the 1D heat equation. The goal is to compute a control that drives (a numerical approximation of) the solution from a prescribed initial state exactly to zero. We extend the earlier contribution of Carthel, Glowinski, and Lions, which is devoted to the computation of controls of minimal square-integrable norm. We start from some constrained extremal problems (involving unbounded weights in time), introduced by Fursikov and Imanuvilov, and we apply appropriate duality techniques. Then, we provide numerical approximations of the associated dual problems, and apply conjugate gradient algorithms. Finally, several experiments are presented, and we highlight the influence of the weights and analyze this approach in terms of robustness and efficiency. Also, the results are compared with others in a previous article of the authors, where primal methods were considered.

Product-Units neural networks for catchment runoff forecasting

In this paper Product-Units neural networks (PUNNs), which probably have never been used within the field of hydrology, are introduced and applied for catchment runoff forecasting in cold climate zones. This type of neural networks, a subclass of higher order neural networks uses product nodes with inputs raised to exponential weights in one layer and well-known summation nodes in another layer. The present paper empirically shows that PUNNs with unbounded weights are difficult to train and do not perform well for catchment runoff forecasting. However, a very good predictive performance may be achieved when the weights are bounded within [−1,1] interval. Several variants of optimization methods, mostly Differential Evolution-based algorithms, and a few approaches enabling good generalization capabilities of neural networks are compared in order to select the appropriate technique for PUNNs training. PUNNs with parameters bounded within [−1,1] interval are shown to outperform Multi-Layer Perceptron neural networks and HBV conceptual model for runoff forecasting case study at Annapolis River, Nova Scotia, Canada. Gradient-based Levenberg–Marquardt algorithm and Evolutionary Computation-based Differential Evolution with Global and Local Neighborhood method turn out to be the most successful among the tested training algorithms. Surprisingly, in the case of Product-Units neural networks with weights bounded within [−1,1] interval using noise injection or early stopping do not improve the results obtained when no method to avoid overfitting is used.

The relationship between DEA efficiency and the type of production function, the degree of homogeneity, and error variability

In this paper, we use simulations to investigate the relationship between data envelopment analysis (DEA) efficiency and major production functions: Cobb-Douglas, the constant elasticity of substitution, and the transcendental logarithmic. Two DEA models were used: a constant return to scale (CCR model), and a variable return to scale (BCC model). Each of the models was investigated in two versions: with bounded and unbounded weights. Two cases were simulated: with and without errors in the production functions estimation. Various degrees of homogeneity (of the production function) were tested, reflecting a constant increasing and decreasing return to scale. With respect to the case with errors, three distribution functions were utilized: uniform, normal, and double exponential. For each distribution, 16 levels of the coefficient of variance (CV) were used. In all the tested cases, two measures were analysed: the percentage of efficient units (from the total number of units), and the average efficiency score. We applied a regression analysis to test the relationship between these two efficiency measures and the above parameters. Overall, we found that the degree of homogeneity has the largest effect on efficiency. Efficiency declines as the errors grow (as reflected by larger CV and of the expansion of the probability distribution function away from the centre). The bounds on the weights tend to smooth the effect, and bring the various DEA versions closer to one other. The type of efficiency measure has similar regression tendencies. Finally, the relationship between the efficiency measures and the explanatory variables is quadratic.

계층적 X-means와 가중 F-measure를 통한 시뮬레이션 모델 검증 기법

기존 대부분의 연구에서 사용하고 있는 시뮬레이션 검증 기법은 통계적 분석기법으로, 총 처리량이나 자원 이용률의 평균 및 분산을 통해 분석하여 왔다. 그러나 이러한 방식은 모델의 개별적인 요소들에 대한 신뢰성을 보장하기 어려웠다. 이를 해결하기 위해 제시된 방법이 가중 F-measure를 사용한 검증이다. 하지만 가중 F-measure는 Tact time 값 하나에 대해 하나의 클래스를 할당하기 때문에 수많은 Tact time 값들을 갖는 복잡한 시스템에 적용하기 어려운 문제를 가지고 있다. 한편, 가중치의 범위가 정해져 있지 않기 때문에 평가기준(Threshold)의 선정에 있어서 어느 정도의 수준이 만족할만한 수준인지 정하기가 어려웠다. 따라서 본 논문에서는 이러한 문제점을 개선하기 위해 군집분석을 적용한 가중 F-measure를 제시한다. 군집의 클래스화를 통해 클래스의 수를 현저히 줄일 수 있고 다양한 시스템으로의 적용 또한 가능해진다. 또한 객관성을 저하시키지 않는 범위 내에서 최소한의 가중치를 부여하는 방식으로 가중치의 범위를 지정하여 검증 방법을 향상시켰다. 이를 입증하기 위해 국내 'L사'의 LCD공정설비를 대상으로 시뮬레이션 모델링 및 환경을 구축하였고, 그 결과를 통해 타당성을 증명하였다. Simulation validation techniques which have been employed in most studies are statistical analysis, which validate a model with mean or variance of throughput and resource utilization as an evaluation object. However, these methods have not been able to ensure the reliability of individual elements of the model well. To overcome the problem, the weighted F-measure method was proposed, but this technique also had some limitations. First, it is difficult to apply the technique to complex system environment with numerous values of interarrival time because it assigns a class to an individual value of interarrival time. In addition, due to unbounded weights, the value of weighted F-measure has no lower bound, so it is difficult to determine its threshold. Therefore, this paper propose weighted F-measure technique with cluster analysis to solve these problems. The classes for the technique are defined by each cluster, which reduces considerable number of classes and enables to apply the technique to various systems. Moreover, we improved the validation technique in the way of assigning minimum bounded weights without any lack of objectivity.

Quadrature formula for approximating the singular integral of Cauchy type with unbounded weight function on the edges

New quadrature formulas (QFs) for evaluating the singular integral (SI) of Cauchy type with unbounded weight function on the edges is constructed. The construction of the QFs is based on the modification of discrete vortices method (MMDV) and linear spline interpolation over the finite interval [ − 1 , 1 ] . It is proved that the constructed QFs converge for any singular point x not coinciding with the end points of the interval [ − 1 , 1 ] . Numerical results are given to validate the accuracy of the QFs. The error bounds are found to be of order O ( h α | ln h | ) and O ( h | ln h | ) in the classes of functions H α ( [ − 1 , 1 ] ) and C 1 ( [ − 1 , 1 ] ) , respectively.

High‐Water Marks: High Risk Appetites? Convex Compensation, Long Horizons, and Portfolio Choice

ABSTRACTWe study the portfolio choice of hedge fund managers who are compensated by high‐water mark contracts. We find that even risk‐neutral managers do not place unbounded weights on risky assets, despite option‐like contracts. Instead, they place a constant fraction of funds in a mean‐variance efficient portfolio and the rest in the riskless asset, acting as would constant relative risk aversion (CRRA) investors. This result is a direct consequence of the in(de)finite horizon of the contract. We show that the risk‐seeking incentives of option‐like contracts rely on combining finite horizons and convex compensation schemes rather than on convexity alone.

On non-effective weights in Orlicz spaces

Given a weight w in Ω ⊂ ∝ N, |Ω| < ∞ and a Young function φ, we consider the weighted modular ∫ Ω ω(f(x))w(x)dx and the resulting weighted Orlicz space L ω(w). For a Young function Ω ∉ Δ 2(∞) we present a necessary and sufficient conditions in order that L ω(w) = L ω(X Ω) up to the equivalence of norms. We find a necessary and sufficient condition for ω in order that there exists an unbounded weight w such that the above equality of spaces holds. By way of applications we simplify criteria from [5] for continuity of the composition operator from L ω into itself when ω Δ 2(∞) and obtain necessary and sufficient condition in order that the composition operator maps L ω. continuously onto L ω.

Polynomial approximations of continuous functions with unbounded sign-sensitive weight

The conditions of arbitrarily exact polynomial approximation of functions continuous on a closed interval with respect to an unbounded sign-sensitive weight are obtained by using divided differences with multiple nodes.

Existence of minimizer of some functionals involving Hardy-type inequalities

We study a class of $p$-laplacian-type problems with various unbounded weights and a forcing term on open subsets of $\mathbb R^N$ or on the positive real axis. To prove the existence of solution, we use variational methods involving concentration-compactness technique and Hardy-type inequalities.

On the computational power of Boolean decision lists

We study the computational power of decision lists over AND-functions versus $$ \hbox{threshold-}\oplus $$ circuits. AND-decision lists are a natural generalization of formulas in disjunctive or conjunctive normal form. We show that, in contrast to CNF- and DNF-formulas, there are functions with small AND-decision lists which need exponential size unbounded weight $$\hbox{threshold-}\oplus $$ circuits. Consequently, it is questionable if the polynomial learning algorithm for DNFs of Jackson (1994), which is based on the efficient simulation of DNFs by polynomial weight $$ \hbox{threshold-}\oplus $$ circuits (Krause & Pudlak 1994), can be successfully applied to functions with small AND-decision lists. A further result is that for all k ? 1 the complexity class defined by polynomial length AC0k-decision lists lies strictly between AC0k+1 and AC0k+2.

Learning Only When Necessary: Better Memories of Correlated Patterns in Networks with Bounded Synapses

Learning in a neuronal network is often thought of as a linear superposition of synaptic modifications induced by individual stimuli. However, since biological synapses are naturally bounded, a linear superposition would cause fast forgetting of previously acquired memories. Here we show that this forgetting can be avoided by introducing additional constraints on the synaptic and neural dynamics. We consider Hebbian plasticity of excitatory synapses. A synapse is modified only if the postsynaptic response does not match the desired output. With this learning rule, the original memory performances with unbounded weights are regained, provided that (1) there is some global inhibition, (2) the learning rate is small, and (3) the neurons can discriminate small differences in the total synaptic input (e.g., by making the neuronal threshold small compared to the total postsynaptic input). We prove in the form of a generalized perceptron convergence theorem that under these constraints, a neuron learns to classify any linearly separable set of patterns, including a wide class of highly correlated random patterns. During the learning process, excitation becomes roughly balanced by inhibition, and the neuron classifies the patterns on the basis of small differences around this balance. The fact that synapses saturate has the additional benefit that nonlinearly separable patterns, such as similar patterns with contradicting outputs, eventually generate a subthreshold response, and therefore silence neurons that cannot provide any information.

Threshold circuit lower bounds on cryptographic functions

In this work, we are interested in non-trivial upper bounds on the spectral norm of binary matrices M from { - 1 , 1 } N × N . It is known that the distributed Boolean function represented by M is hard to compute in various restricted models of computation if the spectral norm is bounded from above by N 1 - ɛ , where ɛ > 0 denotes a fixed constant. For instance, the size of a two-layer threshold circuit (with polynomially bounded weights for the gates in the hidden layer, but unbounded weights for the output gate) grows exponentially fast with n ≔ log N . We prove sufficient conditions on M that imply small spectral norms (and thus high computational complexity in restricted models). Our general results cover specific cases, where the matrix M represents a bit (the least significant bit or other fixed bits) of fundamental functions. Functions like the discrete multiplication and division, as well as cryptographic functions such as the Diffie–Hellman function (IEEE Trans. Inform. Theory 22(6) (1976) 644–654) and the decryption functions of the Pointcheval (Advances in Cryptology—Proceedings of EUROCRYPT ’99, Lecture Notes in Computer Science, Springer, Berlin, 1999, pp. 239–254) and the El Gamal (Advances in Cryptology—CRYPTO ’84, 1984, pp. 10–18) cryptosystems can be addressed by our technique. In order to obtain our results, we make a detour on exponential sums and on spectral norms of matrices with complex entries. This method might be considered interesting in its own right.

Positive solutions for sublinear periodic parabolic problems

Let Ω be a bounded domain in R N . We characterize the set of positive principal eigenvalues for the Dirichlet periodic parabolic problem Lu= λg( x, t, u) in Ω× R , under the assumptions that g is a function such that ξ→ g( x, t, ξ)/ ξ is continuously differentiable and nonincreasing in [0,∞) a.e. (x,t)∈Ω× R satisfying some integrability and positivity conditions. We also prove the uniqueness of the positive solution. As a consequence we obtain a necessary and sufficient condition for the existence of a (unique) positive solution for the periodic parabolic logistic equation with unbounded weights. Existence of positive solutions for a generalization of the logistic equation is also shown.

Computing Boolean functions by polynomials and threshold circuits

We investigate the computational power of threshold—AND circuits versus threshold—XOR circuits. In contrast to the observation that small weight threshold—AND circuits can be simulated by small weight threshold—XOR circuit, we present a function with small size unbounded weight threshold—AND circuits for which all threshold—XOR circuits have exponentially many nodes. This answers the basic question of separating subsets of the hypercube by hypersurfaces induced by sparse real polynomials. We prove our main result by a new lower bound argument for threshold circuits. Finally we show that unbounded weight threshold gates cannot simulate alternation: There are \( AC^{0,3} \)-functions which need exponential size threshold—AND circuits.

Almost linear VC-dimension bounds for piecewise polynomial networks

We compute upper and lower bounds on the VC dimension and pseudo-dimension of feedforward neural networks composed of piecewise polynomial activation functions. We show that if the number of layers is fixed, then the VC dimension and pseudo-dimension grow as WlogW, where W is the number of parameters in the network. This result stands in opposition to the case where the number of layers is unbounded, in which case the VC dimension and pseudo-dimension grow as W2. We combine our results with recently established approximation error rates and determine error bounds for the problem of regression estimation by piecewise polynomial networks with unbounded weights.

Singular Inner Functions and Biinvariant Subspaces for Dissymetric Weighted Shifts

We consider dissymetric weights on Z , i.e., nonincreasing unbounded weightsω:Z→[1, ∞) such thatω(n)=1 forn⩾0 and such that limn→−∞ω(n)1/|n|=1 whereω(n)=supp∈Zω(n+p)/ω(p). A dissymetric weight is said to be quasianalytic if ∑n<0logω(n)/n2=+∞. Denote byΓthe unit circle and byDthe open unit disc. SetL2ω(Γ)={f∈L2(Γ)∣‖f‖ω=(∑n∈Z|f(n)|2·ω2(n))1/2<+∞}.We identify the usual Hardy spaceH2(D) to the set {f∈L2ω(Γ)∣f(n)=0 (n<0)} and we denote bySω:f(eit)→eit·f(eit) the usual shift onL2ω(Γ). IfUis a singular inner function denote byEUthe closure of Span(Snω·U)n∈ZinL2ω(Γ). We show that ifωis a dissymetric weight such thatlogω(−n)n→n→∞∞thenEUis a proper subspace ofL2ω(Γ) for every singular inner functionU. Also ifωis any dissymetric weight then there exists some singular inner functionUsuch thatEUis a proper subspace ofL2ω, so thatSωalways has nontrivial biinvariant subspaces (no nontrivial biinvariant subspaces ofSωwere known so far in the quasianalytic case).

Moving Weighted Averages

AbstractLet ℝ denote the real line. Let {Tt}tєℝ be a measure preserving ergodic flow on a non atomic finite measure space (X, ℱ, μ). A nonnegative function φ on ℝ is called a weight function if ∫ℝ φ(t)dt = 1. Consider the weighted ergodic averagesof a function f X —&gt; ℝ, where {θk} is a sequence of weight functions. Some sufficient and some necessary and sufficient conditions are given for the a.e. convergence of Akf, in particular for a special case in whichwhere φ is a fixed weight function and {(ak, rk)} is a sequence of pairs of real numbers such that rk &gt; 0 for all k. These conditions are obtained by a combination of the methods of Bellow-Jones-Rosenblatt, developed to deal with moving ergodic averages, and the methods of Broise-Déniel-Derriennic, developed to deal with unbounded weight functions.

Invariance, Minimax Sequential Estimation, and Continuous Time Processes

Invariance, Minimax Sequential Estimation, and Continuous Time Processes