Linear Memories Research Articles

Late time tails and nonlinear memories in asymptotically de Sitter spacetimes

We study the propagation of a massless scalar wave in de Sitter spacetime perturbed by an arbitrary central mass. By focusing on the late time limit, this probes the portion of the scalar signal traveling inside the null cone. Unlike in asymptotically flat spacetimes, the amplitude of the scalar field detected by an observer at timelike infinity does not decay back to zero but develops a spacetime constant shift---both at zeroth and at first order in the central mass $M$. This indicates that massless scalar field propagation in asymptotically de Sitter spacetimes exhibits both linear and nonlinear tail-induced memories. On the other hand, for sufficiently late retarded times, the monopole portion of the scalar signal measured at null infinity is found to be amplified relative to its timelike infinity counterpart, by its nonlinear interactions with the gravitation field at first order in $M$.

Capacity-approaching quantum repeaters for quantum communications

In present-day quantum communications, one of the main problems is the lack of a quantum repeater design that can simultaneously secure high rates and long distances. Recent literature has established the end-to-end capacities that are achievable by the most general protocols for quantum and private communication within a quantum network, encompassing the case of a quantum repeater chain. However, whether or not a physical design exists to approach such capacities remains a challenging objective. Driven by this motivation, in this work, we put forward a design for continuous-variable quantum repeaters and show that it can actually achieve the feat. We also show that even in a noisy regime our rates surpass the Pirandola-Laurenza-Ottaviani-Banchi (PLOB) bound. Our repeater setup is developed upon using noiseless linear amplifiers, quantum memories, and continuous-variable Bell measurements. We, furthermore, propose a non-ideal model for continuous-variable quantum memories that we make use of in our design. We then show that potential quantum communications rates would deviate from the theoretical capacities, as one would expect, if the quantum link is too noisy and/or low-quality quantum memories and amplifiers are employed.

The gravitational-wave memory from eccentric binaries

The nonlinear gravitational-wave memory causes a time-varying but nonoscillatory correction to the gravitational-wave polarizations. It arises from gravitational waves that are sourced by gravitational waves. Previous considerations of the nonlinear memory effect have focused on quasicircular binaries. Here, I consider the nonlinear memory from Newtonian orbits with arbitrary eccentricity. Expressions for the waveform polarizations and spin-weighted spherical-harmonic modes are derived for elliptic, hyperbolic, parabolic, and radial orbits. In the hyperbolic, parabolic, and radial cases the nonlinear memory provides a 2.5 post-Newtonian (PN) correction to the leading-order waveforms. This is in contrast to the elliptical and quasicircular cases, where the nonlinear memory corrects the waveform at leading (0PN) order. This difference in PN order arises from the fact that the memory builds up over a short "scattering" time scale in the hyperbolic case, as opposed to a much longer radiation-reaction time scale in the elliptical case. The nonlinear memory corrections presented here complete our knowledge of the leading-order (Peters-Mathews) waveforms for elliptical orbits. These calculations are also relevant for binaries with quasicircular orbits in the present epoch which had, in the past, large eccentricities. Because the nonlinear memory depends sensitively on the past evolution of a binary, I discuss the effect of this early-time eccentricity on the value of the late-time memory in nearly circularized binaries. I also discuss the observability of large "memory jumps" in a binary's past that could arise from its formation in a capture process. Lastly, I provide estimates of the signal-to-noise ratio of the linear and nonlinear memories from hyperbolic and parabolic binaries.

Scalable Quantum Computing with Linear Optics and Quantum Memories

Scalable Quantum Computing with Linear Optics and Quantum Memories

Scalable Quantum Computing with Linear Optics and Quantum Memories

Quantum computing is a powerful application of the laws of quantum physics. It may one day be far more efficient than conventional computation methods at factoring large integers, performing database searches and making quantum physics simulations.

Morphological Scale Spaces and Associative Morphological Memories: Results on Robustness and Practical Applications

Associative Morphological Memories are the analogous construct to Linear Associative Memories defined on the lattice algebra (/Bbb R, +, ∨, ∧). They have excellent recall properties for noiseless patterns. However they suffer from the sensitivity to specific noise models, that can be characterized as erosive and dilative noise. To improve their robustness to general noise we propose a construction method that is based on the extrema point preservation of the Erosion/Dilation Morphological Scale Spaces. Here we report on their application to the tasks of face localization in grayscale images and appearance based visual self-localization of a mobile robot.

Smooth fitting with a method for determining the regularization parameter under the genetic programming algorithm

Abstract This paper deals with the smooth fitting problem under the genetic programming (GP) algorithm. To reduce the computational cost required for evaluating the fitness value of GP trees, numerical weights of GP trees are estimated by adopting both linear associative memories (LAM) and the Hook and Jeeves (HJ) method. The quality of smooth fitting is critically dependent on the choice of the regularization parameter. So, we present a novel method for choosing the regularization parameter. Two numerical examples are given with the comparison of generalized cross-validation (GCV) B-splines.

The correlation memory matrix for parameter estimation

Dynamic systems described by nonlinear differential or difference equations are abound in the literature. Until recently, estimation of system parameters has been mainly carried out using iterative (search) techniques such as conjugate gradient, steepest descent and evolutionary algorithms. These methods vary in terms of their noise immunity and robustness but they all tend to require a considerable number of iterations in order for the solution to converge, especially in the presence of noise. This is a disadvantage for systems requiring real-time control. In continuation of a series of papers utilizing linear associative memories (LAM) for parameter estimation, this paper proposes a non-iterative single-pass technique for estimating these parameters using the well-known correlation memory matrix (CMM), previously used in pattern recognition. The CMM method is used to estimate the parameters of the Lotka–Volterra predator–prey model and the Van der Pol oscillator. It is shown that this class of linear associators is superior to other types of LAM in that it provides equally good parameter estimates while requiring neither matrix inversion nor extensive training. As nonlinear (quadratic) associations are incorporated into the CMM, the method becomes more robust than the linear version.

Optimal bidirectional associative memories

This paper proposes a model for linear optimal bidirectional associative memories (BAMs), in which the design of the weight matrix is based on a least-squares cost functional. The minimization of the functional leads to a Lyapunov-type matrix equation. The mathematical properties of such an optimal bidirectional system are studied in detail. Simulations are conducted for the bidirectional association of pattern pairs with grey levels. It is shown that this optimal design of BAM is effective for memorizing patterns with grey levels.

Nonlinear Estimation with Associative Memories and Machine Evaluation of Derivatives: An Application to Calibrating Spatial Interaction Models

In this paper we apply the theory of linear associative memories in producing initial parameter estimates for nonlinear iterative approaches. We also propose the use of FEED (Fast and Efficient Evaluation of Derivatives) to evaluate partial derivatives of functions encountered in nonlinear estimation. Suggested methods are presented in the context of calibrating spatial interaction models and are illustrated through numerical examples.

The optimal encodings for biased association in linear associative memories

In this paper, optimal encoding schemes for linear associative memories are derived for biased association under both the white-noise and colored-noise situations. Analysis and simulation results all show that the biased encodings thus derived are optimal and superior to existing models in their performance. Together with the Wee–Kohonen unbiased encoding, the study settles the optimality issue of linear associative memories and enhances their practicalities.

Complete memory structures for approximating nonlinear discrete-time mappings

This paper introduces a general structure that is capable of approximating input-output maps of nonlinear discrete-time systems. The structure is comprised of two stages, a dynamical stage followed by a memoryless nonlinear stage. A theorem is presented which gives a simple necessary and sufficient condition for a large set of structures of this form to be capable of modeling a wide class of nonlinear discrete time systems. In particular, we introduce the concept of a "complete memory". A structure with a complete memory dynamical stage and a sufficiently powerful memoryless stage is shown to be capable of approximating arbitrarily wide class of continuous, causal, time invariant, approximately-finite-memory mappings between discrete-time signal spaces. Furthermore, we show that any bounded-input bounded output, time-invariant, causal memory structure has such an approximation capability if and only if it is a complete memory. Several examples of linear and nonlinear complete memories are presented. The proposed complete memory structure provides a template for designing a wide variety of artificial neural networks for nonlinear spatiotemporal processing.

Linear associative memories with optimal rejection to colored input noise

It is known that many linear associative memories are sensitive to input noise. To reduce such noise sensitivity, an approach of designing an optimal-noise-rejection linear associative matrix is presented in this paper. An optimally biased, as well as unbiased solution, in the sense of least-mean-square of recall error for colored input noise, is given. The theoretical lower bound of error for biased or unbiased recall is derived. A recursive computational procedure for the optimal design is proposed. Computer simulation and image recall experiments illustrate a superior noise rejection of the proposed approach.

Noise performance of linear associative memories

The performance of two commonly used linear models of associative memories, generalized inverse (GI) and correlation matrix memory (CMM) is studied analytically in the presence of a new type of noise (training noise due to noisy training patterns). Theoretical expressions are determined for the S/N ratio gain of the GI and CMM memories in the auto-associative and hetero-associative modes of operation. It is found that the GI method performance degrades significantly in the presence of training noise while the CMM method is relatively unaffected by it. The theoretical expressions are plotted and compared with the results obtained from Monte Carlo simulations and the two are found to be in excellent agreement. >

Nonlinear parameter estimation by linear association: Application to a five-parameter passive neuron model

Linear associative memories (LAM) have been intensely used in the areas of pattern recognition and parallel processing for the past two decades. Application of LAM to nonlinear parameter estimation, however, has only been recently attempted. The process consists in converting the nonlinear function in the parameters into a set of linear algebraic equations. The nature of the linearized system and the factors influencing the accuracy of the parameter estimates have not yet been fully investigated. In this paper, LAM is applied to a nonlinear five-parameter model of the neuron. Ill-conditioning, which is often exhibited in LAM, is treated with the method of regularization as well as by the singular value decomposition (SVD). Simulation results indicate that the parameters estimated by LAM exhibit a remarkable robustness against additive white noise in comparison with the classical gradient optimization technique. Moreover, it is shown that regularization can be superior to SVD under certain conditions. Our results suggest that LAM can be used both as a noise reduction technique and as a stand-alone nonlinear parameter estimation algorithm. The comparison between LAM and a gradient technique show that, for this estimation problem, the LAM method can give more reliable estimates. Further improvements in estimation quality may still be achieved by the use of other forms of regularizing functions.

Constraint satisfaction neural networks for image recognition

Constraint satisfaction neural networks (CSNNs) are proposed for image recognition which is the process to identify objects of interest in an image. The image recognition problem is formulated as a constraint satisfaction problem (CSP). The regions resulting from a signal-based image segmentation algorithm can be considered as objects and their entities as labels in the CSP. The proposed neural network is multi-layered and the function of each layer is to identify one physical entity (label) from the input regions (objects). Through cascading these layers, each label can be assigned to the corresponding region(s). Attention is focused on recognizing X-ray CT (computed tomography) images of human heads (e.g. anatomic structure in a medical image). The proposed CSNN is capable of identifying the components of a brain in a preliminarily segmented image. The major components in a human head include bone, ventricle and gyrus. Therefore, three layers are needed for this task. Each layer is constructed based on the principles proposed in Kohonen's self-organizing feature maps and the optimal linear associative memories and their weights are trained by the features of each entity in the prestored model. The experimental results show that the proposed approach is very promising.

Further noise rejection in linear associative memories

It is well known that linear associative memories are sensitive to input noise. In this paper, a new noise rejection method for linear associative memories is proposed. It is shown that with a little cost in recall bias the model is much more resistant to noise than previous models, especially as the number of stored vector pairs approaches the number of the key vector components. Using the singular value decomposition (SVD) of matrices, parameter estimations in white or colour noise environments are discussed.

Optimising synaptic learning rules in linear associative memories

Associative matrix memories with real-valued synapses have been studied in many incarnations. We consider how the signal/noise ratio for associations depends on the form of the learning rule, and we show that a covariance rule is optimal. Two other rules, which have been suggested in the neurobiology literature, are asymptotically optimal in the limit of sparse coding. The results appear to contradict a line of reasoning particularly prevalent in the physics community. It turns out that the apparent conflict is due to the adoption of different underlying models. Ironically, they perform identically at their co-incident optima. We give details of the mathematical results, and discuss some other possible derivations and definitions of the signal/noise ratio.

On recursive calculation of the generalized inverse of a matrix

The generalized inverse of a matrix is an extension of the ordinary square matrix inverse which applies to any matrix (e.g., singular, rectangular). The generalized inverse has numerous important applications such as regression analysis, filtering, optimization and, more recently, linear associative memories. In this latter application known as Distributed Associative Memory, stimulus vectors are associated with response vectors and the result of many associations is spread over the entire memory matrix, which is calculated as the generalized inverse. Addition/deletion of new associations requires recalculation of the generalized inverse, which becomes computationally costly for large systems. A better solution is to calculate the generalized inverse recursively. The proposed algorithm is a modification of the well known algorithm due to Rust et al. [2], originally introduced for nonrecursive computation. We compare our algorithm with Greville's recursive algorithm and conclude that our algorithm provides better numerical stability at the expense of little extra computation time and additional storage.

Linear programming and simple associative memories

Linear programming and simple associative memories