Online Discrimination of Nonlinear Dynamics with Switching Differential Equations
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A Relativistic Extension of Hopfield Neural Networks Via the Mechanical Analogy
A relativistic extension of Hopfield neural networks via the mechanical analogy Adriano Barra,a;b;c Matteo Beccariaa;b and Alberto Fachechia;b aDipartimento di Matematica e Fisica Ennio De Giorgi, Universit`adel Salento, Lecce, Italy bINFN, Istituto Nazionale di Fisica Nucleare, Sezione di Lecce, Italy cGNFM-INdAM, Gruppo Nazionale per la Fisica Matematica, Sezione di Lecce, Italy E-mail: [email protected], [email protected], [email protected] Abstract: We propose a modification of the cost function of the Hopfield model whose salient features shine in its Taylor expansion and result in more than pairwise interactions with alternate signs, suggesting a unified framework for handling both with deep learning and network pruning. In our analysis, we heavily rely on the Hamilton-Jacobi correspon- dence relating the statistical model with a mechanical system. In this picture, our model is nothing but the relativistic extension of the original Hopfield model (whose cost function is a quadratic form in the Mattis magnetization which mimics the non-relativistic Hamiltonian for a free particle). We focus on the low-storage regime and solve the model analytically by taking advantage of the mechanical analogy, thus obtaining a complete characterization of the free energy and the associated self-consistency equations in the thermodynamic limit. On the numerical side, we test the performances of our proposal with MC simulations, showing that the stability of spurious states (limiting the capabilities of the standard Heb- bian -
Optimization by Mean Field Annealing
91 OPTIMIZATION BY MEAN FIELD ANNEALING Griff Bilbro Reinhold Mann Thomas K. Miller ECE Dept. Eng. Physics and Math. Div. ECE Dept. NCSU Oak Ridge N atl. Lab. NCSU Raleigh, NC 27695 Oak Ridge, TN 37831 Raleigh, N C 27695 Wesley. E. Snyder David E. Van den Bout Mark White ECE Dept. ECE Dept. ECE Dept. NCSU NCSU NCSU Raleigh, NC 27695 Raleigh, NC 27695 Raleigh, NC 27695 ABSTRACT Nearly optimal solutions to many combinatorial problems can be found using stochastic simulated annealing. This paper extends the concept of simulated annealing from its original formulation as a Markov process to a new formulation based on mean field theory. Mean field annealing essentially replaces the discrete de grees of freedom in simulated annealing with their average values as computed by the mean field approximation. The net result is that equilibrium at a given temperature is achieved 1-2 orders of magnitude faster than with simulated annealing. A general frame work for the mean field annealing algorithm is derived, and its re lationship to Hopfield networks is shown. The behavior of MFA is examined both analytically and experimentally for a generic combi natorial optimization problem: graph bipartitioning. This analysis indicates the presence of critical temperatures which could be im portant in improving the performance of neural networks. STOCHASTIC VERSUS MEAN FIELD In combinatorial optimization problems, an objective function or Hamiltonian, H(s), is presented which depends on a vector of interacting 3pim, S = {81," .,8N}, in some complex nonlinear way. Stochastic simulated annealing (SSA) (S. Kirk patrick, C. Gelatt, and M. Vecchi (1983)) finds a global minimum of H by com bining gradient descent with a random process. -
7Network Models
7 Network Models 7.1 Introduction Extensive synaptic connectivity is a hallmark of neural circuitry. For ex- ample, a typical neuron in the mammalian neocortex receives thousands of synaptic inputs. Network models allow us to explore the computational potential of such connectivity, using both analysis and simulations. As illustrations, we study in this chapter how networks can perform the fol- lowing tasks: coordinate transformations needed in visually guided reach- ing, selective amplification leading to models of simple and complex cells in primary visual cortex, integration as a model of short-term memory, noise reduction, input selection, gain modulation, and associative mem- ory. Networks that undergo oscillations are also analyzed, with applica- tion to the olfactory bulb. Finally, we discuss network models based on stochastic rather than deterministic dynamics, using the Boltzmann ma- chine as an example. Neocortical circuits are a major focus of our discussion. In the neocor- tex, which forms the convoluted outer surface of the (for example) human brain, neurons lie in six vertical layers highly coupled within cylindrical columns. Such columns have been suggested as basic functional units, cortical columns and stereotypical patterns of connections both within a column and be- tween columns are repeated across cortex. There are three main classes of interconnections within cortex, and in other areas of the brain as well. Feedforward connections bring input to a given region from another re- feedforward, gion located at an earlier stage along a particular processing pathway. Re- recurrent, current synapses interconnect neurons within a particular region that are and top-down considered to be at the same stage along the processing pathway. -
Chapter 14 Stochastic Networks
14 Stochastic Networks 14.1 Variations of the Hopfield model In the previous chapter we showed that Hopfield networks can be used to provide solutions to combinatorial problems that can be expressed as the minimization of an energy function, although without guaranteeing global optimality. Once the weights of the edges have been defined, the network shows spontaneous computational properties. Harnessing this spontaneous dynamics for useful computations requires some way of avoiding falling into local minima of the energy function in such a way that the global minimum is reached. In the case of the eight queens problem, the number of local minima is much higher than the number of global minima and very often the Hopfield network does not stabilize at a correct solution. The issue to be investigated is therefore whether a certain variation of the Hopfield model could achieve better results in combinatorial optimization problems. One possible strategy to improve the global convergence properties of a network consists in increasing the number of paths in search space, in such a way that not only binary but also real-valued states become possible. Contin- uous updates of the network state become possible and the search for minima of the energy function is no longer limited to the corners of a hypercube in state space. All interior points are now legal network states. A continuous acti- vation function also makes it possible to describe more precisely the electrical circuits used to implement Hopfield networks. A second strategy to avoid local minima of the energy function consists in introducing noise into the network dynamics. -
Statistical Computations Underlying the Dynamics of Memory Updating
Statistical Computations Underlying the Dynamics of Memory Updating Samuel J. Gershman1*, Angela Radulescu2, Kenneth A. Norman2, Yael Niv2 1 Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, Massachussetts, United States of America, 2 Department of Psychology and Princeton Neuroscience Institute, Princeton University, Princeton, New Jersey, United States of America Abstract Psychophysical and neurophysiological studies have suggested that memory is not simply a carbon copy of our experience: Memories are modified or new memories are formed depending on the dynamic structure of our experience, and specifically, on how gradually or abruptly the world changes. We present a statistical theory of memory formation in a dynamic environment, based on a nonparametric generalization of the switching Kalman filter. We show that this theory can qualitatively account for several psychophysical and neural phenomena, and present results of a new visual memory experiment aimed at testing the theory directly. Our experimental findings suggest that humans can use temporal discontinuities in the structure of the environment to determine when to form new memory traces. The statistical perspective we offer provides a coherent account of the conditions under which new experience is integrated into an old memory versus forming a new memory, and shows that memory formation depends on inferences about the underlying structure of our experience. Citation: Gershman SJ, Radulescu A, Norman KA, Niv Y (2014) Statistical Computations Underlying the Dynamics of Memory Updating. PLoS Comput Biol 10(11): e1003939. doi:10.1371/journal.pcbi.1003939 Editor: Olaf Sporns, Indiana University, United States of America Received August 19, 2013; Accepted September 26, 2014; Published November 6, 2014 Copyright: ß 2014 Gershman et al. -
Forecast of Consumer Behaviour Based on Neural Networks Models Comparison
ACTA UNIVERSITATIS AGRICULTURAE ET SILVICULTURAE MENDELIANAE BRUNENSIS Volume LX 57 Number 2, 2012 FORECAST OF CONSUMER BEHAVIOUR BASED ON NEURAL NETWORKS MODELS COMPARISON M. Štencl, O. Popelka, J. Šťastný Received: November 30, 2011 Abstract ŠTENCL, POPELKA, ŠŤASTNÝ: Forecast of consumer behaviour based on neural networks models comparison. Acta univ. agric. et silvic. Mendel. Brun., 2012, LX, No. 2, pp. 437–442 The aim of this article is comparison of accuracy level of forecasted values of several artifi cial neural network models. The comparison is performed on datasets of Czech household consumption values. Several statistical models o en resolve this task with more or fewer restrictions. In previous work where models’ input conditions were not so strict and model with missing data was used (the time series didn’t contain many values) we have obtained comparably good results with artifi cial neural networks. Two views – practical and theoretical, motivate the purpose of this study. Forecasting models for medium term prognosis of the main trends of Czech household consumption is part of the faculty research design grant MSM 6215648904/03/02 (Sub-task 5.3) which defi nes the practical purpose. Testing of nonlinear autoregressive artifi cial neural network model compared with feed- forward neural network and radial basis function neural network defi nes the theoretical purpose. The performance metrics of the models were evaluated using a combination of common error metrics, namely Correlation Coeffi cient and Mean Square Error, together with the number of epochs and/or main prediction error. artifi cial neural networks, forecasting methods, customer behaviour Prediction of tendencies in consumer behaviour of statistical learning, but they are built on diff erent of Czech (or any other) households is one of key fundamentals. -
Artificial Neural Networks for Financial Time Series Prediction and Portfolio
Master of Science Thesis in Industrial Engineering and Management Department of Management and Engineering, Linköping University June 2017 Artificial neural networks for financial time series prediction and portfolio optimization Samuel Björklund Tobias Uhlin Credits: 30HP Level: A Advisor: Jörgen Blomvall Department of Management and Engineering, Linköping University Examiner: Ou Tang Department of Management and Engineering, Linköping University ISRN: LIU-IEI-TEK-A--17/02920—SE Abstract Predicting the return of financial times series using traditional technical analysis and widely used economic models such as the capital asset pricing model, has proven to be difficult. Machine learning is a subfield of computer science, a concept that is frequently being used within different domains and recurrently delivers successful results. Artificial neural network is a product from the field of machine learning, a black box model that if properly designed processes data, learns its dynamic and subsequently provides an informative output. The objective of this thesis is to develop artificial neural networks that predict time varying expected return of financial time series with purpose of optimizing portfolio weights. This thesis begins with a review of prevailing characteristics of financial times series, and existing methods for estimating statistical properties. Background on artificial neural network along with empirical case studies of their suitability within the fi- nancial domain is put into a brief discussion vis-a-vis the efficient market hypothesis where potential anomalies is highlighted. Based on the theoretical review, an inter- disciplinary approach between machine learning and finance culminated into a model that estimates the expected return of time series on a quarterly basis. -
Lecture Notes 6 Random Processes • Definition and Simple Examples
Lecture Notes 6 Random Processes • Definition and Simple Examples • Important Classes of Random Processes ◦ IID ◦ Random Walk Process ◦ Markov Processes ◦ Independent Increment Processes ◦ Counting processes and Poisson Process • Mean and Autocorrelation Function • Gaussian Random Processes ◦ Gauss–Markov Process EE 278: Random Processes Page 6–1 Random Process • A random process (RP) (or stochastic process) is an infinite indexed collection of random variables {X(t): t ∈ T }, defined over a common probability space • The index parameter t is typically time, but can also be a spatial dimension • Random processes are used to model random experiments that evolve in time: ◦ Received sequence/waveform at the output of a communication channel ◦ Packet arrival times at a node in a communication network ◦ Thermal noise in a resistor ◦ Scores of an NBA team in consecutive games ◦ Daily price of a stock ◦ Winnings or losses of a gambler EE 278: Random Processes Page 6–2 Questions Involving Random Processes • Dependencies of the random variables of the process ◦ How do future received values depend on past received values? ◦ How do future prices of a stock depend on its past values? • Long term averages ◦ What is the proportion of time a queue is empty? ◦ What is the average noise power at the output of a circuit? • Extreme or boundary events ◦ What is the probability that a link in a communication network is congested? ◦ What is the probability that the maximum power in a power distribution line is exceeded? ◦ What is the probability that a gambler -
On the Mapping Between Hopfield Networks and Restricted Boltzmann
Published as a conference paper at ICLR 2021 ON THE MAPPING BETWEEN HOPFIELD NETWORKS AND RESTRICTED BOLTZMANN MACHINES Matthew Smart Anton Zilman Department of Physics Department of Physics University of Toronto and Institute for Biomedical Engingeering [email protected] University of Toronto [email protected] ABSTRACT Hopfield networks (HNs) and Restricted Boltzmann Machines (RBMs) are two important models at the interface of statistical physics, machine learning, and neu- roscience. Recently, there has been interest in the relationship between HNs and RBMs, due to their similarity under the statistical mechanics formalism. An exact mapping between HNs and RBMs has been previously noted for the special case of orthogonal (“uncorrelated”) encoded patterns. We present here an exact map- ping in the case of correlated pattern HNs, which are more broadly applicable to existing datasets. Specifically, we show that any HN with N binary variables and p < N arbitrary binary patterns can be transformed into an RBM with N binary visible variables and p gaussian hidden variables. We outline the conditions under which the reverse mapping exists, and conduct experiments on the MNIST dataset which suggest the mapping provides a useful initialization to the RBM weights. We discuss extensions, the potential importance of this correspondence for the training of RBMs, and for understanding the performance of deep architectures which utilize RBMs. 1 INTRODUCTION Hopfield networks (HNs) (Hopfield, 1982; Amit, 1989) are a classical neural network architecture that can store prescribed patterns as fixed-point attractors of a dynamical system. In their standard formulation with binary valued units, HNs can be regarded as spin glasses with pairwise interactions Jij that are fully determined by the patterns to be encoded. -
Generalization in a Hopfield Network J.F
Generalization in a Hopfield network J.F. Fontanari To cite this version: J.F. Fontanari. Generalization in a Hopfield network. Journal de Physique, 1990, 51 (21), pp.2421- 2430. 10.1051/jphys:0199000510210242100. jpa-00212540 HAL Id: jpa-00212540 https://hal.archives-ouvertes.fr/jpa-00212540 Submitted on 1 Jan 1990 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. J. Phys. France 51 (1990) 2421-2430 ler NOVEMBRE 1990, 2421 Classification Physics Abstracts 87.10 - 64.60C Generalization in a Hopfield network J. F. Fontanari Instituto de Fisica e Quimica de São Carlos, Universidade de São Paulo, Caixa Postal 369, 13560 São Carlos SP, Brasil (Received 23 April 1990, accepted in final form 12 July 1990) Abstract. 2014 The performance of a Hopfield network in learning an extensive number of concepts having access only to a finite supply of typical data which exemplify the concepts is studied. The minimal number of examples which must be taught to the network in order it starts to create representations for the concepts is calculated analitically. It is shown that the mixture states play a crucial role in the creation of these representations. -
A Novel Neural Network Training Framework with Data Assimilation
A Novel Neural Network Training Framework with Data Assimilation Chong Chen1, *, Qinghui Xing1, Xin Ding1, Yaru Xue1, Tianfu Zhong1 1 College of Information Science and Engineering, China University of Petroleum – Beijing * Email: [email protected] Abstract In recent years, the prosperity of deep learning has revolutionized the Artificial Neural Networks. However, the dependence of gradients and the offline training mechanism in the learning algorithms prevents the ANN for further improvement. In this study, a gradient-free training framework based on data assimilation is proposed to avoid the calculation of gradients. In data assimilation algorithms, the error covariance between the forecasts and observations is used to optimize the parameters. Feedforward Neural Networks (FNNs) are trained by gradient decent, data assimilation algorithms (Ensemble Kalman Filter (EnKF) and Ensemble Smoother with Multiple Data Assimilation (ESMDA)), respectively. ESMDA trains FNN with pre-defined iterations by updating the parameters using all the available observations which can be regard as offline learning. EnKF optimize FNN when new observation available by updating parameters which can be regard as online learning. Two synthetic cases with the regression of a Sine Function and a Mexican Hat function are assumed to validate the effectiveness of the proposed framework. The Root Mean Square Error (RMSE) and coefficient of determination (R2) are used as criteria to assess the performance of different methods. The results show that the proposed training framework performed better than the gradient decent method. The proposed framework provides alternatives for online/offline training the existing ANNs (e.g., Convolutional Neural Networks, Recurrent Neural Networks) without the dependence of gradients. -
CPY Document
Lecture i: Overview 1 Diffusion by discontinous movements In 1827 Robert Brown, observed that suspended pollen grains are in an uninterrupted and irregular "swarming" motion. Brown was a botanist and at first he believed that only organic materials exhibited this agitation. But very soon he extended his observations to particles of inorganic material, such as a ground-up fragment of the Sphin. Through the nineteenth century there was a intermittent discussion concerning the cause of this Brownian motion, and in 1877 DelsailX suggested that the impact of molecules on a macroscopic particle produces observable displacements. In 1905, after nearly a century of debate, Einstein definitively explained this phenomenon (6,7). 1.1 Einstein's derivation of the diffusion equation Our interest here is in Einstein's derivation of the dision equation, which is very different from that of Fourier. We consider one-dimensional Brownian motion by projecting the location of the particle onto a straight line which we call the x-axs. Einstein's assumptions are the following: (i) the particles move independently of one another; (ii) we observe particle positions at time intervals 7 which are much greater than the time intervals between molecular collsions. As a result, the motion in one interval is independent of what happened in the previous interval. In the interval 7 each paricle has a random displacement ß along the x-axs. The probability density function (PDF) of ß is ø(ß). This means that if we observe N ).). 1 paricles for a time 7 then the number of particles which are displaced through a distance which lies between ß and ß + dß is dN = N ø(ß) dß.