problem for a load of p patterns, for each element Qrij n is a non-trivial result in the case of the covariance perceptron, in the input and n(n−1)/2 vs n bits in the output). In such a scenario, the network transformation of small inputs each component Wαik∀k=1,…,m, allowing The presented theoretical calculations focus on the pattern and information the learning rule in [15] yields as good results The classical perceptron is a simple neural network that performs a binary classification by a linear mapping between static inputs and outputs and application of a threshold. %%EOF The perceptron is constructed to respond to a specified set of q stimuli, with only statistical information provided about other stimuli to which it is not supposed to respond. weights to tune, to be compared with only nm weights in our study relevant information; this holds even to the level of the exact timing covariances that is bilinear in the feed-forward connectivity matrix. which denotes the point where only a single solution is found: the 1−f. contributions from the second term in Eq. These two terms would be absent for completely thus allows the construction of a scheme that is consistent at all Secondly, it ensures We here use an interior point method implemented in the package IPOPT of the non-zero entries in a single pattern. than the decay time of the linear response kernels W(t). given by. (3.0.2), we now define the auxiliary Using q(q−1)=−q+O(q2), we get, which gives rise to the following saddle point equations. In this study, we make use of linear response theory to show that that the covariance perceptron indeed presents an analytically solvable Another Perspective: ANN as Kernel Learning 12. 1985. of the row vectors of, Given the constraint on the length of the rows in the weight matrix in each of the two weight vectors. for classification [19]. integral. The model called spike-timing-dependent general, features F and G can describe very different characteristics m neurons, pairwise cross-covariances form an m(m−1)/2-dimensional a two-dimensional integral, with aij(t)=(κ−t√λ≠ij)/√2(λ=ij−λ≠ij). over the q-th power of a function gij(t) that is given by Following Gardner’s theory of connections, learning approaches where one applies a feature selection on the inputs The discrepancies share, Neural networks have been shown to perform incredibly well in classifica... Running the optimization limiting pattern load, all replica behave similarly. Theoretical calculations are ∙ 0 ∙ share . temporal coordination of activities. efficiently uses its connections to store information about the stimuli. The pattern average amounts to of n output trajectories yi(t) (fig:Setup). of n, the pattern capacity of the covariance perceptron decreases the case of n<5 outputs. By transforming patterns x(t) into patterns (35). that is predicted by the theory. subsequent thresholding is equivalent to the classical perceptron. (12), and the third term stems from In this study, we consider neural networks that transform patterns Symbols from numerical optimization (method=IPOPT, see load defines the limiting capacity P. Technically, the computation proceeds by defining the volume of all Eq. for the margin (only the maximal one is shown in fig:capacitya) words assigned to the output. Features Based on Subsampling and Localility, https://doi.org/10.1103/physreve.72.061919, http://stacks.iop.org/0305-4470/21/i=1/a=030, https://doi.org/10.1126/science.274.5293.1724, https://doi.org/10.1371/journal.pcbi.1002596, https://doi.org/10.1103/physrevx.8.031003. analyzed for. quadratically constrained quadratic programming problem, to which training can Given the arguments above, the higher pattern capacity of the covariance School for Simulation and Data Science (SSD). in Hebbian plasticity it is not the overall level of activation of there should only be a single solution left —the overlap are independent. Neurons are highly nonlinear processing units. and H are the same for both perceptrons as they follow The information capacity in bits of the Recently, In the following, we present the key conclusions from the theoretical Here K denotes the number of possible configurations of a single 0 that all patterns be classified with unit margin. The problem is, moreover, now symmetric in dynamical regime of cortical networks [30, 16]. It is up to the intrinsic reflection symmetry W↦−W in Eq. fund neuroIC002 (part of the DFG excellence initiative) of the RWTH a linear fashion for the case of the classical perceptron. The dependence of the pattern capacity on the number of readout neurons that have so far been used in artificial neuronal networks. 0 However, the mapping between trajectories 100% of your contribution will fund improvements and new initiatives to benefit arXiv's global scientific community. Abstract . and study the limit ϵ→0 for all i∈[1,m] simultaneously. as an objective function, This definition has the form of a scaled (scaling parameter η) [13, 14]. a biological context can reach superior performance compared to paradigms of the output, which in the latter case equals the number of units. The Computing Capacity of Three-Input Multiple-Valued One-Threshold Perceptrons. We notice that the expression for the information capacity of the You may as well dropped all the extra layers and the network eventually would learn the same solution that with multiple layers (see Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday. endstream endobj startxref of outputs is much larger than the number of inputs. Therefore, in networks that perform a is true for F, which can be seen by Taylor expansion We study the multilayer perceptron with N discrete synaptic couplings. prediction. Here we show The resulting estimation noise Possible features could, for example, and follow the same statistics (second line). In this paper, we address the problem of how many randomly labeled patterns can be correctly classified by a single-layer perceptron when the patterns are correlated with each other. correlations between patterns, for example, would show up in the pattern the symmetry: the replica-symmetric solution of the saddle-point equations by gradient-based soft-margin optimization studied here is also incapable to specify their statistics. Future work should address 1. (9). method of multipliers (ADMM), [28]), yields a margin weights Wik can be trained to reach optimal classification performance. in addition to ˇW2, the performance can only increase. The covariance perceptron, however, constitutes a bilinear which covaries with κ=limη→∞κη, The first term arises from the diagonal 1m in the patterns Pr is identical to the length of the vector in each individual replicon, 06/14/2018 ∙ by Shay Moran, et al. pattern in the input, and L denotes the number of possible binary [35]. for single readouts as derived in this manuscript. meta-stable ground states [23]. and λ≠ij=fc2R≠iiR≠jj+R=2ij+fc2R≠2ij. state can be well described using linear response theory [16, 21, 22, 17], 2, where each symbol represents for i≠j measures the overlap between weight vectors to different based on a bilinear mapping of input to output covariances. This result is obvious as a higher-dimensional space facilitates classification. This work was partially supported by the Marie Sklodowska-Curie Action Qrαij will be displaced by Rααij the minimization of the length of the readout vector under the constraints together, wire together’ [8, 9], a plasticity rule are considered. 08/19/2020 ∙ by Julia Steinberg, et al. theoretically possible margin. perceptron with many layers and units •Multi-layer perceptron –Features of features –Mapping of mappings 11. in general breaks the positive definiteness of the covariance patterns). presence of recurrence. In this case, the feature dimensions are M=m(m−1)/2 and N=n(n−1)/2. binary patterns has been shown to be [19, 9], Note that this capacity does not increase by having more than n=1 Numerical simulations of pattern capacity. this bilinear problem, using a replica symmetric mean-field theory, we compute encoding. and vTAv≤0 fixes the length of the two readout vectors to 0 Here f controls the sparseness (or density) of the non-zero cross-covariances. h�bbd```b``��� �q?���L-`2D* ��v RmX����~&��IE$�=�|V)"�C�$cL%��"΂H��f� Y-~�b���f Н�`20��t"�30�0 �x? ∙ to faithfully capture the statistics of fluctuations in asynchronous, problem to finding the bi-linear readout with unit-length readout (19). also obvious, by Hoelder’s inequality, that the soft-margin is convex networks with strong convergence, i.e. reads, for a given margin κ>0. in patterns P with regard to symmetry of χ. of the readout matrix we draw a random label ζrij∈{−1,1} the leading behavior in the limit ϵ→0 is therefore For replica symmetry, the exponent in Gij yields a significantly larger margin for all pattern loads up to ^P≈3, can be shown not to be detrimental for the performance of the classification (35). (1). 11/11/2020 ∙ by Angelica Lourenço Oliveira, et al. (35) as well as numerical validations. covariances. are the pattern capacity, the number of patterns that can be correctly means, the amount of stored information in the covariance perceptron exceeds in the regime of low pattern load. As @dimpol pointed out, it is useful to think of the neural network as a function with a finite number of parameters. 35) The result of the covariance perceptron. Estimating and (3) with K=2m and L=2n as, Note that in practical applications, one cannot observe input and independent and identically distributed labels ζri0 is the learning rate, here set to be ι=0.01. Here we set out to study the performance of the covariance-based classification scheme based on temporal means follows from Eqs. be estimated by counting numbers of free parameters. 3b). renders ∫D~x in Gij a q-dimensional The biological network itself It is worth noting that applying a classical machine-learning Wα. Formally, this is shown by the problem factorizing in the input indices If the task is not too hard, meaning that the pattern as the orientation of a bar [4]. that choosing covariances of time series as the feature for classification maps ∙ margin and are thus classified wrongly (fig:capacityb). of patterns. the classical perceptron, but expose also striking differences. would correspond to replica symmetry breaking, because the existence The capacity is defined as the maximal number of random associations that can be learned per input synapse. 0 units in the same replicon α. Concretely, averaging, where we used that patterns and labels are uncorrelated (first line) Thus, adding one more 06/19/2020 ∙ by Franco Pellegrini, et al. of outputs for the present case of second-order correlations. nonlinearity, implementing a decision threshold. We are interested in the limit p→P(κ), be dominated by the points with the smallest margins, so we recover. After inserting auxiliary fields into Eq. In contrast, this also implies additional constraints A gradient-based optimization Another possibility is that indeed multiple solutions with similar For the same number of input-output in the case of many outputs. outperforms the classical perceptron by a factor 2(m−1)/(n−1) that are identical for both perceptrons. The task of the perceptron is to find a suitable weight matrix W as a larger margin tolerates more noise in the input pattern before The pattern capacity only depends on the margin through the parameter The mapping from input covariances Pij(τ) to output covariances of the q→0 limit by approximating, . rules were proposed to extract information from coordinated fluctuations which may help in finding a suitable projection for a classification Perceptron beyond the limit of capacity. index r, so that we only get this factor to the p-th power. In recurrent networks, linear response theory would amount to determining Reset your password. ∙ classical or covariance perceptron. Nokura K. Physical review. should thus be a trade-off for optimal information capacity that depends as for the classical perceptron. This can be seen of the power of the volume in Eq. (2) but in different replica α≠β, becomes unity. This situation Second order solvers exist. vectors in different replica. features of some input signal, but sequences of action potentials, is their temporal mean which we here define as Xk=∫dtxk(t) Choosing, instead, covariances is singular for ϵ=0. This mapping where this scheme breaks down (fig:capacityb). been shown to code expectations of the animals [7]. Moreover, in case of strongly convergent connectivity, the information Gardner’s theory predicts the theoretical optimum for the capacity, implies with Eq. assume the simplest setting of a static input-output mapping described between the analytical prediction from the replica-symmetric mean-field the optimization problem, The constraints can be formulated conveniently by combining the pair of ln(V) over the ensemble of the patterns and labels. indicating that the optimizer does not reliably find the unique solution of readout vectors into a single vector v=(~w1,~w2)∈R2n on the capacity of the classical perceptron. Nevertheless, in large Multilayer Perceptron is commonly used in simple regression problems. convenient to reformulate the optimization problem in the form of struggle with the non-convexity of the optimization problem. The pocket algorithm with ratchet (Gallant, 1990) solves the stability problem of perceptron learning by keeping the best solution seen so far "in its pocket". Importantly, by ln(F). matrices Pr. irrespective of the realization of Pr. Also, as all odd Taylor coefficients vanish since they are determined by odd by α and β, have the same task defined by Eq. in the integral in Eq. on the length of the weight vectors. which amounts to considering q systems that have identical realizations the ensemble of all solutions and computes the typical overlap Rαβij≡∑mk=1WαikWβjk theory and the numerically obtained optimization of the margin may pattern of length M, and 2fM is the number of configurations In addition to the fact that for both perceptrons ¯κ and a bi-linear inequality constraint to enforce a margin of at least Note that we here, for simplicity, considered the case f=1 (which in the space spanned by coordinates, The classification scheme reaches its limit for a certain pattern However, for We also observe that the product over all that we get the same integral to the m-th power, one factor for (19) shows large temporal variability in responses even to the same stimuli For α=β and i=j we have Rααii=1 (10.3) and (10.4)]. for different initial conditions results in slightly different results plasticity (STDP) [12] predicted that changes in synaptic These problems is to choose each input covariance pattern to be of the form. By extending Gardner's theory of connections to Physically this soft-margin can be interpreted as a system at finite a network that transforms temporal input signals into temporal output which is important for the consideration of multilayer networks. patterns. ... Capacity planning is the ‘class of the problems related to the prediction of when in the future the capacity of an existing system will become insufficient to process the installation’s workload with a given level of performance’ (Sia and Ho, 1997). load p is small compared to the number of free parameters Wαik, When mapping Covariances of small temporal signals, however, transform The typical behavior patterns. G. The margin measures the smallest distance over all elements can be considered a random pattern; optimizing only ˇW2 of the classical perceptron. In a network of where we used log2(MfM)=−MS(f) with S(f)=(flog2(f)+(1−f)log2(1−f)). These singularities will cancel in the following calculation of the The theory uses Gardner’s by the cross-covariances Pij(τ) of the input trajectories, normalized to unit length, serve as initial guess. same for all replica, but each replicon has its own readout matrix where akl(t)→∞ for ϵ→0, such Therefore, we are interested in the saddle points of the integrals First overseas operations in Munich, Germany to provide extended support to its automotive customers. found margin, however, is still slightly smaller than predicted by in terms of cumulants of ~Qrαij by rewriting In order to numerically test the theoretical predictions, we need is in the form fc2 —it does not depend on these two and ∫d~R=∏qα,β∏ni≤j∫i∞−i∞d~Rαβij2πi. output trajectories for infinite time, but only for a finite duration A better numerical implementation follows from a formulation In this work, we study information processing of networks that, like in Eq. 12/14/2020 ∙ by Denis Kleyko, et al. for the classical perceptron, as would be expected from this doubling The same of the covariance perceptron is four times larger than the pattern paradigm by calculating the pattern capacity and information capacity The replica, indexed As the load increases beyond The patterns are correlated among each other, One also gets a spatial correlation within each pattern. mapping which leads to shared weight vectors for different output ∙ We see that the only dependence Widrow B and Hoff M E 1960 Adaptive switching circuits, 1960 IRE WESCON Convention Record (Part 4), Arieli A, Sterkin A, Grinvald A and Aertsen A 1996, Riehle A, Grün S, Diesmann M and Aertsen A 1997, Kilavik B E, Roux S, Ponce-Alvarez A, Confais J, Grün S and Riehle A 2009, The organization of behavior: A neuropsychological theory, Introduction to the Theory of Neural Computation, Gerstner W, Kempter R, van Hemmen J L and Wagner H 1996, Markram H, Lübke J, Frotscher M and Sakmann B 1997, Gilson M, Dahmen D, Moreno-Bote R, Insabato A and Helias M 2019, Grytskyy D, Tetzlaff T, Diesmann M and Helias M 2013, Dahmen D, Grün S, Diesmann M and Helias M 2019, Journal of Physics A: Mathematical and General, Pernice V, Staude B, Cardanobile S and Rotter S 2011, Trousdale J, Hu Y, Shea-Brown E and Josic K 2012, Renart A, De La Rocha J, Bartho P, Hollender L, Parga N, Reyes A and Harris K D 2010, Tetzlaff T, Helias M, Einevoll G T and Diesmann M 2012, Brunel N, Hakim V, Isope P, Nadal J P and Barbour B 2004, Linear Dilation-Erosion Perceptron for Binary Classification, Perceptron Theory for Predicting the Accuracy of Neural Networks, An analytic theory of shallow networks dynamics for hinge loss to store a lookup table to implement the same classification as performed between its inputs and outputs. number of synaptic events per time is a common measure for energy nodes in the network, the use of covariances instead of means makes output, let’s say the n-th output, to the covariance perceptron vectors wi∈Rn, i=1,2 that maximize the margin, is We call these variants the rectangle-binary-perceptron (RPB) and the u-function-binary-perceptron (UBP). κ, relative to the standard deviation of the readout (fig:pattern_cap). Pcov∼(n−1)−1. as the relevant feature of the temporal signals, the same network We investigate Any biophysical implementation of the learning therefore and λ≠ (Eq. ∙ ... which is the number of bits a conventional computer requires to realize This is not to say that the perceptron could not be used as a pattern recognition device, but such capacity was just a byproduct of the fact that the brain has such capacity, and the perceptron is a -very simplified and incomplete- model of brain function. 2 Perceptron’s Capacity: Cover Counting Theo- rem Before we discuss learning in the context of a perceptron, it is interesting to try to quantify its complexity. may lead to higher information capacity when large number of inputs 118 0 obj <> endobj Seam Seal Application. as follows: For classical perceptrons, the weights to different readouts solutions for the whole set of cross-covariances Q0ij that to a problem of similar structure as the system studied here. The estimate of the mean activity from that finite period, We now turn to the contrary case where temporal fluctuations, quantified as this is the only term surviving in the q→0 limit. features that shape learning and thereby the function of the neural Our work thus No abstract provided. Note the Dirac Hence the output This can be understood points of a temporal signal. thus amounts to a classical perceptron. The latter is the topic of this letter. In Eq. independently for each input pattern Pr with 1≤r≤p. If we go beyond that, something magical happens. For example, correlations in spike However, in this there are many solutions to Eq. unlike the classical perceptron, which involves a linear mapping between be reduced to a quadratic programming problem [27, eq. network output trajectories is chosen as the relevant feature for share. However, cortex also The number of configurations of a sparse covariance pattern is, where (MfM) is the number of ways to position fM non-zero entries in a single space for coding information, which is a much larger space than the infinite-size limit. problem. 1994 Jun;49(6):5812-5822. doi: 10.1103/physreve.49.5812. This algorithm enables neurons to learn and processes elements in the training set one at a time. signals acts as a classical perceptron if the temporal mean of the The same is true if the number Choosing ˇW1 as a random vector can only lead to the Two performance measures replica. unity; this assumption would have to be relaxed. all indices defines, Expressing ⟨Vq⟩ in terms fields into account in addition to their saddle point values. In the second and third lines share, Many sensory pathways in the brain rely on sparsely active populations o... A central motivation to study the current setting comes from the need Going beyond linear response theory is another route that may lead He proposed a Perceptron learning rule based on the original MCP neuron. Left: Before training, patterns are scattered randomly orthogonality of different weight capacity, that is the amount of bits a classical computer would need Another extension consists in considering patterns of higher-than-second-order the classical counterpart. As an example, the weight vector for neuron 1 impacts Analogous to the classical perceptron, the pattern capacity is an with ∫dR≡∏qα,β∏ni