搜索结果：Perceptron

A Perceptron is a fundamental building block of a neural network. The flexibility and scalability of perceptron make it ubiquitous in building intelligent systems. Studies have shown the efficacy of a single neuron in making intelligent decisions. Here, we examined and compared two perceptrons with distinct mechanisms, and developed a quantum version of one of those perceptrons. As a part of this modeling, we implemented the quantum circuit for an artificial perception, generated a dataset, and simulated the training. Through these experiments, we show that there is an exponential growth advantage and test different qubit versions. Our findings show that this quantum model of an individual perceptron can be used as a pattern classifier. For the second type of model, we provide an understanding to design and simulate a spike-dependent quantum perceptron. Our code is available at https://github.com/ashutosh1919/quantum-perceptron

With the growing interest in quantum machine learning, the perceptron -- a fundamental building block in traditional machine learning -- has emerged as a valuable model for exploring quantum advantages. Two quantum perceptron algorithms based on Grover's search, were developed in arXiv:1602.04799 to accelerate training and improve statistical efficiency in perceptron learning. This paper points out and corrects a mistake in the proof of Theorem 2 in arXiv:1602.04799. Specifically, we show that the probability of sampling from a normal distribution for a $D$-dimensional hyperplane that perfectly classifies the data scales as $Ω(γ^{D})$ instead of $Θ(γ)$, where $γ$ is the margin. We then revisit two well-established linear programming algorithms -- the ellipsoid method and the cutting plane random walk algorithm -- in the context of perceptron learning, and show how quantum search algorithms can be leveraged to enhance the overall complexity. Specifically, both algorithms gain a sub-linear speed-up $O(\sqrt{N})$ in the number of data points $N$ as a result of Grover's algorithm and an additional $O(D^{1.5})$ speed-up is possible for cutting plane random walk algorithm employing quant

In this paper, we introduce the gated perceptron, an enhancement of the conventional perceptron, which incorporates an additional input computed as the product of the existing inputs. This allows the perceptron to capture non-linear interactions between features, significantly improving its ability to classify and regress on complex datasets. We explore its application in both linear and non-linear regression tasks using the Iris dataset, as well as binary and multi-class classification problems, including the PIMA Indian dataset and Breast Cancer Wisconsin dataset. Our results demonstrate that the gated perceptron can generate more distinct decision regions compared to traditional perceptrons, enhancing its classification capabilities, particularly in handling non-linear data. Performance comparisons show that the gated perceptron competes with state-of-the-art classifiers while maintaining a simple architecture.

Motivated by Ridgway's proof of the perceptron algorithm, we study a simple subgradient method for convex inequality systems in Hilbert space. Assuming strict feasibility and bounded subgradients, we establish finite termination for several natural step sizes. We also examine what can go wrong without strict feasibility: finite convergence may fail even for one function, and with several functions the method may converge to a point outside the feasible set. The linear setting recovers the classical perceptron algorithm.

We investigate a quantum perceptron implemented on a quantum circuit using a repeat until method. We evaluate this from the perspective of capacity, one of the performance evaluation measures for perceptions. We assess a Gardner volume, defined as a volume of coefficients of the perceptron that can correctly classify given training examples using the replica method. The model is defined on the quantum circuit. Nevertheless, it is straightforward to assess the capacity using the replica method, which is a standard method in classical statistical mechanics. The reason why we can solve our model by the replica method is the repeat until method, in which we focus on the output of the measurements of the quantum circuit. We find that the capacity of a quantum perceptron is larger than that of a classical perceptron since the quantum one is simple but effectively falls into a highly nonlinear form of the activation function.

Quantum machine learning algorithms could provide significant speed-ups over their classical counterparts; however, whether they could also achieve good generalization remains unclear. Recently, two quantum perceptron models which give a quadratic improvement over the classical perceptron algorithm using Grover's search have been proposed by Wiebe et al. arXiv:1602.04799 . While the first model reduces the complexity with respect to the size of the training set, the second one improves the bound on the number of mistakes made by the perceptron. In this paper, we introduce a hybrid quantum-classical perceptron algorithm with lower complexity and better generalization ability than the classical perceptron. We show a quadratic improvement over the classical perceptron in both the number of samples and the margin of the data. We derive a bound on the expected error of the hypothesis returned by our algorithm, which compares favorably to the one obtained with the classical online perceptron. We use numerical experiments to illustrate the trade-off between computational complexity and statistical accuracy in quantum perceptron learning and discuss some of the key practical issues surroun

The quantum perceptron, the variational circuit, and the Grover algorithm have been proposed as promising components for quantum machine learning. This paper presents a new quantum perceptron that combines the quantum variational circuit and the Grover algorithm. However, this does not guarantee that this quantum variational perceptron with Grover's algorithm (QVPG) will have any advantage over its quantum variational (QVP) and classical counterparts. Here, we examine the performance of QVP and QVP-G by computing their loss function and analyzing their accuracy on the classification task, then comparing these two quantum models to the classical perceptron (CP). The results show that our two quantum models are more efficient than CP, and our novel suggested model QVP-G outperforms the QVP, demonstrating that the Grover can be applied to the classification task and even makes the model more accurate, besides the unstructured search problems.

While multivariate logistic regression classifiers are a great way of implementing collaborative filtering - a method of making automatic predictions about the interests of a user by collecting preferences or taste information from many other users, we can also achieve similar results using neural networks. A recommender system is a subclass of information filtering system that provide suggestions for items that are most pertinent to a particular user. A perceptron or a neural network is a machine learning model designed for fitting complex datasets using backpropagation and gradient descent. When coupled with advanced optimization techniques, the model may prove to be a great substitute for classical logistic classifiers. The optimizations include feature scaling, mean normalization, regularization, hyperparameter tuning and using stochastic/mini-batch gradient descent instead of regular gradient descent. In this use case, we will use the perceptron in the recommender system to fit the parameters i.e., the data from a multitude of users and use it to predict the preference/interest of a particular user.

Determining the capacity $α_c$ of the Binary Perceptron is a long-standing problem. Krauth and Mezard (1989) conjectured an explicit value of $α_c$, approximately equal to .833, and a rigorous lower bound matching this prediction was recently established by Ding and Sun (2019). Regarding the upper bound, Kim and Roche (1998) and Talagrand (1999) independently showed that $α_c$ < .996, while Krauth and Mezard outlined an argument which can be used to show that $α_c$ < .847. The purpose of this expository note is to record a complete proof of the bound $α_c$ < .847. The proof is a conditional first moment method combined with known results on the spherical perceptron

The binary perceptron problem asks us to find a sign vector in the intersection of independently chosen random halfspaces with intercept $-κ$. We analyze the performance of the canonical discrepancy minimization algorithms of Lovett-Meka and Rothvoss/Eldan-Singh for the asymmetric binary perceptron problem. We obtain new algorithmic results in the $κ= 0$ case and in the large-$|κ|$ case. In the $κ\to-\infty$ case, we additionally characterize the storage capacity and complement our algorithmic results with an almost-matching overlap-gap lower bound.

Biological neurons are more powerful than artificial perceptrons, in part due to complex dendritic input computations. Inspired to empower the perceptron with biologically inspired features, we explore the effect of adding and tuning input branching factors along with input dropout. This allows for parameter efficient non-linear input architectures to be discovered and benchmarked. Furthermore, we present a PyTorch module to replace multi-layer perceptron layers in existing architectures. Our initial experiments on MNIST classification demonstrate the accuracy and generalization improvement of dendritic neurons compared to existing perceptron architectures.

The classical Perceptron algorithm of Rosenblatt can be used to find a linear threshold function to correctly classify $n$ linearly separable data points, assuming the classes are separated by some margin $γ> 0$. A foundational result is that Perceptron converges after $Ω(1/γ^{2})$ iterations. There have been several recent works that managed to improve this rate by a quadratic factor, to $Ω(\sqrt{\log n}/γ)$, with more sophisticated algorithms. In this paper, we unify these existing results under one framework by showing that they can all be described through the lens of solving min-max problems using modern acceleration techniques, mainly through optimistic online learning. We then show that the proposed framework also lead to improved results for a series of problems beyond the standard Perceptron setting. Specifically, a) For the margin maximization problem, we improve the state-of-the-art result from $O(\log t/t^2)$ to $O(1/t^2)$, where $t$ is the number of iterations; b) We provide the first result on identifying the implicit bias property of the classical Nesterov's accelerated gradient descent (NAG) algorithm, and show NAG can maximize the margin with an $O(1/t^2)$ rate;

Memristors are low-power memory-holding resistors thought to be useful for neuromophic computing, which can compute via spike-interactions mediated through the device's short-term memory. Using interacting spikes, it is possible to build an AND gate that computes OR at the same time, similarly a full adder can be built that computes the arithmetical sum of its inputs. Here we show how these gates can be understood by modelling the memristors as a novel type of perceptron: one which is sensitive to input order. The memristor's memory can change the input weights for later inputs, and thus the memristor gates cannot be accurately described by a single perceptron, requiring either a network of time-invarient perceptrons or a complex time-varying self-reprogrammable perceptron. This work demonstrates the high functionality of memristor logic gates, and also that the addition of theasholding could enable the creation of a standard perceptron in hardware, which may have use in building neural net chips.

Perceptron is a classic online algorithm for learning a classification function. In this paper, we provide a novel extension of the perceptron algorithm to the learning to rank problem in information retrieval. We consider popular listwise performance measures such as Normalized Discounted Cumulative Gain (NDCG) and Average Precision (AP). A modern perspective on perceptron for classification is that it is simply an instance of online gradient descent (OGD), during mistake rounds, using the hinge loss function. Motivated by this interpretation, we propose a novel family of listwise, large margin ranking surrogates. Members of this family can be thought of as analogs of the hinge loss. Exploiting a certain self-bounding property of the proposed family, we provide a guarantee on the cumulative NDCG (or AP) induced loss incurred by our perceptron-like algorithm. We show that, if there exists a perfect oracle ranker which can correctly rank each instance in an online sequence of ranking data, with some margin, the cumulative loss of perceptron algorithm on that sequence is bounded by a constant, irrespective of the length of the sequence. This result is reminiscent of Novikoff's conver

We propose a focus of attention mechanism to speed up the Perceptron algorithm. Focus of attention speeds up the Perceptron algorithm by lowering the number of features evaluated throughout training and prediction. Whereas the traditional Perceptron evaluates all the features of each example, the Attentive Perceptron evaluates less features for easy to classify examples, thereby achieving significant speedups and small losses in prediction accuracy. Focus of attention allows the Attentive Perceptron to stop the evaluation of features at any interim point and filter the example. This creates an attentive filter which concentrates computation at examples that are hard to classify, and quickly filters examples that are easy to classify.

Algorithmic machine teaching has been studied under the linear setting where exact teaching is possible. However, little is known for teaching nonlinear learners. Here, we establish the sample complexity of teaching, aka teaching dimension, for kernelized perceptrons for different families of feature maps. As a warm-up, we show that the teaching complexity is $Θ(d)$ for the exact teaching of linear perceptrons in $\mathbb{R}^d$, and $Θ(d^k)$ for kernel perceptron with a polynomial kernel of order $k$. Furthermore, under certain smooth assumptions on the data distribution, we establish a rigorous bound on the complexity for approximately teaching a Gaussian kernel perceptron. We provide numerical examples of the optimal (approximate) teaching set under several canonical settings for linear, polynomial and Gaussian kernel perceptrons.

Within the framework of on-line learning, we study the generalization error of an ensemble learning machine learning from a linear teacher perceptron. The generalization error achieved by an ensemble of linear perceptrons having homogeneous or inhomogeneous initial weight vectors is precisely calculated at the thermodynamic limit of a large number of input elements and shows rich behavior. Our main findings are as follows. For learning with homogeneous initial weight vectors, the generalization error using an infinite number of linear student perceptrons is equal to only half that of a single linear perceptron, and converges with that of the infinite case with O(1/K) for a finite number of K linear perceptrons. For learning with inhomogeneous initial weight vectors, it is advantageous to use an approach of weighted averaging over the output of the linear perceptrons, and we show the conditions under which the optimal weights are constant during the learning process. The optimal weights depend on only correlation of the initial weight vectors.

We study a family of Ising perceptron models with $\{0,1\}$-valued activation functions. This includes the classical half-space models, as well as some of the symmetric models considered in recent works. For each of these models we show that the free energy is self-averaging, there is a sharp threshold sequence, and the free energy is universal with respect to the disorder. A prior work of Xu (2019) used very different methods to show a sharp threshold sequence in the half-space Ising perceptron with Bernoulli disorder. Recent works of Perkins--Xu (2021) and Abbe--Li--Sly (2021) determined the sharp threshold and limiting free energy in a symmetric perceptron model. The results of this paper apply in more general settings, and are based on new "add one constraint" estimates extending Talagrand's estimates for the half-space model (1999, 2011).

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. For small inputs, neural networks in a stationary state also perform an effectively linear input-output transformation, but of an entire time series. Choosing the temporal mean of the time series as the feature for classification, the linear transformation of the network with subsequent thresholding is equivalent to the classical perceptron. Here we show that choosing covariances of time series as the feature for classification maps the neural network to what we call a 'covariance perceptron'; a mapping between covariances that is bilinear in terms of weights. By extending Gardner's theory of connections to this bilinear problem, using a replica symmetric mean-field theory, we compute the pattern and information capacities of the covariance perceptron in the infinite-size limit. Closed-form expressions reveal superior pattern capacity in the binary classification task compared to the classical perceptron in the case of a high-dimensional input and low-dimensional output. For less convergent networks, the mean