In this work we introduce a differentiable version of the Compositional Pattern Producing Network, called the DPPN. Unlike a standard CPPN, the topology of a DPPN is evolved but the weights are learned. A Lamarckian algorithm, that combines evolution and learning, produces DPPNs to reconstruct an image. Our main result is that DPPNs can be evolved trained to compress the weights of a denoising autoencoder from 157684 to roughly 200 parameters, while achieving a reconstruction accuracy comparable to a fully connected network with more than two orders of magnitude more parameters. The regularization ability of the DPPN allows it to rediscover (approximate) convolutional network architectures embedded within a fully connected architecture. Such convolutional architectures are the current state of the art for many computer vision applications, so it is satisfying that DPPNs are capable of discovering this structure rather than having to build it in by design. DPPNs exhibit better generalization when tested on the Omniglot dataset after being trained on MNIST, than directly encoded fully connected autoencoders. DPPNs are therefore a new framework for integrating learning and evolution.
In HyperNEAT @cite_19 , CPPNs are used as indirect compressed encodings of the weights of a larger neural network. The inputs to the CPPN are the coordinates of the presynaptic and postsynaptic neuron, and the output is the weight joining those two neurons. If a single CPPN must encode multiple layers of a deeper neural network then there are two possibilities, either an extra input is given signaling which layer of weights the CPPN is outputting @cite_10 , or the CPPN is constrained to always have multiple output nodes, with a specific node outputting the weight for its assigned layer @cite_11 .
@cite_19 @cite_10 @cite_11
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The feedforward multilayer perceptron (MLP) with back-propagation of error is described. Since use of this network requires a set of labeled input-output, as such it cannot be used for segmentation of images when only one image is available. (However, if images to be processed are of similar nature, one can use a set of known images for learning and then use the network for processing of other images.) A self-organizing multilayer neural network architecture suitable for image processing is proposed. The proposed architecture is also a feedforward one with back-propagation of errors; but like MLP it does not require any supervised learning. Each neuron is connected to the corresponding neuron in the previous layer and the set of neighbors of that neuron. The output status of neurons in the output layer is described as a fuzzy set. A fuzziness measure of this fuzzy set is used as a measure of error in the system (instability of the network). Learning rates for various measures of fuzziness have been theoretically and experimentally studied. An application of the proposed network in object extraction from noisy scenes is also demonstrated. A quaternionic extension of feed forward neural network, for processing multi-dimensional signals, is proposed in this paper. This neural network is based on the three layered network with random weights, called Extreme Learning Machines (ELMs), in which iterative least-mean-square algorithms are not required for training networks. All parameters and variables in the proposed network are encoded by quaternions and operations among them follow the quaternion algebra. Neurons in the proposed network are expected to operate multi-dimensional signals as single entities, rather than real-valued neurons deal with each element of signals independently. The performances for the proposed network are evaluated through two types of experiments: classifications and reconstructions for color images in the CIFAR-10 dataset. The experimental results show that the proposed networks are superior in terms of classification accuracies for input images than the conventional (real-valued) networks with similar degrees of freedom. The detailed investigations for operations in the proposed networks are conducted. HighlightsA feedforward neural network for accepting three- or four-dimension is proposed.Quaternions, which are a four-dimensional hypercomlex numbers, are used for encoding neuronal parameters.Its performances are evaluated through the classification and autoencoding for CIFAR-10 dataset. Research in neuroevolution---that is, evolving artificial neural networks (ANNs) through evolutionary algorithms---is inspired by the evolution of biological brains, which can contain trillions of connections. Yet while neuroevolution has produced successful results, the scale of natural brains remains far beyond reach. This article presents a method called hypercube-based NeuroEvolution of Augmenting Topologies (HyperNEAT) that aims to narrow this gap. HyperNEAT employs an indirect encoding called connective compositional pattern-producing networks (CPPNs) that can produce connectivity patterns with symmetries and repeating motifs by interpreting spatial patterns generated within a hypercube as connectivity patterns in a lower-dimensional space. This approach can exploit the geometry of the task by mapping its regularities onto the topology of the network, thereby shifting problem difficulty away from dimensionality to the underlying problem structure. Furthermore, connective CPPNs can represent the same connectivity pattern at any resolution, allowing ANNs to scale to new numbers of inputs and outputs without further evolution. HyperNEAT is demonstrated through visual discrimination and food-gathering tasks, including successful visual discrimination networks containing over eight million connections. The main conclusion is that the ability to explore the space of regular connectivity patterns opens up a new class of complex high-dimensional tasks to neuroevolution. Designing a neural network (NN) to process complex-valued signals is a challenging task since a complex nonlinear activation function (AF) cannot be both analytic and bounded everywhere in the complex plane C . To avoid this difficulty, ‘splitting’, i.e., using a pair of real sigmoidal functions for the real and imaginary components has been the traditional approach. However, this ‘ad hoc’ compromise to avoid the unbounded nature of nonlinear complex functions results in a nowhere analytic AF that performs the error back-propagation (BP) using the split derivatives of the real and imaginary components instead of relying on well-defined fully complex derivatives. In this paper, a fully complex multi-layer perceptron (MLP) structure that yields a simplified complex-valued back-propagation (BP) algorithm is presented. The simplified BP verifies that the fully complex BP weight update formula is the complex conjugate form of real BP formula and the split complex BP is a special case of the fully complex BP. This generalization is possible by employing elementary transcendental functions (ETFs) that are almost everywhere (a.e.) bounded and analytic in C . The properties of fully complex MLP are investigated and the advantage of ETFs over split complex AF is shown in numerical examples where nonlinear magnitude and phase distortions of non-constant modulus modulated signals are successfully restored. Recent advances in deep neural networks (DNNs) owe their success to training algorithms that use backpropagation and gradient-descent. Backpropagation, while highly effective on von Neumann architectures, becomes inefficient when scaling to large networks. Commonly referred to as the weight transport problem, each neuron's dependence on the weights and errors located deeper in the network require exhaustive data movement which presents a key problem in enhancing the performance and energy-efficiency of machine-learning hardware. In this work, we propose a bio-plausible alternative to backpropagation drawing from advances in feedback alignment algorithms in which the error computation at a single synapse reduces to the product of three scalar values, satisfying the three factor rule. Using a sparse feedback matrix, we show that a neuron needs only a fraction of the information previously used by the feedback alignment algorithms to yield results which are competitive with backpropagation. Consequently, memory and compute can be partitioned and distributed whichever way produces the most efficient forward pass so long as a single error can be delivered to each neuron. We evaluate our algorithm using standard data sets, including ImageNet, to address the concern of scaling to challenging problems. Our results show orders of magnitude improvement in data movement and @math improvement in multiply-and-accumulate operations over backpropagation. All the code and results are available under https: github.com bcrafton ssdfa. An important issue in the design and implementation of a neural network is the sensitivity of its output to input and weight perturbations. In this paper, we discuss the sensitivity of the most popular and general feedforward neural networks-multilayer perceptron (MLP). The sensitivity is defined as the mathematical expectation of the output errors of the MLP due to input and weight perturbations with respect to all input and weight values in a given continuous interval. The sensitivity for a single neuron is discussed first and an analytical expression that is a function of the absolute values of input and weight perturbations is approximately derived. Then an algorithm is given to compute the sensitivity for the entire MLP. As intuitively expected, the sensitivity increases with input and weight perturbations, but the increase has an upper bound that is determined by the structural configuration of the MLP, namely the number of neurons per layer and the number of layers. There exists an optimal value for the number of neurons in a layer, which yields the highest sensitivity value. The effect caused by the number of layers is quite unexpected. The sensitivity of a neural network may decrease at first and then almost keeps constant while the number increases. The region-based Convolutional Neural Network (CNN) detectors such as Faster R-CNN or R-FCN have already shown promising results for object detection by combining the region proposal subnetwork and the classification subnetwork together. Although R-FCN has achieved higher detection speed while keeping the detection performance, the global structure information is ignored by the position-sensitive score maps. To fully explore the local and global properties, in this paper, we propose a novel fully convolutional network, named as CoupleNet, to couple the global structure with local parts for object detection. Specifically, the object proposals obtained by the Region Proposal Network (RPN) are fed into the the coupling module which consists of two branches. One branch adopts the position-sensitive RoI (PSRoI) pooling to capture the local part information of the object, while the other employs the RoI pooling to encode the global and context information. Next, we design different coupling strategies and normalization ways to make full use of the complementary advantages between the global and local branches. Extensive experiments demonstrate the effectiveness of our approach. We achieve state-of-the-art results on all three challenging datasets, i.e. a mAP of 82.7 on VOC07, 80.4 on VOC12, and 34.4 on COCO. Codes will be made publicly available. The spiking neural networks (SNNs) are the third generation of neural networks and perform remarkably well in cognitive tasks such as pattern recognition. The spike emitting and information processing mechanisms found in biological cognitive systems motivate the application of the hierarchical structure and temporal encoding mechanism in spiking neural networks, which have exhibited strong computational capability. However, the hierarchical structure and temporal encoding approach require neurons to process information serially in space and time respectively, which reduce the training efficiency significantly. For training the hierarchical SNNs, most existing methods are based on the traditional back-propagation algorithm, inheriting its drawbacks of the gradient diffusion and the sensitivity on parameters. To keep the powerful computation capability of the hierarchical structure and temporal encoding mechanism, but to overcome the low efficiency of the existing algorithms, a new training algorithm, the Normalized Spiking Error Back Propagation (NSEBP) is proposed in this paper. In the feedforward calculation, the output spike times are calculated by solving the quadratic function in the spike response model instead of detecting postsynaptic voltage states at all time points in traditional algorithms. Besides, in the feedback weight modification, the computational error is propagated to previous layers by the presynaptic spike jitter instead of the gradient decent rule, which realizes the layer-wised training. Furthermore, our algorithm investigates the mathematical relation between the weight variation and voltage error change, which makes the normalization in the weight modification applicable. Adopting these strategies, our algorithm outperforms the traditional SNN multi-layer algorithms in terms of learning efficiency and parameter sensitivity, that are also demonstrated by the comprehensive experimental results in this paper. Multilayer Neural Networks (MNNs) are commonly trained using gradient descent-based methods, such as BackPropagation (BP). Inference in probabilistic graphical models is often done using variational Bayes methods, such as Expectation Propagation (EP). We show how an EP based approach can also be used to train deterministic MNNs. Specifically, we approximate the posterior of the weights given the data using a "mean-field" factorized distribution, in an online setting. Using online EP and the central limit theorem we find an analytical approximation to the Bayes update of this posterior, as well as the resulting Bayes estimates of the weights and outputs. Despite a different origin, the resulting algorithm, Expectation BackPropagation (EBP), is very similar to BP in form and efficiency. However, it has several additional advantages: (1) Training is parameter-free, given initial conditions (prior) and the MNN architecture. This is useful for large-scale problems, where parameter tuning is a major challenge. (2) The weights can be restricted to have discrete values. This is especially useful for implementing trained MNNs in precision limited hardware chips, thus improving their speed and energy efficiency by several orders of magnitude. We test the EBP algorithm numerically in eight binary text classification tasks. In all tasks, EBP outperforms: (1) standard BP with the optimal constant learning rate (2) previously reported state of the art. Interestingly, EBP-trained MNNs with binary weights usually perform better than MNNs with continuous (real) weights - if we average the MNN output using the inferred posterior. To reduce the significant redundancy in deep Convolutional Neural Networks (CNNs), most existing methods prune neurons by only considering the statistics of an individual layer or two consecutive layers (e.g., prune one layer to minimize the reconstruction error of the next layer), ignoring the effect of error propagation in deep networks. In contrast, we argue that for a pruned network to retain its predictive power, it is essential to prune neurons in the entire neuron network jointly based on a unified goal: minimizing the reconstruction error of important responses in the "final response layer" (FRL), which is the second-to-last layer before classification. Specifically, we apply feature ranking techniques to measure the importance of each neuron in the FRL, formulate network pruning as a binary integer optimization problem, and derive a closed-form solution to it for pruning neurons in earlier layers. Based on our theoretical analysis, we propose the Neuron Importance Score Propagation (NISP) algorithm to propagate the importance scores of final responses to every neuron in the network. The CNN is pruned by removing neurons with least importance, and it is then fine-tuned to recover its predictive power. NISP is evaluated on several datasets with multiple CNN models and demonstrated to achieve significant acceleration and compression with negligible accuracy loss. Abstract In this work, we propose a technique that utilizes a fully convolutional network (FCN) to localize image splicing attacks. We first evaluated a single-task FCN (SFCN) trained only on the surface label. Although the SFCN is shown to provide superior performance over existing methods, it still provides a coarse localization output in certain cases. Therefore, we propose the use of a multi-task FCN (MFCN) that utilizes two output branches for multi-task learning. One branch is used to learn the surface label, while the other branch is used to learn the edge or boundary of the spliced region. We trained the networks using the CASIA v2.0 dataset, and tested the trained models on the CASIA v1.0, Columbia Uncompressed, Carvalho, and the DARPA NIST Nimble Challenge 2016 SCI datasets. Experiments show that the SFCN and MFCN outperform existing splicing localization algorithms, and that the MFCN can achieve finer localization than the SFCN. Neuromorphic algorithms, which are comprised of highly complex, large-scale networks of artificial neurons, are increasingly used for a variety of recognition, classification, search and vision tasks. However, their computational and energy requirements can be quite high, and hence their energy-efficient implementation is of great interest. We propose a new approach to design energy-efficient hardware implementations of large-scale neural networks (NNs) using approximate computing. Our work is motivated by the observations that (i) NNs are used in applications where less-than-perfect results are acceptable, and often inevitable, and (ii) they are highly resilient to inexactness in many (but not all) of their constituent computations. We make two key contributions. First, we propose a method to transform any given NN into an Approximate Neural Network (AxNN). This is performed by (i) adapting the backpropagation technique, which is commonly used to train these networks, to quantify the impact of approximating each neuron to the overall network quality (e.g., classification accuracy), and (ii) selectively approximating those neurons that impact network quality the least. Further, we make the key observation that training is a naturally error-healing process that can be used to mitigate the impact of approximations to neurons. Therefore, we incrementally retrain the network with the approximations in-place, reclaiming a significant portion of the quality ceded by approximations. As a second contribution, we propose a programmable and quality-configurable neuromorphic processing engine (qcNPE), which utilizes arrays of specialized processing elements that execute neuron computations with dynamically configurable accuracies and can be used to execute AxNNs from diverse applications. We evaluated the proposed approach by constructing AXNNs for 6 recognition applications (ranging in complexity from 12–47,818 neurons and 160–3,155,968 connections) and executing them on two different platforms - qcNPE implementation containing 272 processing elements in 45nm technology and a commodity Intel Xeon server. Our results demonstrate 1.14X–1.92X energy benefits for virtually no loss (< 0.5 ) in output quality, and even higher improvements (upto 2.3X) when some loss (upto 7.5 ) in output quality is acceptable. CANDECOMP PARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank . In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. To address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian treatment by incorporating a sparsity-inducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters, resulting in automatic rank determination. To learn the model, we develop an efficient deterministic Bayesian inference algorithm, which scales linearly with data size. Our method is characterized as a tuning parameter-free approach, which can effectively infer underlying multilinear factors with a low-rank constraint, while also providing predictive distributions over missing entries. Extensive simulations on synthetic data illustrate the intrinsic capability of our method to recover the ground-truth of CP rank and prevent the overfitting problem, even when a large amount of entries are missing. Moreover, the results from real-world applications, including image inpainting and facial image synthesis, demonstrate that our method outperforms state-of-the-art approaches for both tensor factorization and tensor completion in terms of predictive performance. Optimizing deep neural networks (DNNs) often suffers from the ill-conditioned problem. We observe that the scaling-based weight space symmetry property in rectified nonlinear network will cause this negative effect. Therefore, we propose to constrain the incoming weights of each neuron to be unit-norm, which is formulated as an optimization problem over Oblique manifold. A simple yet efficient method referred to as projection based weight normalization (PBWN) is also developed to solve this problem. PBWN executes standard gradient updates, followed by projecting the updated weight back to Oblique manifold. This proposed method has the property of regularization and collaborates well with the commonly used batch normalization technique. We conduct comprehensive experiments on several widely-used image datasets including CIFAR-10, CIFAR-100, SVHN and ImageNet for supervised learning over the state-of-the-art convolutional neural networks, such as Inception, VGG and residual networks. The results show that our method is able to improve the performance of DNNs with different architectures consistently. We also apply our method to Ladder network for semi-supervised learning on permutation invariant MNIST dataset, and our method outperforms the state-of-the-art methods: we obtain test errors as 2.52 , 1.06 , and 0.91 with only 20, 50, and 100 labeled samples, respectively. To reduce the significant redundancy in deep Convolutional Neural Networks (CNNs), most existing methods prune neurons by only considering statistics of an individual layer or two consecutive layers (e.g., prune one layer to minimize the reconstruction error of the next layer), ignoring the effect of error propagation in deep networks. In contrast, we argue that it is essential to prune neurons in the entire neuron network jointly based on a unified goal: minimizing the reconstruction error of important responses in the "final response layer" (FRL), which is the second-to-last layer before classification, for a pruned network to retrain its predictive power. Specifically, we apply feature ranking techniques to measure the importance of each neuron in the FRL, and formulate network pruning as a binary integer optimization problem and derive a closed-form solution to it for pruning neurons in earlier layers. Based on our theoretical analysis, we propose the Neuron Importance Score Propagation (NISP) algorithm to propagate the importance scores of final responses to every neuron in the network. The CNN is pruned by removing neurons with least importance, and then fine-tuned to retain its predictive power. NISP is evaluated on several datasets with multiple CNN models and demonstrated to achieve significant acceleration and compression with negligible accuracy loss. Based on the assumption that there exists a neural network that efficiently represents a set of Boolean functions between all binary inputs and outputs, we propose a process for developing and deploying neural networks whose weight parameters, bias terms, input, and intermediate hidden layer output signals, are all binary-valued, and require only basic bit logic for the feedforward pass. The proposed Bitwise Neural Network (BNN) is especially suitable for resource-constrained environments, since it replaces either floating or fixed-point arithmetic with significantly more efficient bitwise operations. Hence, the BNN requires for less spatial complexity, less memory bandwidth, and less power consumption in hardware. In order to design such networks, we propose to add a few training schemes, such as weight compression and noisy backpropagation, which result in a bitwise network that performs almost as well as its corresponding real-valued network. We test the proposed network on the MNIST dataset, represented using binary features, and show that BNNs result in competitive performance while offering dramatic computational savings. Spiking neural networks (SNNs) can potentially offer an efficient way of doing inference because the neurons in the networks are sparsely activated and computations are event-driven. Previous work showed that simple continuous-valued deep Convolutional Neural Networks (CNNs) can be converted into accurate spiking equivalents. These networks did not include certain common operations such as max-pooling, softmax, batch-normalization and Inception-modules. This paper presents spiking equivalents of these operations therefore allowing conversion of nearly arbitrary CNN architectures. We show conversion of popular CNN architectures, including VGG-16 and Inception-v3, into SNNs that produce the best results reported to date on MNIST, CIFAR-10 and the challenging ImageNet dataset. SNNs can trade off classification error rate against the number of available operations whereas deep continuous-valued neural networks require a fixed number of operations to achieve their classification error rate. From the examples of LeNet for MNIST and BinaryNet for CIFAR-10, we show that with an increase in error rate of a few percentage points, the SNNs can achieve more than 2x reductions in operations compared to the original CNNs. This highlights the potential of SNNs in particular when deployed on power-efficient neuromorphic spiking neuron chips, for use in embedded applications. A key step in insect olfaction is the transformation of a dense representation of odors in a small population of neurons - projection neurons (PNs) of the antennal lobe - into a sparse representation in a much larger population of neurons - Kenyon cells (KCs) of the mushroom body. What computational purpose does this transformation serve? We propose that the PN-KC network implements an online clustering algorithm which we derive from the k-means cost function. The vector of PN-KC synaptic weights converging onto a given KC represents the corresponding cluster centroid. KC activities represent attribution indices, i.e. the degree to which a given odor presentation is attributed to each cluster. Remarkably, such clustering view of the PN-KC circuit naturally accounts for several of its salient features. First, attribution indices are nonnegative thus rationalizing rectification in KCs. Second, the constraint on the total sum of attribution indices for each presentation is enforced by a Lagrange multiplier identified with the activity of a single inhibitory interneuron reciprocally connected with KCs. Third, the soft-clustering version of our algorithm reproduces observed sparsity and overcompleteness of the KC representation which may optimize supervised classification downstream. Conventional convolutional neural networks use either a linear or a nonlinear filter to extract features from an image patch (region) of spatial size @math (typically, @math is small and is equal to @math , e.g., @math is 5 or 7). Generally, the size of the filter is equal to the size @math of the input patch. We argue that the representational ability of equal-size strategy is not strong enough. To overcome the drawback, we propose to use subpatch filter whose spatial size @math is smaller than @math . The proposed subpatch filter consists of two subsequent filters. The first one is a linear filter of spatial size @math and is aimed at extracting features from spatial domain. The second one is of spatial size @math and is used for strengthening the connection between different input feature channels and for reducing the number of parameters. The subpatch filter convolves with the input patch and the resulting network is called a subpatch network. Taking the output of one subpatch network as input, we further repeat constructing subpatch networks until the output contains only one neuron in spatial domain. These subpatch networks form a new network called the cascaded subpatch network (CSNet). The feature layer generated by CSNet is called the csconv layer. For the whole input image, we construct a deep neural network by stacking a sequence of csconv layers. Experimental results on five benchmark data sets demonstrate the effectiveness and compactness of the proposed CSNet. For example, our CSNet reaches a test error of 5.68 on the CIFAR10 data set without model averaging. To the best of our knowledge, this is the best result ever obtained on the CIFAR10 data set. This paper presents a synaptic weight association training (SWAT) algorithm for spiking neural networks (SNNs). SWAT merges the Bienenstock-Cooper-Munro (BCM) learning rule with spike timing dependent plasticity (STDP). The STDP BCM rule yields a unimodal weight distribution where the height of the plasticity window associated with STDP is modulated causing stability after a period of training. The SNN uses a single training neuron in the training phase where data associated with all classes is passed to this neuron. The rule then maps weights to the classifying output neurons to reflect similarities in the data across the classes. The SNN also includes both excitatory and inhibitory facilitating synapses which create a frequency routing capability allowing the information presented to the network to be routed to different hidden layer neurons. A variable neuron threshold level simulates the refractory period. SWAT is initially benchmarked against the nonlinearly separable Iris and Wisconsin Breast Cancer datasets. Results presented show that the proposed training algorithm exhibits a convergence accuracy of 95.5 and 96.2 for the Iris and Wisconsin training sets, respectively, and 95.3 and 96.7 for the testing sets, noise experiments show that SWAT has a good generalization capability. SWAT is also benchmarked using an isolated digit automatic speech recognition (ASR) system where a subset of the TI46 speech corpus is used. Results show that with SWAT as the classifier, the ASR system provides an accuracy of 98.875 for training and 95.25 for testing.