Recurrent correlation neural networks (RCNNs), introduced by Chiueh and Goodman as an improved version of the bipolar correlation-based Hopfield neural network, can be used to implement high-capacity associative memories. In this paper, we extend the bipolar RCNNs for processing hypercomplex-valued data. Precisely, we present the mathematical background for a broad class of hypercomplex-valued RCNNs. Then, we provide the necessary conditions which ensure that a hypercomplex-valued RCNN always settles at an equilibrium using either synchronous or asynchronous update modes. Examples with bipolar, complex, hyperbolic, quaternion, and octonion-valued RCNNs are given to illustrate the theoretical results. Finally, computational experiments confirm the potential application of hypercomplex-valued RCNNs as associative memories designed for the storage and recall of gray-scale images.
Federal Institute of Education, Science and Technology of Espírito Santo（圣埃斯皮里图联邦教育、科学与技术学院）；University of Campinas（坎皮纳斯大学）
In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex number systems. Real-part associative hypercomplex number systems generalize the well-known Cayley-Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as parti…
This paper aims to establish a framework for extreme learning machines (ELMs) on general hypercomplex algebras. Hypercomplex neural networks are machine learning models that feature higher-dimension numbers as parameters, inputs, and outputs. Firstly, we review broad hypercomplex algebras and show a framework to operate in these algebras through real-valued linear algebra operations in a robust manner. We proceed to explore a handful of well-known four-dimensional examples. Then, we propose the hypercomplex-valued ELMs and derive their learning using a hypercomplex-valued least-squares problem…
Hypercomplex-valued neural networks, including quaternion-valued neural networks, can treat multi-dimensional data as a single entity. In this paper, we present the quaternion-valued recurrent projection neural networks (QRPNNs). Briefly, QRPNNs are obtained by combining the non-local projection learning with the quaternion-valued recurrent correlation neural network (QRCNNs). We show that QRPNNs overcome the cross-talk problem of QRCNNs. Thus, they are appropriate to implement associative memories. Furthermore, computational experiments reveal that QRPNNs exhibit greater storage capacity and …
Sapienza University of Rome（罗马大学）；Amazon Web Services AI（亚马逊云服务人工智能部门）
Hypercomplex neural networks have proven to reduce the overall number of parameters while ensuring valuable performance by leveraging the properties of Clifford algebras. Recently, hypercomplex linear layers have been further improved by involving efficient parameterized Kronecker products. In this paper, we define the parameterization of hypercomplex convolutional layers and introduce the family of parameterized hypercomplex neural networks (PHNNs) that are lightweight and efficient large-scale models. Our method grasps the convolution rules and the filter organization directly from data with…
Escuela Superior Politécnica del Litoral（滨海理工学院）；Universidade Estadual de Campinas（坎皮纳斯州立大学）
The field of neural networks has seen significant advances in recent years with the development of deep and convolutional neural networks. Although many of the current works address real-valued models, recent studies reveal that neural networks with hypercomplex-valued parameters can better capture, generalize, and represent the complexity of multidimensional data. This paper explores the quaternion-valued convolutional neural network application for a pattern recognition task from medicine, namely, the diagnosis of acute lymphoblastic leukemia. Precisely, we compare the performance of real-va…
ETH Zurich（苏黎世联邦理工学院）；Tri-Institutional Training Program in Computational Biology and Medicine, Weill Cornell Medical（威尔康奈尔医学院计算生物学与医学三机构联合培养项目）
Generative Adversarial Networks (GANs) have shown remarkable success as a framework for training models to produce realistic-looking data. In this work, we propose a Recurrent GAN (RGAN) and Recurrent Conditional GAN (RCGAN) to produce realistic real-valued multi-dimensional time series, with an emphasis on their application to medical data. RGANs make use of recurrent neural networks in the generator and the discriminator. In the case of RCGANs, both of these RNNs are conditioned on auxiliary information. We demonstrate our models in a set of toy datasets, where we show visually and quantitat…
Complex-valued neural networks (CVNNs) are an emerging field of research in neural networks due to their potential representational properties for audio, image, and physiological signals. It is common in signal processing to transform sequences of real values to the complex domain via a set of complex basis functions, such as the Fourier transform. We show how CVNNs can be used to learn complex representations of real valued time-series data. We present methods and results using a framework that can compose holomorphic and non-holomorphic functions in a multi-layer network using a theoretical …
This paper presents the recurrent estimation of distributions (RED) for modeling real-valued data in a semiparametric fashion. RED models make two novel uses of recurrent neural networks (RNNs) for density estimation of general real-valued data. First, RNNs are used to transform input covariates into a latent space to better capture conditional dependencies in inputs. After, an RNN is used to compute the conditional distributions of the latent covariates. The resulting model is efficient to train, compute, and sample from, whilst producing normalized pdfs. The effectiveness of RED is shown via…
Learning to predict solutions to real-valued combinatorial graph problems promises efficient approximations. As demonstrated based on the NP-hard edge clique cover number, recurrent neural networks (RNNs) are particularly suited for this task and can even outperform state-of-the-art heuristics. However, the theoretical framework for estimating real-valued RNNs is understood only poorly. As our primary contribution, this is the first work that upper bounds the sample complexity for learning real-valued RNNs. While such derivations have been made earlier for feed-forward and convolutional neural…
Institute for Computer Science（计算机科学研究所）；University of Bonn（波恩大学）；School of Computing（计算学院）；National University of Singapore（新加坡国立大学）
Complex numbers have long been favoured for digital signal processing, yet complex representations rarely appear in deep learning architectures. RNNs, widely used to process time series and sequence information, could greatly benefit from complex representations. We present a novel complex gated recurrent cell, which is a hybrid cell combining complex-valued and norm-preserving state transitions with a gating mechanism. The resulting RNN exhibits excellent stability and convergence properties and performs competitively on the synthetic memory and adding task, as well as on the real-world tasks…
Recurrent Neural Networks (RNNs) are designed to handle sequential data but suffer from vanishing or exploding gradients. Recent work on Unitary Recurrent Neural Networks (uRNNs) have been used to address this issue and in some cases, exceed the capabilities of Long Short-Term Memory networks (LSTMs). We propose a simpler and novel update scheme to maintain orthogonal recurrent weight matrices without using complex valued matrices. This is done by parametrizing with a skew-symmetric matrix using the Cayley transform. Such a parametrization is unable to represent matrices with negative one eige…
Networks of spiking neurons and Winner-Take-All spiking circuits (WTA-SNNs) can detect information encoded in spatio-temporal multi-valued events. These are described by the timing of events of interest, e.g., clicks, as well as by categorical numerical values assigned to each event, e.g., like or dislike. Other use cases include object recognition from data collected by neuromorphic cameras, which produce, for each pixel, signed bits at the times of sufficiently large brightness variations. Existing schemes for training WTA-SNNs are limited to rate-encoding solutions, and are hence able to de…
Despite multiple efforts made towards adopting complex-valued deep neural networks (DNNs), it remains an open question whether complex-valued DNNs are generally more effective than real-valued DNNs for monaural speech enhancement. This work is devoted to presenting a critical assessment by systematically examining complex-valued DNNs against their real-valued counterparts. Specifically, we investigate complex-valued DNN atomic units, including linear layers, convolutional layers, long short-term memory (LSTM), and gated linear units. By comparing complex- and real-valued versions of fundamenta…
Oregon State University（俄勒冈州立大学）；Google Brain（谷歌大脑）
Recurrent neural networks (RNNs) are an effective representation of control policies for a wide range of reinforcement and imitation learning problems. RNN policies, however, are particularly difficult to explain, understand, and analyze due to their use of continuous-valued memory vectors and observation features. In this paper, we introduce a new technique, Quantized Bottleneck Insertion, to learn finite representations of these vectors and features. The result is a quantized representation of the RNN that can be analyzed to improve our understanding of memory use and general behavior. We pr…
Long Short-Term Memory (LSTM) is one of the most widely used recurrent structures in sequence modeling. It aims to use gates to control information flow (e.g., whether to skip some information or not) in the recurrent computations, although its practical implementation based on soft gates only partially achieves this goal. In this paper, we propose a new way for LSTM training, which pushes the output values of the gates towards 0 or 1. By doing so, we can better control the information flow: the gates are mostly open or closed, instead of in a middle state, which makes the results more interpr…
Mila；Université de Montréal（蒙特利尔大学）；McGill University（麦吉尔大学）
In this paper, we unravel a fundamental connection between weighted finite automata~(WFAs) and second-order recurrent neural networks~(2-RNNs): in the case of sequences of discrete symbols, WFAs and 2-RNNs with linear activation functions are expressively equivalent. Motivated by this result, we build upon a recent extension of the spectral learning algorithm to vector-valued WFAs and propose the first provable learning algorithm for linear 2-RNNs defined over sequences of continuous input vectors. This algorithm relies on estimating low rank sub-blocks of the so-called Hankel tensor, from whi…
Unitary Evolution Recurrent Neural Networks (uRNNs) have three attractive properties: (a) the unitary property, (b) the complex-valued nature, and (c) their efficient linear operators. The literature so far does not address -- how critical is the unitary property of the model? Furthermore, uRNNs have not been evaluated on large tasks. To study these shortcomings, we propose the complex evolution Recurrent Neural Networks (ceRNNs), which is similar to uRNNs but drops the unitary property selectively. On a simple multivariate linear regression task, we illustrate that dropping the constraints im…
Université d’Avignon（阿维尼翁大学）；Orkis（奥基斯公司）；MILA（米拉研究所）；Université de Montréal（蒙特利尔大学）；McGill University（麦吉尔大学）；Element AI（元素人工智能公司）
Recurrent neural networks (RNNs) are powerful architectures to model sequential data, due to their capability to learn short and long-term dependencies between the basic elements of a sequence. Nonetheless, popular tasks such as speech or images recognition, involve multi-dimensional input features that are characterized by strong internal dependencies between the dimensions of the input vector. We propose a novel quaternion recurrent neural network (QRNN), alongside with a quaternion long-short term memory neural network (QLSTM), that take into account both the external relations and these in…
This paper proposes a Bitwise Gated Recurrent Unit (BGRU) network for the single-channel source separation task. Recurrent Neural Networks (RNN) require several sets of weights within its cells, which significantly increases the computational cost compared to the fully-connected networks. To mitigate this increased computation, we focus on the GRU cells and quantize the feedforward procedure with binarized values and bitwise operations. The BGRU network is trained in two stages. The real-valued weights are pretrained and transferred to the bitwise network, which are then incrementally binarize…
Montreal Institute for Learning Algorithms (MILA)（蒙特利尔学习算法研究所（MILA））；Microsoft Research（微软研究院）；Element AI（元素人工智能公司）；Ecole Polytechnique（巴黎综合理工学院）
At present, the vast majority of building blocks, techniques, and architectures for deep learning are based on real-valued operations and representations. However, recent work on recurrent neural networks and older fundamental theoretical analysis suggests that complex numbers could have a richer representational capacity and could also facilitate noise-robust memory retrieval mechanisms. Despite their attractive properties and potential for opening up entirely new neural architectures, complex-valued deep neural networks have been marginalized due to the absence of the building blocks require…
Recurrent neural networks (RNN) are at the core of modern automatic speech recognition (ASR) systems. In particular, long-short term memory (LSTM) recurrent neural networks have achieved state-of-the-art results in many speech recognition tasks, due to their efficient representation of long and short term dependencies in sequences of inter-dependent features. Nonetheless, internal dependencies within the element composing multidimensional features are weakly considered by traditional real-valued representations. We propose a novel quaternion long-short term memory (QLSTM) recurrent neural netw…
Recurrent Graph Neural Networks and Arithmetic Circuits
Institut für Theoretische Informatik, Leibniz Universität Hannover, Germany（汉诺威莱布尼茨大学理论计算机研究所）；School of Computing Science, University of Glasgow, UK（格拉斯哥大学计算机科学学院）
A Recurrent Neural Network (RNN) for audio synthesis is trained by augmenting the audio input with information about signal characteristics such as pitch, amplitude, and instrument. The result after training is an audio synthesizer that is played like a musical instrument with the desired musical characteristics provided as continuous parametric control. The focus of this paper is on conditioning data-driven synthesis models with real-valued parameters, and in particular, on the ability of the system a) to generalize and b) to be responsive to parameter values and sequences not seen during tra…
Several speech processing systems have demonstrated considerable performance improvements when deep complex neural networks (DCNN) are coupled with self-attention (SA) networks. However, the majority of DCNN-based studies on speech dereverberation that employ self-attention do not explicitly account for the inter-dependencies between real and imaginary features when computing attention. In this study, we propose a complex-valued T-F attention (TFA) module that models spectral and temporal dependencies by computing two-dimensional attention maps across time and frequency dimensions. We validate…
Compared to conventional artificial neurons that produce dense and real-valued responses, biologically-inspired spiking neurons transmit sparse and binary information, which can also lead to energy-efficient implementations. Recent research has shown that spiking neural networks can be trained like standard recurrent neural networks using the surrogate gradient method. They have shown promising results on speech command recognition tasks. Using the same technique, we show that they are scalable to large vocabulary continuous speech recognition, where they are capable of replacing LSTMs in the …
Université d'Avignon（阿维尼翁大学）；Université de Montréal（蒙特利尔大学）；McGill University（麦吉尔大学）；Orkis（奥基斯公司）；Element AI（Element AI公司）
Recently, the connectionist temporal classification (CTC) model coupled with recurrent (RNN) or convolutional neural networks (CNN), made it easier to train speech recognition systems in an end-to-end fashion. However in real-valued models, time frame components such as mel-filter-bank energies and the cepstral coefficients obtained from them, together with their first and second order derivatives, are processed as individual elements, while a natural alternative is to process such components as composed entities. We propose to group such elements in the form of quaternions and to process thes…
University of Wisconsin Madison（威斯康星大学麦迪逊分校）；University of California, Berkeley（加州大学伯克利分校）
Efforts are underway to study ways via which the power of deep neural networks can be extended to non-standard data types such as structured data (e.g., graphs) or manifold-valued data (e.g., unit vectors or special matrices). Often, sizable empirical improvements are possible when the geometry of such data spaces are incorporated into the design of the model, architecture, and the algorithms. Motivated by neuroimaging applications, we study formulations where the data are {\em sequential manifold-valued measurements}. This case is common in brain imaging, where the samples correspond to symme…
NTT Communication Science Laboratories（NTT通信科学研究所）
An optical neural network (ONN) is a promising system due to its high-speed and low-power operation. Its linear unit performs a multiplication of an input vector and a weight matrix in optical analog circuits. Among them, a circuit with a multiple-layered structure of programmable Mach-Zehnder interferometers (MZIs) can realize a specific class of unitary matrices with a limited number of MZIs as its weight matrix. The circuit is effective for balancing the number of programmable MZIs and ONN performance. However, it takes a lot of time to learn MZI parameters of the circuit with a conventiona…
