E. Paxon Frady

Alpha Renner, Lazar Supic, Andreea Danielescu et al. · eth-zurich

Analyzing a visual scene by inferring the configuration of a generative model is widely considered the most flexible and generalizable approach to scene understanding. Yet, one major problem is the computational challenge of the inference procedure, involving a combinatorial search across object identities and poses. Here we propose a neuromorphic solution exploiting three key concepts: (1) a computational framework based on Vector Symbolic Architectures (VSA) with complex-valued vectors; (2) the design of Hierarchical Resonator Networks (HRN) to factorize the non-commutative transforms translation and rotation in visual scenes; (3) the design of a multi-compartment spiking phasor neuron model for implementing complex-valued resonator networks on neuromorphic hardware. The VSA framework uses vector binding operations to form a generative image model in which binding acts as the equivariant operation for geometric transformations. A scene can, therefore, be described as a sum of vector products, which can then be efficiently factorized by a resonator network to infer objects and their poses. The HRN features a partitioned architecture in which vector binding is equivariant for horizontal and vertical translation within one partition and for rotation and scaling within the other partition. The spiking neuron model allows mapping the resonator network onto efficient and low-power neuromorphic hardware. Our approach is demonstrated on synthetic scenes composed of simple 2D shapes undergoing rigid geometric transformations and color changes. A companion paper demonstrates the same approach in real-world application scenarios for machine vision and robotics.

Garrick Orchard, E. Paxon Frady, Daniel Ben Dayan Rubin et al.

The biologically inspired spiking neurons used in neuromorphic computing are nonlinear filters with dynamic state variables -- very different from the stateless neuron models used in deep learning. The next version of Intel's neuromorphic research processor, Loihi 2, supports a wide range of stateful spiking neuron models with fully programmable dynamics. Here we showcase advanced spiking neuron models that can be used to efficiently process streaming data in simulation experiments on emulated Loihi 2 hardware. In one example, Resonate-and-Fire (RF) neurons are used to compute the Short Time Fourier Transform (STFT) with similar computational complexity but 47x less output bandwidth than the conventional STFT. In another example, we describe an algorithm for optical flow estimation using spatiotemporal RF neurons that requires over 90x fewer operations than a conventional DNN-based solution. We also demonstrate promising preliminary results using backpropagation to train RF neurons for audio classification tasks. Finally, we show that a cascade of Hopf resonators - a variant of the RF neuron - replicates novel properties of the cochlea and motivates an efficient spike-based spectrogram encoder.

E. Paxon Frady, Denis Kleyko, Christopher J. Kymn et al.

Vector space models for symbolic processing that encode symbols by random vectors have been proposed in cognitive science and connectionist communities under the names Vector Symbolic Architecture (VSA), and, synonymously, Hyperdimensional (HD) computing. In this paper, we generalize VSAs to function spaces by mapping continuous-valued data into a vector space such that the inner product between the representations of any two data points represents a similarity kernel. By analogy to VSA, we call this new function encoding and computing framework Vector Function Architecture (VFA). In VFAs, vectors can represent individual data points as well as elements of a function space (a reproducing kernel Hilbert space). The algebraic vector operations, inherited from VSA, correspond to well-defined operations in function space. Furthermore, we study a previously proposed method for encoding continuous data, fractional power encoding (FPE), which uses exponentiation of a random base vector to produce randomized representations of data points and fulfills the kernel properties for inducing a VFA. We show that the distribution from which elements of the base vector are sampled determines the shape of the FPE kernel, which in turn induces a VFA for computing with band-limited functions. In particular, VFAs provide an algebraic framework for implementing large-scale kernel machines with random features, extending Rahimi and Recht, 2007. Finally, we demonstrate several applications of VFA models to problems in image recognition, density estimation and nonlinear regression. Our analyses and results suggest that VFAs constitute a powerful new framework for representing and manipulating functions in distributed neural systems, with myriad applications in artificial intelligence.

Denis Kleyko, E. Paxon Frady, Friedrich T. Sommer

Various non-classical approaches of distributed information processing, such as neural networks, computation with Ising models, reservoir computing, vector symbolic architectures, and others, employ the principle of collective-state computing. In this type of computing, the variables relevant in a computation are superimposed into a single high-dimensional state vector, the collective-state. The variable encoding uses a fixed set of random patterns, which has to be stored and kept available during the computation. Here we show that an elementary cellular automaton with rule 90 (CA90) enables space-time tradeoff for collective-state computing models that use random dense binary representations, i.e., memory requirements can be traded off with computation running CA90. We investigate the randomization behavior of CA90, in particular, the relation between the length of the randomization period and the size of the grid, and how CA90 preserves similarity in the presence of the initialization noise. Based on these analyses we discuss how to optimize a collective-state computing model, in which CA90 expands representations on the fly from short seed patterns - rather than storing the full set of random patterns. The CA90 expansion is applied and tested in concrete scenarios using reservoir computing and vector symbolic architectures. Our experimental results show that collective-state computing with CA90 expansion performs similarly compared to traditional collective-state models, in which random patterns are generated initially by a pseudo-random number generator and then stored in a large memory.

E. Paxon Frady, Denis Kleyko, Friedrich T. Sommer

Symbolic reasoning and neural networks are often considered incompatible approaches. Connectionist models known as Vector Symbolic Architectures (VSAs) can potentially bridge this gap. However, classical VSAs and neural networks are still considered incompatible. VSAs encode symbols by dense pseudo-random vectors, where information is distributed throughout the entire neuron population. Neural networks encode features locally, often forming sparse vectors of neural activation. Following Rachkovskij (2001); Laiho et al. (2015), we explore symbolic reasoning with sparse distributed representations. The core operations in VSAs are dyadic operations between vectors to express variable binding and the representation of sets. Thus, algebraic manipulations enable VSAs to represent and process data structures in a vector space of fixed dimensionality. Using techniques from compressed sensing, we first show that variable binding between dense vectors in VSAs is mathematically equivalent to tensor product binding between sparse vectors, an operation which increases dimensionality. This result implies that dimensionality-preserving binding for general sparse vectors must include a reduction of the tensor matrix into a single sparse vector. Two options for sparsity-preserving variable binding are investigated. One binding method for general sparse vectors extends earlier proposals to reduce the tensor product into a vector, such as circular convolution. The other method is only defined for sparse block-codes, block-wise circular convolution. Our experiments reveal that variable binding for block-codes has ideal properties, whereas binding for general sparse vectors also works, but is lossy, similar to previous proposals. We demonstrate a VSA with sparse block-codes in example applications, cognitive reasoning and classification, and discuss its relevance for neuroscience and neural networks.

E. Paxon Frady, Spencer Kent, Bruno A. Olshausen et al.

The ability to encode and manipulate data structures with distributed neural representations could qualitatively enhance the capabilities of traditional neural networks by supporting rule-based symbolic reasoning, a central property of cognition. Here we show how this may be accomplished within the framework of Vector Symbolic Architectures (VSA) (Plate, 1991; Gayler, 1998; Kanerva, 1996), whereby data structures are encoded by combining high-dimensional vectors with operations that together form an algebra on the space of distributed representations. In particular, we propose an efficient solution to a hard combinatorial search problem that arises when decoding elements of a VSA data structure: the factorization of products of multiple code vectors. Our proposed algorithm, called a resonator network, is a new type of recurrent neural network that interleaves VSA multiplication operations and pattern completion. We show in two examples -- parsing of a tree-like data structure and parsing of a visual scene -- how the factorization problem arises and how the resonator network can solve it. More broadly, resonator networks open the possibility to apply VSAs to myriad artificial intelligence problems in real-world domains. A companion paper (Kent et al., 2020) presents a rigorous analysis and evaluation of the performance of resonator networks, showing it out-performs alternative approaches.

E. Paxon Frady, Garrick Orchard, David Florey et al.

Neuromorphic computing applies insights from neuroscience to uncover innovations in computing technology. In the brain, billions of interconnected neurons perform rapid computations at extremely low energy levels by leveraging properties that are foreign to conventional computing systems, such as temporal spiking codes and finely parallelized processing units integrating both memory and computation. Here, we showcase the Pohoiki Springs neuromorphic system, a mesh of 768 interconnected Loihi chips that collectively implement 100 million spiking neurons in silicon. We demonstrate a scalable approximate k-nearest neighbor (k-NN) algorithm for searching large databases that exploits neuromorphic principles. Compared to state-of-the-art conventional CPU-based implementations, we achieve superior latency, index build time, and energy efficiency when evaluated on several standard datasets containing over 1 million high-dimensional patterns. Further, the system supports adding new data points to the indexed database online in O(1) time unlike all but brute force conventional k-NN implementations.

Denis Kleyko, Mansour Kheffache, E. Paxon Frady et al.

The deployment of machine learning algorithms on resource-constrained edge devices is an important challenge from both theoretical and applied points of view. In this article, we focus on resource-efficient randomly connected neural networks known as Random Vector Functional Link (RVFL) networks since their simple design and extremely fast training time make them very attractive for solving many applied classification tasks. We propose to represent input features via the density-based encoding known in the area of stochastic computing and use the operations of binding and bundling from the area of hyperdimensional computing for obtaining the activations of the hidden neurons. Using a collection of 121 real-world datasets from the UCI Machine Learning Repository, we empirically show that the proposed approach demonstrates higher average accuracy than the conventional RVFL. We also demonstrate that it is possible to represent the readout matrix using only integers in a limited range with minimal loss in the accuracy. In this case, the proposed approach operates only on small n-bits integers, which results in a computationally efficient architecture. Finally, through hardware Field-Programmable Gate Array (FPGA) implementations, we show that such an approach consumes approximately eleven times less energy than that of the conventional RVFL.

Spencer J. Kent, E. Paxon Frady, Friedrich T. Sommer et al.

We develop theoretical foundations of Resonator Networks, a new type of recurrent neural network introduced in Frady et al. (2020) to solve a high-dimensional vector factorization problem arising in Vector Symbolic Architectures. Given a composite vector formed by the Hadamard product between a discrete set of high-dimensional vectors, a Resonator Network can efficiently decompose the composite into these factors. We compare the performance of Resonator Networks against optimization-based methods, including Alternating Least Squares and several gradient-based algorithms, showing that Resonator Networks are superior in several important ways. This advantage is achieved by leveraging a combination of nonlinear dynamics and "searching in superposition," by which estimates of the correct solution are formed from a weighted superposition of all possible solutions. While the alternative methods also search in superposition, the dynamics of Resonator Networks allow them to strike a more effective balance between exploring the solution space and exploiting local information to drive the network toward probable solutions. Resonator Networks are not guaranteed to converge, but within a particular regime they almost always do. In exchange for relaxing this guarantee of global convergence, Resonator Networks are dramatically more effective at finding factorizations than all alternative approaches considered.

E. Paxon Frady, Friedrich T. Sommer

Information coding by precise timing of spikes can be faster and more energy-efficient than traditional rate coding. However, spike-timing codes are often brittle, which has limited their use in theoretical neuroscience and computing applications. Here, we propose a novel type of attractor neural network in complex state space, and show how it can be leveraged to construct spiking neural networks with robust computational properties through a phase-to-timing mapping. Building on Hebbian neural associative memories, like Hopfield networks, we first propose threshold phasor associative memory (TPAM) networks. Complex phasor patterns whose components can assume continuous-valued phase angles and binary magnitudes can be stored and retrieved as stable fixed points in the network dynamics. TPAM achieves high memory capacity when storing sparse phasor patterns, and we derive the energy function that governs its fixed point attractor dynamics. Second, through simulation experiments we show how the complex algebraic computations in TPAM can be approximated by a biologically plausible network of integrate-and-fire neurons with synaptic delays and recurrently connected inhibitory interneurons. The fixed points of TPAM in the complex domain are commensurate with stable periodic states of precisely timed spiking activity that are robust to perturbation. The link established between rhythmic firing patterns and complex attractor dynamics has implications for the interpretation of spike patterns seen in neuroscience, and can serve as a framework for computation in emerging neuromorphic devices.

Ashish Kapoor, E. Paxon Frady, Stefanie Jegelka et al.

We introduce and study methods for inferring and learning from correspondences among neurons. The approach enables alignment of data from distinct multiunit studies of nervous systems. We show that the methods for inferring correspondences combine data effectively from cross-animal studies to make joint inferences about behavioral decision making that are not possible with the data from a single animal. We focus on data collection, machine learning, and prediction in the representative and long-studied invertebrate nervous system of the European medicinal leech. Acknowledging the computational intractability of the general problem of identifying correspondences among neurons, we introduce efficient computational procedures for matching neurons across animals. The methods include techniques that adjust for missing cells or additional cells in the different data sets that may reflect biological or experimental variation. The methods highlight the value harnessing inference and learning in new kinds of computational microscopes for multiunit neurobiological studies.