Variable Binding for Sparse Distributed Representations: Theory and Applications

Principled neuromorphic reservoir computing

Reservoir computing advances the intriguing idea that a nonlinear recurrent neural circuit-the reservoir-can encode spatio-temporal input signals to enable efficient ways to perform tasks like classification or regression. However, recently the idea of a monolithic reservoir network that simultaneously buffers input signals and expands them into nonlinear features has been challenged. A representation scheme in which memory buffer and expansion into higher-order polynomial features can be configured separately has been shown to significantly outperform traditional reservoir computing in prediction of multivariate time-series. Here we propose a configurable neuromorphic representation scheme that provides competitive performance on prediction, but with significantly better scaling properties than directly materializing higher-order features as in prior work. Our approach combines the use of randomized representations from traditional reservoir computing with mathematical principles for approximating polynomial kernels via such representations. While the memory buffer can be realized with standard reservoir networks, computing higher-order features requires networks of 'Sigma-Pi' neurons, i.e., neurons that enable both summation as well as multiplication of inputs. Finally, we provide an implementation of the memory buffer and Sigma-Pi networks on Loihi 2, an existing neuromorphic hardware platform.

Hyperseed: Unsupervised Learning With Vector Symbolic Architectures

Motivated by recent innovations in biologically inspired neuromorphic hardware, this article presents a novel unsupervised machine learning algorithm named Hyperseed that draws on the principles of vector symbolic architectures (VSAs) for fast learning of a topology preserving feature map of unlabeled data. It relies on two major operations of VSA, binding and bundling. The algorithmic part of Hyperseed is expressed within the Fourier holographic reduced representations (FHRR) model, which is specifically suited for implementation on spiking neuromorphic hardware. The two primary contributions of the Hyperseed algorithm are few-shot learning and a learning rule based on single vector operation. These properties are empirically evaluated on synthetic datasets and on illustrative benchmark use cases, IRIS classification, and a language identification task using the n -gram statistics. The results of these experiments confirm the capabilities of Hyperseed and its applications in neuromorphic hardware.

Hyperdimensional computing with holographic and adaptive encoder

Introduction

Brain-inspired computing has become an emerging field, where a growing number of works focus on developing algorithms that bring machine learning closer to human brains at the functional level. As one of the promising directions, Hyperdimensional Computing (HDC) is centered around the idea of having holographic and high-dimensional representation as the neural activities in our brains. Such representation is the fundamental enabler for the efficiency and robustness of HDC. However, existing HDC-based algorithms suffer from limitations within the encoder. To some extent, they all rely on manually selected encoders, meaning that the resulting representation is never adapted to the tasks at hand.

Methods

In this paper, we propose FLASH, a novel hyperdimensional learning method that incorporates an adaptive and learnable encoder design, aiming at better overall learning performance while maintaining good properties of HDC representation. Current HDC encoders leverage Random Fourier Features (RFF) for kernel correspondence and enable locality-preserving encoding. We propose to learn the encoder matrix distribution via gradient descent and effectively adapt the kernel for a more suitable HDC encoding.

Results

Our experiments on various regression datasets show that tuning the HDC encoder can significantly boost the accuracy, surpassing the current HDC-based algorithm and providing faster inference than other baselines, including RFF-based kernel ridge regression.

Discussion

The results indicate the importance of an adaptive encoder and customized high-dimensional representation in HDC.

Simple and complex cells revisited: toward a selectivity-invariance model of object recognition

This paper presents a theoretical perspective on modeling ventral stream processing by revisiting the computational abstraction of simple and complex cells. In parallel to David Marr's vision theory, we organize the new perspective into three levels. At the computational level, we abstract simple and complex cells into space partitioning and composition in a topological space based on the redundancy exploitation hypothesis of Horace Barlow. At the algorithmic level, we present a hierarchical extension of sparse coding by exploiting the manifold constraint in high-dimensional space (i.e., the blessing of dimensionality). The resulting over-parameterized models for object recognition differ from existing hierarchical models by disentangling the objectives of selectivity and invariance computation. It is possible to interpret our hierarchical construction as a computational implementation of cortically local subspace untangling for object recognition and face representation, which are closely related to exemplar-based and axis-based coding in the medial temporal lobe. At the implementation level, we briefly discuss two possible implementations based on asymmetric sparse autoencoders and divergent spiking neural networks.

Efficient Decoding of Compositional Structure in Holistic Representations

We investigate the task of retrieving information from compositional distributed representations formed by hyperdimensional computing/vector symbolic architectures and present novel techniques that achieve new information rate bounds. First, we provide an overview of the decoding techniques that can be used to approach the retrieval task. The techniques are categorized into four groups. We then evaluate the considered techniques in several settings that involve, for example, inclusion of external noise and storage elements with reduced precision. In particular, we find that the decoding techniques from the sparse coding and compressed sensing literature (rarely used for hyperdimensional computing/vector symbolic architectures) are also well suited for decoding information from the compositional distributed representations. Combining these decoding techniques with interference cancellation ideas from communications improves previously reported bounds (Hersche et al., 2021) of the information rate of the distributed representations from 1.20 to 1.40 bits per dimension for smaller codebooks and from 0.60 to 1.26 bits per dimension for larger codebooks.

Optimal Quadratic Binding for Relational Reasoning in Vector Symbolic Neural Architectures

Binding operation is fundamental to many cognitive processes, such as cognitive map formation, relational reasoning, and language comprehension. In these processes, two different modalities, such as location and objects, events and their contextual cues, and words and their roles, need to be bound together, but little is known about the underlying neural mechanisms. Previous work has introduced a binding model based on quadratic functions of bound pairs, followed by vector summation of multiple pairs. Based on this framework, we address the following questions: Which classes of quadratic matrices are optimal for decoding relational structures? And what is the resultant accuracy? We introduce a new class of binding matrices based on a matrix representation of octonion algebra, an eight-dimensional extension of complex numbers. We show that these matrices enable a more accurate unbinding than previously known methods when a small number of pairs are present. Moreover, numerical optimization of a binding operator converges to this octonion binding. We also show that when there are a large number of bound pairs, however, a random quadratic binding performs, as well as the octonion and previously proposed binding methods. This study thus provides new insight into potential neural mechanisms of binding operations in the brain.

On separating long- and short-term memories in hyperdimensional computing

Operations on high-dimensional, fixed-width vectors can be used to distribute information from several vectors over a single vector of the same width. For example, a set of key-value pairs can be encoded into a single vector with multiplication and addition of the corresponding key and value vectors: the keys are bound to their values with component-wise multiplication, and the key-value pairs are combined into a single superposition vector with component-wise addition. The superposition vector is, thus, a memory which can then be queried for the value of any of the keys, but the result of the query is approximate. The exact vector is retrieved from a codebook (a.k.a. item memory), which contains vectors defined in the system. To perform these operations, the item memory vectors and the superposition vector must be the same width. Increasing the capacity of the memory requires increasing the width of the superposition and item memory vectors. In this article, we demonstrate that in a regime where many (e.g., 1,000 or more) key-value pairs are stored, an associative memory which maps key vectors to value vectors requires less memory and less computing to obtain the same reliability of storage as a superposition vector. These advantages are obtained because the number of storage locations in an associate memory can be increased without increasing the width of the vectors in the item memory. An associative memory would not replace a superposition vector as a medium of storage, but could augment it, because data recalled from an associative memory could be used in algorithms that use a superposition vector. This would be analogous to how human working memory (which stores about seven items) uses information recalled from long-term memory (which is much larger than the working memory). We demonstrate the advantages of an associative memory experimentally using the storage of large finite-state automata, which could model the storage and recall of state-dependent behavior by brains.

Associative memory of structured knowledge

A long standing challenge in biological and artificial intelligence is to understand how new knowledge can be constructed from known building blocks in a way that is amenable for computation by neuronal circuits. Here we focus on the task of storage and recall of structured knowledge in long-term memory. Specifically, we ask how recurrent neuronal networks can store and retrieve multiple knowledge structures. We model each structure as a set of binary relations between events and attributes (attributes may represent e.g., temporal order, spatial location, role in semantic structure), and map each structure to a distributed neuronal activity pattern using a vector symbolic architecture scheme.We then use associative memory plasticity rules to store the binarized patterns as fixed points in a recurrent network. By a combination of signal-to-noise analysis and numerical simulations, we demonstrate that our model allows for efficient storage of these knowledge structures, such that the memorized structures as well as their individual building blocks (e.g., events and attributes) can be subsequently retrieved from partial retrieving cues. We show that long-term memory of structured knowledge relies on a new principle of computation beyond the memory basins. Finally, we show that our model can be extended to store sequences of memories as single attractors.

Cellular Automata Can Reduce Memory Requirements of Collective-State Computing

Various nonclassical approaches of distributed information processing, such as neural networks, reservoir computing (RC), vector symbolic architectures (VSAs), and others, employ the principle of collective-state computing. In this type of computing, the variables relevant in 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. In this article, we show that an elementary cellular automaton with rule 90 (CA90) enables the 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 RC and VSAs. 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 pseudorandom number generator and then stored in a large memory.

A comparison of vector symbolic architectures

Vector Symbolic Architectures combine a high-dimensional vector space with a set of carefully designed operators in order to perform symbolic computations with large numerical vectors. Major goals are the exploitation of their representational power and ability to deal with fuzziness and ambiguity. Over the past years, several VSA implementations have been proposed. The available implementations differ in the underlying vector space and the particular implementations of the VSA operators. This paper provides an overview of eleven available VSA implementations and discusses their commonalities and differences in the underlying vector space and operators. We create a taxonomy of available binding operations and show an important ramification for non self-inverse binding operations using an example from analogical reasoning. A main contribution is the experimental comparison of the available implementations in order to evaluate (1) the capacity of bundles, (2) the approximation quality of non-exact unbinding operations, (3) the influence of combining binding and bundling operations on the query answering performance, and (4) the performance on two example applications: visual place- and language-recognition. We expect this comparison and systematization to be relevant for development of VSAs, and to support the selection of an appropriate VSA for a particular task. The implementations are available.