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1. Resonator Networks, 1: An Efficient Solution for Factoring High-Dimensional, Distributed Representations of Data Structures

   https://doi.org/https://doi.org/10.1162/neco_a_01331  

   Frady, E. Paxon; Kent, Spencer J.; Olshausen, Bruno A.; Sommer, Friedrich (December 2020, Neural computation)

   null (Ed.)

   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 (VSAs) (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 codevectors. 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 of applying VSAs to myriad artificial intelligence problems in real-world domains. The companion article in this issue (Kent, Frady, Sommer, & Olshausen, 2020) presents a rigorous analysis and evaluation of the performance of resonator networks, showing it outperforms alternative approaches. 

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2. Vector Symbolic Architectures as computing framework for nanoscale hardware

   Kleyko, Denis; Davies, Mike; Frady, E. Paxon; Kanerva, Pentti; Kent, Spencer J.; Olshausen, Bruno A.; Osipov, Evgeny; Rabaey, Jan M.; Rachkovskij, Dmirti A.; Rahimi, Abbas; et al (October 2022, Proceedings of the IEEE)

   This article reviews recent progress in the development of the computing framework Vector Symbolic Architectures (also known as Hyperdimensional Computing). This framework is well suited for implementation in stochastic, nanoscale hardware and it naturally expresses the types of cognitive operations required for Artificial Intelligence (AI). We demonstrate in this article that the ring-like algebraic structure of Vector Symbolic Architectures offers simple but powerful operations on highdimensional vectors that can support all data structures and manipulations relevant in modern computing. In addition, we illustrate the distinguishing feature of Vector Symbolic Architectures, “computing in superposition,” which sets it apart from conventional computing. This latter property opens the door to efficient solutions to the difficult combinatorial search problems inherent in AI applications. Vector Symbolic Architectures are Turing complete, as we show, and we see them acting as a framework for computing with distributed representations in myriad AI settings. This paper serves as a reference for computer architects by illustrating techniques and philosophy of VSAs for distributed computing and relevance to emerging computing hardware, such as neuromorphic computing. 

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3. Variable Binding for Sparse Distributed Representations: Theory and Applications

   https://doi.org/10.1109/TNNLS.2021.3105949  

   Frady, Edward Paxon; Kleyko, Denis; Sommer, Friedrich T. (September 2021, IEEE Transactions on Neural Networks and Learning Systems)

   null (Ed.)

   Variable binding is a cornerstone of symbolic reasoning and cognition. But how binding can be implemented in connectionist models has puzzled neuroscientists, cognitive psychologists, and neural network researchers for many decades. One type of connectionist model that naturally includes a binding operation is vector symbolic architectures (VSAs). In contrast to other proposals for variable binding, the binding operation in VSAs is dimensionality-preserving, which enables representing complex hierarchical data structures, such as trees, while avoiding a combinatoric expansion of dimensionality. Classical VSAs encode symbols by dense randomized vectors, in which information is distributed throughout the entire neuron population. By contrast, in the brain, features are encoded more locally, by the activity of single neurons or small groups of neurons, often forming sparse vectors of neural activation. Following Laiho et al. (2015), we explore symbolic reasoning with a special case of sparse distributed representations. Using techniques from compressed sensing, we first show that variable binding in classical VSAs is mathematically equivalent to tensor product binding between sparse feature vectors, another well-known binding operation which increases dimensionality. This theoretical result motivates us to study two dimensionality-preserving binding methods that include a reduction of the tensor matrix into a single sparse vector. One binding method for general sparse vectors uses random projections, the other, block-local circular convolution, is defined for sparse vectors with block structure, sparse block-codes. Our experiments reveal that block-local circular convolution binding has ideal properties, whereas random projection based binding also works, but is lossy. We demonstrate in example applications that a VSA with block-local circular convolution and sparse block-codes reaches similar performance as classical VSAs. Finally, we discuss our results in the context of neuroscience and neural networks. 

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4. Error Resilient Hyperdimensional Computing Using Hypervector Encoding and Cross-Clustering

   https://doi.org/10.1109/VTS60656.2024.10538955  

   Mejri, Mohamed; Amarnath, Chandramouli; Chatterjee, Abhijit (April 2024, IEEE)

   Emerging brain-inspired hyperdimensional computing (HDC) algorithms are vulnerable to timing and soft errors in associative memory used to store high-dimensional data representations. Such errors can significantly degrade HDC performance. A key challenge is error correction after an error in computation is detected. This work presents two novel error resilience frameworks for hyperdimensional computing systems. The first, called the checksum hypervector encoding (CHE) framework, relies on creation of a single additional hypervector that is a checksum of all the class hypervectors of the HDC system. For error resilience, elementwise validation of the checksum property is performed and those elements across all class vectors for which the property fails are removed from consideration. For an HDC system with K class hypervectors of dimension D, the second cross-hypervector clustering (CHC) framework clusters D, Kdimensional vectors consisting of the i-th element of each of the K HDC class hypervectors, 1 ≤ i ≤ K. Statistical properties of these vector clusters are checked prior to each hypervector query and all the elements of all K-dimensional vectors corresponding to statistical outlier vectors are removed as before. The choice of which framework to use is dictated by the complexity of the dataset to classify. Up to three orders of magnitude better resilience to errors than the state-of-the-art across multiple HDC high-dimensional encoding (representation) systems is demonstrated. 

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5. Error Resilient Hyperdimensional Computing Using Hypervector Encoding and Cross-Clustering

   Mejri, M; Amarnath, C; Chatterjee, A (April 2025, IEEE)

   Emerging brain-inspired hyperdimensional computing (HDC) algorithms are vulnerable to timing and soft errors in associative memory used to store high-dimensional data representations. Such errors can significantly degrade HDC performance. A key challenge is error correction after an error in computation is detected. This work presents two novel error resilience frameworks for hyperdimensional computing systems. The first, called the checksum hypervector encoding (CHE) framework, relies on creation of a single additional hypervector that is a checksum of all the class hypervectors of the HDC system. For error resilience, elementwise validation of the checksum property is performed and those elements across all class vectors for which the property fails are removed from consideration. For an HDC system with K class hypervectors of dimension D, the second cross-hypervector clustering (CHC) framework clusters D, K-dimensional vectors consisting of the i-th element of each of the K HDC class hypervectors, 1 ≤ i ≤ K. Statistical properties of these vector clusters are checked prior to each hypervector query and all the elements of all K-dimensional vectors corresponding to statistical outlier vectors are removed as before. The choice of which framework to use is dictated by the complexity of the dataset to classify. Up to three orders of magnitude better resilience to errors than the state-of-the-art across multiple HDC high-dimensional encoding (representation) systems is demonstrated. 

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