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We consider the problem of training a neural network to store a set of patterns with maximal noise robustness. A solution, in terms of optimal weights and state update rules, is derived by training each individual neuron to perform either kernel classification or interpolation with a minimum weight norm. By applying this method to feed-forward and recurrent networks, we derive optimal models, termed kernel memory networks, that include, as special cases, many of the hetero-and auto-associative memory models that have been proposed over the past years, such as modern Hopfield networks and Kanerva's sparse distributed memory. We modify Kanerva's model and demonstrate a simple way to design a kernel memory network that can store an exponential number of continuous-valued patterns with a finite basin of attraction. The framework of kernel memory networks offers a simple and intuitive way to understand the storage capacity of previous memory models, and allows for new biological interpretations in terms of dendritic non-linearities and synaptic cross-talk.

The Case That A.I. Is Thinking ChatGPT does not have an inner life. Yet it seems to know what it's talking about. How convincing does the illusion of understanding have to be before you stop calling it an illusion? Dario Amodei, the C.E.O. of the artificial-intelligence company Anthropic, has been predicting that an A.I. "smarter than a Nobel Prize winner" in such fields as biology, math, engineering, and writing might come online by 2027. He envisions millions of copies of a model whirring away, each conducting its own research: a "country of geniuses in a datacenter." In June, Sam Altman, of OpenAI, wrote that the industry was on the cusp of building "digital superintelligence." "The 2030s are likely going to be wildly different from any time that has come before," he asserted. Meanwhile, the A.I. tools that most people currently interact with on a day-to-day basis are reminiscent of Clippy, the onetime Microsoft Office "assistant" that was actually more of a gadfly. A Zoom A.I. tool suggests that you ask it "What are some meeting icebreakers?" or instruct it to "Write a short message to share gratitude." Siri is good at setting reminders but not much else. A friend of mine saw a button in Gmail that said "Thank and tell anecdote." When he clicked it, Google's A.I. invented a funny story about a trip to Turkey that he never took. The rushed and uneven rollout of A.I. has created a fog in which it is tempting to conclude that there is nothing to see here--that it's all hype. There is, to be sure, plenty of hype: Amodei's timeline is science-fictional.

Kanerva (2014) suggested that it would be possible to construct a complete Lisp out of a vector-symbolic architecture. We present the general form of a vector-symbolic representation of the five Lisp elementary functions, lambda expressions, and other auxiliary functions, found in the Lisp 1.5 specification (McCarthy, 1960), which is near minimal and sufficient for Turing-completeness. Our specific implementation uses holographic reduced representations (Plate, 1995), with a lookup table cleanup memory. Lisp, as all Turing-complete languages, is a Cartesian closed category (nLab authors, 2024), unusual in its proximity to the mathematical abstraction. We discuss the mathematics, the purpose, and the significance of demonstrating vector-symbolic architectures' Cartesian-closedness, as well as the importance of explicitly including cleanup memories in the specification of the architecture.

We consider the problem of training a neural network to store a set of patterns with maximal noise robustness. A solution, in terms of optimal weights and state update rules, is derived by training each individual neuron to perform either kernel classification or interpolation with a minimum weight norm. By applying this method to feed-forward and recurrent networks, we derive optimal models, termed kernel memory networks, that include, as special cases, many of the hetero- and auto-associative memory models that have been proposed over the past years, such as modern Hopfield networks and Kanerva's sparse distributed memory. We modify Kanerva's model and demonstrate a simple way to design a kernel memory network that can store an exponential number of continuous-valued patterns with a finite basin of attraction. The framework of kernel memory networks offers a simple and intuitive way to understand the storage capacity of previous memory models, and allows for new biological interpretations in terms of dendritic non-linearities and synaptic cross-talk.

We present a novel, simple, fast, and efficient approach for semi-supervised learning on graphs. The proposed approach takes advantage of hyper-dimensional computing which encodes data samples using random projections into a high dimensional space (HD space for short). Specifically, we propose a Hyper-dimensional Graph Learning (HDGL) algorithm that leverages the injectivity property of the node representations of a family of graph neural networks. HDGL maps node features to the HD space and then uses HD operators such as bundling and binding to aggregate information from the local neighborhood of each node. Results of experiments with widely used benchmark data sets show that HDGL achieves predictive performance that is competitive with the state-of-the-art deep learning methods, without the need for computationally expensive training.

The information capacity of Kanerva's Sparse, Distributed Memory (SDM) and Hopfield-type neural networks is investigated. Under the approximations used here, it is shown that the to(cid:173) tal information stored in these systems is proportional to the number connections in the net(cid:173) work. The proportionality constant is the same for the SDM and HopJreld-type models in(cid:173) dependent of the particular model, or the order of the model. The approximations are checked numerically. This same analysis can be used to show that the SDM can store se(cid:173) quences of spatiotemporal patterns, and the addition of time-delayed connections allows the retrieval of context dependent temporal patterns.

A new viewpoint of the processing performed by Kanerva's sparse distributed memory (SDM) is presented. In conditions of near- or over- capacity, where the associative-memory behavior of the mod(cid:173) el breaks down, the processing performed by the model can be inter(cid:173) preted as that of a statistical predictor. Mathematical results are presented which serve as the framework for a new statistical view(cid:173) point of sparse distributed memory and for which the standard for(cid:173) mulation of SDM is a special case. This viewpoint suggests possi(cid:173) ble enhancements to the SDM model, including a procedure for improving the predictiveness of the system based on Holland's work with'Genetic Algorithms', and a method for improving the capacity of SDM even when used as an associative memory.

Kanerva's sparse distributed memory (SDM) is an associative-memo(cid:173) ry model based on the mathematical properties of high-dimensional binary address spaces. Holland's genetic algorithms are a search tech(cid:173) nique for high-dimensional spaces inspired by evolutionary processes of DNA. "Genetic Memory" is a hybrid of the above two systems, in which the memory uses a genetic algorithm to dynamically recon(cid:173) figure its physical storage locations to reflect correlations between the stored addresses and data. For example, when presented with raw weather station data, the Genetic Memory discovers specific fea(cid:173) tures in the weather data which correlate well with upcoming rain, and reconfigures the memory to utilize this information effectively. This architecture is designed to maximize the ability of the system to scale-up to handle real-world problems.

This is Part II of the two-part comprehensive survey devoted to a computing framework most commonly known under the names Hyperdimensional Computing and Vector Symbolic Architectures (HDC/VSA). Both names refer to a family of computational models that use high-dimensional distributed representations and rely on the algebraic properties of their key operations to incorporate the advantages of structured symbolic representations and vector distributed representations. Holographic Reduced Representations is an influential HDC/VSA model that is well-known in the machine learning domain and often used to refer to the whole family. However, for the sake of consistency, we use HDC/VSA to refer to the area. Part I of this survey covered foundational aspects of the area, such as historical context leading to the development of HDC/VSA, key elements of any HDC/VSA model, known HDC/VSA models, and transforming input data of various types into high-dimensional vectors suitable for HDC/VSA. This second part surveys existing applications, the role of HDC/VSA in cognitive computing and architectures, as well as directions for future work. Most of the applications lie within the machine learning/artificial intelligence domain, however we also cover other applications to provide a thorough picture. The survey is written to be useful for both newcomers and practitioners.