EAGER: Hyperdimensional computing with geometric algebra

在现代大数据时代，一个关键的挑战是发现隐藏在高度冗余、看似无关、不完整甚至损坏的数据集中的有用信息。这些信息通常包含在数据的高维空间中隐藏的某些低维结构中，或者可能仅依赖于数据的一小部分。如何高效、自动地提取这些信息仍然是一个有待解决的问题。该项目将两个新兴的研究领域——超维（HD）计算和几何代数（GA）——结合在一起，通过研究数据表示和数据的内在几何，从一个新的角度来解决这个问题。这项研究也是第一次系统地探索在分析和发现大规模传感数据中的模式时使用复数的高维泛化的潜力。这项研究的成功可以帮助推进其他机器学习模型的能力，比如深度神经网络，它目前主要基于实数。它还将一种主要在物理界知名的强大数学工具（GA）带入了机器学习界。高清计算是一种受大脑启发的机器学习和人工智能框架，它基于将数量或符号表示为高维向量，并通过简单的操作来操纵向量。在研究人员最近的工作中，研究表明，通过在高清计算中使用复值向量，可以以这样一种方式对图像进行编码，即可以通过高清向量的因数分解有效地识别模式。在这个方向上，他们正在探索将复数推广到任何n维空间的几何代数的使用。本研究的核心内容为：(1)探索将数据映射到几何代数空间的方法；(2)研究如何将几何代数与高清计算操作相结合；(3)将这些方法应用于多麦克风语音识别或分布式感知等实际应用领域，评估其有效性和计算效率。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In the modern era of big data, a crucial challenge is to discover useful information that is buried in highly redundant, seemingly irrelevant, incomplete, or even corrupted data sets. Such information is often contained in certain low-dimensional structures hidden within the high-dimensional space of the data, or may only depend on a small subset of the data. How to extract this information efficiently and automatically remains an open problem. This project brings together two emerging areas of research — hyperdimensional (HD) computing and geometric algebra (GA) — to tackle this problem from a new stand point by investigating the data representation and the intrinsic geometry of the data. This research is also the first in a systematic quest to uncover the potential of using the high-dimensional generalization of complex numbers in analyzing and discovering patterns in large-scale sensing data. The success of this research can help advance the capability of other machine learning models, such as deep neural networks, which are mostly based on real numbers today. It also brings a powerful mathematical tool (GA) which is mainly known in the physics community into the machine learning community.HD computing is a brain-inspired framework for machine learning and artificial intelligence that is based on representing quantities or symbols as high-dimensional vectors and manipulating vectors with simple operations. In recent work by the investigators, it was shown that by using complex-valued vectors in HD computing it is possible to encode images in such a way that patterns can be effectively recognized by a factorization of HD vectors. To build on this direction, they are exploring the use of geometric algebras which generalize complex numbers to any n-dimensional space. The following thrusts form the core of this research: (1) explore ways of mapping data into the geometric algebra space; (2) investigate how to integrate geometric algebra with the operations of HD computing; (3) apply these methods to real application domains such as multi-microphone speech recognition or distributed sensing to evaluate their efficacy and computational efficiency.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.