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This article introduces HODLR2D, a new hierarchical low-rank representation for a class of dense matrices arising out of $N$ body problems in two dimensions. Using this new hierarchical framework, we propose a new fast matrix-vector product…

Quantum error-correcting codes (QECCs) can eliminate the negative effects of quantum noise, the major obstacle to the execution of quantum algorithms. However, realizing practical quantum error correction (QEC) requires resolving many…

In this paper we present two new approaches to efficiently solve large-scale compressed sensing problems. These two ideas are independent of each other and can therefore be used either separately or together. We consider all possibilities.…

The sequence reconstruction problem, introduced by Levenshtein in 2001, considers a communication setting in which a sender transmits a codeword and the receiver observes K independent noisy versions of this codeword. In this work, we study…

Large Language Models (LLMs) have achieved remarkable success but face significant computational and memory challenges, particularly due to their extensive output vocabularies. The final linear projection layer, mapping hidden states to…

Vector Symbolic Architectures (VSAs) are a powerful framework for representing compositional reasoning. They lend themselves to neural-network implementations, allowing us to create neural networks that can perform cognitive functions, like…

Efficient transformer variants with linear time complexity have been developed to mitigate the quadratic computational overhead of the vanilla transformer. Among them are low-rank projection methods such as Linformer and kernel-based…

The Volterra signature extends the classical path signature by incorporating general matrix-valued kernel into its iterated integral structure, yielding a flexible notion of memory for time series. Its components can be viewed as successive…

Neural speech codecs aim to compress input signals into minimal bits while maintaining content quality in a low-latency manner. However, existing neural codecs often trade model complexity for reconstruction performance. These codecs…

Vector quantization is a technique in machine learning that discretizes continuous representations into a set of discrete vectors. It is widely employed in tokenizing data representations for large language models, diffusion models, and…

Large Language Models (LLMs) have demonstrated strong capabilities across various domains, with recent advancements in challenging reasoning tasks such as mathematics and programming. However, solving reasoning tasks often requires an LLM…

Second-order optimization algorithms exhibit excellent convergence properties for training deep learning models, but often incur significant computation and memory overheads. This can result in lower training efficiency than the first-order…

We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra…

Vector is the general format of input data of most algorithms. Designing unitary operation to load all information of vector into quantum registers of quantum CPU from classical memory is called quantum loading scheme (QLS). QLS assembles…

Fast, scalable decoding architectures that operate in a block-wise parallel fashion across space and time are essential for real-time fault-tolerant quantum computing. We introduce a scalable AI-based pre-decoder for the surface code that…

The high-resolution time series classification problem is essential due to the increasing availability of detailed temporal data in various domains. To tackle this challenge effectively, it is imperative that the state-of-the-art attention…

In recent years, a class of dictionaries have been proposed for multidimensional (tensor) data representation that exploit the structure of tensor data by imposing a Kronecker structure on the dictionary underlying the data. In this work, a…

Vector Symbolic Architectures (VSAs) have emerged as a novel framework for enabling interpretable machine learning algorithms equipped with the ability to reason and explain their decision processes. The basic idea is to represent discrete…

Matrices with the displacement structures of circulant, Toeplitz, and Hankel types as well as matrices with structures generalizing these types are omnipresent in computations of sciences and engineering. In this paper, we present efficient…

Deep Learning neural networks are pervasive, but traditional computer architectures are reaching the limits of being able to efficiently execute them for the large workloads of today. They are limited by the von Neumann bottleneck: the high…