Related papers: Compressed Modular Matrix Multiplication
In this paper, we introduce the Maximum Matrix Contraction problem, where we aim to contract as much as possible a binary matrix in order to maximize its density. We study the complexity and the polynomial approximability of the problem.…
Boundary integral equations lead to dense system matrices when discretized, yet they are data-sparse. Using the $\mathcal{H}$-matrix format, this sparsity is exploited to achieve $\mathcal{O}(N\log N)$ complexity for storage and…
A number theoretic algorithm is given for writing gauge theory amplitudes in a compact manner. It is possible to write down all details of the complete $L$ loop amplitude with two integers, or a complex integer. However, a more symmetric…
Neural networks are among the state-of-the-art techniques for language modeling. Existing neural language models typically map discrete words to distributed, dense vector representations. After information processing of the preceding…
The Gram matrix of a matrix $A$ is defined as $AA^T$ (or $A^T\!A$). Computing the Gram matrix is an important operation in many applications, such as linear regression with the least squares method, where the explicit solution formula…
For a composite $n$ and an odd $c$ with $c$ not dividing $n$, the number of solutions to the equation $n+a\equiv b\mod c$ with $a,b$ quadratic residues modulus $c$ is calculated. We establish a direct relation with those modular solutions…
We study the existence of the product of two weighted modulation spaces. For this purpose we discuss two different strategies. The more simple one allows transparent proofs in various situations. However, our second method allows a closer…
Reranking, the process of refining the output of a first-stage retriever, is often considered computationally expensive, especially with Large Language Models. Borrowing from recent advances in document compression for RAG, we reduce the…
Compressing neural nets is an active research problem, given the large size of state-of-the-art nets for tasks such as object recognition, and the computational limits imposed by mobile devices. We give a general formulation of model…
We demonstrate a multiplication method based on numbers represented as set of polynomial radix 2 indices stored as an integer list. The 'polynomial integer index multiplication' method is a set of algorithms implemented in python code. We…
In a previous paper, we described a computer program called Qubiter which can decompose an arbitrary unitary matrix into elementary operations of the type used in quantum computation. In this paper, we describe a method of reducing the…
Recurrent neural networks have proved to be an effective method for statistical language modeling. However, in practice their memory and run-time complexity are usually too large to be implemented in real-time offline mobile applications.…
Counting distinct permutations with replacement, especially when involving multiple subwords, is a longstanding challenge in combinatorial analysis, with critical applications in cryptography, bioinformatics, and statistical modeling. This…
In a paper published in 1981, Sch\"onhage showed that large total matrix multiplications can be reduced to powers of partial matrix multiplication tensors, which correspond to the bilinear computation task of multiplying matrices with some…
Decimal multiplication is the task of multiplying two numbers in base $10^N.$ Specifically, we focus on the number-theoretic transform (NTT) family of algorithms. Using only portable techniques, we achieve a 3x-5x speedup over the mpdecimal…
We study the reformulation of integer linear programs by means of a mixed integer linear program with fewer integer variables. Such reformulations can be solved efficiently with mixed integer linear programming techniques. We exhibit…
Various grammar compression algorithms have been proposed in the last decade. A grammar compression is a restricted CFG deriving the string deterministically. An efficient grammar compression develops a smaller CFG by finding duplicated…
Boolean matrix factorization (BMF) approximates a given binary input matrix as the product of two smaller binary factors. As opposed to binary matrix factorization which uses standard arithmetic, BMF uses the Boolean OR and Boolean AND…
This paper describes several new improvements of modular arithmetic and how to exploit them in order to gain more efficient implementations of commonly used algorithms, especially in cryptographic applications. We further present a new…
This paper presents an efficient technique for matrix-vector and vector-transpose-matrix multiplication in distributed-memory parallel computing environments, where the matrices are unstructured, sparse, and have a substantially larger…