Related papers: Monotonicity of Multi-Term Floating-Point Adders

Modern speech processing systems rely on self-attention. Unfortunately, token mixing with self-attention takes quadratic time in the length of the speech utterance, slowing down inference and training and increasing memory consumption.…

Windowed recurrences are sliding window calculations where a function is applied iteratively across the window of data, and are ubiquitous throughout the natural, social, and computational sciences. In this monograph we explore the…

Large neural networks spend most computation on floating point tensor multiplications. In this work, we find that a floating point multiplier can be approximated by one integer adder with high precision. We propose the linear-complexity…

Non-trivial analysis problems require posets with infinite ascending and descending chains. In order to compute reasonably precise post-fixpoints of the resulting systems of equations, Cousot and Cousot have suggested accelerated fixpoint…

In these introductory lectures we discuss classes of presently known nested sums, associated iterated integrals, and special constants which hierarchically appear in the evaluation of massless and massive Feynman diagrams at higher loops.…

We present algorithms to perform modular polynomial multiplication or modular dot product efficiently in a single machine word. We pack polynomials into integers and perform several modular operations with machine integer or floating point…

This paper describes a new accumulate-and-add multiplication algorithm. The method partitions one of the operands and re-combines the results of computations done with each of the partitions. The resulting design turns-out to be both…

Numerical data processing is a key task across different fields of computer technology use. However, even simple summation of values is not precise due to the floating point representation use. This paper presents a practical algorithm for…

Moment polytopes of tensors, the study of which is deeply rooted in invariant theory, representation theory and symplectic geometry, have found relevance in numerous places, from quantum information (entanglement polytopes) and algebraic…

The Euler--Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum $\sum_{k=0}^{n-1} f(k)$ of values of a function $f$ by a linear combination of a corresponding integral of…

Ootomo, Ozaki, and Yokota [Int. J. High Perform. Comput. Appl., 38 (2024), p. 297-313] have proposed a strategy to recast a floating-point matrix multiplication in terms of integer matrix products. The factors A and B are split into integer…

This paper introduces a checksum algorithm that provides a new point in the performance/complexity/effectiveness checksum tradeoff space. It has better fault detection properties than single-sum and dual-sum modular addition checksums. It…

Streaming generation models are utilized across fields, with the Transducer architecture being popular in industrial applications. However, its input-synchronous decoding mechanism presents challenges in tasks requiring non-monotonic…

In this paper, we propose a scalable approximate multiplier design, scaleTRIM, that approximates the multiplication operation using fitted linear functions, also referred to as linearization. We show that multiplication operations can be…

This paper deals with simultaneously fast and in-place algorithms for formulae where the result has to be linearly accumulated: some output variables are also input variables, linked by a linear dependency. Fundamental examples include the…

In the renormalisation analysis of critical phenomena in quasi-periodic systems, a fundamental role is often played by fixed points of functional recurrences of the form \begin{equation*} f_{n}(x) = \sum_{i=1}^\ell a_i(x) f_{n_i}…

The aim of this article is to present two different primal-dual methods for solving structured monotone inclusions involving parallel sums of compositions of maximally monotone operators with linear bounded operators. By employing some…

In this paper, we propose a mixed-precision convolution unit architecture which supports different integer and floating point (FP) precisions. The proposed architecture is based on low-bit inner product units and realizes higher precision…

Cosine similarity normalizes both sides; dot product normalizes neither. We propose a 2x2 framework that independently controls query-side and document-side normalization, exposing two intermediate variants (QNorm, DNorm) that have not been…

Many domains, from deep learning to finance, require compounding real numbers over long sequences, often leading to catastrophic numerical underflow or overflow. We introduce generalized orders of magnitude (GOOMs), a principled extension…