Related papers: Floating-Point Arithmetic on Round-to-Nearest Repr…
Unary representation is straightforward, error tolerant and requires simple logic while its latency is a concern. On the other hand, positional representation (like binary) is compact and requires less space, but it is sensitive to errors.…
Many modern search domains comprise high-dimensional vectors of floating point numbers derived from neural networks, in the form of embeddings. Typical embeddings range in size from hundreds to thousands of dimensions, making the size of…
We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments…
Here we present an analytic approximation for the entropy of floating-point numbers, along with bounds on the error of this approximation. It is well-known that the differential entropy is tightly linked to the discrete entropy of a…
We introduce a fixed point iteration process built on optimization of a linear function over a compact domain. We prove the process always converges to a fixed point and explore the set of fixed points in various convex sets. In particular,…
Over the last few years, neural networks have started penetrating safety critical systems to take decisions in robots, rockets, autonomous driving car, etc. A problem is that these critical systems often have limited computing resources.…
Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis…
We propose a new instruction (FPADDRE) that computes the round-off error in floating-point addition. We explain how this instruction benefits high-precision arithmetic operations in applications where double precision is not sufficient.…
Floating-point accumulation networks (FPANs) are key building blocks used in many floating-point algorithms, including compensated summation and double-double arithmetic. FPANs are notoriously difficult to analyze, and algorithms using…
A new mathematical notation is proposed for the iteration of functions. It facilitates the application of the iteration of functions in mathematical and logical expressions, definitions of sets, and formulations of algorithms. Illustrations…
Given the current trend of increasing size and complexity of machine learning architectures, it has become of critical importance to identify new approaches to improve the computational efficiency of model training. In this context, we…
Due to the limited number of bits in floating-point or fixed-point arithmetic, rounding is a necessary step in many computations. Although rounding methods can be tailored for different applications, round-off errors are generally…
In this work, approximate eight-bit floating-point operations performed using simple integer operations is discussed. For two-bit mantissa formats, faithful rounding can always be obtained for the considered operations. For all operations,…
An improvement on precision of recursive function simulation in IEEE floating point standard is presented. It is shown that the average of rounding towards negative infinite and rounding towards positive infinite yields a better result than…
Nowadays, parallel computing is ubiquitous in several application fields, both in engineering and science. The computations rely on the floating-point arithmetic specified by the IEEE754 Standard. In this context, an elementary brick of…
Weighted model counting computes the sum of the rational-valued weights associated with the satisfying assignments for a Boolean formula, where the weight of an assignment is given by the product of the weights assigned to the positive and…
This paper presents a framework for rigid point-set registration and merging using a robust continuous data representation. Our point-set representation is constructed by training a one-class support vector machine with a Gaussian radial…
We consider the error due to a single bit-flip in a floating point number. We assume IEEE 754 double precision arithmetic, which encodes binary floating point numbers in a 64-bit word. We assume that the bit-flip happens randomly so it has…
Neural networks successfully capture the computational power of the human brain for many tasks. Similarly inspired by the brain architecture, Nearest Neighbor (NN) representations is a novel approach of computation. We establish a firmer…
Recurrent neural networks are widely used in speech and language processing. Due to dependency on the past, standard algorithms for training these models, such as back-propagation through time (BPTT), cannot be efficiently parallelised.…