An overflow free fixed-point eigenvalue decomposition algorithm: Case study of dimensionality reduction in hyperspectral images

Abstract

Abstract-We consider the problem of enabling robust range estimation of eigenvalue decomposition (EVD) algorithm for a reliable fixed-point design. The simplicity of fixed-point circuitry has always been so tempting to implement EVD algorithms in fixed-point arithmetic. Working towards an effective fixed-point design, integer bit-width allocation is a significant step which has a crucial impact on accuracy and hardware efficiency. This paper investigates the shortcomings of the existing range estimation methods while deriving bounds for the variables of the EVD algorithm. In light of the circumstances, we introduce a range estimation approach based on vector and matrix norm properties together with a scaling procedure that maintains all the assets of an analytical method. The method could derive robust and tight bounds for the variables of EVD algorithm. The bounds derived using the proposed approach remain same for any input matrix and are also independent of the number of iterations or size of the problem. Some benchmark hyperspectral data sets have been used to evaluate the efficiency of the proposed technique. It was found that by the proposed range estimation approach, all the variables generated during the computation of Jacobi EVD is bounded within ±1.

Stochastic rounding and reduced-precision fixed-point arithmetic for solving neural ordinary differential equations

“…On the other hand, changing the numerical type from floating-to fixed-point, significantly reduces the range of representable values, increasing the potential for the under/overflow of the data types. In some tasks, these range issues can be dealt with safely by analysing the algorithm in various novel or problem-specific ways, whereas in other tasks it can be a very difficult to generalize effectively across all use cases [2]. There are other approaches apart from floating-and fixed-point arithmetics that are worth mentioning: posit arithmetic [3] is a completely new format proposed to replace floats and is based on the principles of interval arithmetic and tapered arithmetic [4] (dynamically sized exponent and significand fields which optimize the relative accuracy of the floating-point format in some specific range of real numbers rather than having the same relative accuracy across the whole range); bfloat16, with hardware support in recent Intel processors [5], is simply a single-precision floating-point type with the 16 bottom bits dropped for hardware and memory efficiency; flexpoint [6], an efficient combination of fixed-and floating-point also by Intel; and various approaches to transforming floating-point using, for example the logarithmic number system in which multiplication becomes addition [7].…”