Loop and data transformations for sparse matrix code

Abstract

This paper introduces three new compiler transformations for representing and transforming sparse matrix computations and their data representations. In cooperation with run-time inspection, our compiler derives transformed matrix representations and associated transformed code to implement a variety of representations targeting different architecture platforms. This systematic approach to combining code and data transformations on sparse computations, which extends a polyhedral transformation and code generation framework, permits the compiler to compose these transformations with other transformations to generate code that is on average within 5% and often exceeds manually-tuned, highperformance sparse matrix libraries CUSP and OSKI. Additionally, the compiler-generated inspector codes are on average 1.5× faster than OSKI and perform comparably to CUSP, respectively.

Composable, sound transformations of nested recursion and loops

“…As mentioned above, these approaches focus on dense loops over dense arrays, so are not applicable to our domain. There has been work done in the past to generalize the loop-based model to handle non-affine loop bounds and subscripts using symbolic expressions [23,34], and to handle sparse matrices and arrays [30][31][32]35], but these approaches still only target loops, and hence do not generalize to the recursive constructs we consider. To break down the transformations a little more concretely, first, the method outer was strip mined (Section 5.3.4) to break it into two loops, outer1 (at line 1) and outer2 (at line 16); outer2 performs groups of 4 iterations from outer1.…”

A polyhedral compilation framework for loops with dynamic data-dependent bounds

“…Programmers may use different formats to store a sparse matrix, among which we consider four representations: CSR, Block CSR (BCSR), Diagonal (DIA) and ELLPACK (ELL) [27]. Our experiment in this subsection target the benchmarks used in [24], with our own modifications to suit the syntactic constraints of our framework. We first consider the CSR representation.…”

“…We first consider the CSR representation. The other three representations can be modeled with a make-dense transformation, as proposed by [24], followed by a series of loop and data transformations. BCSR is the blocked version of CSR, its parallel version is the same as that of CSR, after tiling with PPCG.…”

“…Manually-tuned libraries [2,4,7,18,19,27] are a commonly used approach, but it is tedious to implement and tune for each representation and target architecture. A polyhedral framework that can handle non-affine subscripts has a greater potential to achieve transformations and optimizations on sparse matrix computations, as illustrated by Venkat et al [24].…”

“…Our method goes beyond the state of the art [5,17,24] in fully automating the process, specializing the code generation algorithm to the case of dynamic counted loops, and avoiding the introduction of spurious loop-carried dependences or resorting to speculative execution. We conduct experiments on representative irregular computations, including dynamic programming, computer vision, finite element methods, and sparse matrix linear algebra.…”