A Dynamically Scheduled HLS Flow in MLIR

In High-Level Synthesis (HLS), we consider abstractions that span from software to hardware and target heterogeneous architectures. Therefore, managing the complexity introduced by this is key to implementing good, maintainable, and extendible HLS compilers. Traditionally, HLS flows have been built on top of software compilation infrastructure such as LLVM, with hardware aspects of the flow existing peripherally to the core of the compiler. Through this work, we aim to show that MLIR, a compiler infrastructure with a focus on domain-specific intermediate representations (IR), is a better infrastructure for HLS compilers. Using MLIR, we define HLS and hardware abstractions as first-class citizens of the compiler, simplifying analysis, transformations, and optimization. To demonstrate this, we present a C-to-RTL, dynamically scheduled HLS flow. We find that our flow generates circuits comparable to those of an equivalent LLVM-based HLS compiler. Notably, we achieve this while lacking key optimization passes typically found in HLS compilers and through the use of an experimental front-end. To this end, we show that significant improvements in the generated RTL are but low-hanging fruit, requiring engineering effort to attain. We believe that our flow is more modular and more extendible than comparable open-source HLS compilers and is thus a good candidate as a basis for future research. Apart from the core HLS flow, we provide MLIR-based tooling for C-to-RTL cosimulation and visual debugging, with the ultimate goal of building an MLIR-based HLS infrastructure that will drive innovation in the field

Banach frames for multivariate α-modulation spaces

AbstractThe α-modulation spaces Mp,qs,α(Rd), α∈[0,1], form a family of spaces that include the Besov and modulation spaces as special cases. This paper is concerned with construction of Banach frames for α-modulation spaces in the multivariate setting. The frames constructed are unions of independent Riesz sequences based on tensor products of univariate brushlet functions, which simplifies the analysis of the full frame. We show that the multivariate α-modulation spaces can be completely characterized by the Banach frames constructed

On the equivalence of brushlet and wavelet bases

AbstractWe prove that the Meyer wavelet basis and a class of brushlet systems associated with exponential type partitions of the frequency axis form a family of equivalent (unconditional) bases for the Besov and Triebel–Lizorkin function spaces. This equivalence is then used to obtain new results on nonlinear approximation with brushlets in Triebel–Lizorkin spaces

Approximation with brushlet systems

AbstractWe consider an orthonormal basis for L2(R) consisting of functions that are well localized in the spatial domain and have compact support in the frequency domain. The construction is based on smooth local cosine bases and is inspired by Meyer and Coifman's brushlets, which are local exponentials in the frequency domain. For brushlet bases associated with an exponential-type partition of the frequency axis, we show that the system constitutes an unconditional basis for Lp(R), 1<p<∞, Bqs(Lp(R)), 1<p,q<∞, s>0, and that the norm in these spaces can be expressed by the expansion coefficients. In Lp(R), we construct greedy brushlet-type bases and derive Jackson and Bernstein inequalities. Finally, we investigate a natural bivariate extension leading to ridgelet-type bases for L2(R2)