To achieve peak performance on modern GPUs, one must balance two frames of mind: issuing instructions to individual threads to control their behavior, while simultaneously tracking the convergence of many threads acting in concert to perform collective operations like Tensor Core instructions. The tension between these two mindsets makes modular programming error prone. Functions that encapsulate collective operations, despite being called per-thread, must be executed cooperatively by groups of threads. In this work, we introduce Prism, a new GPU language that restores modularity while still giving programmers the low-level control over collective operations necessary for high performance. Our core idea is typed perspectives, which materialize, at the type level, the granularity at which the programmer is controlling the behavior of threads. We describe the design of Prism, implement a compiler for it, and lay its theoretical foundations in a core calculus called Bundl. We implement state-of-the-art GPU kernels in Prism and find that it offers programmers the safety guarantees needed to confidently write modular code without sacrificing performance.

We present a new pure functional language and type system with borrows with lifetimes, and a corresponding fully-automatic type inference procedure. Inference provides users the performance benefits of borrows with lifetimes without requiring user annotation. If the user’s program cannot be typed, inference inserts a handful of reference count operations so that it can be typed. We provide a heap semantics for our borrowing language and prove a soundness theorem, guaranteeing that well-typed programs do not violate memory safety. We implement our memory management strategy as part of the Morphic language stack and compare it to Perceus, a state-of-the-art reference-counting technique based on linear type inference. We find that our system is able to eliminate almost all reference count operations across a range of programs, reducing reference count increments by 75–100% on all benchmarks with reference count increments under the baseline. As a result, we achieve a 1 . 48× geomean speedup overall on all benchmarks.

In this paper, we introduce a verified framework for defining and composing sparse tensor formats. We extend the ATL tensor language and scheduling framework, which formerly could only express dense tensor kernels. We define a levelized abstraction to describe per-dimension tensor formats via their encoding routines, access and iteration functions, and formal properties enforcing soundness of the sparse structures as representations of the original dense tensors. Using this abstraction, we compositionally define format-agnostic, multidimensional compression and decompression functions that are used to express the top-level soundness theorem for these abstract sparse tensor formats. We then use this soundness theorem as an adjoint-pair rewrite theorem to introduce sparse data structures and iteration into a dense tensor kernel via the existing scheduling-rewrite framework of ATL. Overall, we are able to start with a program computing over dense operands and derive a proven semantically equivalent, optimized program computing over sparse structures. We further prove a minimal set of instances of the level-format abstraction, which can be composed and passed as parameters to compression to capture a broad range of canonical, multidimensional tensor-compression formats.

Trees can accelerate queries that search or aggregate values over large collections. They achieve this by storing metadata that enables quick pruning (or inclusion) of subtrees when predicates on that metadata can prove that none (or all) of the data in a subtree affect the query result. Existing systems implement this pruning logic manually for each query predicate and data structure. We generalize and mechanize this class of optimization. Our method derives conditions for when subtrees can be pruned (or included wholesale), expressed in terms of the metadata available at each node. We efficiently generate these conditions using symbolic interval analysis, extended with new rules to handle geometric predicates (e.g., intersection, containment). Additionally, our compiler fuses compound queries (e.g., reductions on filters) into a single tree traversal. These techniques enable the automatic derivation of generalized single-index and dual-index tree joins that support a wide class of join predicates beyond standard equality and range predicates. The generated traversals match the behavior of expert-written code that implements query-specific traversals, and can asymptotically outperform the linear scans and nested-loop joins that existing systems fall back to when hand-written cases do not apply.

We present a GPU-based system for automatic differentiation (AD) of functions defined on triangle meshes, designed to exploit the locality and sparsity in mesh-based computation. Our system evaluates derivatives using perelement forward-mode AD, confining all computation to registers and shared memory and assembling global gradients, sparse Jacobians, and sparse Hessians directly on the GPU. By avoiding global computation graphs, intermediate buffers, and device-host synchronization, our approach minimizes memory traffic and enables efficient differentiation under both static and dynamically changing sparsity. Our programming model lets users express energy terms over mesh neighborhoods, while our system automatically manages parallel execution, derivative propagation, sparse assembly, and matrix-free operations such as Hessian-vector products. Our system supports both scalar- and vector-valued objectives, dynamic interaction-driven sparsity updates, and seamless integration with external GPU sparse linear solvers. We evaluate our system on applications including elastic and cloth simulation, surface parameterization, mesh smoothing, frame field design, ARAP deformation, and spherical manifold optimization. Across these tasks, our system consistently outperforms state-of-the-art differentiation frameworks, including PyTorch, JAX, Warp, Dr.JIT, EnzymeAD, and Thallo. We demonstrate speedups across a range of solver types, from Newton and Gauss-Newton for nonlinear least squares to L-BFGS and gradient descent, and across different derivative usage modes, including Hessian-vector products as well as full sparse Hessian and Jacobian construction. Our system is available as open source at https://github.com/owensgroup/RXMesh.

Computer graphics applications are sophisticated heterogeneous programs that orchestrate numerous components, including shaders, pipelines, resource descriptors, and command streams. In modern graphics workflows, communication between these components is mediated by resources such as shader I/O, storage buffers, and resource bindings. To make this communication correct, developers must uphold resource contracts. These contracts cover agreements on type, layout, binding locations, synchronization requirements, and related properties. In standard practice, these contracts are satisfied implicitly through conventions. As a result, contract breaches are often detected too late, at runtime by validation layers, or worse, as visual artifacts in rendered output. In this work, we present RCGP 1 , a system that mechanistically enforces resource contracts between components of a graphics program. RCGP formalizes contracts , which are centralized through single-source-of-truth resource declarations, as statically introspectable units that can be verified. We map objects from distinct program components to modules that import and export contracts. Modules are then composed into larger components via combinators , which check compatibility between the corresponding contracts and compute the residual contracts for the result. Finally, RCGP upholds these contracts through a set of witnesses , which encode the necessary promises. Using a prototype implementation in C++ and Vulkan, we demonstrate that this formulation converts common classes of late failures, such as descriptor-type mismatches, layout drift, and missing synchronization barriers, into early, actionable compile-time diagnostics.

User-schedulable languages (USLs) help programmers productively optimize programs by providing safe means of transforming them. Current USLs are designed to give programmers exactly the control they want, while automating all other concerns. However, there is no universal answer for what performance-conscious programmers want to control, how they want to control it, and what they want to automate, even in relatively narrow domains. We claim that USLs should, instead, be designed to grow. We present Exo 2, a scheduling language that enables users to define new scheduling operations externally to the compiler. By composing a set of trusted, fine-grained primitives, users can safely write their own scheduling library to build up desired automation. We identify actions (ways of modifying code), inspection (ways of interrogating code), and references (ways of pointing to code) as essential for any user-extensible USL. We fuse these ideas into a new mechanism called Cursors that enables the creation of scheduling libraries in user code. We demonstrate libraries that amortize scheduling effort across more than 80 high-performance kernels, reducing total scheduling code by an order of magnitude and delivering performance competitive with state-of-the-art implementations on three different platforms.

Decoding with autoregressive language models traditionally occurs sequentially, generating one token after another. Recent attempts to introduce parallelism require a pre-determined structure in the generated content to implement parallel generation, such as by pattern-matching on bullet points. In this work, we present a new technique to automate parallel generation by dynamically exploiting the semantic independence of generation outputs to implement asynchronous decoding. We introduce an annotation language Pasta-Lang for language models to initiate asynchronous decoding at inference time. We also develop an accompanying Pasta-Lang interpreter that performs on-the-fly asynchronous decoding, effectively implementing parallel generation and speeding up inference. We present an instruction-finetuning dataset with Pasta-Lang-annotated responses for teaching LLMs to annotate semantic independence with Pasta-Lang as well as the methodology for creating the dataset. Our evaluation shows using the interpreter with a Pasta-Lang-equipped model achieves significant speedup while maintaining the same generation quality.

Large language model inference is both memory-intensive and time-consuming, often requiring distributed algorithms to efficiently scale. Various model parallelism strategies are used in multi-gpu training and inference to partition computation across multiple devices, reducing memory load and computation time. However, using model parallelism necessitates communication of information between GPUs, which has been a major bottleneck and limits the gains obtained by scaling up the number of devices. We introduce Ladder Residual, a simple architectural modification applicable to all residual-based models that enables straightforward overlapping that effectively hides the latency of communication. Our insight is that in addition to systems optimization, one can also redesign the model architecture to decouple communication from computation. While Ladder Residual can allow communication-computation decoupling in conventional parallelism patterns, we focus on Tensor Parallelism in this paper, which is particularly bottlenecked by its heavy communication. For a Transformer model with 70B parameters, applying Ladder Residual to all its layers can achieve 29% end-to-end wall clock speed up at inference time with TP sharding over 8 devices. We refer the resulting Transformer model as the Ladder Transformer. We train a 1B and 3B Ladder Transformer from scratch and observe comparable performance to a standard dense transformer baseline. We also show that it is possible to convert parts of the Llama-3.1 8B model to our Ladder Residual architecture with minimal accuracy degradation by only retraining for 3B tokens.

Subroutines are essential building blocks in software design: users encapsulate common functionality in libraries and write applications by composing calls to subroutines. Unfortunately, performance may be lost at subroutine boundaries due to reduced locality and increased memory consumption. Operator fusion helps recover the performance lost at composition boundaries. Previous solutions fuse operators by manually rewriting code into monolithic fused subroutines, or by relying on heavy-weight compilers to generate code that performs fusion. Both approaches require a semantic understanding of the entire computation, breaking the decoupling necessary for modularity and reusability of subroutines. In this work, we attempt to identify the minimal ingredients required to fuse computations, enabling composition of subroutines without sacrificing performance or modularity. We find that, unlike previous approaches that require a semantic understanding of the computation, most opportunities for fusion require understanding only data production and consumption patterns. Exploiting this insight, we add fusion on top of black-box subroutines by proposing a lightweight enrichment of subroutine declarations to expose data-dependence patterns. We implement our approach in a system called Fern, and demonstrate Fern’s benefits by showing that it is competitive with state-of-the-art, high-performance libraries with manually fused operators, can fuse across library and domain boundaries for unforeseen workloads, and can deliver speedups of up to 5× over unfused code.

The human ability to think about thinking (“theory of mind”) is a fundamental object of study in many disciplines. In recent decades, researchers across these disciplines have converged on a rich computational paradigm for modeling theory of mind, grounded in recursive probabilistic reasoning. However, practitioners often find programming in this paradigm challenging: first, because thinking-about-thinking is confusing for programmers, and second, because models are slow to run. This paper presents memo , a new domain-specific probabilistic programming language that overcomes these challenges: first, by providing specialized syntax and semantics for theory of mind, and second, by taking a unique approach to inference that scales well on modern hardware via array programming. memo enables practitioners to write dramatically faster models with much less code, and has already been adopted by several research groups.

Impossible objects, geometric constructions that humans can perceive but that cannot exist in real life, have been a topic of intrigue in visual arts, perception, and graphics, yet no satisfying computer representation of such objects exists. Previous work embeds impossible objects in 3D, cutting them or twisting/bending them in the depth axis. Cutting an impossible object changes its local geometry at the cut, which can hamper downstream graphics applications, such as smoothing, while bending makes it difficult to relight the object. Both of these can invalidate geometry operations, such as distance computation. As an alternative, we introduce Meschers, meshes capable of representing impossible constructions akin to those found in M.C. Escher's woodcuts. Our representation has a theoretical foundation in discrete exterior calculus and supports the use-cases above, as we demonstrate in a number of example applications. Moreover, because we can do discrete geometry processing on our representation, we can inverse-render impossible objects. We also compare our representation to cut and bend representations of impossible objects.

We study how down-scaling large language model (LLM) size impacts LLM capabilities. We begin by measuring the effects of weight pruning – a popular technique for reducing model size – on the two abilities of LLMs: (a) recalling facts presented during pre-training and (b) processing information presented in context. Surprisingly, we find that existing pruning techniques affect these two abilities of LLMs differently. For example, pruning more than 30% of weights significantly decreases an LLM’s ability to recall facts presented during pre-training. Yet pruning 60-70% of weights largely preserves an LLM’s ability to process information in-context, ranging from retrieving answers based on information presented in context to learning parameterized functions such as a linear classifier based on a few examples. Moderate pruning impairs LLM’s ability to recall facts learnt from pre-training. However, its effect on model’s ability to process information presented in context is much less pronounced. The said disparate effects similarly arise when replacing the original model with a smaller dense one with reduced width and depth. This similarity suggests that model size reduction in general underpins the said disparity.

Key-value (KV) caching plays an essential role in accelerating decoding for transformer-based autoregressive large language models (LLMs). However, the amount of memory required to store the KV cache can become prohibitive at long sequence lengths and large batch sizes. Since the invention of the transformer, two of the most effective interventions discovered for reducing the size of the KV cache have been Multi-Query Attention (MQA) and its generalization, Grouped-Query Attention (GQA). MQA and GQA both modify the design of the attention block so that multiple query heads can share a single key/value head, reducing the number of distinct key/value heads by a large factor while only minimally degrading accuracy. In this paper, we show that it is possible to take Multi-Query Attention a step further by also sharing key and value heads between adjacent layers, yielding a new attention design we call Cross-Layer Attention (CLA). With CLA, we find that it is possible to reduce the size of the KV cache by another $2\times$ while maintaining nearly the same accuracy as unmodified MQA. In experiments training 1B- and 3B-parameter models from scratch, we demonstrate that CLA provides a Pareto improvement over the memory/accuracy tradeoffs which are possible with traditional MQA, potentially enabling future models to operate at longer sequence lengths and larger batch sizes than would otherwise be possible.

SA Conference Papers ’24, December 03–06, 2024, Tokyo, Japan

Producing efficient array code is crucial in high-performance domains like image processing and machine learning. It requires the ability to control factors like compute intensity and locality by reordering computations into different stages and granularities with respect to where they are stored. However, traditional pure, functional tensor languages struggle to do so. In a previous publication, we introduced ATL as a pure, functional tensor language capable of systematically decoupling compute and storage order via a set of high-level combinators known as reshape operators. Reshape operators are a unique functional-programming construct since they manipulate storage location in the generated code by modifying the indices that appear on the left-hand sides of storage expressions. We present a formal correctness proof for an implementation of the compilation algorithm, marking the first verification of a lowering algorithm targeting imperative loop nests from a source functional language that enables separate control of compute and storage ordering. One of the core difficulties of this proof required properly formulating the complex invariants to ensure that these storage-index remappings were well-formed. Notably, this exercise revealed a soundness bug in the original published compilation algorithm regarding the truncation reshape operators. Our fix is a new type system that captures safety conditions that were previously implicit and enables us to prove compiler correctness for well-typed source programs. We evaluate this type system and compiler implementation on a range of common programs and optimizations, including but not limited to those previously studied to demonstrate performance comparable to established compilers like Halide.

Computations in physical simulation, computer graphics, and probabilistic inference often require the differentiation of discontinuous processes due to contact, occlusion, and changes at a point in time. Popular differentiable programming languages, such as PyTorch and JAX, ignore discontinuities during differentiation. This is incorrect for parametric discontinuities —conditionals containing at least one real-valued parameter and at least one variable of integration. We introduce Potto, the first differentiable first-order programming language to soundly differentiate parametric discontinuities. We present a denotational semantics for programs and program derivatives and show the two accord. We describe the implementation of Potto, which enables separate compilation of programs. Our prototype implementation overcomes previous compile-time bottlenecks achieving an 88.1x and 441.2x speed up in compile time and a 2.5x and 7.9x speed up in runtime, respectively, on two increasingly large image stylization benchmarks. We showcase Potto by implementing a prototype differentiable renderer with separately compiled shaders.

Modern vector processors support a wide variety of instructions for fixed-point digital signal processing. These instructions support a proliferation of rounding, saturating, and type conversion modes, and are often fused combinations of more primitive operations. While these are common idioms in fixed-point signal processing, it is difficult to use these operations in portable code. It is challenging for programmers to write down portable integer arithmetic in a C-like language that corresponds exactly to one of these instructions, and even more challenging for compilers to recognize when these instructions can be used. Our system, Pitchfork, defines a portable fixed-point intermediate representation, FPIR, that captures common idioms in fixed-point code. FPIR can be used directly by programmers experienced with fixed-point, or Pitchfork can automatically lift from integer operations into FPIR using a term-rewriting system (TRS) composed of verified manual and automatically-synthesized rules. Pitchfork then lowers from FPIR into target-specific fixed-point instructions using a set of target-specific TRSs. We show that this approach improves runtime performance of portably-written fixed-point signal processing code in Halide, across a range of benchmarks, by geomean 1.31x on x86 with AVX2, 1.82x on ARM Neon, and 2.44x on Hexagon HVX compared to a standard LLVM-based compiler flow, while maintaining or improving existing compile times.

A single panel of a comic book can say a lot: it can depict not only where the characters currently are, but also their motions, their motivations, their emotions, and what they might do next. More generally, humans routinely infer complex sequences of past and future events from a *static snapshot* of a *dynamic scene*, even in situations they have never seen before. In this paper, we model how humans make such rapid and flexible inferences. Building on a long line of work in cognitive science, we offer a Monte Carlo algorithm whose inferences correlate well with human intuitions in a wide variety of domains, while only using a small, cognitively-plausible number of samples. Our key technical insight is a surprising connection between our inference problem and Monte Carlo path tracing, which allows us to apply decades of ideas from the computer graphics community to this seemingly-unrelated theory of mind task.

We introduce SLANG.D, an extension to the Slang shading language that incorporates first-class automatic differentiation support. The new shading language allows us to transform a Direct3D-based path tracer to be fully differentiable with minor modifications to existing code. SLANG.D enables a shared ecosystem between machine learning frameworks and pre-existing graphics hardware API-based rendering systems, promoting the interchange of components and ideas across these two domains. Our contributions include a differentiable type system designed to ensure type safety and semantic clarity in codebases that blend differentiable and non-differentiable code, language primitives that automatically generate both forward and reverse gradient propagation methods, and a compiler architecture that generates efficient derivative propagation shader code for graphics pipelines. Our compiler supports differentiating code that involves arbitrary control-flow, dynamic dispatch, generics and higher-order differentiation, while providing developers flexible control of checkpointing and gradient aggregation strategies for best performance. Our system allows us to differentiate an existing real-time path tracer, Falcor, with minimal change to its shader code. We show that the compiler-generated derivative kernels perform as efficiently as handwritten ones. In several benchmarks, the SLANG.D code achieves significant speedup when compared to prior automatic differentiation systems.

Machine knitting is a well-established fabrication technique for complex soft objects, and both companies and researchers have developed tools for generating machine knitting patterns. However, existing representations for machine knitted objects are incomplete (do not cover the complete domain of machine knittable objects) or overly specific (do not account for symmetries and equivalences among knitting instruction sequences). This makes it difficult to define correctness in machine knitting, let alone verify the correctness of a given program or program transformation. The major contribution of this work is a formal semantics for knitout, a low-level Domain Specific Language for knitting machines. We accomplish this by using what we call the fenced tangle , which extends concepts from knot theory to allow for a mathematical definition of knitting program equivalence that matches the intuition behind knit objects. Finally, using this formal representation, we prove the correctness of a sequence of rewrite rules; and demonstrate how these rewrite rules can form the foundation for higher-level tasks such as compiling a program for a specific machine and optimizing for time/reliability, all while provably generating the same knit object under our proposed semantics. By establishing formal definitions of correctness, this work provides a strong foundation for compiling and optimizing knit programs.

Working with any gradient-based machine learning algorithm involves the tedious task of tuning the optimizer's hyperparameters, such as its step size. Recent work has shown how the step size can itself be optimized alongside the model parameters by manually deriving expressions for "hypergradients" ahead of time.We show how to automatically compute hypergradients with a simple and elegant modification to backpropagation. This allows us to easily apply the method to other optimizers and hyperparameters (e.g. momentum coefficients). We can even recursively apply the method to its own hyper-hyperparameters, and so on ad infinitum. As these towers of optimizers grow taller, they become less sensitive to the initial choice of hyperparameters. We present experiments validating this for MLPs, CNNs, and RNNs. Finally, we provide a simple PyTorch implementation of this algorithm (see http://people.csail.mit.edu/kach/gradient-descent-the-ultimate-optimizer).

High-performance kernel libraries are critical to exploiting accelerators and specialized instructions in many applications. Because compilers are difficult to extend to support diverse and rapidly-evolving hardware targets, and automatic optimization is often insufficient to guarantee state-of-the-art performance, these libraries are commonly still coded and optimized by hand, at great expense, in low-level C and assembly. To better support development of high-performance libraries for specialized hardware, we propose a new programming language, Exo, based on the principle of exocompilation: externalizing target-specific code generation support and optimization policies to user-level code. Exo allows custom hardware instructions, specialized memories, and accelerator configuration state to be defined in user libraries. It builds on the idea of user scheduling to externalize hardware mapping and optimization decisions. Schedules are defined as composable rewrites within the language, and we develop a set of effect analyses which guarantee program equivalence and memory safety through these transformations. We show that Exo enables rapid development of state-of-the-art matrix-matrix multiply and convolutional neural network kernels, for both an embedded neural accelerator and x86 with AVX-512 extensions, in a few dozen lines of code each.

We present a lightweight Coq framework for optimizing tensor kernels written in a pure, functional array language. Optimizations rely on user scheduling using series of verified, semantics-preserving rewrites. Unusually for compilation targeting imperative code with arrays and nested loops, all rewrites are source-to-source within a purely functional language. Our language comprises a set of core constructs for expressing high-level computation detail and a set of what we call reshape operators, which can be derived from core constructs but trigger low-level decisions about storage patterns and ordering. We demonstrate that not only is this system capable of deriving the optimizations of existing state-of-the-art languages like Halide and generating comparably performant code, it is also able to schedule a family of useful program transformations beyond what is reachable in Halide.

We present a method to automatically synthesize efficient, high-quality demosaicking algorithms, across a range of computational budgets, given a loss function and training data. It performs a multi-objective, discrete-continuous optimization which simultaneously solves for the program structure and parameters that best tradeoff computational cost and image quality. We design the method to exploit domain-specific structure for search efficiency. We apply it to several tasks, including demosaicking both Bayer and Fuji X-Trans color filter patterns, as well as joint demosaicking and super-resolution. In a few days on 8 GPUs, it produces a family of algorithms that significantly improves image quality relative to the prior state-of-the-art across a range of computational budgets from 10 s to 1000 s of operations per pixel (1 dB–3 dB higher quality at the same cost, or 8.5–200× higher throughput at same or better quality). The resulting programs combine features of both classical and deep learning-based demosaicking algorithms into more efficient hybrid combinations, which are bandwidth-efficient and vectorizable by construction. Finally, our method automatically schedules and compiles all generated programs into optimized SIMD code for modern processors.

DNN accelerators are often developed and evaluated in isolation without considering the cross-stack, system-level effects in real-world environments. This makes it difficult to appreciate the impact of Systemon-Chip (SoC) resource contention, OS overheads, and programming-stack inefficiencies on overall performance/energy-efficiency. To address this challenge, we present Gemmini, an open-source, full-stack DNN accelerator generator. Gemmini generates a wide design-space of efficient ASIC accelerators from a flexible architectural template, together with flexible programming stacks and full SoCs with shared resources that capture system-level effects. Gemmini-generated accelerators have also been fabricated, delivering up to three orders-of-magnitude speedups over high-performance CPUs on various DNN benchmarks.

Writing high-performance image processing code is challenging and labor-intensive. To address this, we propose a programming support method called “guided optimization.” Guided optimization provides programmers a set of valid optimization options and interactive feedback about their current choices, which enables them to comprehend and efficiently optimize image processing code without the time-consuming trial-and-error process of traditional text editors. We implemented a proof-of-concept system, Roly-poly, which integrates guided optimization, program visualization, and schedule cost estimation to support the comprehension and development of efficient image processing code. We conducted a user study with novice Halide programmers and confirmed that Roly-poly and its guided optimization was informative, increased productivity, and resulted in higher-performing schedules in less time.

We present a new algorithm to quickly generate high-performance GPU implementations of complex imaging and vision pipelines, directly from high-level Halide algorithm code. It is fully automatic, requiring no schedule templates or hand-optimized kernels. We address the scalability challenge of extending search-based automatic scheduling to map large real-world programs to the deep hierarchies of memory and parallelism on GPU architectures in reasonable compile time. We achieve this using (1) a two-phase search algorithm that first ‘freezes’ decisions for the lowest cost sections of a program, allowing relatively more time to be spent on the important stages, (2) a hierarchical sampling strategy that groups schedules based on their structural similarity, then samples representatives to be evaluated, allowing us to explore a large space with few samples, and (3) memoization of repeated partial schedules, amortizing their cost over all their occurrences. We guide the process with an efficient cost model combining machine learning, program analysis, and GPU architecture knowledge. We evaluate our method’s performance on a diverse suite of real-world imaging and vision pipelines. Our scalability optimizations lead to average compile time speedups of 49x (up to 530x). We find schedules that are on average 1.7x faster than existing automatic solutions (up to 5x), and competitive with what the best human experts were able to achieve in an active effort to beat our automatic results.

We present a novel programming language design that attempts to combine the clarity and safety of high-level functional languages with the efficiency and parallelism of low-level numerical languages. We treat arrays as eagerly-memoized functions on typed index sets, allowing abstract function manipulations, such as currying, to work on arrays. In contrast to composing primitive bulk-array operations, we argue for an explicit nested indexing style that mirrors application of functions to arguments. We also introduce a fine-grained typed effects system which affords concise and automatically-parallelized in-place updates. Specifically, an associative accumulation effect allows reverse-mode automatic differentiation of in-place updates in a way that preserves parallelism. Empirically, we benchmark against the Futhark array programming language, and demonstrate that aggressive inlining and type-driven compilation allows array programs to be written in an expressive, "pointful" style with little performance penalty.

Emerging research in computer graphics, inverse problems, and machine learning requires us to differentiate and optimize parametric discontinuities. These discontinuities appear in object boundaries, occlusion, contact, and sudden change over time. In many domains, such as rendering and physics simulation, we differentiate the parameters of models that are expressed as integrals over discontinuous functions. Ignoring the discontinuities during differentiation often has a significant impact on the optimization process. Previous approaches either apply specialized hand-derived solutions, smooth out the discontinuities, or rely on incorrect automatic differentiation. We propose a systematic approach to differentiating integrals with discontinuous integrands, by developing a new differentiable programming language. We introduce integration as a language primitive and account for the Dirac delta contribution from differentiating parametric discontinuities in the integrand. We formally define the language semantics and prove the correctness and closure under the differentiation, allowing the generation of gradients and higher-order derivatives. We also build a system, Teg, implementing these semantics. Our approach is widely applicable to a variety of tasks, including image stylization, fitting shader parameters, trajectory optimization, and optimizing physical designs.

Datacenter disaggregation provides numerous benefits to both the datacenter operator and the application designer. However switching from the server-centric model to a disaggregated model requires developing new programming abstractions that can achieve high performance while benefiting from the greater elasticity. To explore the limits of datacenter disaggregation, we study an application area that near-maximally benefits from current server-centric datacenters: dense linear algebra. We build NumPyWren, a system for linear algebra built on a disaggregated serverless programming model, and LAmbdaPACK, a companion domain-specific language designed for serverless execution of highly parallel linear algebra algorithms. We show that, for a number of linear algebra algorithms such as matrix multiply, singular value decomposition, Cholesky decomposition, and QR decomposition, NumPyWren's performance (completion time) is within a factor of 2 of optimized server-centric MPI implementations, and has up to 15% greater compute efficiency (total CPU-hours), while providing fault tolerance.

We investigate the connections between neural networks and simple building blocks in kernel space. In particular, using well established feature space tools such as direct sum, averaging, and moment lifting, we present an algebra for creating “compositional” kernels from bags of features. We show that these operations correspond to many of the building blocks of “neural tangent kernels (NTK)”. Experimentally, we show that there is a correlation in test error between neural network architectures and the associated kernels. We construct a simple neural network architecture using only 3x3 convolutions, 2x2 average pooling, ReLU, and optimized with SGD and MSE loss that achieves 96% accuracy on CIFAR10, and whose corresponding compositional kernel achieves 90% accuracy. We also use our constructions to investigate the relative performance of neural networks, NTKs, and compositional kernels in the small dataset regime. In particular, we find that compositional kernels outperform NTKs and neural networks outperform both kernel methods.

We introduce a differentiable rasterizer that bridges the vector graphics and raster image domains, enabling powerful raster-based loss functions, optimization procedures, and machine learning techniques to edit and generate vector content. We observe that vector graphics rasterization is differentiable after pixel prefiltering. Our differentiable rasterizer offers two prefiltering options: an analytical prefiltering technique and a multisampling anti-aliasing technique. The analytical variant is faster but can suffer from artifacts such as conflation. The multisampling variant is still efficient, and can render high-quality images while computing unbiased gradients for each pixel with respect to curve parameters. We demonstrate that our rasterizer enables new applications, including a vector graphics editor guided by image metrics, a painterly rendering algorithm that fits vector primitives to an image by minimizing a deep perceptual loss function, new vector graphics editing algorithms that exploit well-known image processing methods such as seam carving, and deep generative models that generate vector content from raster-only supervision under a VAE or GAN training objective.

We present a new algorithm to automatically schedule Halide programs for high-performance image processing and deep learning. We significantly improve upon the performance of previous methods, which considered a limited subset of schedules. We define a parameterization of possible schedules much larger than prior methods and use a variant of beam search to search over it. The search optimizes runtime predicted by a cost model based on a combination of new derived features and machine learning. We train the cost model by generating and featurizing hundreds of thousands of random programs and schedules. We show that this approach operates effectively with or without autotuning. It produces schedules which are on average almost twice as fast as the existing Halide autoscheduler without autotuning, or more than twice as fast with, and is the first automatic scheduling algorithm to significantly outperform human experts on average.

This paper presents Dexter, a new tool that automatically translates image processing functions from a low-level general-purpose language to a high-level domain-specific language (DSL), allowing them to leverage cross-platform optimizations enabled by DSLs. Rather than building a classical syntax-driven compiler to do this translation, Dexter leverages recent advances in program synthesis and program verification, along with a new domain-specific synthesis algorithm, to translate C++ image processing code to the Halide DSL, while guaranteeing semantic equivalence. This new synthesis algorithm scales and generalizes to much larger and more complex functions than prior work, including the ability to handle tiling, conditionals, and multi-stage pipelines in the original low-level code. To demonstrate the effectiveness of our approach, we evaluate Dexter using real-world image processing functions from Adobe Photoshop, a widely used multi-platform image processing program. Our results show that Dexter can translate 264 out of 353 functions in our test set, with the original implementations ranging from 20 to 150 lines of code. By leveraging Halide's advanced auto-scheduling capabilities, we get median speedups of 7.03× and 4.52× for Dexter-translated functions as compared to the original implementations on Intel and ARM architectures, respectively.

3D visual computing data are often spatially sparse. To exploit such sparsity, people have developed hierarchical sparse data structures, such as multi-level sparse voxel grids, particles, and 3D hash tables. However, developing and using these high-performance sparse data structures is challenging, due to their intrinsic complexity and overhead. We propose Taichi , a new data-oriented programming language for efficiently authoring, accessing, and maintaining such data structures. The language offers a high-level, data structure-agnostic interface for writing computation code. The user independently specifies the data structure. We provide several elementary components with different sparsity properties that can be arbitrarily composed to create a wide range of multi-level sparse data structures. This decoupling of data structures from computation makes it easy to experiment with different data structures without changing computation code, and allows users to write computation as if they are working with a dense array. Our compiler then uses the semantics of the data structure and index analysis to automatically optimize for locality, remove redundant operations for coherent accesses, maintain sparsity and memory allocations, and generate efficient parallel and vectorized instructions for CPUs and GPUs. Our approach yields competitive performance on common computational kernels such as stencil applications, neighbor lookups, and particle scattering. We demonstrate our language by implementing simulation, rendering, and vision tasks including a material point method simulation, finite element analysis, a multigrid Poisson solver for pressure projection, volumetric path tracing, and 3D convolution on sparse grids. Our computation-data structure decoupling allows us to quickly experiment with different data arrangements, and to develop high-performance data structures tailored for specific computational tasks. With 1 1 0 th as many lines of code, we achieve 4.55× higher performance on average, compared to hand-optimized reference implementations.

Gradient-based optimization has enabled dramatic advances in computational imaging through techniques like deep learning and nonlinear optimization. These methods require gradients not just of simple mathematical functions, but of general programs which encode complex transformations of images and graphical data. Unfortunately, practitioners have traditionally been limited to either hand-deriving gradients of complex computations, or composing programs from a limited set of coarse-grained operators in deep learning frameworks. At the same time, writing programs with the level of performance needed for imaging and deep learning is prohibitively difficult for most programmers. We extend the image processing language Halide with general reverse-mode automatic differentiation (AD), and the ability to automatically optimize the implementation of gradient computations. This enables automatic computation of the gradients of arbitrary Halide programs, at high performance, with little programmer effort. A key challenge is to structure the gradient code to retain parallelism. We define a simple algorithm to automatically schedule these pipelines, and show how Halide's existing scheduling primitives can express and extend the key AD optimization of "checkpointing." Using this new tool, we show how to easily define new neural network layers which automatically compile to high-performance GPU implementations, and how to solve nonlinear inverse problems from computational imaging. Finally, we show how differentiable programming enables dramatically improving the quality of even traditional, feed-forward image processing algorithms, blurring the distinction between classical and deep methods.

Specialized image processing accelerators are necessary to deliver the performance and energy efficiency required by important applications in computer vision, computational photography, and augmented reality. But creating, “programming,” and integrating this hardware into a hardware/software system is difficult. We address this problem by extending the image processing language Halide so users can specify which portions of their applications should become hardware accelerators, and then we provide a compiler that uses this code to automatically create the accelerator along with the “glue” code needed for the user’s application to access this hardware. Starting with Halide not only provides a very high-level functional description of the hardware but also allows our compiler to generate a complete software application, which accesses the hardware for acceleration when appropriate. Our system also provides high-level semantics to explore different mappings of applications to a heterogeneous system, including the flexibility of being able to change the throughput rate of the generated hardware. We demonstrate our approach by mapping applications to a commercial Xilinx Zynq system. Using its FPGA with two low-power ARM cores, our design achieves up to 6× higher performance and 38× lower energy compared to the quad-core ARM CPU on an NVIDIA Tegra K1, and 3.5× higher performance with 12× lower energy compared to the K1’s 192-core GPU.

Many graphics and vision problems can be expressed as non-linear least squares optimizations of objective functions over visual data, such as images and meshes. The mathematical descriptions of these functions are extremely concise, but their implementation in real code is tedious, especially when optimized for real-time performance on modern GPUs in interactive applications. In this work, we propose a new language, Opt, 1 for writing these objective functions over image- or graph-structured unknowns concisely and at a high level. Our compiler automatically transforms these specifications into state-of-the-art GPU solvers based on Gauss-Newton or Levenberg-Marquardt methods. Opt can generate different variations of the solver, so users can easily explore tradeoffs in numerical precision, matrix-free methods, and solver approaches. In our results, we implement a variety of real-world graphics and vision applications. Their energy functions are expressible in tens of lines of code and produce highly optimized GPU solver implementations. These solvers are competitive in performance with the best published hand-tuned, application-specific GPU solvers, and orders of magnitude beyond a general-purpose auto-generated solver.

The Halide image processing language has proven to be an effective system for authoring high-performance image processing code. Halide programmers need only provide a high-level strategy for mapping an image processing pipeline to a parallel machine (a schedule ), and the Halide compiler carries out the mechanical task of generating platform-specific code that implements the schedule. Unfortunately, designing high-performance schedules for complex image processing pipelines requires substantial knowledge of modern hardware architecture and code-optimization techniques. In this paper we provide an algorithm for automatically generating high-performance schedules for Halide programs. Our solution extends the function bounds analysis already present in the Halide compiler to automatically perform locality and parallelism-enhancing global program transformations typical of those employed by expert Halide developers. The algorithm does not require costly (and often impractical) auto-tuning, and, in seconds, generates schedules for a broad set of image processing benchmarks that are performance-competitive with, and often better than, schedules manually authored by expert Halide developers on server and mobile CPUs, as well as GPUs.

Computational photography systems are becoming increasingly diverse, while computational resources---for example on mobile platforms---are rapidly increasing. As diverse as these camera systems may be, slightly different variants of the underlying image processing tasks, such as demosaicking, deconvolution, denoising, inpainting, image fusion, and alignment, are shared between all of these systems. Formal optimization methods have recently been demonstrated to achieve state-of-the-art quality for many of these applications. Unfortunately, different combinations of natural image priors and optimization algorithms may be optimal for different problems, and implementing and testing each combination is currently a time-consuming and error-prone process. ProxImaL is a domain-specific language and compiler for image optimization problems that makes it easy to experiment with different problem formulations and algorithm choices. The language uses proximal operators as the fundamental building blocks of a variety of linear and nonlinear image formation models and cost functions, advanced image priors, and noise models. The compiler intelligently chooses the best way to translate a problem formulation and choice of optimization algorithm into an efficient solver implementation. In applications to the image processing pipeline, deconvolution in the presence of Poisson-distributed shot noise, and burst denoising, we show that a few lines of ProxImaL code can generate highly efficient solvers that achieve state-of-the-art results. We also show applications to the nonlinear and nonconvex problem of phase retrieval.

Image processing algorithms implemented using custom hardware or FPGAs of can be orders-of-magnitude more energy efficient and performant than software. Unfortunately, converting an algorithm by hand to a hardware description language suitable for compilation on these platforms is frequently too time consuming to be practical. Recent work on hardware synthesis of high-level image processing languages demonstrated that a single-rate pipeline of stencil kernels can be synthesized into hardware with provably minimal buffering. Unfortunately, few advanced image processing or vision algorithms fit into this highly-restricted programming model. In this paper, we present Rigel, which takes pipelines specified in our new multi-rate architecture and lowers them to FPGA implementations. Our flexible multi-rate architecture supports pyramid image processing, sparse computations, and space-time implementation tradeoffs. We demonstrate depth from stereo, Lucas-Kanade, the SIFT descriptor, and a Gaussian pyramid running on two FPGA boards. Our system can synthesize hardware for FPGAs with up to 436 Megapixels/second throughput, and up to 297x faster runtime than a tablet-class ARM CPU.

With existing programming tools, writing high-performance simulation code is labor intensive and requires sacrificing readability and portability. The alternative is to prototype simulations in a high-level language like Matlab, thereby sacrificing performance. The Matlab programming model naturally describes the behavior of an entire physical system using the language of linear algebra. However, simulations also manipulate individual geometric elements, which are best represented using linked data structures like meshes. Translating between the linked data structures and linear algebra comes at significant cost, both to the programmer and to the machine. High-performance implementations avoid the cost by rephrasing the computation in terms of linked or index data structures, leaving the code complicated and monolithic, often increasing its size by an order of magnitude. In this article, we present Simit, a new language for physical simulations that lets the programmer view the system both as a linked data structure in the form of a hypergraph and as a set of global vectors, matrices, and tensors depending on what is convenient at any given time. Simit provides a novel assembly construct that makes it conceptually easy and computationally efficient to move between the two abstractions. Using the information provided by the assembly construct, the compiler generates efficient in-place computation on the graph. We demonstrate that Simit is easy to use: a Simit program is typically shorter than a Matlab program; that it is high performance: a Simit program running sequentially on a CPU performs comparably to hand-optimized simulations; and that it is portable: Simit programs can be compiled for GPUs with no change to the program, delivering 4 to 20× speedups over our optimized CPU code.

Highly optimized programs are prone to bit rot, where performance quickly becomes suboptimal in the face of new hardware and compiler techniques. In this paper we show how to automatically lift performance-critical stencil kernels from a stripped x86 binary and generate the corresponding code in the high-level domain-specific language Halide. Using Halide’s state-of-the-art optimizations targeting current hardware, we show that new optimized versions of these kernels can replace the originals to rejuvenate the application for newer hardware. The original optimized code for kernels in stripped binaries is nearly impossible to analyze statically. Instead, we rely on dynamic traces to regenerate the kernels. We perform buffer structure reconstruction to identify input, intermediate and output buffer shapes. We abstract from a forest of concrete dependency trees which contain absolute memory addresses to symbolic trees suitable for high-level code generation. This is done by canonicalizing trees, clustering them based on structure, inferring higher-dimensional buffer accesses and finally by solving a set of linear equations based on buffer accesses to lift them up to simple, high-level expressions. Helium can handle highly optimized, complex stencil kernels with input-dependent conditionals. We lift seven kernels from Adobe Photoshop giving a 75% performance improvement, four kernels from IrfanView, leading to 4.97× performance, and one stencil from the miniGMG multigrid benchmark netting a 4.25× improvement in performance. We manually rejuvenated Photoshop by replacing eleven of Photoshop’s filters with our lifted implementations, giving 1.12× speedup without affecting the user experience.

Cloud image processing is often proposed as a solution to the limited computing power and battery life of mobile devices: it allows complex algorithms to run on powerful servers with virtually unlimited energy supply. Unfortunately, this overlooks the time and energy cost of uploading the input and downloading the output images. When transfer overhead is accounted for, processing images on a remote server becomes less attractive and many applications do not benefit from cloud offloading. We aim to change this in the case of image enhancements that preserve the overall content of an image. Our key insight is that, in this case, the server can compute and transmit a description of the transformation from input to output, which we call a transform recipe. At equivalent quality, our recipes are much more compact than JPEG images: this reduces the client's download. Furthermore, recipes can be computed from highly compressed inputs which significantly reduces the data uploaded to the server. The client reconstructs a high-fidelity approximation of the output by applying the recipe to its local high-quality input. We demonstrate our results on 168 images and 10 image processing applications, showing that our recipes form a compact representation for a diverse set of image filters. With an equivalent transmission budget, they provide higher-quality results than JPEG-compressed input/output images, with a gain of the order of 10 dB in many cases. We demonstrate the utility of recipes on a mobile phone by profiling the energy consumption and latency for both local and cloud computation: a transform recipe-based pipeline runs 2--4x faster and uses 2--7x less energy than local or naive cloud computation.

Program autotuning has been shown to achieve better or more portable performance in a number of domains. However, autotuners themselves are rarely portable between projects, for a number of reasons: using a domain-informed search space representation is critical to achieving good results; search spaces can be intractably large and require advanced machine learning techniques; and the landscape of search spaces can vary greatly between different problems, sometimes requiring domain specific search techniques to explore efficiently.

Specialized image signal processors (ISPs) exploit the structure of image processing pipelines to minimize memory bandwidth using the architectural pattern of line-buffering , where all intermediate data between each stage is stored in small on-chip buffers. This provides high energy efficiency, allowing long pipelines with tera-op/sec. image processing in battery-powered devices, but traditionally requires painstaking manual design in hardware. Based on this pattern, we present Darkroom, a language and compiler for image processing. The semantics of the Darkroom language allow it to compile programs directly into line-buffered pipelines, with all intermediate values in local line-buffer storage, eliminating unnecessary communication with off-chip DRAM. We formulate the problem of optimally scheduling line-buffered pipelines to minimize buffering as an integer linear program. Finally, given an optimally scheduled pipeline, Darkroom synthesizes hardware descriptions for ASIC or FPGA, or fast CPU code. We evaluate Darkroom implementations of a range of applications, including a camera pipeline, low-level feature detection algorithms, and deblurring. For many applications, we demonstrate gigapixel/sec. performance in under 0.5mm 2 of ASIC silicon at 250 mW (simulated on a 45nm foundry process), real-time 1080p/60 video processing using a fraction of the resources of a modern FPGA, and tens of megapixels/sec. of throughput on a quad-core x86 processor.

Trends in both consumer and high performance computing are bringing not only more cores, but also increased heterogeneity among the computational resources within a single machine. In many machines, one of the greatest computational resources is now their graphics coprocessors (GPUs), not just their primary CPUs. But GPU programming and memory models differ dramatically from conventional CPUs, and the relative performance characteristics of the different processors vary widely between machines. Different processors within a system often perform best with different algorithms and memory usage patterns, and achieving the best overall performance may require mapping portions of programs across all types of resources in the machine.

Image processing pipelines combine the challenges of stencil computations and stream programs. They are composed of large graphs of different stencil stages, as well as complex reductions, and stages with global or data-dependent access patterns. Because of their complex structure, the performance difference between a naive implementation of a pipeline and an optimized one is often an order of magnitude. Efficient implementations require optimization of both parallelism and locality, but due to the nature of stencils, there is a fundamental tension between parallelism, locality, and introducing redundant recomputation of shared values.

3D printing hardware is rapidly scaling up to output continuous mixtures of multiple materials at increasing resolution over ever larger print volumes. This poses an enormous computational challenge: large high-resolution prints comprise trillions of voxels and petabytes of data and simply modeling and describing the input with spatially varying material mixtures at this scale is challenging. Existing 3D printing software is insufficient; in particular, most software is designed to support only a few million primitives, with discrete material choices per object. We present OpenFab, a programmable pipeline for synthesis of multi-material 3D printed objects that is inspired by RenderMan and modern GPU pipelines. The pipeline supports procedural evaluation of geometric detail and material composition, using shader-like fablets , allowing models to be specified easily and efficiently. We describe a streaming architecture for OpenFab; only a small fraction of the final volume is stored in memory and output is fed to the printer with little startup delay. We demonstrate it on a variety of multi-material objects.

Using existing programming tools, writing high-performance image processing code requires sacrificing readability, portability, and modularity. We argue that this is a consequence of conflating what computations define the algorithm , with decisions about storage and the order of computation. We refer to these latter two concerns as the schedule , including choices of tiling, fusion, recomputation vs. storage, vectorization, and parallelism. We propose a representation for feed-forward imaging pipelines that separates the algorithm from its schedule, enabling high-performance without sacrificing code clarity. This decoupling simplifies the algorithm specification: images and intermediate buffers become functions over an infinite integer domain, with no explicit storage or boundary conditions. Imaging pipelines are compositions of functions. Programmers separately specify scheduling strategies for the various functions composing the algorithm, which allows them to efficiently explore different optimizations without changing the algorithmic code. We demonstrate the power of this representation by expressing a range of recent image processing applications in an embedded domain specific language called Halide, and compiling them for ARM, x86, and GPUs. Our compiler targets SIMD units, multiple cores, and complex memory hierarchies. We demonstrate that it can handle algorithms such as a camera raw pipeline, the bilateral grid, fast local Laplacian filtering, and image segmentation. The algorithms expressed in our language are both shorter and faster than state-of-the-art implementations.

We propose a generalized approach to decoupling shading from visibility sampling in graphics pipelines, which we call decoupled sampling . Decoupled sampling enables stochastic supersampling of motion and defocus blur at reduced shading cost, as well as controllable or adaptive shading rates which trade off shading quality for performance. It can be thought of as a generalization of multisample antialiasing (MSAA) to support complex and dynamic mappings from visibility to shading samples, as introduced by motion and defocus blur and adaptive shading. It works by defining a many-to-one hash from visibility to shading samples, and using a buffer to memoize shading samples and exploit reuse across visibility samples. Decoupled sampling is inspired by the Reyes rendering architecture, but like traditional graphics pipelines, it shades fragments rather than micropolygon vertices, decoupling shading from the geometry sampling rate. Also unlike Reyes, decoupled sampling only shades fragments after precise computation of visibility, reducing overshading. We present extensions of two modern graphics pipelines to support decoupled sampling: a GPU-style sort-last fragment architecture, and a Larrabee-style sort-middle pipeline. We study the architectural implications of decoupled sampling and blur, and derive end-to-end performance estimates on real applications through an instrumented functional simulator. We demonstrate high-quality motion and defocus blur, as well as variable and adaptive shading rates.

Volumetric effects such as beams of light through participating media are an important component in the appearance of the natural world. Many such effects can be faithfully modeled by a single scattering medium. In the presence of shadows, rendering these effects can be prohibitively expensive: current algorithms are based on ray marching, i.e., integrating the illumination scattered towards the camera along each view ray, modulated by visibility to the light source at each sample. Visibility must be determined for each sample using shadow rays or shadow-map lookups. We observe that in a suitably chosen coordinate system, the visibility function has a regular structure that we can exploit for significant acceleration compared to brute force sampling. We propose an efficient algorithm based on partial sum trees for computing the scattering integrals in a single-scattering homogeneous medium. On a CPU, we achieve speedups of 17--120x over ray marching.

We present an automated approach for high-quality preview of feature-film rendering during lighting design. Similar to previous work, we use a deep-framebuffer shaded on the GPU to achieve interactive performance. Our first contribution is to generate the deep-framebuffer and corresponding shaders automatically through data-flow analysis and compilation of the original scene. Cache compression reduces automatically-generated deep-framebuffers to reasonable size for complex production scenes and shaders. We also propose a new structure, the indirect framebuffer , that decouples shading samples from final pixels and allows a deep-framebuffer to handle antialiasing, motion blur and transparency efficiently. Progressive refinement enables fast feedback at coarser resolution. We demonstrate our approach in real-world production.