Moment invariants are a powerful tool for the generation of rotation-invariant descriptors needed for many applications in pattern detection, classification, and machine learning. A set of invariants is optimal if it is complete, independent, and robust against degeneracy in the input. In this paper, we show that the current state of the art for the generation of these bases of moment invariants, despite being robust against moment tensors being identically zero, is vulnerable to a degeneracy that is common in real-world applications, namely spherical functions. We show how to overcome this vulnerability by combining two popular moment invariant approaches: one based on spherical harmonics and one based on Cartesian tensor algebra.
It is known that a linear system with a system matrix A constitutes a Hamiltonian system with a quadratic Hamiltonian if and only if A is a Hamiltonian matrix. This provides a straightforward method to verify whether a linear system is Hamiltonian or whether a given Hamiltonian function corresponds to a linear system. These techniques fundamentally rely on the properties of Hamiltonian matrices. Building on recent advances in tensor algebra, this paper generalizes such results to a broad class of polynomial systems. As the systems of interest can be naturally represented in tensor forms, we name them tensor-based polynomial systems. Our main contribution is that we formally define Hamiltonian cubical tensors and characterize their properties. Crucially, we demonstrate that a tensor-based polynomial system is a Hamiltonian system with a polynomial Hamiltonian if and only if all associated system tensors are Hamiltonian cubical tensors-a direct parallel to the linear case. Additionally, we establish a computationally tractable stability criterion for tensor-based polynomial Hamiltonian systems. Finally, we validate all theoretical results through numerical examples and provide a further intuitive discussion.
Integrated sensing and communication (ISAC) is envisioned be to one of the paradigms upon which next-generation mobile networks will be built, extending localization and tracking capabilities, as well as giving birth to environment-aware wireless access. A key aspect of sensing integration is parameter estimation, which involves extracting information about the surrounding environment, such as the direction, distance, and velocity of various objects within. This is typically of a high-dimensional nature, which leads to significant computational complexity, if performed jointly across multiple sensing dimensions, such as space, frequency, and time. Additionally, due to the incorporation of sensing on top of the data transmission, the time window available for sensing is likely to be short, resulting in an estimation problem where only a single snapshot is accessible. In this work, we propose PLAIN, a tensor-based estimation architecture that flexibly scales with multiple sensing dimensions and can handle high dimensionality, limited measurement time, and super-resolution requirements. It consists of three stages: a compression stage, where the high dimensional input is converted into lower dimensionality, without sacrificing resolution; a decoupled estimation stage, where the parameters across the different dimensions are estimated in parallel with low complexity; an input-based fusion stage, where the decoupled parameters are fused together to form a paired multidimensional estimate. We investigate the performance of the architecture for different configurations and compare it against practical sequential and joint estimation baselines, as well as theoretical bounds. Our results show that PLAIN, using tools from tensor algebra, subspace-based processing, and compressed sensing, can scale flexibly with dimensionality, while operating with low complexity and maintaining super-resolution.
Transformers have become the leading choice in natural language processing over other deep learning architectures. This trend has also permeated the field of time series analysis, especially for long-horizon forecasting, showcasing promising results both in performance and running time. In this paper, we introduce Local Attention Mechanism (LAM), an efficient attention mechanism tailored for time series analysis. This mechanism exploits the continuity properties of time series to reduce the number of attention scores computed. We present an algorithm for implementing LAM in tensor algebra that runs in time and memory O(nlogn), significantly improving upon the O(n^2) time and memory complexity of traditional attention mechanisms. We also note the lack of proper datasets to evaluate long-horizon forecast models. Thus, we propose a novel set of datasets to improve the evaluation of models addressing long-horizon forecasting challenges. Our experimental analysis demonstrates that the vanilla transformer architecture magnified with LAM surpasses state-of-the-art models, including the vanilla attention mechanism. These results confirm the effectiveness of our approach and highlight a range of future challenges in long-sequence time series forecasting.
Tensor algebra is a crucial component for data-intensive workloads such as machine learning and scientific computing. As the complexity of data grows, scientists often encounter a dilemma between the highly specialized dense tensor algebra and efficient structure-aware algorithms provided by sparse tensor algebra. In this paper, we introduce DASTAC, a framework to propagate the tensors's captured high-level structure down to low-level code generation by incorporating techniques such as automatic data layout compression, polyhedral analysis, and affine code generation. Our methodology reduces memory footprint by automatically detecting the best data layout, heavily benefits from polyhedral optimizations, leverages further optimizations, and enables parallelization through MLIR. Through extensive experimentation, we show that DASTAC achieves 1 to 2 orders of magnitude speedup over TACO, a state-of-the-art sparse tensor compiler, and StructTensor, a state-of-the-art structured tensor algebra compiler, with a significantly lower memory footprint.
Bayesian optimization is a powerful method for automating tuning of compilers. The complex landscape of autotuning provides a myriad of rarely considered structural challenges for black-box optimizers, and the lack of standardized benchmarks has limited the study of Bayesian optimization within the domain. To address this, we present CATBench, a comprehensive benchmarking suite that captures the complexities of compiler autotuning, ranging from discrete, conditional, and permutation parameter types to known and unknown binary constraints, as well as both multi-fidelity and multi-objective evaluations. The benchmarks in CATBench span a range of machine learning-oriented computations, from tensor algebra to image processing and clustering, and uses state-of-the-art compilers, such as TACO and RISE/ELEVATE. CATBench offers a unified interface for evaluating Bayesian optimization algorithms, promoting reproducibility and innovation through an easy-to-use, fully containerized setup of both surrogate and real-world compiler optimization tasks. We validate CATBench on several state-of-the-art algorithms, revealing their strengths and weaknesses and demonstrating the suite's potential for advancing both Bayesian optimization and compiler autotuning research.
Kernel orchestration is the task of mapping the computation defined in different operators of a deep neural network (DNN) to the execution of GPU kernels on modern hardware platforms. Prior approaches optimize kernel orchestration by greedily applying operator fusion, which fuses the computation of multiple operators into a single kernel, and miss a variety of optimization opportunities in kernel orchestration. This paper presents Korch, a tensor program optimizer that discovers optimal kernel orchestration strategies for tensor programs. Instead of directly fusing operators, Korch first applies operator fission to decompose tensor operators into a small set of basic tensor algebra primitives. This decomposition enables a diversity of fine-grained, inter-operator optimizations. Next, Korch optimizes kernel orchestration by formalizing it as a constrained optimization problem, leveraging an off-the-shelf binary linear programming solver to discover an optimal orchestration strategy, and generating an executable that can be directly deployed on modern GPU platforms. Evaluation on a variety of DNNs shows that Korch outperforms existing tensor program optimizers by up to 1.7x on V100 GPUs and up to 1.6x on A100 GPUs. Korch is publicly available at https://github.com/humuyan/Korch.
We consider a stochastic volatility model where the dynamics of the volatility are given by a possibly infinite linear combination of the elements of the time extended signature of a Brownian motion. First, we show that the model is remarkably universal, as it includes, but is not limited to, the celebrated Stein-Stein, Bergomi, and Heston models, together with some path-dependent variants. Second, we derive the joint characteristic functional of the log-price and integrated variance provided that some infinite dimensional extended tensor algebra valued Riccati equation admits a solution. This allows us to price and (quadratically) hedge certain European and path-dependent options using Fourier inversion techniques. We highlight the efficiency and accuracy of these Fourier techniques in a comprehensive numerical study.
Discovering governing equations from data is important to many scientific and engineering applications. Despite promising successes, existing methods are still challenged by data sparsity and noise issues, both of which are ubiquitous in practice. Moreover, state-of-the-art methods lack uncertainty quantification and/or are costly in training. To overcome these limitations, we propose a novel equation discovery method based on Kernel learning and BAyesian Spike-and-Slab priors (KBASS). We use kernel regression to estimate the target function, which is flexible, expressive, and more robust to data sparsity and noises. We combine it with a Bayesian spike-and-slab prior -- an ideal Bayesian sparse distribution -- for effective operator selection and uncertainty quantification. We develop an expectation-propagation expectation-maximization (EP-EM) algorithm for efficient posterior inference and function estimation. To overcome the computational challenge of kernel regression, we place the function values on a mesh and induce a Kronecker product construction, and we use tensor algebra to enable efficient computation and optimization. We show the advantages of KBASS on a list of benchmark ODE and PDE discovery tasks.
In past few decades, tensor algebra also known as multi-linear algebra has been developed and customized as a tool to be used for various engineering applications. In particular, with the help of a special form of tensor contracted product, known as the Einstein Product and its properties, many of the known concepts from Linear Algebra could be extended to a multi-linear setting. This enables to define the notions of multi-linear system theory where the input, output signals and the system are multi-domain in nature. This paper provides an overview of tensor algebra tools which can be seen as an extension of linear algebra, at the same time highlighting the difference and advantages that the multi-linear setting brings forth. In particular, the notion of tensor inversion, tensor singular value and tensor Eigenvalue decomposition using the Einstein product is explained. In addition, this paper also introduces the notion of contracted convolution in both discrete and continuous multi-linear system tensors. Tensor Networks representation of various tensor operations is also presented. Also, application of tensor tools in developing transceiver schemes for multi-domain communication systems, with an example of MIMO CDMA systems, is presented. Thus this paper acts as an entry point tutorial for graduate students whose research involves multi-domain or multi-modal signals and systems.
Despite the omnipresence of tensors and tensor operations in modern deep learning, the use of tensor mathematics to formally design and describe neural networks is still under-explored within the deep learning community. To this end, we introduce the Graph Tensor Network (GTN) framework, an intuitive yet rigorous graphical framework for systematically designing and implementing large-scale neural learning systems on both regular and irregular domains. The proposed framework is shown to be general enough to include many popular architectures as special cases, and flexible enough to handle data on any and many data domains. The power and flexibility of the proposed framework is demonstrated through real-data experiments, resulting in improved performance at a drastically lower complexity costs, by virtue of tensor algebra.
A universal kernel is constructed whose sections approximate any causal and time-invariant filter in the fading memory category with inputs and outputs in a finite-dimensional Euclidean space. This kernel is built using the reservoir functional associated with a state-space representation of the Volterra series expansion available for any analytic fading memory filter. It is hence called the Volterra reservoir kernel. Even though the state-space representation and the corresponding reservoir feature map are defined on an infinite-dimensional tensor algebra space, the kernel map is characterized by explicit recursions that are readily computable for specific data sets when employed in estimation problems using the representer theorem. We showcase the performance of the Volterra reservoir kernel in a popular data science application in relation to bitcoin price prediction.
Analytics of financial data is inherently a Big Data paradigm, as such data are collected over many assets, asset classes, countries, and time periods. This represents a challenge for modern machine learning models, as the number of model parameters needed to process such data grows exponentially with the data dimensions; an effect known as the Curse-of-Dimensionality. Recently, Tensor Decomposition (TD) techniques have shown promising results in reducing the computational costs associated with large-dimensional financial models while achieving comparable performance. However, tensor models are often unable to incorporate the underlying economic domain knowledge. To this end, we develop a novel Graph-Regularized Tensor Regression (GRTR) framework, whereby knowledge about cross-asset relations is incorporated into the model in the form of a graph Laplacian matrix. This is then used as a regularization tool to promote an economically meaningful structure within the model parameters. By virtue of tensor algebra, the proposed framework is shown to be fully interpretable, both coefficient-wise and dimension-wise. The GRTR model is validated in a multi-way financial forecasting setting and compared against competing models, and is shown to achieve improved performance at reduced computational costs. Detailed visualizations are provided to help the reader gain an intuitive understanding of the employed tensor operations.
Boosting the runtime performance of deep neural networks (DNNs) is critical due to their wide adoption in real-world tasks. Existing approaches to optimizing the tensor algebra expression of a DNN only consider expressions representable by a fixed set of predefined operators, missing possible optimization opportunities between general expressions. We propose OLLIE, the first derivation-based tensor program optimizer. OLLIE optimizes tensor programs by leveraging transformations between general tensor algebra expressions, enabling a significantly larger expression search space that includes those supported by prior work as special cases. OLLIE uses a hybrid derivation-based optimizer that effectively combines explorative and guided derivations to quickly discover highly optimized expressions. Evaluation on seven DNNs shows that OLLIE can outperform existing optimizers by up to 2.73$\times$ (1.46$\times$ on average) on an A100 GPU and up to 2.68$\times$ (1.51$\times$) on a V100 GPU, respectively.
We present tntorch, a tensor learning framework that supports multiple decompositions (including Candecomp/Parafac, Tucker, and Tensor Train) under a unified interface. With our library, the user can learn and handle low-rank tensors with automatic differentiation, seamless GPU support, and the convenience of PyTorch's API. Besides decomposition algorithms, tntorch implements differentiable tensor algebra, rank truncation, cross-approximation, batch processing, comprehensive tensor arithmetics, and more.
In recent years, many accelerators have been proposed to efficiently process sparse tensor algebra applications (e.g., sparse neural networks). However, these proposals are single points in a large and diverse design space. The lack of systematic description and modeling support for these sparse tensor accelerators impedes hardware designers from efficient and effective design space exploration. This paper first presents a unified taxonomy to systematically describe the diverse sparse tensor accelerator design space. Based on the proposed taxonomy, it then introduces Sparseloop, the first fast, accurate, and flexible analytical modeling framework to enable early-stage evaluation and exploration of sparse tensor accelerators. Sparseloop comprehends a large set of architecture specifications, including various dataflows and sparse acceleration features (e.g., elimination of zero-based compute). Using these specifications, Sparseloop evaluates a design's processing speed and energy efficiency while accounting for data movement and compute incurred by the employed dataflow as well as the savings and overhead introduced by the sparse acceleration features using stochastic tensor density models. Across representative accelerators and workloads, Sparseloop achieves over 2000 times faster modeling speed than cycle-level simulations, maintains relative performance trends, and achieves 0.1% to 8% average error. With a case study, we demonstrate Sparseloop's ability to help reveal important insights for designing sparse tensor accelerators (e.g., it is important to co-design orthogonal design aspects).
We present LoopStack, a domain specific compiler stack for tensor operations, composed of a frontend, LoopTool, and an efficient optimizing code generator, LoopNest. This stack enables us to compile entire neural networks and generate code targeting the AVX2, AVX512, NEON, and NEONfp16 instruction sets while incorporating optimizations often missing from other machine learning compiler backends. We evaluate our stack on a collection of full neural networks and commonly used network blocks as well as individual operators, and show that LoopStack generates machine code that matches and frequently exceeds the performance of in state-of-the-art machine learning frameworks in both cases. We also show that for a large collection of schedules LoopNest's compilation is orders of magnitude faster than LLVM, while resulting in equal or improved run time performance. Additionally, LoopStack has a very small memory footprint - a binary size of 245KB, and under 30K lines of effective code makes it ideal for use on mobile and embedded devices.
The strong demand for efficient and performant deployment of Deep Learning (DL) applications prompts the rapid development of a rich DL ecosystem. To keep up with this fast advancement, it is crucial for modern DL frameworks to efficiently integrate a variety of optimized tensor algebra libraries and runtimes as their backends and generate the fastest possible executable using these backends. However, current DL frameworks require significant manual effort and expertise to integrate every new backend while failing to unleash its full potential. Given the fast-evolving nature of the DL ecosystem, this manual approach often slows down continuous innovations across different layers; it prevents hardware vendors from the fast deployment of their cutting-edge libraries, DL framework developers must repeatedly adjust their hand-coded rules to accommodate new versions of libraries, and machine learning practitioners need to wait for the integration of new technologies and often encounter unsatisfactory performance. In this paper, we propose Collage, a DL framework that offers seamless integration of DL backends. Collage provides an expressive backend registration interface that allows users to precisely specify the capability of various backends. By leveraging the specifications of available backends, Collage automatically searches for an optimized backend placement strategy for a given workload and execution environment. Our evaluation shows that Collage outperforms the best existing framework for each hardware by $1.26\times$, $1.43\times$, $1.40\times$ on average on NVIDIA's RTX 2070 GPU, V100 GPU, and Intel's Xeon 8259CL CPU, respectively. Collage has been open-sourced and deployed in Apache TVM.
Outlier detection (OD) is a key learning task for finding rare and deviant data samples, with many time-critical applications such as fraud detection and intrusion detection. In this work, we propose TOD, the first tensor-based system for efficient and scalable outlier detection on distributed multi-GPU machines. A key idea behind TOD is decomposing complex OD applications into a small collection of basic tensor algebra operators. This decomposition enables TOD to accelerate OD computations by leveraging recent advances in deep learning infrastructure in both hardware and software. Moreover, to deploy memory-intensive OD applications on modern GPUs with limited on-device memory, we introduce two key techniques. First, provable quantization speeds up OD computations and reduces its memory footprint by automatically performing specific floating-point operations in lower precision while provably guaranteeing no accuracy loss. Second, to exploit the aggregated compute resources and memory capacity of multiple GPUs, we introduce automatic batching, which decomposes OD computations into small batches for both sequential execution on a single GPU and parallel execution on multiple GPUs. TOD supports a diverse set of OD algorithms. Extensive evaluation on 11 real and 3 synthetic OD datasets shows that TOD is on average 10.9x faster than the leading CPU-based OD system PyOD (with a maximum speedup of 38.9x), and can handle much larger datasets than existing GPU-based OD systems. In addition, TOD allows easy integration of new OD operators, enabling fast prototyping of emerging and yet-to-discovered OD algorithms.
Nonnegative Tucker decomposition (NTD), a tensor decomposition model, has received increased interest in the recent years because of its ability to blindly extract meaningful patterns, in particular in Music Information Retrieval. Nevertheless, existing algorithms to compute NTD are mostly designed for the Euclidean loss. This work proposes a multiplicative updates algorithm to compute NTD with the beta-divergence loss, often considered a better loss for audio processing. We notably show how to implement efficiently the multiplicative rules using tensor algebra. Finally, we show on a music structure analysis task that unsupervised NTD fitted with beta-divergence loss outperforms earlier results obtained with the Euclidean loss.