Grey Ballard

Grey Ballard, Koby Hayashi, Ramakrishnan Kannan

The CP tensor decomposition is a low-rank approximation of a tensor. We present a distributed-memory parallel algorithm and implementation of an alternating optimization method for computing a CP decomposition of dense tensor data that can enforce nonnegativity of the computed low-rank factors. The principal task is to parallelize the matricized-tensor times Khatri-Rao product (MTTKRP) bottleneck subcomputation. The algorithm is computation efficient, using dimension trees to avoid redundant computation across MTTKRPs within the alternating method. Our approach is also communication efficient, using a data distribution and parallel algorithm across a multidimensional processor grid that can be tuned to minimize communication. We benchmark our software on synthetic as well as hyperspectral image and neuroscience dynamic functional connectivity data, demonstrating that our algorithm scales well to 100s of nodes (up to 4096 cores) and is faster and more general than the currently available parallel software.

Aditya Devarakonda, Grey Ballard

In this work, we design, analyze, and optimize sequential and shared-memory parallel algorithms for partitioned local depths (PaLD). Given a set of data points and pairwise distances, PaLD is a method for identifying strength of pairwise relationships based on relative distances, enabling the identification of strong ties within dense and sparse communities even if their sizes and within-community absolute distances vary greatly. We design two algorithmic variants that perform community structure analysis through triplet comparisons of pairwise distances. We present theoretical analyses of computation and communication costs and prove that the sequential algorithms are communication optimal, up to constant factors. We introduce performance optimization strategies that yield sequential speedups of up to $29\times$ over a baseline sequential implementation and parallel speedups of up to $19.4\times$ over optimized sequential implementations using up to $32$ threads on an Intel multicore CPU.

Arvind K. Saibaba, Bhisham Dev Verma, Grey Ballard

Randomization has emerged as a powerful set of tools for large-scale matrix and tensor decompositions. Randomized algorithms involve computing sketches with random matrices. A prevalent approach is to take the random matrix as a standard Gaussian random matrix, for which the theory is well developed. However, this approach has the drawback that the cost of generating and multiplying by the random matrix can be prohibitively expensive. Khatri-Rao random projections (KRPs), obtained by sketching with Khatri-Rao products of random matrices, offer a viable alternative and are much cheaper to generate. However, the theoretical guarantees of using KRPs are much more pessimistic compared to their accuracy observed in practice. We attempt to close this gap by obtaining improved analysis of the use of KRPs in matrix and tensor low-rank decompositions. We propose and analyze a new algorithm for low-rank approximations of block-structured matrices (e.g., block Hankel) using KRPs. We also show how to accelerate tensor computations in the Tucker format using KRPs and give theoretical guarantees of the resulting low-rank approximations. Numerical experiments on synthetic and real-world tensors show the computational benefits of the proposed methods.

Rudi Smith, Mirjeta Pasha, Andrés Galindo-Olarte et al.

We present efficient, sketching-based methods for the summation of tensors in Tucker format. Leveraging the algebraic structure of Khatri-Rao and Kronecker products, our approach enables compressed arithmetic on Tucker tensors while controlling rank growth and computational cost. The proposed sketching framework avoids the explicit formation of large intermediate tensors, instead operating directly on the factor matrices and core tensors to produce accurate low-rank approximations of tensor sums. Furthermore, we analyze the computational complexity and the theoretical approximation properties of the proposed methodology. Numerical experiments demonstrate the effectiveness of our approach on four problems: two synthetic test cases, a parameter-dependent elliptic equation (commonly referred to as the cookie problem) solved via GMRES, and a one-dimensional linear transport problem discretized via high-order discontinuous Galerkin methods, where repeated tensor summation arises as a core computational bottleneck. Across these examples, the sketching-based summation achieves substantial computational savings while preserving high accuracy relative to direct summation and re-compression.

Koby Hayashi, Sinan G. Aksoy, Grey Ballard et al.

Symmetric Nonnegative Matrix Factorization (SymNMF) is a technique in data analysis and machine learning that approximates a symmetric matrix with a product of a nonnegative, low-rank matrix and its transpose. To design faster and more scalable algorithms for SymNMF we develop two randomized algorithms for its computation. The first algorithm uses randomized matrix sketching to compute an initial low-rank approximation to the input matrix and proceeds to rapidly compute a SymNMF of the approximation. The second algorithm uses randomized leverage score sampling to approximately solve constrained least squares problems. Many successful methods for SymNMF rely on (approximately) solving sequences of constrained least squares problems. We prove theoretically that leverage score sampling can approximately solve nonnegative least squares problems to a chosen accuracy with high probability. Additionally, we prove sampling complexity results for previously proposed hybrid sampling techniques which deterministically include high leverage score rows. This hybrid scheme is crucial for obtaining speeds ups in practice. Finally we demonstrate that both methods work well in practice by applying them to graph clustering tasks on large real world data sets. These experiments show that our methods approximately maintain solution quality and achieve significant speed ups for both large dense and large sparse problems.

Chao Wang, Grey Ballard, Robert Plemmons et al.

We consider the problem of joint three-dimensional (3D) localization and material classification of unresolved space debris using a multispectral rotating point spread function (RPSF). The use of RPSF allows one to estimate the 3D locations of point sources from their rotated images acquired by a single 2D sensor array, since the amount of rotation of each source image about its x, y location depends on its axial distance z. Using multi-spectral images, with one RPSF per spectral band, we are able not only to localize the 3D positions of the space debris but also classify their material composition. We propose a three-stage method for achieving joint localization and classification. In Stage 1, we adopt an optimization scheme for localization in which the spectral signature of each material is assumed to be uniform, which significantly improves efficiency and yields better localization results than possible with a single spectral band. In Stage 2, we estimate the spectral signature and refine the localization result via an alternating approach. We process classification in the final stage. Both Poisson noise and Gaussian noise models are considered, and the implementation of each is discussed. Numerical tests using multispectral data from NASA show the efficiency of our three-stage approach and illustrate the improvement of point source localization and spectral classification from using multiple bands over a single band.

Ramakrishnan Kannan, Grey Ballard, Haesun Park

Non-negative matrix factorization (NMF) is the problem of determining two non-negative low rank factors $W$ and $H$, for the given input matrix $A$, such that $A \approx W H$. NMF is a useful tool for many applications in different domains such as topic modeling in text mining, background separation in video analysis, and community detection in social networks. Despite its popularity in the data mining community, there is a lack of efficient parallel algorithms to solve the problem for big data sets. The main contribution of this work is a new, high-performance parallel computational framework for a broad class of NMF algorithms that iteratively solves alternating non-negative least squares (NLS) subproblems for $W$ and $H$. It maintains the data and factor matrices in memory (distributed across processors), uses MPI for interprocessor communication, and, in the dense case, provably minimizes communication costs (under mild assumptions). The framework is flexible and able to leverage a variety of NMF and NLS algorithms, including Multiplicative Update, Hierarchical Alternating Least Squares, and Block Principal Pivoting. Our implementation allows us to benchmark and compare different algorithms on massive dense and sparse data matrices of size that spans for few hundreds of millions to billions. We demonstrate the scalability of our algorithm and compare it with baseline implementations, showing significant performance improvements. The code and the datasets used for conducting the experiments are available online.

Grey Ballard, Austin R. Benson, Alex Druinsky et al.

Fast algorithms for matrix multiplication, namely those that perform asymptotically fewer scalar operations than the classical algorithm, have been considered primarily of theoretical interest. Apart from Strassen's original algorithm, few fast algorithms have been efficiently implemented or used in practical applications. However, there exist many practical alternatives to Strassen's algorithm with varying performance and numerical properties. Fast algorithms are known to be numerically stable, but because their error bounds are slightly weaker than the classical algorithm, they are not used even in cases where they provide a performance benefit. We argue in this paper that the numerical sacrifice of fast algorithms, particularly for the typical use cases of practical algorithms, is not prohibitive, and we explore ways to improve the accuracy both theoretically and empirically. The numerical accuracy of fast matrix multiplication depends on properties of the algorithm and of the input matrices, and we consider both contributions independently. We generalize and tighten previous error analyses of fast algorithms and compare their properties. We discuss algorithmic techniques for improving the error guarantees from two perspectives: manipulating the algorithms, and reducing input anomalies by various forms of diagonal scaling. Finally, we benchmark performance and demonstrate our improved numerical accuracy.