arXiv subjects

Rio Yokota

Publications and source records attributed to Rio Yokota.

At least 73 records · Page 4Linked to original sources

arXiv2022

Scalable Linear Time Dense Direct Solver for 3-D Problems Without Trailing Sub-Matrix Dependencies

Factorization of large dense matrices are ubiquitous in engineering and data science applications, e.g. preconditioners for iterative boundary integral solvers, frontal matrices in sparse multifrontal solvers, and computing the determinant of covariance matrices. HSS and $\mathcal{H}^2$-matrices are hierarchical low-rank matrix formats that can reduce the complexity of factorizing such dense matrices from $\mathcal{O}(N^3)$ to $\mathcal{O}(N)$. For HSS matrices, it is possible to remove the dependency on the trailing matrices during Cholesky/LU factorization, which results in a highly parallel algorithm. However, the weak admissibility of HSS causes the rank of off-diagonal blocks to grow for 3-D problems, and the method is no longer $\mathcal{O}(N)$. On the other hand, the strong admissibility of $\mathcal{H}^2$-matrices allows it to handle 3-D problems in $\mathcal{O}(N)$, but introduces a dependency on the trailing matrices. In the present work, we pre-compute the fill-ins and integrate them into the shared basis, which allows us to remove the dependency on trailing-matrices even for $\mathcal{H}^2$-matrices. Comparisons with a block low-rank factorization code LORAPO showed a maximum speed up of 4,700x for a 3-D problem with complex geometry.

math.NA↗

arXiv2022

Parallel QR Factorization of Block Low-Rank Matrices

We present two new algorithms for Householder QR factorization of Block Low-Rank (BLR) matrices: one that performs block-column-wise QR, and another that is based on tiled QR. We show how the block-column-wise algorithm exploits BLR structure to achieve arithmetic complexity of $\mathcal{O}(mn)$, while the tiled BLR-QR exhibits $\mathcal{O}(mn^{1.5})$ complexity. However, the tiled BLR-QR has finer task granularity that allows parallel task-based execution on shared memory systems. We compare the block-column-wise BLR-QR using fork-join parallelism with tiled BLR-QR using task-based parallelism. We also compare these two implementations of Householder BLR-QR with a block-column-wise Modified Gram-Schmidt (MGS) BLR-QR using fork-join parallelism, and a state-of-the-art vendor-optimized dense Householder QR in Intel MKL. For a matrix of size 131k $\times$ 65k, all BLR methods are more than an order of magnitude faster than the dense QR in MKL. Our methods are also robust to ill-conditioning and produce better orthogonal factors than the existing MGS-based method. On a CPU with 64 cores, our parallel tiled Householder and block-column-wise Householder algorithms show a speedup of 50 and 37 times, respectively.

math.NA↗

arXiv2022

OPIRL: Sample Efficient Off-Policy Inverse Reinforcement Learning via Distribution Matching

Inverse Reinforcement Learning (IRL) is attractive in scenarios where reward engineering can be tedious. However, prior IRL algorithms use on-policy transitions, which require intensive sampling from the current policy for stable and optimal performance. This limits IRL applications in the real world, where environment interactions can become highly expensive. To tackle this problem, we present Off-Policy Inverse Reinforcement Learning (OPIRL), which (1) adopts off-policy data distribution instead of on-policy and enables significant reduction of the number of interactions with the environment, (2) learns a stationary reward function that is transferable with high generalization capabilities on changing dynamics, and (3) leverages mode-covering behavior for faster convergence. We demonstrate that our method is considerably more sample efficient and generalizes to novel environments through the experiments. Our method achieves better or comparable results on policy performance baselines with significantly fewer interactions. Furthermore, we empirically show that the recovered reward function generalizes to different tasks where prior arts are prone to fail.

cs.LG↗

arXiv2021

RePOSE: Fast 6D Object Pose Refinement via Deep Texture Rendering

We present RePOSE, a fast iterative refinement method for 6D object pose estimation. Prior methods perform refinement by feeding zoomed-in input and rendered RGB images into a CNN and directly regressing an update of a refined pose. Their runtime is slow due to the computational cost of CNN, which is especially prominent in multiple-object pose refinement. To overcome this problem, RePOSE leverages image rendering for fast feature extraction using a 3D model with a learnable texture. We call this deep texture rendering, which uses a shallow multi-layer perceptron to directly regress a view-invariant image representation of an object. Furthermore, we utilize differentiable Levenberg-Marquardt (LM) optimization to refine a pose fast and accurately by minimizing the feature-metric error between the input and rendered image representations without the need of zooming in. These image representations are trained such that differentiable LM optimization converges within few iterations. Consequently, RePOSE runs at 92 FPS and achieves state-of-the-art accuracy of 51.6% on the Occlusion LineMOD dataset - a 4.1% absolute improvement over the prior art, and comparable result on the YCB-Video dataset with a much faster runtime. The code is available at https://github.com/sh8/repose.

cs.CV↗

arXiv2021

Quantum Turbulence Coupled with Externally Driven Normal-Fluid Turbulence in Superfluid $^4$He

The coupled dynamics of quantum turbulence (QT) and normal-fluid turbulence (NFT) have been a central challenge in quantum hydrodynamics, since it is expected to cause the unsolved T2 state of QT. We numerically studied the coupled dynamics of the two turbulences in thermal counterflow. NFT is driven by external forces to control its turbulent intensity, and the fast multipole method accelerates the calculation of QT. We show that NFT enhances QT via mutual friction. The vortex line density $L$ of the QT satisfies the statistical law $L^{1/2} \approx γV_{ns}$ with the counterflow velocity $V_{ns}$. The obtained $γ$ agrees with the experiment of T2 state, validating the idea that the T2 state is caused by NFT. We propose a theoretical insight into the relation between the two turbulences.

cond-mat.supr-con↗

arXiv2020

Epipolar-Guided Deep Object Matching for Scene Change Detection

This paper describes a viewpoint-robust object-based change detection network (OBJ-CDNet). Mobile cameras such as drive recorders capture images from different viewpoints each time due to differences in camera trajectory and shutter timing. However, previous methods for pixel-wise change detection are vulnerable to the viewpoint differences because they assume aligned image pairs as inputs. To cope with the difficulty, we introduce a deep graph matching network that establishes object correspondence between an image pair. The introduction enables us to detect object-wise scene changes without precise image alignment. For more accurate object matching, we propose an epipolar-guided deep graph matching network (EGMNet), which incorporates the epipolar constraint into the deep graph matching layer used in OBJCDNet. To evaluate our network's robustness against viewpoint differences, we created synthetic and real datasets for scene change detection from an image pair. The experimental results verified the effectiveness of our network.

cs.CV↗

arXiv2020

Scalable and Practical Natural Gradient for Large-Scale Deep Learning

Large-scale distributed training of deep neural networks results in models with worse generalization performance as a result of the increase in the effective mini-batch size. Previous approaches attempt to address this problem by varying the learning rate and batch size over epochs and layers, or ad hoc modifications of batch normalization. We propose Scalable and Practical Natural Gradient Descent (SP-NGD), a principled approach for training models that allows them to attain similar generalization performance to models trained with first-order optimization methods, but with accelerated convergence. Furthermore, SP-NGD scales to large mini-batch sizes with a negligible computational overhead as compared to first-order methods. We evaluated SP-NGD on a benchmark task where highly optimized first-order methods are available as references: training a ResNet-50 model for image classification on ImageNet. We demonstrate convergence to a top-1 validation accuracy of 75.4% in 5.5 minutes using a mini-batch size of 32,768 with 1,024 GPUs, as well as an accuracy of 74.9% with an extremely large mini-batch size of 131,072 in 873 steps of SP-NGD.

cs.LG↗

arXiv2019

Practical Deep Learning with Bayesian Principles

Bayesian methods promise to fix many shortcomings of deep learning, but they are impractical and rarely match the performance of standard methods, let alone improve them. In this paper, we demonstrate practical training of deep networks with natural-gradient variational inference. By applying techniques such as batch normalisation, data augmentation, and distributed training, we achieve similar performance in about the same number of epochs as the Adam optimiser, even on large datasets such as ImageNet. Importantly, the benefits of Bayesian principles are preserved: predictive probabilities are well-calibrated, uncertainties on out-of-distribution data are improved, and continual-learning performance is boosted. This work enables practical deep learning while preserving benefits of Bayesian principles. A PyTorch implementation is available as a plug-and-play optimiser.

stat.ML↗

arXiv2019

Effect of Mixed Precision Computing on H-Matrix Vector Multiplication in BEM Analysis

Hierarchical Matrix (H-matrix) is an approximation technique which splits a target dense matrix into multiple submatrices, and where a selected portion of submatrices are low-rank approximated. The technique substantially reduces both time and space complexity of dense matrix vector multiplication, and hence has been applied to numerous practical problems. In this paper, we aim to accelerate the H-matrix vector multiplication by introducing mixed precision computing, where we employ both binary64 (FP64) and binary32 (FP32) arithmetic operations. We propose three methods to introduce mixed precision computing to H-matrix vector multiplication, and then evaluate them in a boundary element method (BEM) analysis. The numerical tests examine the effects of mixed precision computing, particularly on the required simulation time and rate of convergence of the iterative (BiCG-STAB) linear solver. We confirm the effectiveness of the proposed methods.

cs.MS↗

arXiv2019

Large-Scale Distributed Second-Order Optimization Using Kronecker-Factored Approximate Curvature for Deep Convolutional Neural Networks

Large-scale distributed training of deep neural networks suffer from the generalization gap caused by the increase in the effective mini-batch size. Previous approaches try to solve this problem by varying the learning rate and batch size over epochs and layers, or some ad hoc modification of the batch normalization. We propose an alternative approach using a second-order optimization method that shows similar generalization capability to first-order methods, but converges faster and can handle larger mini-batches. To test our method on a benchmark where highly optimized first-order methods are available as references, we train ResNet-50 on ImageNet. We converged to 75% Top-1 validation accuracy in 35 epochs for mini-batch sizes under 16,384, and achieved 75% even with a mini-batch size of 131,072, which took only 978 iterations.

cs.LG↗

arXiv2018

Extreme Scale FMM-Accelerated Boundary Integral Equation Solver for Wave Scattering

Algorithmic and architecture-oriented optimizations are essential for achieving performance worthy of anticipated energy-austere exascale systems. In this paper, we present an extreme scale FMM-accelerated boundary integral equation solver for wave scattering, which uses FMM as a matrix-vector multiplication inside the GMRES iterative method. Our FMM Helmholtz kernels treat nontrivial singular and near-field integration points. We implement highly optimized kernels for both shared and distributed memory, targeting emerging Intel extreme performance HPC architectures. We extract the potential thread- and data-level parallelism of the key Helmholtz kernels of FMM. Our application code is well optimized to exploit the AVX-512 SIMD units of Intel Skylake and Knights Landing architectures. We provide different performance models for tuning the task-based tree traversal implementation of FMM, and develop optimal architecture-specific and algorithm aware partitioning, load balancing, and communication reducing mechanisms to scale up to 6,144 compute nodes of a Cray XC40 with 196,608 hardware cores. With shared memory optimizations, we achieve roughly 77% of peak single precision floating point performance of a 56-core Skylake processor, and on average 60% of peak single precision floating point performance of a 72-core KNL. These numbers represent nearly 5.4x and 10x speedup on Skylake and KNL, respectively, compared to the baseline scalar code. With distributed memory optimizations, on the other hand, we report near-optimal efficiency in the weak scalability study with respect to both the logarithmic communication complexity as well as the theoretical scaling complexity of FMM. In addition, we exhibit up to 85% efficiency in strong scaling. We compute in excess of 2 billion DoF on the full-scale of the Cray XC40 supercomputer.

cs.PF↗

arXiv2017

Communication Reducing Algorithms for Distributed Hierarchical N-Body Problems with Boundary Distributions

Reduction of communication and efficient partitioning are key issues for achieving scalability in hierarchical $N$-Body algorithms like FMM. In the present work, we propose four independent strategies to improve partitioning and reduce communication. First of all, we show that the conventional wisdom of using space-filling curve partitioning may not work well for boundary integral problems, which constitute about 50% of FMM's application user base. We propose an alternative method which modifies orthogonal recursive bisection to solve the cell-partition misalignment that has kept it from scaling previously. Secondly, we optimize the granularity of communication to find the optimal balance between a bulk-synchronous collective communication of the local essential tree and an RDMA per task per cell. Finally, we take the dynamic sparse data exchange proposed by Hoefler et al. and extend it to a hierarchical sparse data exchange, which is demonstrated at scale to be faster than the MPI library's MPI_Alltoallv that is commonly used.

cs.DC↗

arXiv2016

A Matrix-free Preconditioner for the Helmholtz Equation based on the Fast Multipole Method

Fast multipole methods (FMM) were originally developed for accelerating $N$-body problems for particle-based methods. FMM is more than an $N$-body solver, however. Recent efforts to view the FMM as an elliptic Partial Differential Equation (PDE) solver have opened the possibility to use it as a preconditioner for a broader range of applications. FMM can solve Helmholtz problems with optimal $\mathcal{O}(N \log N)$ complexity, has compute-bound inner kernels, and highly asynchronous communication patterns. The combination of these features makes FMM an interesting candidate as a preconditioner for sparse solvers on architectures of the future. The use of FMM as a preconditioner allows us to use lower order multipole expansions than would be required as a solver because individual solves need not be accurate. This reduces the amount of computation and communication significantly and makes the time-to-solution competitive with state-of-the-art preconditioners. Furthermore, the high asynchronicity of FMM allows it to scale to much larger core counts than factorization-based and multilevel methods. We describe our tests in reproducible details with freely available codes.

math.NA↗

arXiv2016

Multi-Level Restricted Maximum Likelihood Covariance Estimation and Kriging for Large Non-Gridded Spatial Datasets

We develop a multi-level restricted Gaussian maximum likelihood method for estimating the covariance function parameters and computing the best unbiased predictor. Our approach produces a new set of multi-level contrasts where the deterministic parameters of the model are filtered out thus enabling the estimation of the covariance parameters to be decoupled from the deterministic component. Moreover, the multi-level covariance matrix of the contrasts exhibit fast decay that is dependent on the smoothness of the covariance function. Due to the fast decay of the multi-level covariance matrix coefficients only a small set is computed with a level dependent criterion. We demonstrate our approach on problems of up to 512,000 observations with a Matern covariance function and highly irregular placements of the observations. In addition, these problems are numerically unstable and hard to solve with traditional methods.

stat.CO↗

arXiv2016

Fast Multipole Method as a Matrix-Free Hierarchical Low-Rank Approximation

There has been a large increase in the amount of work on hierarchical low-rank approximation methods, where the interest is shared by multiple communities that previously did not intersect. This objective of this article is two-fold; to provide a thorough review of the recent advancements in this field from both analytical and algebraic perspectives, and to present a comparative benchmark of two highly optimized implementations of contrasting methods for some simple yet representative test cases. We categorize the recent advances in this field from the perspective of compute-memory tradeoff, which has not been considered in much detail in this area. Benchmark tests reveal that there is a large difference in the memory consumption and performance between the different methods.

math.NA↗

arXiv2016

Fast Multipole Preconditioners for Sparse Matrices Arising from Elliptic Equations

Among optimal hierarchical algorithms for the computational solution of elliptic problems, the Fast Multipole Method (FMM) stands out for its adaptability to emerging architectures, having high arithmetic intensity, tunable accuracy, and relaxable global synchronization requirements. We demonstrate that, beyond its traditional use as a solver in problems for which explicit free-space kernel representations are available, the FMM has applicability as a preconditioner in finite domain elliptic boundary value problems, by equipping it with boundary integral capability for satisfying conditions at finite boundaries and by wrapping it in a Krylov method for extensibility to more general operators. Here, we do not discuss the well developed applications of FMM to implement matrix-vector multiplications within Krylov solvers of boundary element methods. Instead, we propose using FMM for the volume-to-volume contribution of inhomogeneous Poisson-like problems, where the boundary integral is a small part of the overall computation. Our method may be used to precondition sparse matrices arising from finite difference/element discretizations, and can handle a broader range of scientific applications. Compared with multigrid methods, it is capable of comparable algebraic convergence rates down to the truncation error of the discretized PDE, and it offers potentially superior multicore and distributed memory scalability properties on commodity architecture supercomputers. Compared with other methods exploiting the low rank character of off-diagonal blocks of the dense resolvent operator, FMM-preconditioned Krylov iteration may reduce the amount of communication because it is matrix-free and exploits the tree structure of FMM. We describe our tests in reproducible detail with freely available codes and outline directions for further extensibility.

math.NA↗

arXiv2014

Communication Complexity of the Fast Multipole Method and its Algebraic Variants

A combination of hierarchical tree-like data structures and data access patterns from fast multipole methods and hierarchical low-rank approximation of linear operators from H-matrix methods appears to form an algorithmic path forward for efficient implementation of many linear algebraic operations of scientific computing at the exascale. The combination provides asymptotically optimal computational and communication complexity and applicability to large classes of operators that commonly arise in scientific computing applications. A convergence of the mathematical theories of the fast multipole and H-matrix methods has been underway for over a decade. We recap this mathematical unification and describe implementation aspects of a hybrid of these two compelling hierarchical algorithms on hierarchical distributed-shared memory architectures, which are likely to be the first to reach the exascale. We present a new communication complexity estimate for fast multipole methods on such architectures. We also show how the data structures and access patterns of H-matrices for low-rank operators map onto those of fast multipole, leading to an algebraically generalized form of fast multipole that compromises none of its architecturally ideal properties.

cs.DC↗

arXiv2014

Asynchronous Execution of the Fast Multipole Method Using Charm++

Fast multipole methods (FMM) on distributed mem- ory have traditionally used a bulk-synchronous model of com- municating the local essential tree (LET) and overlapping it with computation of the local data. This could be perceived as an extreme case of data aggregation, where the whole LET is communicated at once. Charm++ allows a much finer control over the granularity of communication, and has a asynchronous execution model that fits well with the structure of our FMM code. Unlike previous work on asynchronous fast N-body methods such as ChaNGa and PEPC, the present work performs a direct comparison against the traditional bulk-synchronous approach and the asynchronous approach using Charm++. Furthermore, the serial performance of our FMM code is over an order of magnitude better than these previous codes, so it is much more challenging to hide the overhead of Charm++.

cs.DC↗