Clustered Low-Rank Tensor Format: Introduction and Application to Fast Construction of Hartree–Fock Exchange

Abstract

Clustered Low Rank (CLR) framework for block-sparse and block-low-rank tensor representation and computation is described. The CLR framework depends on 2 parameters that control precision: one controlling the CLR block rank truncation and another that controls screening of small contributions in arithmetic operations on CLR tensors. As these parameters approach zero CLR representation and arithmetic become exact. There are no other ad-hoc heuristics, such as domains. Use of the CLR format for the order-2 and order-3 tensors that appear in the context of density fitting (DF) evaluation of the Hartree-Fock (exact) exchange significantly reduced the storage and computational complexities below their standard O N 3 and O N 4 figures. Even for relatively small systems and realistic basis sets CLR-based DF HF becomes more efficient than the standard DF approach, and significantly more efficient than the conventional non-DF HF, while negligibly affecting molecular energies and properties.

From NWChem to NWChemEx: Evolving with the Computational Chemistry Landscape

“…Other lower-level programming styles, such as functional-style iteration over tensor blocks to explicit loops over tile indices and direct byte-level access to the data, are also supported to provide experts with the ability to compose arbitrary algorithms over general sparse tensorial data structures. 213 TA has been designed to support efficient execution on modern and future hardware of all scales, from a single multi-core machine to a cluster of multi-core, multi-GPU nodes, to leadership-class supercomputers. To maximize the concurrency and hide latency, which is crucial for alleviating the load imbalance and lower computation-to-communication ratio of the irregular sparse tensor algebra, TA has an asynchronous, dataflow-style core.…”

Distributed-memory multi-GPU block-sparse tensor contraction for electronic structure

“…The ABCD term was evaluated using the AO-based formalism [26]. The input tensor T representing its initial state in the coupled-cluster simulation was evaluated in AO basis using the Laplace transform approximation, with the occupied orbitals localized and both occupied and AO basis clustered to group spatially-close orbitals together [29]; the clustering defines tiling of the corresponding index ranges. The CPU-only implementation in MPQC evaluates tensor V on the fly, as needed; due to the lack of publicly-available efficient kernels for direct evaluation of AO integrals on GPUs (such kernels are under development by some of us) the GPU benchmarks used blocksparse V with the actual sparsity pattern determined by the CPU-only code but the tiles filled with random data.…”

“…To evaluate the impact of the tiling on performance, we consider three representative tilings of the index ranges. Since the k-means-based clustering algorithm that determines the range tilings is quasirandom [29] and cannot ensure uniform tiling (this would necessarily violate locality in all practical applications), these tilings are generated by specifying the target number of clusters for each index range. Table 1 summarizes the difference between the three different tilings, from the most fine-grained one (v 1 ) to the most coarse-grained one (v 3 ).…”

Scalable task-based algorithm for multiplication of block-rank-sparse matrices

“…To recover data sparsity in tensors appearing in quantum physics applications we developed the Clustered Low-Rank (CLR) representation [23] that is a general, hierarchy-free compressed tensor format. In this representation, each MIJ block of matrix M is approximated by a low-rank decomposition of the form MIJ ≈ XW † , where for a given MIJ ∈ R m×n of rank r, X is m × r and W is n × r. X and W were constructed from a rank-revealing QR decomposition, [26] MIJ P = QR.…”