Modewise Johnson–Lindenstrauss embeddings for nuclear many-body theory

Abstract

In the present work, we initiate a program that explores modewise Johnson–Lindenstrauss embeddings (JLEs) as a tool to reduce the computational cost and memory requirements of (nuclear) many-body methods. These embeddings are randomized projections of high-dimensional data tensors onto low-dimensional subspaces that preserve structural features like norms and inner products. An appealing feature of randomized embedding techniques is that they allow for the oblivious and incremental compression of large tensors, e.g., the nuclear Hamiltonian or wave functions amplitudes, into significantly smaller random sketches that still allow for the accurate calculation of ground-state energies and other observables. In particular, the oblivious nature of randomized JLE techniques makes it possible to compress a tensor without knowing in advance exactly what observables one might want to approximate at a later time. This opens the door for the use of tensors that are much too large to store in memory, e.g., untruncated three-nucleon forces in current approaches, or complete two- plus three-nucleon Hamiltonians in large, symmetry-unrestricted bases. Such compressed Hamiltonians can be stored and used later on with relative ease. As a first step, we perform a detailed analysis of a JLE’s impact on the second-order Many-Body Perturbation Theory (MBPT) corrections for nuclear ground-state observables like the energy and the radius, noting that these will be the dominant corrections in a well-behaved perturbative expansion, and highly important implicit contributions even in nonperturbative approaches. Numerical experiments for a wide range of closed-shell nuclei, model spaces and state-of-the-art nuclear interactions demonstrate the validity and potential of the proposed approach: We can compress nuclear Hamiltonians hundred- to thousandfold while only incurring mean relative errors of 1% or less in ground-state observables. Importantly, we show that JLEs capture the relevant physical information contained in the highly structured Hamiltonian tensor despite their random characteristics. In addition to the significant storage savings, the achieved compressions imply multiple order-of-magnitude reductions in computational effort when the compressed Hamiltonians are used in higher-order MBPT or nonperturbative many-body methods.

Appendix A: Additional results

For completeness, we compile detailed results for energy corrections in additional nuclei in this appendix.

Fig. 13

Contributions of each angular momentum channel to the mean relative error $\overline{\varDelta E^{(2)}}/|E^{(2)}$ as a function of $e_\text {max}$, for fixed compression $c_\textrm{tot}\le 10^{-3}$. All calculations were performed with the EM1.8/2.0 interaction. Our favored two-stage JL embedding (RFD + RFD)_real (cf. Sect. 5.2) was used for compression, and 200 trials were carried out

Figure 13 shows the angular-momentum channel breakdown of the mean relative error of $E^{(2)}$ for different basis sizes in the nuclei ${}^{40,48}\textrm{Ca}$. The observed trends match what we found and discussed for ${}^{16}\textrm{O}$ and ${}^{132}\textrm{Sn}$ via Fig. 6.

Figure 14 extends the results for $\overline{\varDelta E^{(2)}}(J)/|E^{(2)|}$ of Fig. 7 for nuclei beyond ${}^{16}\textrm{O}$. The mean relative errors grow exponentially as we decrease $c_\textrm{tot}$ over a wide range, with some allowance for fluctuations due to the random sampling performed by the JLE (cf. Sect. 5.3.2).

Fig. 14

Contributions of each angular momentum channel to the mean relative error $\overline{\varDelta E^{(2)}}/|E^{(2)}|$ as a function of the compression $c_\textrm{tot}$. Calculations were performed with the EM1.8/2.0 interaction in an $e_\text {max}=14$ basis, and a two-stage JL embedding (RFD + RFD)_real with $c_2=0.2$ has been used. 200 trials were performed