论文标题
有效构建张量网络的规范多核近似
Efficient construction of canonical polyadic approximations of tensor networks
论文作者
论文摘要
我们考虑为张量网络而不是单个张量构建规范多核（CP）分解的问题。我们说明如何通过在CP因子优化过程中利用其结构来降低网络的近似CP表示的复杂性。对于通过近似的广义平方根（SQ）分解（例如密度拟合或（透视）Cholesky，由2个订单-3张量近似于2个订单-3张量近似的订单-4库仑相互作用张量。构建4路CP分解的复杂性从$ \ Mathcal {o}（n^4 r_ r_ \ text {cp}）$（对于非及可拟合的库仑张量）到$ \ mathcal {o}（n^3 r_ \ text {cp} $ n $ n with with with with with went $ r_ \ text {cp} $分别是基础和CP等级。这降低了与精确多体电子结构相关的2体相互作用张量的CP近似值的成本，对于此处研究的最多36个原子的系统，最多2个数量级。库仑相互作用张量的完整4路CP近似比使用CP代理SQ因子的已知方法（也以$ \ Mathcal {o}（n^3 r_ \ text {cp}）$获得的已知方法更为准确，例如ealgebraic peceospect pretspearts andercontractactractactactactractactractactractactractactractration。 CP分解的SQ因子也可以作为4向CP因子的强大初始猜测。
We consider the problem of constructing a canonical polyadic (CP) decomposition for a tensor network, rather than a single tensor. We illustrate how it is possible to reduce the complexity of constructing an approximate CP representation of the network by leveraging its structure in the course of the CP factor optimization. The utility of this technique is demonstrated for the order-4 Coulomb interaction tensor approximated by 2 order-3 tensors via an approximate generalized square-root (SQ) factorization, such as density fitting or (pivoted) Cholesky. The complexity of constructing a 4-way CP decomposition is reduced from $\mathcal{O}(n^4 R_\text{CP})$ (for the non-approximated Coulomb tensor) to $\mathcal{O}(n^3 R_\text{CP})$ for the SQ-factorized tensor, where $n$ and $R_\text{CP}$ are the basis and CP ranks, respectively. This reduces the cost of constructing the CP approximation of 2-body interaction tensors of relevance to accurate many-body electronic structure by up to 2 orders of magnitude for systems with up to 36 atoms studied here. The full 4-way CP approximation of the Coulomb interaction tensor is shown to be more accurate than the known approaches utilizing CP-decomposed SQ factors (also obtained at the $\mathcal{O}(n^3 R_\text{CP})$ cost), such as the algebraic pseudospectral and tensor hypercontraction approaches. The CP decomposed SQ factors can also serve as a robust initial guess for the 4-way CP factors.
