Convergence theory and principled subspace enrichment for one-site DMRG-type alternating optimization in tensor train format with adaptive ranks

Computing the extreme eigenpair of a high-dimensional Hermitian matrix in tensor train / matrix product state (TT/MPS) format is dominated by one-site alternating schemes with subspace enrichment: White's corrected one-site DMRG (DMRG1c), its successor DMRG3S, and the alternating minimal energy (AMEn) algorithm. For symmetric positive definite linear systems, residual-based AMEn with adaptively grown ranks is provably globally convergent with a steepest-descent-type rate, but no comparable theory exists for the non-convex Rayleigh-quotient (eigenvalue) setting, for the cheap enrichment surrogates used in production codes, or for the Gram-averaging enrichment of DMRG1c, whose mixing weight has only heuristics and whose SVD truncation of system-only approximations of $p=Ax$ can lose the target direction, by a counterexample construction of the source paper. The problem asks for rigorous foundations of this family: a global convergence theory with quantitative per-sweep progress bounds that account for rank growth and truncation errors (or explicit counterexamples to convergence), a theoretically grounded rule for the mixing weight that vanishes at a controlled rate near convergence, and an enrichment-and-truncation step with a uniform guarantee of retaining the target direction for broad Hamiltonian classes (or a proof of impossibility).

Background

Quantum lattice systems with $d$ sites of local dimension $n$ have a Hilbert space of dimension $n^d$, so a wavefunction cannot be stored explicitly for $d\gtrsim 30$. The density matrix renormalization group (DMRG), introduced by S. R. White in 1992, computes ground states by exploiting the fact that the relevant states admit a data-sparse representation, now called the matrix product state (MPS) in physics and the tensor train (TT) decomposition in numerical linear algebra: the eigenvector $x$ is written as $x(i_1,\ldots,i_d)=\sum_{\alpha_1=1}^{r_1}\cdots\sum_{\alpha_{d-1}=1}^{r_{d-1}}x^{(1)}_{\alpha_1}(i_1)\,x^{(2)}_{\alpha_1\alpha_2}(i_2)\cdots x^{(d)}_{\alpha_{d-1}}(i_d),$ where each site tensor (TT core) $x^{(k)}$ carries one physical index $i_k$ and auxiliary bond indices $\alpha$; the maximal bond dimensions $r_k$ are the TT ranks (bond dimensions), and storage is $O(dnr^2)$ instead of $n^d$. The ground state is the smallest eigenpair of a Hermitian matrix $A$ (the Hamiltonian, itself given in a tensor-product operator format), equivalently the global minimizer of the Rayleigh quotient $J_A(x)=(x,Ax)/(x,x)$, which is a non-convex objective.

DMRG minimizes $J_A$ by alternating over one site at a time. Since the TT format is linear in each core, $x=X_{\neq k}x^{(k)}$ with the frame matrix $X_{\neq k}$ built from the other cores, each microstep reduces to a small Hermitian eigenproblem for the reduced matrix $A_k=X_{\neq k}^{*}AX_{\neq k}$. Optimizing two neighboring sites jointly (two-site DMRG) is the standard method for one-dimensional short-range systems, but stagnates for long-range interactions or periodic chains, where two sites carry too little information about the environment; one-site DMRG is cheaper but gets trapped, because the subspace searched in each microstep is fixed by the current basis of the other sites. For fixed TT ranks and convex objectives, local convergence of such alternating schemes is understood (Rohwedder and Uschmajew, 2013); in practice, however, the ranks of the target eigenstate are not known in advance and are grown adaptively during the iteration, and for the Rayleigh quotient no convergence theory of comparable strength exists.

The practical cure is subspace enrichment: after optimizing a site, auxiliary directions are injected into the basis of the neighboring site. White's corrected one-site algorithm (DMRG1c, 2005) targets the power-iteration vector $p=Ax$ through system-only surrogates $p_\gamma=(A^<_\gamma\otimes I)x$, obtained by splitting the operator into left (system) and right (environment) tensor blocks and discarding the environment part for speed; the surrogate directions are merged with the current basis $X$ by Gram averaging, $G=XX^{*}+aSS^{*}$ (with $S$ collecting the enrichment and $a$ a mixing weight), and the resulting enlarged subspace is truncated back by a singular value decomposition (SVD) keeping the dominant part. The alternating minimal energy (AMEn) algorithm instead enriches with the gradient direction $z=Ax-J_A(x)x$ (orthogonal to $x$, as in steepest descent), approximated in TT format to a prescribed tolerance, and merges the bases directly by zero-padding. Dolgov and Savostyanov (ENUMATH 2013) compared the two schemes on the spin-1 periodic Heisenberg chain (a periodic chain of $d=100$ spin-1 quantum spins with nearest-neighbor isotropic exchange coupling, a standard hard benchmark for DMRG-type methods) and identified the unresolved foundations of exactly this family: first, convergence theory for alternating schemes with variable (adapted) ranks is lacking; second, the mixing weight $a$ must vanish near convergence, but no principled recipe for it is known; third, the SVD truncation of the system-only surrogates can lose the target direction: for every choice of the enrichment weights and the current iterate, the source constructs a Hamiltonian for which every nontrivial truncation of the enriched subspace discards the system-related component of $p=Ax$ - the coefficients of that component are determined by the environment blocks $A^>_\gamma$ that the surrogates discard. For symmetric positive definite linear systems the same authors proved that residual-based AMEn with adaptively grown ranks converges globally with a steepest-descent-type rate (the proof requires each approximate enrichment vector to keep a positive inner product with the true residual, i.e., sufficiently small truncation tolerances). This theory covers neither the eigenvalue problem, nor nonsymmetric systems, nor the cheap enrichment surrogates that later production methods run: the strictly single-site DMRG with subspace expansion (DMRG3S) substitutes $\alpha$-scaled partial-projection terms for the exact residual, and the zero-site method DMRG0 uses parameter-free variational approximations of Lanczos or Jacobi-Davidson corrections (the update directions produced by classical iterative Krylov-subspace eigensolvers), both without theorems. Consequently, the most widely used tensor eigensolver family is applied successfully in practice while its convergence behavior, its mixing parameters, and its truncation rules remain without theoretical foundation.

Problem Statement

For the class of one-site, subspace-enriched alternating TT/MPS schemes that compute the extreme eigenpair of a Hermitian matrix $A$ given in tensor-product operator format with adaptively grown TT ranks - including White's DMRG1c with Gram-averaging enrichment $G=XX^{*}+aSS^{*}$, the DMRG3S subspace expansion with its $\alpha$-scaled partial-projection terms, and AMEn with (approximate) residual-based enrichment - establish the following jointly supporting rigorous foundations for the family. (1) A convergence theory: either a proof of global convergence with a quantitative per-sweep progress or rate bound that accounts for rank growth and for the errors introduced by TT truncation of the enrichment vectors, with explicit assumptions on truncation tolerances and on the schedule by which ranks may be increased, or an explicit counterexample instance (Hamiltonian, initial state, algorithm parameters) demonstrating non-convergence for a scheme with effective, nondegenerate enrichment: an instance in which the mixing weight is zero, or so small relative to the truncation tolerance that the truncated enrichment correction vanishes, reproduces the documented trapping of unenriched one-site DMRG and is not a counterexample within the intended class. The already-settled symmetric-positive-definite linear-system case with accurate residual enrichment does not by itself answer this; the target is the non-convex Rayleigh-quotient setting and the cheaper enrichment surrogates that production codes actually run. (2) A principled rule for the mixing weight (the scale $a$ in $G=XX^{*}+aSS^{*}$, or $\alpha$ in the DMRG3S expansion) that provably balances the $O(\sqrt{a})$ perturbation of the current iterate against the enlargement of the search subspace and vanishes at a controlled rate near convergence, or a proof that no such rule exists within the averaging class. (3) An enrichment-and-truncation step that, when only system-only approximations $p_\gamma=(A^<_\gamma\otimes I)x$ of the target vector $p=Ax$ are computed, either carries a guarantee that the system-related component of the target direction is retained for an explicitly stated class of Hamiltonians in which the discarded environment blocks $A^>_\gamma$ genuinely enter $p=Ax$ (excluding the degenerate case in which the surrogates reproduce $p$ exactly), uniformly over all current iterates and admissible enrichment weights, with a retention error that vanishes with the truncation tolerance; or is accompanied by a rigorous proof of the impossibility assertion made informally in the source paper - that for every choice of the enrichment weights and the current iterate there is a Hamiltonian for which every nontrivial truncation of the enriched subspace discards the system-related component of $p=Ax$ - in a form that rules out any such uniform retention guarantee beyond degenerate classes. The three deliverables are jointly required. Under the positive branch of deliverable (1), a scheme with a proven per-sweep rate also pins down the admissible mixing weights and truncation tolerances, which is exactly what principled rank-growth and truncation-tolerance schedules - still chosen by hand in production codes - require; a counterexample under its negative branch settles deliverable (1) alone and leaves (2) and (3) in force.

References

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