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REVIEW 3 major objections 5 minor 86 references

An accelerated linear method for optimizing non-linear wavefunctions in variational Monte Carlo

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper claims that the linear method for optimizing variational Monte Carlo wavefunctions can shed its memory bottleneck by solving its generalized eigenvalue problem iteratively with Jacobi-Davidson, without ever constructing the…

desk verdict A genuine algorithmic advance for large-parameter VMC optimization, with a reproducibility gap around the nonsymmetric sampled Hamiltonian that needs fixing before the solver details can be trusted. read the letter →

arxiv 1908.04423 v1 pith:5KWSELX5 submitted 2019-08-12 cond-mat.str-el physics.chem-ph

Iliya Sabzevari , Ankit Mahajan , Sandeep Sharma This is my paper · ORCID
classification cond-mat.str-elphysics.chem-ph
keywords variationalMonteCarlolinearmethodJacobi-DavidsonwavefunctionoptimizationJastrowfactorAMSGradgeneralizedeigenvalueproblemHubbardmodel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Variational Monte Carlo optimizes wavefunctions that depend non-linearly on parameters, and the linear method is an effective optimizer for this task but has been limited by the need to store the Hamiltonian and overlap matrices at each step. This paper claims that applying the Jacobi-Davidson iterative eigenvector solver to the linear method's generalized eigenvalue problem removes that memory bottleneck, because both matrices can be applied to vectors using stored Monte Carlo samples without ever being constructed. The result is a direct linear method whose per-processor memory and time scale almost linearly with the number of parameters, enabling optimizations of wavefunctions with up to 57,722 parameters. A hybrid schedule that starts with the cheap first-order AMSGrad optimizer and switches to the direct linear method near convergence reaches energies matching high-accuracy benchmarks for systems including the Hubbard model and iron(II) porphyrin.

What carries the argument

The load-bearing identity is the outer-product decomposition of the linear-method matrices: with $\bar g_i[n]$ and $\bar h_i[n]$ vanishing for index zero, equations (24) and (25) express $\bar{\mathbf{S}}$ and $\bar{\mathbf{H}}$ as sums of outer products of the sampled vectors. This turns each matrix-vector product into a few dot products and vector additions. The other half of the machinery is Jacobi-Davidson, an iterative eigensolver that builds a small subspace, solves a projected eigenproblem, and corrects its current best vector by approximately solving a preconditioned equation with the Jacobi orthogonal component correction; the paper uses subspace size 25 with restarts of 5 vectors and loose tolerances for fast progress.

What would settle it

Measure the autocorrelation time of $g_i[n]$, $h_i[n]$, and $E_L[n]$ as the system size grows for a fixed ansatz family; if $n_{\mathrm{corr}}$ does not grow linearly with $N$, the $O(N^4)$ scaling argument in Section III breaks. Alternatively, run the optimizer on a system with a short, saturating correlation length and check whether per-iteration memory or time departs from the reported $O(n_s n_{\mathrm{var}}^{1.16}/n_{\mathrm{proc}})$ scaling.

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Extended reading notes

Core claim

The central claim is that the linear method's step can be obtained by solving $\bar{\mathbf{H}}\,\delta p = E\,\bar{\mathbf{S}}\,\delta p$ with the Jacobi-Davidson algorithm, using only matrix-vector products. The paper shows that $\bar{\mathbf{H}}$ and $\bar{\mathbf{S}}$ can be written as sums of outer products of the sampled primitives $g_i[n]=\langle n|\Psi_i\rangle/\langle n|\Psi\rangle$, $h_i[n]=\langle n|\hat H|\Psi_i\rangle/\langle n|\Psi\rangle$, and the local energy $E_L[n]$, so their action on a vector costs $O(n_s n_{\mathrm{var}})$ rather than the $O(n_s n_{\mathrm{var}}^2)$ needed to build the matrices. Storing these primitives instead of the matrices gives a memory per processor of $O(n_s n_{\mathrm{var}}/n_{\mathrm{proc}})$, and sampling only every correlation length reduces both memory and time by a factor $n_{\mathrm{corr}}$. Empirically the solver scales as $O(n_s n_{\mathrm{var}}^{1.16}/n_{\mathrm{proc}})$, and the paper reports optimizations of a 98-site Hubbard model with 57,722 parameters, with converged energies matching reference results.

Load-bearing premise

The claimed $\mathcal O(N^4)$ overall scaling rests on the assumption that the sampled primitives are serially correlated with a correlation length that grows linearly with system size, so that sampling every $\mathcal O(N)$ Monte Carlo moves cuts storage and computation by a factor $n_{\mathrm{corr}}$; if the correlation length grows more slowly, memory and CPU costs grow faster than claimed.

Editorial extensions

If this is right

  • The linear method can now be applied to wavefunctions with tens of thousands of parameters; the paper demonstrates 57,722 parameters on the 98-site Hubbard model.
  • Per-iteration wall time for the linear method drops by roughly an order of magnitude compared with explicitly building and diagonalizing the matrices.
  • Because the matrices are never stored, memory per processor shrinks as parallelism grows, making the optimizer progressively more attractive on many-core machines.
  • Starting with AMSGrad far from the minimum and switching to the direct linear method near convergence reaches lower energies in comparable or less wall time than either approach alone.
  • For real-space Jastrow wavefunctions, the direct linear method is reported as the optimizer of choice, since first-order methods converge inconsistently on those systems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the near-linear solver scaling persists beyond 57,722 parameters, the practical ceiling for this optimization strategy shifts from linear algebra to the cost of computing the primitives $g_i$ and $h_i$, so hand-coded analytic gradients could extend the method further.
  • The same matrix-free recipe, outer-product-stored samples plus an iterative eigensolver, should transfer to other high-dimensional stochastic optimizations, including neural-network quantum states, where the overlap matrix would otherwise be prohibitively large.
  • The bias check protocol, running many unbiased SGD steps around the converged minimum and checking the energy histogram, is a reusable diagnostic for any stochastic second-order optimizer.
  • A direct testable extension is to record the measured correlation length $n_{\mathrm{corr}}$ for each system; the claimed $O(N^4)$ scaling assumes it grows linearly with system size, and publishing these values would let others verify the scaling claim without reimplementing the code.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript presents an accelerated linear method (aLM) for optimizing non-linearly parametrized variational Monte Carlo wavefunctions. The Hamiltonian and overlap matrices of the linear method are rewritten as sums of outer products of sampled primitives (Eqs. 24-25), so that their action on a vector can be computed in O(ns nvar) time without building and storing the matrices. The generalized eigenvalue problem is then solved with a Jacobi-Davidson iterative solver, and AMSGrad is used far from the minimum before switching to the linear method. The method is applied to real-space Jastrow wavefunctions for Be and Ne, orbital-space Jastrow-GHF wavefunctions for C2, H50, a 98-site Hubbard model, and Fe(II) porphyrin, with up to 57,722 parameters. Energies agree with exact diagonalization, SHCI, and GFMC references. The authors report an empirical solver scaling of O(ns nvar^1.16/nproc) and argue for an overall O(N^4) scaling.

Significance. The central idea of avoiding explicit matrix storage by combining the outer-product representation with an iterative eigensolver is conceptually clean and, if correctly implemented, would remove the main memory bottleneck that limits the linear method to roughly 10^4 parameters. The algebraic identities in Eqs. (24)-(26) are straightforward, and the benchmark results against SHCI and GFMC are credible. The paper also includes a sensible check for stochastic bias using long SGD runs after convergence. These are real strengths. The main uncertainties are the correctness of the iterative solve for the nonsymmetric sampled Hamiltonian and the scope of the scaling claims, which are not fully established by the presented data.

major comments (3)
  1. [Section II.C, Eqs. (30)-(32)] The sampled Hamiltonian is explicitly stated to be nonsymmetric in Section II.B after Eq. (11), yet the Jacobi-Davidson correction equation F = P†(A - rho B)P is claimed to be positive semi-definite and to require only a few conjugate-gradient iterations. For a nonsymmetric A, F is not Hermitian, so conjugate gradient is not mathematically justified and positive semi-definiteness is not defined in the usual sense. The paper does not state any symmetrization of Hbar, such as replacing A by (A + A^T)/2 or averaging Gr and Gc, before entering the Davidson iterations. Because every reported aLM optimization passes through this solve, this missing specification is load-bearing for the central algorithmic claim.
  2. [Section IV.E, Fig. 4 and Section III] The empirical scaling measurement covers only the Jacobi-Davidson solver time, not the end-to-end optimization, while the abstract claims that the method 'improves the scaling of the algorithm' with respect to the number of parameters. Please either provide an end-to-end scaling benchmark or explicitly restrict the claim to the solver. In addition, the O(N^4) scaling in Section III depends on the assumption that the Monte Carlo correlation length grows linearly with system size. That assumption is stated without supporting measurements, and the observed correlation length of about 1.5 for FeP in Section IV.G shows that the correlation length is not universally large; if it grows more slowly than linearly, the number of stored primitives ns and the direct-method cost grow accordingly.
  3. [Section IV, Tables I-VI] The paper's motivation is the memory bottleneck of the linear method, but no memory measurements are reported. Timings alone do not directly demonstrate the memory advantage of aLM over LM. A comparison of peak memory per process for aLM versus LM, at least for the larger systems, would make the central claim concrete and is needed to support the abstract's statement that the method 'greatly lowers the cost'.
minor comments (5)
  1. [Throughout] There are several typos and formatting issues: 'wav efunctions' in the title, 'symmerty' in Section II.A, and 'Lowdin' should be 'Löwdin' in Section IV.D and the references.
  2. [Section II.C, Davidson algorithm step 2] The equation A' = V†A V = V†VA is ambiguous because V†VA could be read as (V†V)A; please write V†(VA) or define VA = A V explicitly before the step.
  3. [Tables I and II] The comparison energies for Be and Ne are exact diagonalization in aug-cc-pVQZ and aug-cc-pV5Z Gaussian bases, while the VMC calculations use Slater-type bases. The text should state explicitly that these are different basis sets, which explains why the VMC energy can lie below the quoted CI reference.
  4. [Section III, line search] The line-search procedure with slave wavefunctions and Eq. (37) is described, but the statement that 'the median value step size' is used is not completely specified; please clarify how the median is selected among the five candidate step sizes.
  5. [Fig. 4] The y-axis label 'Avg. time/(ns/nproc)' and the caption 'per Monte Carlo sample per process' should be reconciled, and the axis label should state explicitly that the horizontal axis is the number of parameters nvar.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the aLM derivation is self-contained and its results are benchmarked against external references.

full rationale

The paper's central contribution is an algorithmic reformulation: instead of building the LM Hamiltonian and overlap matrices, the Jacobi-Davidson solver acts on vectors via outer-product forms of Eqs. (24)-(26). This is derived from the standard linear method equations, which are explicitly stated and attributed to prior community work rather than assumed from the authors' own results. The claimed memory and CPU improvements follow from the structural fact that matrix-vector products cost O(ns nvar) rather than O(ns nvar^2), and from the stated sampling assumption about correlation length. No parameter of the method is fitted to the benchmark energies; the empirical scaling exponent 1.16 is a measured performance characteristic, not an input to the physics. The self-citations to refs. 8 and 23 provide the underlying VMC framework and wavefunction ansatz, but the core Jacobi-Davidson/direct linear method is independently derived and could stand without those citations. The optimized energies are compared with external SHCI and GFMC references, so the numerical claims are not validated by the paper's own inputs. The reviewer-identified issue that the sampled nonsymmetric H may invalidate the conjugate-gradient solve is a correctness or reproducibility concern, not a circularity: it does not make any output equivalent to an input by construction. Accordingly, no circular step can be quoted, and the appropriate score is 0.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The method introduces no new physical entities. The algorithmic improvements rest on standard mathematics plus domain assumptions about sampling correlation, primitive evaluation costs, and the stabilizing effect of a diagonal shift.

free parameters (5)
  • Initial diagonal shift and decay factor for H trust radius = 0.1, decay 0.65
    Used in all aLM calculations to stabilize the linear method; chosen by hand, not derived. Sensitivity to these values is not reported.
  • Umrigar-Toulouse nonlinear scaling parameter xi = 0.5
    Fixed for all calculations in this work; affects the scaling of parameter updates for non-linear parameters.
  • AMSGrad hyperparameters (alpha, beta1, beta2) = 0.01, 0.1, 0.01 (aggressive); 0.001, 0.1, 0.001 (start of aLM)
    Standard optimizer settings from the machine learning literature; chosen for performance, not fitted to target energies.
  • Davidson subspace sizes Vmax, Vrestart = 25, 5
    Reported to work reasonably well for matrices of any dimension; chosen by hand.
  • Line search step multipliers = 0.01, 0.05, 0.1, 0.5, 1.0
    Used in correlated sampling to select the step size; arbitrary grid chosen by the authors.
assumptions (5)
  • standard math The variational principle: the expectation value of the Hamiltonian for any wavefunction is an upper bound to the true ground state energy.
    Basis for the optimization objective in the introduction.
  • domain assumption The Monte Carlo estimates of the matrices H and S converge to the exact averages, and the resulting stochastic noise in the eigenvector is unbiased enough to be mitigated.
    The paper tests for bias with a final SGD run on the Hubbard model (Section IV.F), but does not prove an unbiased estimator.
  • domain assumption The primitives EL[n], gi[n], and hi[n] can be evaluated at the claimed O(N^2) and O(N^3) costs, including the reverse-mode algorithmic differentiation of the local energy.
    For real space, details are deferred to a forthcoming publication; for orbital space, the O(N^3) AD cost is stated with a large prefactor.
  • domain assumption The correlation length of the sampled primitives grows linearly with system size, justifying the O(N) downsampling and the claimed O(N^4) total scaling.
    Stated as an assumption in Section III; the scaling plot measures only solver time, not this correlation property.
  • ad hoc to paper The Jacobi-Davidson method with Jacobi orthogonal component correction converges for the non-symmetric, noisy generalized eigenvalue problem encountered here, with a diagonal shift acting as a trust radius.
    The diagonal shift and tolerance schedule are heuristics; no convergence analysis is provided for the stochastic case.

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Cite this review

Pith. "Pith review of An accelerated linear method for optimizing non-linear wavefunctions in variational Monte Carlo." pith.science (2026). https://pith.science/paper/5KWSELX5

@misc{pith2026190804423,
  author       = {Pith},
  title        = {Pith review of: An accelerated linear method for optimizing non-linear wavefunctions in variational Monte Carlo},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5KWSELX5}},
  note         = {Machine review of arXiv:1908.04423}
}
read the original abstract

Although the linear method is one of the most robust algorithms for optimizing non-linearly parametrized wavefunctions in variational Monte Carlo, it suffers from a memory bottleneck due to the fact at each optimization step a generalized eigenvalue problem is solved in which the Hamiltonian and overlap matrices are stored in memory. Here we demonstrate that by applying the Jacobi-Davidson algorithm, one can solve the generalized eigenvalue problem iteratively without having to build and store the matrices in question. The resulting direct linear method greatly lowers the cost and improves the scaling of the algorithm with respect to the number of parameters. To further improve the efficiency of optimization for wavefunctions with a large number of parameters, we use the first order method AMSGrad far from the minimum as it is very inexpensive, and only switch to the direct linear method near the end of the optimization where methods such as AMSGrad have long convergence tails. We apply this improved optimizer to various wavefunctions with both real and orbital space Jastrow factors for atomic systems such as Beryllium and Neon, molecular systems such as the Carbon dimer and Iron(II) Porphyrin, and model systems such as the Hubbard model and Hydrogen chains.

Figures

Figures reproduced from arXiv: 1908.04423 by the authors.

Figure 2
Figure 2. FIG. 2. Optimization curves for orbital space Carbon Dimer [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Optimization curves for orbital space Carbon Dimer [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Average wall time of Jacobi-Davidson solver per [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

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Reference graph

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