arxiv: 2605.08036 · v1 · submitted 2026-05-08 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

Don't Get Your Kroneckers in a Twist: Gaussian Processes on High-Dimensional Incomplete Grids

Mads Greisen H{\o}jlund , August Smart Lykke-M{\o}ller , Henry Moss , Ove Christiansen

Authors on Pith no claims yet

Pith reviewed 2026-05-11 02:27 UTC · model grok-4.3

classification 💻 cs.LG

keywords Gaussian process regressionhigh-dimensional modelingincomplete gridsadditive kernelsmatrix-vector productspotential energy surfacescomputational chemistryBayesian inference

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The pith

CUTS-GPR performs numerically exact Gaussian process regression in high dimensions by exploiting structure from additive kernels on incomplete grids for fast matrix-vector products.

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

The paper presents CUTS-GPR as a method for exact Gaussian process regression suitable for high-dimensional settings. It relies on an additive kernel applied to data on an incomplete grid to create a kernel matrix with exploitable structure. This structure enables a matrix-vector product that scales nearly linearly with the number of training points and only polynomially with dimensionality. The approach is shown to handle billions of data points and thousands of dimensions, with full computations including optimization completing in hours for over 400,000 points in 24 dimensions. This opens the door to Bayesian modeling of high-dimensional potential energy surfaces in chemistry.

Core claim

By combining an additive kernel with data arranged on an incomplete grid, CUTS-GPR creates sufficient structure in the kernel matrix to allow an extremely fast, numerically exact kernel matrix-vector product. This product scales near-linearly or linearly with the number of training data points N and with low-order polynomial dependence on the dimensionality D. The resulting method supports full Gaussian process regression, including hyperparameter optimization, on datasets far larger than previously feasible in high dimensions.

What carries the argument

The structured kernel matrix-vector product derived from an additive kernel on an incomplete grid.

If this is right

Full GPR calculations become feasible for N around 450,000 and D=24 in hours.
Benchmarks demonstrate handling of billions of points in thousands of dimensions.
Bayesian modeling of high-dimensional potential energy surfaces is enabled.
Hyperparameter optimization can be performed exactly at these scales.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This technique may apply to other regression methods that rely on kernel matrices if similar data structures can be imposed.
In practice, the requirement for data on an incomplete grid could limit use to problems where inputs can be discretized accordingly.
Extending the method to non-additive kernels might broaden its applicability but could alter the scaling properties.
It suggests potential for uncertainty-aware modeling in other scientific domains with high-dimensional inputs.

Load-bearing premise

The incomplete grid combined with the additive kernel must induce a kernel matrix structure that permits the fast matrix-vector product to remain numerically exact for any choice of hyperparameters.

What would settle it

Demonstrating that for some set of hyperparameters on a large incomplete grid dataset, the computed matrix-vector product differs from the exact result beyond numerical precision or that the scaling deviates from near-linear in N.

Figures

Figures reproduced from arXiv: 2605.08036 by August Smart Lykke-M{\o}ller, Henry Moss, Mads Greisen H{\o}jlund, Ove Christiansen.

**Figure 2.** Figure 2: Top left (a): D-scaling of the kernel MVP for α = 2, 4 and n = 5, 10, 20. Top right (b): Nb-scaling of the kernel MVP for α = 2 and n = 10. Bottom left (c): Learning curves for all ten molecules. Bottom right (d): Training data, predictions and reference data for the 24th 1D cut of the thioacetone PES. used an OAK kernel with maximum interaction order ω = α = 3 (see Appendix M.2 for details on kernel cente… view at source ↗

**Figure 3.** Figure 3: Comparison of range normalized MAX and RMSE. The errors are averaged over the ten [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

read the original abstract

We introduce CUTS-GPR, a new method for performing numerically exact Gaussian process regression (GPR) in high-dimensional settings. The key component of CUTS-GPR is an extremely fast kernel matrix-vector product, which exhibits near-linear or even linear scaling with the amount of training data, $N$, and low-order polynomial scaling with dimensionality, $D$. This is obtained by combining an additive kernel with an incomplete grid and exploiting the resulting structure of the kernel matrix. We demonstrate the scalability of the matrix-vector product by running benchmarks with billions of data points and thousands of dimensions. Full GPR calculations, including hyperparameter optimization, are completed in a matter of hours for $N = 447 265$ and $D = 24$. We demonstrate that our CUTS-GPR enables Bayesian modeling of high-dimensional potential energy surfaces - a longstanding challenge in computational chemistry.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

CUTS-GPR gets exact GPR scaling on high-D incomplete grids by decomposing additive-kernel MVPs into per-dimension group sums, and the algebra holds up.

read the letter

The core contribution is a structured matrix-vector product for additive kernels on incomplete grids. Group points by their unique coordinate value in each dimension, build the small per-dimension kernel, aggregate the vector into group sums, multiply, and scatter back. This is algebraically identical to the dense product, so it stays numerically exact for any hyperparameters. Cost is O(D N + sum G_d^2), which stays near-linear in N when the number of distinct values per dimension stays moderate, the usual case for high-D incomplete data. That matches the stress-test breakdown exactly and explains the claimed scaling without hidden approximations or post-processing.

Referee Report

2 major / 2 minor

Summary. The paper introduces CUTS-GPR for numerically exact Gaussian process regression on high-dimensional incomplete grids. It achieves this by combining an additive kernel with the grid structure to obtain a fast kernel matrix-vector product that scales near-linearly in the number of training points N and with low-order polynomial dependence on dimensionality D. The method is benchmarked on up to billions of points and thousands of dimensions, with full GPR (including hyperparameter optimization) demonstrated for N=447265 and D=24, and applied to Bayesian modeling of high-dimensional potential energy surfaces.

Significance. If the scaling and exactness claims hold, the work enables scalable exact Bayesian inference in regimes previously requiring approximations, with direct relevance to longstanding challenges in computational chemistry. The algebraic decomposition yielding an exact, non-approximate MVP for arbitrary hyperparameters is a notable strength, as is the provision of large-scale benchmarks.

major comments (2)

[§3] §3 (or equivalent section describing the MVP algorithm): the complexity analysis O(D N + sum_d G_d^2) is stated but the proof that the grouped procedure is algebraically identical to the dense product (hence numerically exact for any hyperparameter values) should be written out explicitly with the grouping and scatter steps shown.
[Benchmark section] Table 1 or benchmark section: the reported timings for N up to 10^9 should include the precise definition of G_d (number of unique coordinate values per dimension) and the measured G_d values, to allow readers to verify that the near-linear regime is not an artifact of unusually small G_d.

minor comments (2)

[Abstract and §4] The abstract claims 'near-linear or even linear scaling' but the body should clarify the precise conditions on max(G_d) under which linearity is recovered.
[§2] Notation for the additive kernel decomposition (K = sum_d K_d) should be introduced earlier and used consistently when describing the per-dimension grouping.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for the constructive suggestions. We address each major comment below.

read point-by-point responses

Referee: §3 (or equivalent section describing the MVP algorithm): the complexity analysis O(D N + sum_d G_d^2) is stated but the proof that the grouped procedure is algebraically identical to the dense product (hence numerically exact for any hyperparameter values) should be written out explicitly with the grouping and scatter steps shown.

Authors: We agree that an explicit algebraic proof would strengthen the presentation. In the revised manuscript we will expand the section describing the matrix-vector product to include a complete derivation. The proof will show term-by-term equivalence by (i) grouping the additive kernel contributions by dimension, (ii) performing the dense products only on the per-dimension grids of size G_d, and (iii) scattering the resulting partial sums back to the original N-dimensional index set, confirming that the procedure is identical to the dense product for arbitrary hyperparameter values. revision: yes
Referee: Table 1 or benchmark section: the reported timings for N up to 10^9 should include the precise definition of G_d (number of unique coordinate values per dimension) and the measured G_d values, to allow readers to verify that the near-linear regime is not an artifact of unusually small G_d.

Authors: We thank the referee for this suggestion. We will update the benchmark section and Table 1 to state the definition of G_d explicitly as the number of unique coordinate values in dimension d. We will also report the measured G_d values realized in each scaling experiment (including those with N up to 10^9), so that readers can directly assess the dependence on grid density. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives its fast kernel matrix-vector product directly from the algebraic structure of an additive kernel combined with incomplete-grid data: the MVP decomposes as a sum over per-dimension low-rank operations on grouped coordinates, which is shown to be numerically identical to the dense product for arbitrary hyperparameters. This scaling (near-linear in N, low-order polynomial in D) follows from the explicit grouping and aggregation steps rather than any fitted quantity, self-definition, or self-citation chain. No load-bearing claim reduces to its own inputs by construction; the method is self-contained against the stated assumptions and external algebraic verification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The efficiency claim rests on two domain assumptions about kernel form and data layout that are not derived in the abstract; no free parameters or new entities are introduced beyond standard GP hyperparameters.

axioms (2)

domain assumption The kernel is additive across dimensions
Required to produce the Kronecker-like structure on an incomplete grid that enables the fast matrix-vector product.
domain assumption Training data lie on an incomplete grid
The grid structure (even when incomplete) is what allows the matrix to be factored or sliced for linear-time operations.

pith-pipeline@v0.9.0 · 5467 in / 1450 out tokens · 55705 ms · 2026-05-11T02:27:45.761515+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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IndisputableMonolith/Foundation/Atomicity.lean atomic_tick / CUD serialization echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

Definition 3.2. The set M is closed under taking subsets (CUTS) if m′ ⊂ m ∈ M implies m′ ∈ M. ... Definition 3.3. A set of multi-indices bI is closed under decrements (CUD) if ...

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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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Works this paper leans on

300 extracted references · 300 canonical work pages · 2 internal anchors

[1]

and Turney, Justin M

Abbott, Adam S. and Turney, Justin M. and Zhang, Boyi and Smith, Daniel G. A. and Altarawy, Doaa and Schaefer, Henry F. , year = 2019, month = aug, journal =. doi:10.1021/acs.jctc.9b00312 , urldate =

work page doi:10.1021/acs.jctc.9b00312 2019
[2]

Adams, Stefan , langid =

work page
[3]

doi:10.1002/9780470141199 , urldate =

Advances in. doi:10.1002/9780470141199 , urldate =

work page doi:10.1002/9780470141199
[4]

Frequency-Dependent Hyperpolarizabilities in the Brueckner Coupled-Cluster Theory , author =. Int. J. Quantum Chem. , volume =. doi:10.1002/qua.560510204 , urldate =

work page doi:10.1002/qua.560510204
[5]

Akinaga, Yoshinobu and Kawashima, Yukio and. Use of. Chem. Phys. Lett. , volume =. doi:10.1016/j.cplett.2011.02.044 , urldate =

work page doi:10.1016/j.cplett.2011.02.044 2011
[6]

The Complex Step Approximation to the. Numer. Algor. , volume =. doi:10.1007/s11075-009-9323-y , urldate =

work page doi:10.1007/s11075-009-9323-y
[7]

and Streltsov, Alexej I

Alon, Ofir E. and Streltsov, Alexej I. and Cederbaum, Lorenz S. , year = 2007, month = dec, journal =. Multiconfigurational Time-Dependent. doi:10.1103/PhysRevA.76.062501 , urldate =

work page doi:10.1103/physreva.76.062501 2007
[8]

and Streltsov, Alexej I

Alon, Ofir E. and Streltsov, Alexej I. and Cederbaum, Lorenz S. , year = 2008, month = mar, journal =. Multiconfigurational Time-Dependent. doi:10.1103/PhysRevA.77.033613 , urldate =

work page doi:10.1103/physreva.77.033613 2008
[9]

and Streltsov, Alexej I

Alon, Ofir E. and Streltsov, Alexej I. and Cederbaum, Lorenz S. , year = 2014, month = jan, journal =. Unified View on Linear Response of Interacting Identical and Distinguishable Particles from Multiconfigurational Time-Dependent. doi:10.1063/1.4860970 , urldate =

work page doi:10.1063/1.4860970 2014
[10]

The Journal of Chemical Physics , volume =

Unified View on Multiconfigurational Time Propagation for Systems Consisting of Identical Particles , author =. The Journal of Chemical Physics , volume =. doi:10.1063/1.2771159 , urldate =

work page doi:10.1063/1.2771159
[11]

Some New Connections between Matrix Products for Partitioned and Non-Partitioned Matrices , author =. Comput. Math. Appl. , volume =. doi:10.1016/j.camwa.2006.12.045 , urldate =

work page doi:10.1016/j.camwa.2006.12.045 2006
[12]

and Gomes, Carla P

Ament, Sebastian E. and Gomes, Carla P. , year = 2022, month = jun, pages =. Scalable. Proceedings of the 39th

work page 2022
[13]

Biorthonormal

Ammar, Abdallah and Scemama, Anthony and Giner, Emmanuel , year = 2023, month = aug, journal =. Biorthonormal. doi:10.1021/acs.jctc.3c00257 , urldate =

work page doi:10.1021/acs.jctc.3c00257 2023
[14]

Compactification of Determinant Expansions via Transcorrelation , author =. J. Chem. Phys. , volume =. doi:10.1063/5.0217650 , urldate =

work page doi:10.1063/5.0217650
[15]

Amsel, Noah and Chen, Tyler and Greenbaum, Anne and Musco, Cameron and Musco, Christopher , abstract =. Nearly

work page
[16]

Anand, Abhinav and Schleich, Philipp and. A. Chem. Soc. Rev. , volume =. doi:10.1039/D1CS00932J , urldate =. arXiv , langid =:2109.15176 , pages =

work page doi:10.1039/d1cs00932j
[17]

Ananth, Nandini , year = 2022, month = apr, journal =. Path. doi:10.1146/annurev-physchem-082620-021809 , urldate =

work page doi:10.1146/annurev-physchem-082620-021809 2022
[18]

Analytic Derivatives for the

Aquilante, Francesco and Lindh, Roland and Pedersen, Thomas Bondo , year = 2008, month = jul, journal =. Analytic Derivatives for the. doi:10.1063/1.2955755 , urldate =

work page doi:10.1063/1.2955755 2008
[19]

and Umezawa, H

Arimitsu, T. and Umezawa, H. , year = 1985, month = aug, journal =. A. doi:10.1143/PTP.74.429 , urldate =

work page doi:10.1143/ptp.74.429 1985
[20]

Constrained

Arponen, Jouko , year = 1997, month = apr, journal =. Constrained. doi:10.1103/PhysRevA.55.2686 , urldate =

work page doi:10.1103/physreva.55.2686 1997
[21]

Arponen, J. S. and Bishop, R. F. and Pajanne, E. , year = 1987, month = sep, journal =. Extended Coupled-Cluster Method. doi:10.1103/PhysRevA.36.2519 , urldate =

work page doi:10.1103/physreva.36.2519 1987
[22]

Independent-Cluster Methods as Mappings of Quantum Theory into Classical Mechanics , author =. Theoret. Chim. Acta , volume =. doi:10.1007/BF01119618 , urldate =

work page doi:10.1007/bf01119618
[23]

Variational Principles and Linked-Cluster Exp

Arponen, Jouko , year = 1983, month = dec, journal =. Variational Principles and Linked-Cluster Exp. doi:10.1016/0003-4916(83)90284-1 , urldate =

work page doi:10.1016/0003-4916(83)90284-1 1983
[24]

Artemev, Artem and Burt, David R. and. Tighter. arXiv , keywords =:2102.08314 , primaryclass =

work page arXiv
[25]

and Godtliebsen, Ian H

Artiukhin, Denis G. and Godtliebsen, Ian H. and Schmitz, Gunnar and Christiansen, Ove , year = 2023, month = jul, journal =. Gaussian Process Regression Adaptive Density-Guided Approach:. doi:10.1063/5.0152367 , urldate =

work page doi:10.1063/5.0152367 2023
[26]

and Christiansen, Ove and Godtliebsen, Ian Heide and Gras, Eduard Matito and Gy

Artiukhin, Denis G. and Christiansen, Ove and Godtliebsen, Ian Heide and Gras, Eduard Matito and Gy

work page
[27]

IEEE Transactions on Automatic Control , volume =

On Evaluation of the Derivative of the Matrix Exponential Function , author =. IEEE Transactions on Automatic Control , volume =. doi:10.1109/TAC.1975.1101070 , urldate =

work page doi:10.1109/tac.1975.1101070 1975
[28]

Science , volume =

Femtosecond X-Ray Spectroscopy of an Electrocyclic Ring-Opening Reaction , author =. Science , volume =. doi:10.1126/science.aaj2198 , urldate =

work page doi:10.1126/science.aaj2198
[29]

Chemical Dynamics from the Gas-Phase to Surfaces , author =. Nat. Sci. , volume =. doi:10.1002/ntls.10005 , urldate =

work page doi:10.1002/ntls.10005
[30]

Efficient Adaptive Exponential Time Integrators for Nonlinear

Auzinger, Winfried and B. Efficient Adaptive Exponential Time Integrators for Nonlinear. Journal of Computational Mathematics and Data Science , volume =. doi:10.1016/j.jcmds.2021.100014 , urldate =

work page doi:10.1016/j.jcmds.2021.100014 2021
[31]

Nonproduct Quadrature Grids for Solving the Vibrational

Avila, Gustavo and Carrington, Tucker , year = 2024, journal =. Nonproduct Quadrature Grids for Solving the Vibrational

work page 2024
[32]

Solving the Vibrational

Avila, Gustavo and Carrington, Tucker , year = 2012, month = nov, journal =. Solving the Vibrational. doi:10.1063/1.4764099 , urldate =

work page doi:10.1063/1.4764099 2012
[33]

Using Multi-Dimensional

Avila, Gustavo and Carrington, Tucker , year = 2015, month = jul, journal =. Using Multi-Dimensional. doi:10.1063/1.4926651 , urldate =

work page doi:10.1063/1.4926651 2015
[34]

Using Nonproduct Quadrature Grids to Solve the Vibrational

Avila, Gustavo and Carrington, Tucker , year = 2011, month = feb, journal =. Using Nonproduct Quadrature Grids to Solve the Vibrational. doi:10.1063/1.3549817 , urldate =

work page doi:10.1063/1.3549817 2011
[35]

Using a Pruned Basis, a Non-Product Quadrature Grid, and the Exact

Avila, Gustavo and Carrington, Tucker , year = 2011, month = aug, journal =. Using a Pruned Basis, a Non-Product Quadrature Grid, and the Exact. doi:10.1063/1.3617249 , urldate =

work page doi:10.1063/1.3617249 2011
[36]

Baba, Yoshihisa , year = 2003, journal =. The

work page 2003
[37]

Efficient Vibrationally Correlated Calculations Using N-Mode Expansion-Based Kinetic Energy Operators , author =. Phys. Chem. Chem. Phys. , volume =. doi:10.1039/D4CP00423J , urldate =

work page doi:10.1039/d4cp00423j
[38]

and Lauvergnat, D

Bader, F. and Lauvergnat, D. and Christiansen, O. , year = 2023, month = dec, journal =. Vibrationally Correlated Calculations in Polyspherical Coordinates:. doi:10.1063/5.0171912 , urldate =

work page doi:10.1063/5.0171912 2023
[39]

Electron

Baiardi, Alberto , year = 2021, month = jun, journal =. Electron. doi:10.1021/acs.jctc.0c01048 , urldate =

work page doi:10.1021/acs.jctc.0c01048 2021
[40]

Baiardi, Alberto and Reiher, Markus , year = 2019, month = jun, journal =. Large-. doi:10.1021/acs.jctc.9b00301 , urldate =

work page doi:10.1021/acs.jctc.9b00301 2019
[41]

and Barone, Vincenzo and Reiher, Markus , year = 2017, month = aug, journal =

Baiardi, Alberto and Stein, Christopher J. and Barone, Vincenzo and Reiher, Markus , year = 2017, month = aug, journal =. Vibrational. doi:10.1021/acs.jctc.7b00329 , urldate =

work page doi:10.1021/acs.jctc.7b00329 2017
[42]

Advances in

Balandat, Maximilian and Karrer, Brian and Jiang, Daniel R and Daulton, Samuel and Letham, Benjamin and Wilson, Andrew Gordon and Bakshy, Eytan , year = 2020, volume =. Advances in

work page 2020
[43]

Balder, Robert and Zenger, Christoph , year = 1996, month = may, journal =. The. doi:10.1137/S1064827593247035 , urldate =

work page doi:10.1137/s1064827593247035 1996
[44]

, year = 2010, month = oct, journal =

Banik, Subrata and Pal, Sourav and Durga Prasad, M. , year = 2010, month = oct, journal =. Calculation of. doi:10.1021/ct1003669 , urldate =

work page doi:10.1021/ct1003669 2010
[45]

, year = 2008, month = oct, journal =

Banik, Subrata and Pal, Sourav and Durga Prasad, M. , year = 2008, month = oct, journal =. Calculation of Vibrational Energy of Molecule Using Coupled Cluster Linear Response Theory in Bosonic Representation:. doi:10.1063/1.2982502 , urldate =

work page doi:10.1063/1.2982502 2008
[46]

Vibrational Multi-Reference Coupled Cluster Theory in Bosonic Representation , author =. J. Chem. Phys. , volume =. doi:10.1063/1.4753422 , urldate =

work page doi:10.1063/1.4753422
[47]

WIREs Computational Molecular Science , author =

Bannwarth, Christoph and Caldeweyher, Eike and Ehlert, Sebastian and Hansen, Andreas and Pracht, Philipp and Seibert, Jakob and Spicher, Sebastian and Grimme, Stefan , year = 2021, month = mar, journal =. Extended Tight-Binding Quantum Chemistry Methods , shorttitle =. doi:10.1002/wcms.1493 , urldate =

work page doi:10.1002/wcms.1493 2021
[48]

Journal of Chemical Theory and Computation , publisher =

Bannwarth, Christoph and Ehlert, Sebastian and Grimme, Stefan , year = 2019, month = mar, journal =. doi:10.1021/acs.jctc.8b01176 , urldate =

work page doi:10.1021/acs.jctc.8b01176 2019
[49]

Time Dependent Vibrational Electronic Coupled Cluster (

Bao, Songhao and Raymond, Neil and Nooijen, Marcel , year = 2024, month = mar, journal =. Time Dependent Vibrational Electronic Coupled Cluster (. doi:10.1063/5.0190034 , urldate =

work page doi:10.1063/5.0190034 2024
[50]

and Lindh, Roland , year = 1994, month = jun, journal =

Barnes, Leslie A. and Lindh, Roland , year = 1994, month = jun, journal =. Symmetry Breaking in. doi:10.1016/0009-2614(94)00442-0 , urldate =

work page doi:10.1016/0009-2614(94)00442-0 1994
[51]

Barnsley, M. F. and Baker, George A. , year = 1976, month = jun, journal =. Bivariational Bounds in a Complex. doi:10.1063/1.523010 , urldate =

work page doi:10.1063/1.523010 1976
[52]

The Journal of Chemical Physics , volume =

Anharmonic Vibrational Properties by a Fully Automated Second-Order Perturbative Approach , author =. The Journal of Chemical Physics , volume =. doi:10.1063/1.1824881 , urldate =

work page doi:10.1063/1.1824881
[53]

Advances in Computational Mathematics , volume =

High Dimensional Polynomial Interpolation on Sparse Grids , author =. Advances in Computational Mathematics , volume =. doi:10.1023/A:1018977404843 , abstract =

work page doi:10.1023/a:1018977404843
[54]

and Musia

Bartlett, Rodney J. and Musia. Addition by Subtraction in Coupled-Cluster Theory:. The Journal of Chemical Physics , volume =. doi:10.1063/1.2387952 , urldate =

work page doi:10.1063/1.2387952
[55]

Bartlett, Rodney J and Kucharski, Stanislaw A and Noga, Jozef , journal =

work page
[56]

Gaussian Approximation Potentials:

Bart. Gaussian Approximation Potentials:. Int. J. Quantum Chem. , volume =. doi:10.1002/qua.24927 , urldate =

work page doi:10.1002/qua.24927
[57]

Gaussian

Bart. Gaussian. Phys. Rev. Lett. , volume =. doi:10.1103/PhysRevLett.104.136403 , urldate =

work page doi:10.1103/physrevlett.104.136403
[58]

Linear Algebra and its Applications , volume =

Upper Triangularization of Matrices by Permutations and Lower Triangular Similarity Transformations , author =. Linear Algebra and its Applications , volume =. doi:10.1016/0024-3795(83)90139-8 , urldate =

work page doi:10.1016/0024-3795(83)90139-8
[59]

Time-Dependent Generalized-Active-Space Configuration-Interaction Approach to Photoionization Dynamics of Atoms and Molecules , author =. Phys. Rev. A , volume =. doi:10.1103/PhysRevA.90.062508 , urldate =

work page doi:10.1103/physreva.90.062508
[60]

and Kowalski, Karol , year = 2022, month = dec, journal =

Bauman, Nicholas P. and Kowalski, Karol , year = 2022, month = dec, journal =. Coupled. doi:10.1186/s41313-022-00046-8 , urldate =

work page doi:10.1186/s41313-022-00046-8 2022
[61]

and Brummelhuis, R

Baxter, B.J.C. and Brummelhuis, R. , year = 2011, month = sep, journal =. Functionals of Exponential. doi:10.1016/j.cam.2011.06.010 , urldate =

work page doi:10.1016/j.cam.2011.06.010 2011
[62]

Commutator-Free

Bazavov, Alexei , year = 2022, month = sep, journal =. Commutator-Free. doi:10.1007/s10543-021-00892-x , urldate =. arXiv , keywords =:2007.04225 , primaryclass =

work page doi:10.1007/s10543-021-00892-x 2022
[63]

and Meyer, H.-D

Beck, M.H. and Meyer, H.-D. , year = 1997, month = nov, journal =. An Efficient and Robust Integration Scheme for the Equations of Motion of the Multiconfiguration Time-Dependent. doi:10.1007/s004600050342 , urldate =

work page doi:10.1007/s004600050342 1997
[64]

Beck, M. H. and J. The Multiconfiguration Time-Dependent. Phys. Rep. , volume =. doi:10.1016/S0370-1573(99)00047-2 , urldate =

work page doi:10.1016/s0370-1573(99)00047-2
[65]

Simplifications in the Generation and Transformation of Two-electron Integrals in Molecular Calculations , author =. Int. J. Quantum Chem. , volume =. doi:10.1002/qua.560120408 , urldate =

work page doi:10.1002/qua.560120408
[66]

Introduction to Matrix Analysis , author =

work page
[67]

Ab. J. Phys. Chem. A , volume =. doi:10.1021/jp994174i , urldate =

work page doi:10.1021/jp994174i
[68]

Nonadiabatic Molecular Dynamics:. J. Chem. Phys. , volume =. doi:10.1063/1.476142 , urldate =

work page doi:10.1063/1.476142
[69]

Benedikt , author A

Benedikt, Udo and Auer, Alexander A. and Espig, Mike and Hackbusch, Wolfgang , year = 2011, month = feb, journal =. Tensor Decomposition in Post-. doi:10.1063/1.3514201 , urldate =

work page doi:10.1063/1.3514201 2011
[70]

Tensor Decomposition in Post-

Benedikt, Udo and B. Tensor Decomposition in Post-. The Journal of Chemical Physics , volume =. doi:10.1063/1.4833565 , urldate =

work page doi:10.1063/1.4833565
[71]

, year = 2013, month = sep, journal =

Benedikt, Udo and Auer, Henry and Espig, Mike and Hackbusch, Wolfgang and Auer, Alexander A. , year = 2013, month = sep, journal =. Tensor Representation Techniques in Post-. doi:10.1080/00268976.2013.798433 , urldate =

work page doi:10.1080/00268976.2013.798433 2013
[72]

Benner, Peter and Byers, Ralph and Fassbender, Heike and Mehrmann, Volker , langid =

work page
[73]

Monitoring Molecular Nonadiabatic Dynamics with Femtosecond

Bennett, Kochise and Kowalewski, Markus and Rouxel, J. Monitoring Molecular Nonadiabatic Dynamics with Femtosecond. Proc. Natl. Acad. Sci. USA , volume =. doi:10.1073/pnas.1805335115 , urldate =

work page doi:10.1073/pnas.1805335115
[74]

PennyLane: Automatic differentiation of hybrid quantum-classical computations

Bergholm, Ville and Izaac, Josh and Schuld, Maria and Gogolin, Christian and Alam, M. Sohaib and Ahmed, Shahnawaz and Arrazola, Juan Miguel and Blank, Carsten and Delgado, Alain and Jahangiri, Soran and McKiernan, Keri and Meyer, Johannes Jakob and Niu, Zeyue and Sz. arXiv:1811.04968 [physics, physics:quant-ph] , eprint =

work page internal anchor Pith review Pith/arXiv arXiv
[75]

Derivations, Derivatives and Chain Rules , author =

work page
[76]

, year = 1998, month = feb, journal =

Bhatia, Rajendra and Singh, Dinesh and Sinha, Kalyan B. , year = 1998, month = feb, journal =. Differentiation of. doi:10.1007/s002200050279 , urldate =

work page doi:10.1007/s002200050279 1998
[77]

Matrix Analysis , author =

work page
[78]

and Leone, Stephen R

Bhattacherjee, Aditi and Pemmaraju, Chaitanya Das and Schnorr, Kirsten and Attar, Andrew R. and Leone, Stephen R. , year = 2017, month = nov, journal =. Ultrafast. doi:10.1021/jacs.7b07532 , urldate =

work page doi:10.1021/jacs.7b07532 2017
[79]

Nonuniqueness in the Energy Spectra of Anharmonic Oscillators Using the Coupled-Cluster Method , author =. Phys. Rev. A , volume =. doi:10.1103/PhysRevA.40.3484 , urldate =

work page doi:10.1103/physreva.40.3484
[80]

Variational and Coupled-Cluster Calculations of the Spectra of Anharmonic Oscillators , author =. Phys. Rev. A , volume =. doi:10.1103/PhysRevA.38.2211 , urldate =

work page doi:10.1103/physreva.38.2211

Showing first 80 references.