{"id":"8b7c47de-d570-4d10-9239-0b53ca1876a7","arxiv_id":"1909.02206","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A stable incremental SVD method and a hierarchical low-rank construction represent long-range pairwise interactions as sums of O(log N log(N/epsilon)) constant-bond-dimension MPOs/PEPOs.","lead":"The paper develops two approaches for efficiently representing long-range pairwise interactions, like the Coulomb interaction, in tensor network operators. It shows that Coulomb-type operators can be written as sums of a modest number of small tensor networks, without assuming translation invariance.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 4.4 asserts global 2-norm error control via epsilon' = C epsilon/N without an error-accumulation proof; this underpins the O(log N log(N/epsilon)) accuracy claim.","rationale":"The reader's weakest_assumption correctly identifies the missing error-accumulation proof in Section 4.4, and I agree that this is the most load-bearing gap in the paper's main quantitative claim. The paper presents the hierarchical low-rank method as the route with a priori guarantees, yet the step connecting blockwise truncation thresholds to global accuracy is asserted in a single sentence with no accompanying analysis. A concrete numerical reconstruction of the global coefficient matrix from the truncated blocks would directly test whether the epsilon' = C epsilon/N choice actually delivers the claimed epsilon accuracy; this is the natural experiment the paper itself does not report, since its numerical sections only test blockwise ranks and not the assembled global error. I do not elevate the ISVD's admitted lack of an error bound to the same level: that limitation is clearly disclosed in Section 7, and the central O(log N log(N/epsilon)) bound depends on the hierarchical construction, not on ISVD. The reader's verdict of CONDITIONAL is therefore appropriate: the concern is addressable via a focused numerical check or an error-accumulation proof, and it does not currently warrant rejection, but the promised accuracy guarantee is not yet established. My agreement is partial because the reader's weakest_assumption bundles the ISVD issue together with the hierarchical error-accumulation issue, whereas I view the latter as the single load-bearing concern for the central claim.","tokens_in":25379,"tokens_out":26246,"duration_ms":264323,"concrete_test":"Implement the 1D H-matrix construction exactly as in Section 4.4 for N = 2^L with L = 4,...,10, using the Coulomb coefficient matrix V(i,j) = 1/|i-j| and truncation epsilon' = C epsilon/N (e.g., epsilon = 1e-4, C = 1). Assemble the global approximation by summing the truncated block approximations from all levels and types, explicitly forming the full coefficient matrix, and compute the relative 2-norm error ||V_approx - V||_2 / ||V||_2. If the measured error exceeds epsilon, or grows proportionally to N epsilon' rather than remaining O(epsilon), the asserted error-control step in Section 4.4 fails. Repeat for a 2D grid for N = 16, 32, 64 with the H-matrix PEPO construction to validate the 36ML count and its stated accuracy.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim relies on the hierarchical low-rank construction, and Section 4.4's error control is its weakest link. The authors truncate each off-diagonal block at a blockwise SVD threshold epsilon' = C epsilon/N and assert, without proof, that the global 2-norm error of the coefficient matrix is below epsilon. The coefficient matrix is assembled from O(N log N) independently truncated blocks (plus symmetric counterparts), and the spectral norm of a sum of such localized perturbations is not bounded by the maximum blockwise error: blocks sharing rows or columns can add constructively, and the accumulation over hierarchical levels needs an argument. The paper provides none, and the H-matrix case (Section 4.4) is the only route with a claimed a priori bound: Section 7 explicitly admits that the modified ISVD has no theoretical error bound. If the true global error scales like N epsilon' instead of O(epsilon), the claimed accuracy fails for the stated M = O(log(N/epsilon)); at best the bound would need revision, and at worst the advertised O(log N log(N/epsilon)) representation with accuracy epsilon is unsupported. The connection between the coefficient-matrix 2-norm error and the operator error in the high-dimensional tensor space is also not established, so the abstract's 'accuracy of the low-rank truncation' is ambiguous. This is an internally missing proof rather than a disagreement with consensus, and it is testable.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two strategies for constructing efficient MPO/PEPO representations of long-range pairwise interactions, with the Coulomb interaction as the main example. The first strategy formalizes the observation that many existing MPO constructions rely on an upper-triangular low-rank (UTLR) property of the coefficient matrix, casts the construction as an ill-conditioned matrix completion problem, and introduces a modified incremental SVD (ISVD) method that avoids explicit computation of the exponentially growing lower-triangular part. The second strategy uses hierarchical low-rank matrix formats: the HODLR format for 1D MPOs and the H-matrix format for 1D MPOs and 2D PEPOs. The authors claim that for the Coulomb interaction the resulting representation is a linear combination of O(log(N) log(N/epsilon)) MPOs/PEPOs, each with constant bond dimension, and that neither algorithm requires translation invariance. Numerical experiments in quasi-1D and 2D systems are presented in support of the scaling claims.","tokens_in":25636,"tokens_out":8273,"duration_ms":90637,"significance":"If the main quantitative claim is established, the hierarchical low-rank construction is a valuable contribution: it would give the first a priori bounded construction for non-translation-invariant long-range interactions in both MPO and PEPO formats, with an explicit logarithmic scaling in system size and accuracy. The paper is also useful for clearly identifying the UTLR property as the common mechanism behind earlier exponential-fitting and SVD-compression MPO methods. The numerical experiments are consistent with the claimed scaling, and the authors are honest in Section 7 about the absence of an error bound for the modified ISVD. However, the advertised accuracy guarantee for the hierarchical route currently rests on an unproved error-accumulation assertion, and the relation between the coefficient-matrix error and the operator error is not specified. The central result is therefore plausible but not yet fully supported.","major_comments":[{"comment":"The global error control for the H-matrix construction is asserted rather than proved. The text states that choosing the blockwise SVD threshold epsilon' = C epsilon/N ensures a coefficient-matrix 2-norm error below epsilon, and this leads to M = O(log(N/epsilon)). But the coefficient matrix is assembled from O(N log N) independently truncated blocks plus symmetric counterparts, and the spectral norm of a sum of localized perturbations is not bounded by the maximum blockwise error. At a fixed level the interaction blocks are disjoint and could be analyzed as block-diagonal perturbations, but the accumulation over hierarchical levels and the role of the factor N are not explained. This step is load-bearing because the abstract's O(log(N) log(N/epsilon)) accuracy claim depends on it. Please provide an explicit error-accumulation proof, or clearly state the result as a numerical observation with a modified claim.","section":"Section 4.4, after Eq. (4.4)"},{"comment":"The conclusion states that for the Coulomb interaction the MPO rank from the modified ISVD method can be bounded by log(N/epsilon), while the same section admits that no theoretical error bound exists for the method. The non-translation-invariant UTLR route is therefore supported only by numerical evidence. Please either supply an analysis of the incremental completion error or rephrase the claim in Section 7 and in the introduction as a numerical observation rather than a bound.","section":"Section 7"},{"comment":"The accuracy measure is ambiguous. The construction in Section 4.4 controls the 2-norm error of the coefficient matrix V, while the abstract refers to 'epsilon as the accuracy of the low-rank truncation' without specifying whether this is the coefficient-matrix error or the operator error. Since the operator is \\hat V = \\sum_{i<j} V(i,j) \\hat n_i \\hat n_j and the local operators \\hat n_i are not orthonormal in general, the operator-norm error is not automatically equal to the coefficient-matrix 2-norm error; a dimension-dependent factor can enter. The scaling claim may still be correct, but the norm used for epsilon needs to be stated precisely throughout, and the conversion between the two errors needs to be justified if the abstract's accuracy language is retained.","section":"Section 4.4 and Abstract"}],"minor_comments":[{"comment":"The inequality in Definition 4.1 says ||A(I,J) - \\tilde A_{I,J}||_2 <= epsilon ||\\tilde A(I,J)||_2; for consistency with Definition 3.1 and with the surrounding text, the right-hand side should be ||A(I,J)||_2.","section":"Section 4.2, Definition 4.1"},{"comment":"The sentence 'The matrix completion problem (3.5) has several features' refers to the MPO tensor displayed in Eq. (3.5); the reference should be to the optimization problem in Eq. (3.4).","section":"Section 3, after Eq. (3.4)"},{"comment":"There is a typo: 'Sill we divide the interactions' should be 'Still we divide the interactions'.","section":"Section 4.4"},{"comment":"The axis labels and legend in Fig. 6.3(a) are garbled in the current text ('number of MPOs 1 2 3 rel 2-norm error 1e 4'); the curves, axes, and threshold values should be labeled unambiguously.","section":"Figure 6.3(a)"},{"comment":"The notation W_p is reused for matrices of different sizes in the algorithmic description after the 'abuse of notation' remark; for reproducibility it would help to define the dimensions and the submatrix convention explicitly in each algorithm.","section":"Algorithms 1 and 2"},{"comment":"The counts 78, 12, and 8 interactions in the 2D H-matrix construction are stated without derivation; a short counting argument for the reduction from 120 pairwise box interactions to the listed numbers would make the constant prefactor transparent.","section":"Section 4.5"}],"recommendation":"major_revision","confidential_remarks":"The reader's stress-test concern about Section 4.4 is valid and is the main reason for my recommendation. The hierarchical construction is explicit and numerically promising, but the advertised logarithmic accuracy bound is currently unsupported by a proof of error accumulation. I would encourage the editor to request either a proof of the epsilon' = C epsilon/N step, with the norm conversion clarified, or a substantive reframing of the claim as a numerical observation. The modified ISVD is presented honestly as lacking an error bound, but the conclusion overstates the resulting rank bound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new piece here is the observation that exponential fitting and SVD compression for long-range MPOs are both solving the same upper-triangular low-rank completion problem, plus the hierarchical low-rank construction that turns HODLR/H-matrix approximations into sums of constant-bond-dimension MPOs/PEPOs with O(log N log(N/epsilon)) terms. That second piece is concrete, new, and directly useful for DMRG and PEPS codes. The modified ISVD is a stable practical algorithm, and the paper is honest that it lacks a theoretical error bound.\n\nThe UTLR framing is a real organizing insight, not just a relabeling. The hierarchical construction is explicit: the block decomposition, the FSM rules, and the bond-dimension counting are all clear, and the numerical tests match the claimed scaling for both translation-invariant and non-translation-invariant systems. Credit is also due for comparing against exponential fitting rather than just asserting superiority, and for admitting in Section 7 that the ISVD route is not yet proven.\n\nThe soft spots are real but mostly addressable. Section 4.4 asserts epsilon' = C epsilon/N without proving how local SVD truncation errors accumulate into the global 2-norm error of the coefficient matrix. The stress-test note worries this is load-bearing. I think the concern is legitimate but not fatal: a standard H-matrix partition covers each matrix entry once, so a max-block error bound should be available, but the authors do not provide that argument, and the overlapping structure in their equations (4.3)-(4.4) makes the disjointness nontrivial. The asymptotic O(log N log(N/epsilon)) claim survives even a linear-in-N accumulation, so the advertised complexity is not obviously wrong, but the proof is missing. Similarly, the equivalence to Stoudenmire-White is asserted rather than demonstrated, and the connection between coefficient-matrix 2-norm and the high-dimensional operator error is left unstated. The ISVD route has no a priori error bound, which the authors acknowledge. None of these are fatal; they are gaps a careful revision can close.\n\nThis paper is for researchers building tensor-network codes for Coulomb-type interactions who need a non-translation-invariant, a priori bounded representation. It deserves serious peer review: the central hierarchical construction is likely correct, the numerics are credible, and the UTLR observation will be cited. I would send it to a referee and ask for the Section 4.4 error analysis and the operator-norm connection to be addressed before acceptance.","headline":"A genuinely useful unification of MPO compression ideas and a new hierarchical construction with a plausible O(log N log(N/epsilon)) bound, held back by a few unproved error-accumulation claims that should be fixed in revision.","tokens_in":26153,"tokens_out":3380,"would_cite":true,"duration_ms":42452,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A69","41A99","65Z05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Long-range pairwise interactions compress into O(log N · log(N/ε)) tensor-network operators of constant bond dimension, without any translation-invariance assumption; all efficient MPO constructions are traced to one upper-triangular…","keywords":["matrix product operator","projected entangled-pair operator","upper-triangular low-rank matrix","matrix completion","incremental singular value decomposition","hierarchical low-rank matrix","Coulomb interaction","H-matrix"],"falsifier":"Run the hierarchical construction on a coefficient matrix engineered so that every off-diagonal block has the same low-rank structure and all truncation residuals point in the same direction; double the system size while holding the per-block threshold at $C\\epsilon/N$ and measure the global 2-norm error. If the global error grows faster than linearly with the number of blocks, the $\\mathcal{O}(\\log(N)\\log(N/\\epsilon))$ accuracy claim fails. A weaker test for the ISVD route: push the paper's quasi-1D non-translation-invariant experiment to larger $N$ and check whether the achieved MPO rank stays bounded by $\\log(N/\\epsilon)$; the paper provides no proof that it does.","tokens_in":25167,"feed_emoji":"⚛️","tokens_out":22202,"duration_ms":185226,"temperature":0.7,"pith_summary":"This paper claims that long-range pairwise interactions — above all the Coulomb interaction — can be stored in tensor-network operators far more cheaply than a term-by-term representation would suggest. For the Coulomb interaction, the central result is that the operator decomposes into a linear combination of $\\mathcal{O}(\\log N \\cdot \\log(N/\\epsilon))$ matrix product operators or projected entangled-pair operators, each with a constant internal channel count (bond dimension), where $N$ is the system size and $\\epsilon$ the target accuracy. The paper also identifies the structural reason behind all efficient existing constructions: the coefficient matrix is full-rank, but its upper-triangular half alone is approximately low-rank, which turns the MPO problem into a numerically delicate matrix-completion task solved by a stabilized incremental singular value decomposition. Neither construction assumes translation invariance, so both extend to realistic systems such as molecules with arbitrary basis sets. If the accuracy claims hold, tensor network simulations of interacting quantum systems lose their long-range bottleneck.","feed_headline":"Coulomb forces fit in O(log N log(N/ε)) constant-rank operators","feed_subtitle":"Hierarchical low-rank matrices and stabilized incremental SVD cover Coulomb interactions without translation invariance.","key_machinery":"The paper runs on two matrix structures plus a stabilization step. The first structure is the upper-triangular low-rank (UTLR) property (Definition 3.1): a symmetric matrix $A\\in\\mathbb{R}^{N\\times N}$ is UTLR if for every $\\epsilon>0$ there is a rank-$r$ matrix, $r \\lesssim \\log(N/\\epsilon)$, that approximates $A$ to relative accuracy $\\epsilon$ on every block of upper-triangular entries, even though $A$ itself is full-rank; this property converts MPO construction into the matrix-completion problem $\\min_{\\mathrm{rank}(\\tilde V)\\le M} \\|P_\\Omega(\\tilde V)-P_\\Omega(V)\\|_F^2$ over the upper triangle. The second structure is hierarchical low-rank blocking: in the HODLR format an interval interacts only with its sibling (weak admissibility), while in the H-matrix format interacting boxes must be separated by a distance comparable to their size (strong admissibility); each admissible block is compressed by truncated SVD, and the rank bound $M = \\mathcal{O}(\\log(N/\\epsilon))$ for the Coulomb kernel comes from the classical fast-multipole separation estimate for $1/r$. The assembly step that turns compressed blocks into operators is the finite-state-machine rule for sums of non-overlapping rank-one operators, which produces an MPO or PEPO of bond dimension 5; summing $L = \\log N$ levels of blocks gives the total count $\\mathcal{O}(LM)$. The stabilization step is the added QR factorization in the modified incremental SVD, which keeps the row-by-row matrix-completion updates well-conditioned even though the singular values of the completed matrix grow exponentially with $N$.","core_discovery":"On its own terms the paper makes one diagnostic and one constructive claim. The diagnostic claim is that every efficient existing method for building a matrix product operator for a long-range pairwise interaction — exponential fitting by finite-state machines and the sliced-basis SVD compression method alike — exploits the same hidden structure: the coefficient matrix $V\\in\\mathbb{R}^{N\\times N}$ is full-rank, yet for any block of upper-triangular indices the block can be matched to relative accuracy $\\epsilon$ by a rank-$r$ matrix with $r \\lesssim \\log(N/\\epsilon)$. This 'upper-triangular low-rank' (UTLR) view recasts MPO construction as a matrix completion problem, and the paper's modified incremental SVD (an incremental SVD with an added QR factorization) solves it without ever forming entries that grow exponentially with system size; the MPO it produces is equivalent to the one from the sliced-basis compression method. The constructive claim is that hierarchical low-rank matrix formats give a representation valid for both MPOs and PEPOs: each off-diagonal block of $V$ is compressed by truncated SVD to rank $\\mathcal{O}(\\log(N/\\epsilon))$, each compressed block contributes a few rank-one 'non-overlapping' operators, and the finite-state-machine assembly of those operators gives an MPO or PEPO of bond dimension 5. With $L = \\log N$ levels, the Coulomb interaction is thereby written as a linear combination of $\\mathcal{O}(\\log N \\cdot \\log(N/\\epsilon))$ such operators, each of constant bond dimension, with an a priori error bound inherited from fast-multipole separation estimates for the $1/r$ kernel. Both routes are stated and demonstrated without assuming the interaction is translation-invariant.","pith_inferences":["Any interaction kernel with a separation property — screened Coulomb, dipole–dipole, Yukawa, or other Green's functions — should inherit the same $\\mathcal{O}(\\log N \\cdot \\log(N/\\epsilon))$ representation, since the rank argument rests only on blockwise separability of the $1/r$ kernel; running the H-matrix construction on such kernels would test this directly.","The upper-triangular low-rank view suggests a recipe for compact MPOs beyond pairwise interactions: any operator whose coefficient array is full-rank but admits a low-rank extension on its triangular support can be fed to the modified incremental SVD; three-body interactions are the obvious next test.","The missing error-accumulation proof is empirically checkable: holding the per-block threshold at $C\\epsilon/N$ while doubling $N$, a flat global error would validate linear accumulation, while growth with the number of levels would force a smaller threshold and a larger constant in the complexity bound.","The paper notes the sum-of-constant-rank-operators form is parallel-friendly but does not quantify the payoff; if combined with parallel time-evolution algorithms, the practical cost of simulating Coulomb-interacting systems could drop from quadratic in $N$ to near-linear with a logarithmic-factor overhead."],"forward_implications":["One-dimensional systems with Coulomb or other long-range pairwise potentials obtain matrix product operators of rank $\\mathcal{O}(\\log(N/\\epsilon))$ from the modified incremental SVD, with accuracy comparable to — and in the paper's tests slightly better than — exponential fitting, even when the interaction is not translation-invariant.","Two-dimensional lattice systems obtain PEPO representations of the Coulomb interaction as linear combinations of $\\mathcal{O}(\\log N \\cdot \\log(N/\\epsilon))$ operators of constant bond dimension with an a priori error bound, a guarantee the earlier correlation-function PEPO approach did not offer.","Because the hierarchical representation is a sum of independent constant-rank operators, the operator can be applied to a state term by term, a structure the paper notes is naturally suited to parallel computation.","Numerical experiments in the paper show the per-block rank of the H-matrix representation grows only mildly with system size (maximal rank 13 at $16\\times16$ up to 15 at $128\\times128$ for threshold $10^{-5}$), consistent with the logarithmic bound.","The two routes distribute error differently: the modified ISVD concentrates error at the far upper-right corner of the coefficient matrix, while the HODLR and H-matrix formats spread it across blocks, as shown in the paper's error-distribution plots."],"supporting_citations":[{"why":"Supplies the hierarchical off-diagonal low-rank (HODLR) matrix format that Section 4.3 converts into MPOs.","marker":"[1]"},{"why":"Gives the exponential-sum (quadrature) approximation of the Coulomb kernel whose error bound fixes the rank M = O(log(N/ε)) in the MPO constructions.","marker":"[3]"},{"why":"Supplies the sharp exponential-sum approximation of 1/x on [1,∞), the estimate behind the logarithmic rank bound.","marker":"[6]"},{"why":"The incremental SVD algorithm for matrices with missing data that the modified ISVD builds on; the paper adds a QR factorization for stability.","marker":"[8]"},{"why":"Provides the finite-state-machine description of MPOs used to assemble sums of non-overlapping rank-one operators into bond-dimension-5 MPOs and PEPOs.","marker":"[12]"},{"why":"The fast multipole separation estimate that bounds the off-diagonal Coulomb rank by O(log(1/ε)), underpinning the H-matrix PEPO rank.","marker":"[16]"},{"why":"Defines the weak-admissibility criterion and the block-concatenation trick that set the HODLR interaction lists and reduce the constant prefactor.","marker":"[18]"},{"why":"The correlation-function PEPO, the previous efficient PEPO for long-range interactions; the hierarchical method is contrasted with it and adds an a priori error bound it lacked.","marker":"[24]"},{"why":"The sliced-basis SVD compression method whose MPO the modified ISVD reproduces; the equivalence anchors the paper's claim that existing methods all exploit the UTLR property.","marker":"[33]"}],"fun_headline_variants":["Coulomb forces: O(log N log(N/ε)) constant-bond MPO/PEPO","Upper-triangular low-rank turns Coulomb MPO into matrix completion","UTLR unlocks Coulomb MPOs without translation invariance","Hierarchical low-rank gives Coulomb MPO/PEPO in O(log N log(N/ε))"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The efficiency bound assumes that the tiny errors made when each matrix block is squeezed separately do not pile up as the number of blocks grows; the paper sets the per-block tolerance to $C\\epsilon/N$ to make this plausible but gives no proof that the errors accumulate only linearly — and for the incremental-SVD route it openly states that no theoretical error bound exists.","fun_headline_variants_meta":{"raw":{"variants":["Coulomb forces: O(log N log(N/ε)) constant-bond MPO/PEPO","Upper-triangular low-rank turns Coulomb MPO into matrix completion","UTLR unlocks Coulomb MPOs without translation invariance","Hierarchical low-rank gives Coulomb MPO/PEPO in O(log N log(N/ε))"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001007,"raw_usage":{"total_tokens":4402,"prompt_tokens":1234,"completion_tokens":3168,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":850,"completion_tokens_details":{"reasoning_tokens":3097}},"tokens_in":850,"tokens_out":3168,"duration_ms":25483,"temperature":1.0,"reasoning_tokens":3097,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:56:41.620406+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the hierarchical construction on a coefficient matrix engineered so that every off-diagonal block has the same low-rank structure and all truncation residuals point in the same direction; double the system size while holding the per-block threshold at $C\\epsilon/N$ and measure the global 2-norm error. If the global error grows faster than linearly with the number of blocks, the $\\mathcal{O}(\\log(N)\\log(N/\\epsilon))$ accuracy claim fails. A weaker test for the ISVD route: push the paper's quasi-1D non-translation-invariant experiment to larger $N$ and check whether the achieved MPO rank stays bounded by $\\log(N/\\epsilon)$; the paper provides no proof that it does.","supporting_citations":[{"cited_title":"Ambikasaran and E","cited_arxiv_id":null,"evidence_quote":"Supplies the hierarchical off-diagonal low-rank (HODLR) matrix format that Section 4.3 converts into MPOs."},{"cited_title":"Beylkin and M","cited_arxiv_id":null,"evidence_quote":"Gives the exponential-sum (quadrature) approximation of the Coulomb kernel whose error bound fixes the rank M = O(log(N/ε)) in the MPO constructions."},{"cited_title":"Braess and W","cited_arxiv_id":null,"evidence_quote":"Supplies the sharp exponential-sum approximation of 1/x on [1,∞), the estimate behind the logarithmic rank bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The incremental SVD algorithm for matrices with missing data that the modified ISVD builds on; the paper adds a QR factorization for stability."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the finite-state-machine description of MPOs used to assemble sums of non-overlapping rank-one operators into bond-dimension-5 MPOs and PEPOs."},{"cited_title":"Greengard and V","cited_arxiv_id":null,"evidence_quote":"The fast multipole separation estimate that bounds the off-diagonal Coulomb rank by O(log(1/ε)), underpinning the H-matrix PEPO rank."},{"cited_title":"Hackbusch, B","cited_arxiv_id":null,"evidence_quote":"Defines the weak-admissibility criterion and the block-concatenation trick that set the HODLR interaction lists and reduce the constant prefactor."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The correlation-function PEPO, the previous efficient PEPO for long-range interactions; the hierarchical method is contrasted with it and adds an a priori error bound it lacked."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The sliced-basis SVD compression method whose MPO the modified ISVD reproduces; the equivalence anchors the paper's claim that existing methods all exploit the UTLR property."}],"review_version":1}