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arxiv: 2606.06093 · v1 · pith:4WARQ4Y2new · submitted 2026-06-04 · 🧮 math.NA · cs.NA· physics.comp-ph

A tensor-train multidimensional inverse Laplace transform

Martin Mikkelsen , Michael Kastoryano This is my paper

Pith reviewed 2026-06-28 00:08 UTC · model grok-4.3

classification 🧮 math.NA cs.NAphysics.comp-ph
keywords tensor traininverse Laplace transformmultidimensional quadraturelow-rank approximationnumerical inversionWishart distributionGamma distribution
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The pith

Tensor-train formulation computes multidimensional inverse Laplace transforms in polynomial time under low-rank assumptions.

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

The paper develops a method to numerically invert Laplace transforms in high dimensions by representing the function on a complex quadrature grid as a tensor train. This allows the inversion to be performed through tensor contractions rather than direct summation over an exponentially large grid. A sympathetic reader would care because standard methods become intractable as dimension increases, while this approach scales polynomially if the bond dimensions stay bounded. The method is demonstrated on several multivariate distributions with error estimates.

Core claim

The multidimensional inverse Laplace transform can be formulated using a tensor-train approximation of the transformed function on the complex quadrature grid, followed by inversion via a sequence of tensor contractions, reducing computational cost from exponential to polynomial in the dimension when bond dimensions remain bounded.

What carries the argument

The tensor-train (TT) approximation on the quadrature grid, where the inversion is carried out by successive tensor contractions.

If this is right

  • Computation becomes feasible for higher-dimensional problems in applied mathematics and physics.
  • Error estimations are available due to the approximation properties of tensor trains.
  • The approach applies to distributions such as multivariate normal-inverse Gaussian, Wishart, and correlated Gamma-type.
  • Only a small number of tunable parameters are needed for the method.

Where Pith is reading between the lines

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

  • Similar tensor-train techniques could extend to other integral transforms that suffer from the curse of dimensionality.
  • Testing on problems where low-rank structure breaks down would reveal the method's limits.
  • Integration with existing low-rank tensor libraries might accelerate adoption in scientific computing.

Load-bearing premise

The transformed function on the complex quadrature grid admits a low-rank tensor-train representation whose bond dimensions remain bounded.

What would settle it

A counterexample distribution where the required tensor-train bond dimensions grow exponentially with the number of dimensions would show that the polynomial scaling does not hold in general.

Figures

Figures reproduced from arXiv: 2606.06093 by Martin Mikkelsen, Michael Kastoryano.

Figure 1
Figure 1. Figure 1: Tensor network diagram of a tensor train with [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Structure of the 3d-dimensional TT for ˜f. Cores are grouped in triplets, one per dimension k, with physical indices (pk, jk, tk) (dashed boxes). Intra-triplet bonds capture correlations among the quadrature indices within a single dimension; inter-triplet bonds (between boxes) carry correlations across dimensions. Boundary bonds r0 = r3d = 1 are omitted. 4.2 Contracting the quadrature indices Using the TT… view at source ↗
Figure 3
Figure 3. Figure 3: Contraction scheme for stage 2 of the algo [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows results for a four-dimensional normal-inverse Gaussian distribution with skew￾ness parameter β = [0.35, −0.2, 0.15, 0.25]. The density is evaluated on a one-dimensional slice along the third component X3, with the remain￾ing components fixed at their marginal means. Because the closed-form density (29) is avail￾able, we use it as the reference and report rel￾ative errors in the lower panel. We compar… view at source ↗
Figure 5
Figure 5. Figure 5: Maximum TT bond dimension χ as a function of the correlation parameter ρ for a d = 4 MNIG distri￾bution (left), with the corresponding covariance matrix Σ shown at ρ = 0 (centre) and ρ = 0.95 (right). To isolate the effect of correlation, α is adjusted at each ρ so that γ = p α2 − β⊤Σβ remains constant. The bond dimension grows substantially with ρ. correlation parameters ρ ∈ {0.0, 0.2, 0.25, 0.3}. We see … view at source ↗
Figure 6
Figure 6. Figure 6: Relative error of the TT inversion versus max [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Cost-accuracy comparison between TT inver [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: TT inversion applied to the d = 4 Wishart Laplace transform (39) with ν = d + 2 and correlation parameter ρ = 0.15. The density is evaluated along the slice tk = µ for k ̸= 3, where µ = νΣ11 = 6. Monte Carlo KDE estimates for N = 105 , 106 , 107 and 108 are shown in dashed blue, orange, green and red, respectively. Parameters: A = 26.8, ℓ = 12, m = n = 10. The lower panel shows the relative difference betw… view at source ↗
Figure 10
Figure 10. Figure 10: Relative error of the TT inversion versus max [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Schematic of the correlated Gamma factor [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: shows the density along the Λ3 axis with all other components held at their marginal means. The TT inversion closely matches the an￾alytical reference, while the KDE estimator with N = 108 samples still shows a visible discrepancy near the mode — a consequence of the O(N −2/9 ) pointwise convergence rate for d = 5. In the second instance we take d = 5 and K = 8, with loading matrix W = [I5×5 | 0.3 · 15×3]… view at source ↗
Figure 13
Figure 13. Figure 13: Correlated Gamma-type model, d = 5, K = 8, with factors Yk ∼ Gamma(2, 1) and loading matrix W = [I5×5 | 0.3 · 15×3] (five idiosyncratic plus three common factors). No closed-form density ex￾ists for this case. The TT inversion (solid black, inver￾sion parameters A = 25, ℓ = 5, nE = 10, me = 12) with χ = 250) is compared against KDE with N = 105 (blue), 106 (orange), 107 (green), and 108 (red) sam￾ples, ev… view at source ↗
Figure 16
Figure 16. Figure 16: Pairwise mutual information I(Xi ; , Xj ) for the d = 5 MNIG distribution, computed from the nor￾malized TT representation via quadrature on the one￾and two-dimensional marginals. Pairs coupled more strongly through the covariance structure exhibit larger mutual information. 8 Discussion The main observation of this work is that the multidimensional inverse Laplace transform ad￾mits a separable tensor-net… view at source ↗
Figure 17
Figure 17. Figure 17: The 2D MNIG naïve inversion algorithm t1 5 10 15 t2 5 10 15 Naive 2D inversion 0.01 0.02 0.03 0.04 0.05 t1 5 10 15 t2 5 10 15 TT inversion 0.01 0.02 0.03 0.04 0.05 t1 5 10 15 t2 5 10 15 |TT − naive| 2.0×10⁻¹³ 4.0×10⁻¹³ 6.0×10⁻¹³ [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The 2D Wishart naïve inversion algorithm [PITH_FULL_IMAGE:figures/full_fig_p020_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The 2D correlated Gamma model compared to the naïve inversion algorithm [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
read the original abstract

Laplace transforms and their numerical inverses arise throughout applied mathematics, physics, finance, and probability theory. Numerical inversion, however, quickly becomes intractable in high dimensions because the number of quadrature evaluations grows exponentially with dimension. We develop a tensor train (TT) formulation of the multidimensional inverse Laplace transform. The method constructs a TT approximation of the transformed function on the complex quadrature grid and then performs the inversion through a sequence of tensor contractions. Under suitable low-rank assumptions, this reduces the computational cost from exponential to polynomial in the dimension, provided that the relevant bond dimensions remain bounded. The method has only a small number of tunable parameters and admits error estimations. We demonstrate its performance in numerical experiments, benchmarked against Monte Carlo estimates and exact references, for multivariate normal-inverse Gaussian, Wishart, and correlated Gamma-type distributions.

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.

Referee Report

2 major / 2 minor

Summary. The paper develops a tensor-train (TT) formulation of the multidimensional inverse Laplace transform. It constructs a TT approximation to the transformed function evaluated on a complex quadrature grid and performs the inversion via a sequence of tensor contractions. Under the assumption that the relevant TT bond dimensions remain bounded, the approach is claimed to reduce the cost from exponential to polynomial in the dimension. The method is equipped with a small number of tunable parameters and error estimates, and is demonstrated numerically on the multivariate normal-inverse Gaussian, Wishart, and correlated Gamma distributions, with comparisons to Monte Carlo and exact references.

Significance. If the low-rank TT structure with dimension-independent bond dimensions holds for the target distributions, the work would supply a practical, parameter-light route to high-dimensional inverse Laplace transforms that appear in probability, finance, and physics. The explicit error estimates and the reduction to tensor contractions are concrete strengths; the numerical benchmarks against independent references further support usability when the rank condition is met.

major comments (2)
  1. [Abstract / §1] Abstract and §1: The central claim that the method achieves polynomial scaling in dimension rests on the transformed function admitting a TT representation whose bond dimensions remain bounded independently of d. No general argument, a priori bound, or scaling analysis is supplied showing that the ranks stay controlled for the Laplace transforms of the multivariate normal-inverse Gaussian, Wishart, or correlated Gamma distributions as dimension grows. This assumption is load-bearing for the complexity reduction.
  2. [Numerical experiments] Numerical experiments section: While benchmarks against Monte Carlo and exact references are reported, the experiments do not include a systematic study of TT rank growth versus dimension d for the cited distributions. Without such data it is impossible to confirm that the polynomial-cost regime is actually attained.
minor comments (2)
  1. [§2] Notation for the quadrature nodes and weights in the complex plane should be introduced with an explicit reference to the one-dimensional inversion formula being discretized.
  2. [§3] The statement that the method 'admits error estimations' would be strengthened by a short theorem or proposition collecting the relevant bounds rather than leaving them implicit in the text.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, clarifying the scope of our claims and indicating where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract / §1] Abstract and §1: The central claim that the method achieves polynomial scaling in dimension rests on the transformed function admitting a TT representation whose bond dimensions remain bounded independently of d. No general argument, a priori bound, or scaling analysis is supplied showing that the ranks stay controlled for the Laplace transforms of the multivariate normal-inverse Gaussian, Wishart, or correlated Gamma distributions as dimension grows. This assumption is load-bearing for the complexity reduction.

    Authors: The manuscript explicitly qualifies the complexity claim as holding 'under suitable low-rank assumptions' and 'provided that the relevant bond dimensions remain bounded.' We do not claim or attempt to prove that the TT ranks are bounded independently of dimension for arbitrary distributions or even for the specific families considered; the paper develops the TT formulation of the inverse Laplace transform and shows how it yields polynomial cost when the low-rank structure is present. The numerical experiments provide supporting evidence that moderate ranks are observed for the tested distributions and dimensions. A general theoretical bound on rank growth is an interesting open question but is outside the scope of this work. revision: no

  2. Referee: [Numerical experiments] Numerical experiments section: While benchmarks against Monte Carlo and exact references are reported, the experiments do not include a systematic study of TT rank growth versus dimension d for the cited distributions. Without such data it is impossible to confirm that the polynomial-cost regime is actually attained.

    Authors: We agree that a systematic examination of TT rank growth versus dimension would make the numerical validation more complete. In the revised manuscript we will add figures and tables reporting the observed TT ranks (maximum bond dimension) as a function of dimension d for each of the three distribution families, using the same parameter settings as the existing experiments. This will directly illustrate whether the ranks remain controlled in the regimes where the method is demonstrated. revision: yes

standing simulated objections not resolved
  • A general argument or a priori bound demonstrating that the TT bond dimensions remain bounded independently of dimension for the Laplace transforms of the normal-inverse Gaussian, Wishart, and correlated Gamma distributions.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper develops a TT-based numerical method for multidimensional inverse Laplace transforms by constructing a low-rank approximation on a quadrature grid followed by tensor contractions. The claimed complexity reduction is explicitly conditional on the external assumption that bond dimensions remain bounded, which is not derived or fitted within the paper itself. No equations reduce a result to its own inputs by construction, no parameters are fitted and then relabeled as predictions, and no load-bearing steps rely on self-citations or imported uniqueness theorems. The approach rests on standard tensor-train algebra and quadrature, making the central formulation independent of the target result.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review is based on abstract only; no explicit free parameters, axioms, or invented entities are stated beyond the general low-rank assumption and tunable parameters mentioned at high level.

free parameters (1)
  • bond dimensions
    Abstract states the method has a small number of tunable parameters and that bond dimensions must remain bounded for polynomial scaling.
axioms (1)
  • domain assumption Existence of low-rank TT structure on the quadrature grid
    Central claim is conditioned on suitable low-rank assumptions.

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