A tensor-train multidimensional inverse Laplace transform

This paper introduces a tensor-train formulation for the multidimensional inverse Laplace transform that overcomes the curse of dimensionality by reducing computational complexity from exponential to polynomial through low-rank tensor approximations and contractions, demonstrating its efficacy on various multivariate distributions.

The Gist

Imagine you are trying to solve a massive, multi-dimensional puzzle. In the world of mathematics and finance, this puzzle is called the Inverse Laplace Transform.

Here is the problem: You have a "shadow" of a complex shape (a mathematical function that describes probabilities, like how likely a stock is to crash or how a chemical reaction behaves). You know the shadow perfectly, but you need to reconstruct the original 3D object from it.

In one dimension, this is like unrolling a single piece of string. It's tricky, but doable. But in high dimensions (like 5, 10, or 20 variables at once), the problem explodes. Traditional methods try to check every single possible combination of variables to rebuild the picture. If you have 5 variables and just 100 points to check for each, you need to calculate $100^5$ (10 billion) points. If you have 10 variables, you need $100^{10}$ points—a number so huge it would take a supercomputer longer than the age of the universe to finish. This is known as the "curse of dimensionality."

The Solution: The Tensor Train

The authors of this paper, Martin Mikkelsen and Michael Kastoryano, found a clever shortcut. They realized that many of these complex mathematical "shadows" aren't actually messy and chaotic; they have a hidden, simple structure.

They used a technique called Tensor Train (TT) decomposition. Think of a Tensor Train like a train of connected train cars.

• Instead of trying to store the entire massive puzzle as one giant, unwieldy block, they break it down into a sequence of small, manageable cars (called "cores").
• Each car only needs to know how it connects to the car before it and the car after it.
• If the puzzle has a "low-rank" structure (meaning the variables aren't all chaotically dependent on each other), you can represent the whole massive puzzle with just a few small cars.

How the Method Works

1. The Map (The Shadow): First, they look at the "shadow" (the Laplace transform) on a complex grid. Instead of writing down every single number on this grid, they use a smart algorithm (called TT-cross interpolation) to figure out the pattern. They build their "train" of small cars that, when linked together, perfectly recreate the shadow.
2. The Inversion (Rebuilding): Once the train is built, they perform the "inversion" (turning the shadow back into the object). Instead of doing a massive calculation for the whole train at once, they simply "contract" the train. They push the math through the cars one by one, like a wave moving down the line.
3. The Result: Because the train cars are small, this process is incredibly fast. Instead of taking billions of years, it takes minutes.

What They Tested

The authors tested this "train" method on three specific types of complex probability puzzles used in finance and physics:

• Normal-Inverse Gaussian: A model often used for things that have "fat tails" (extreme events happen more often than a standard bell curve predicts).
• Wishart Distribution: Used to model how different variables move together (correlations), common in portfolio risk.
• Correlated Gamma Models: Used in credit risk to model how defaults in different parts of a portfolio might happen together.

The Results

They compared their new "train" method against the old standard: Monte Carlo simulation.

• Monte Carlo is like trying to guess the shape of a mountain by throwing millions of darts at a wall and seeing where they land. To get a clear picture, you need billions of darts.
• The Tensor Train method was like having a blueprint. It reconstructed the mountain with high precision using a tiny fraction of the "darts" (computational effort).

In their experiments, the Tensor Train method was able to reconstruct these complex 4D and 5D shapes with high accuracy, while the Monte Carlo method was either too slow or too blurry (noisy) to be useful at the same cost.

What You Can Do With the Result

Once the authors built this "train" representation of the probability density, they didn't just stop there. Because the result is a structured train of cars, they could easily ask specific questions without rebuilding the whole thing:

• Marginals: "What does the shape look like if we only look at variable X?" (They just disconnected the other cars).
• Conditionals: "What is the shape of X if we know Y is greater than 5?" (They adjusted the connection between the cars).
• Mutual Information: "How much do variable X and variable Y depend on each other?" (They calculated the connection strength between the cars).

The Bottom Line

This paper introduces a way to solve a mathematically impossible problem (inverting high-dimensional transforms) by realizing the data has a hidden, simple structure. By treating the problem like a connected train of small cars rather than a giant block of data, they turned a task that was computationally impossible into one that is fast, accurate, and practical for real-world finance and physics problems.

Limitations
The method works best when the variables aren't too tightly tangled. If the variables are extremely correlated (like a train where every car is glued to every other car), the "cars" get too big, and the method loses its speed advantage. However, for the types of problems they tested, it worked beautifully.

Technical Summary

Technical Summary: A Tensor-Train Multidimensional Inverse Laplace Transform

Problem Statement
The numerical inversion of Laplace transforms is a fundamental task in applied mathematics, physics, finance, and probability theory, often required to recover probability densities from their transforms. While numerous established methods exist for one-dimensional inversion (e.g., Post–Widder, Talbot-type, quadrature methods), these approaches face a "curse of dimensionality" in high-dimensional settings. Direct numerical inversion typically requires a number of function evaluations that grows exponentially with the dimension $d$, rendering it computationally intractable for dimensions beyond two or three. This is particularly problematic for multivariate distributions in finance and stochastic modeling where the inverse problem involves high-dimensional oscillatory complex sums.

Methodology
The authors propose a novel formulation of the multidimensional inverse Laplace transform using Tensor-Train (TT) decompositions. The method leverages the specific structure of the inversion formula developed by Choudhury, Lucantoni, and Whitt [10], which expresses the multidimensional inverse as a recursive composition of one-dimensional inversion operators acting along each coordinate.

The proposed algorithm proceeds in two main stages:

1. TT Construction on the Quadrature Grid:
   The transformed function $\tilde{f}(s)$ is evaluated on a complex quadrature grid defined by indices $(p_k, j_k, t_k)$ for each dimension $k$. Instead of storing the full tensor of evaluations, the authors approximate $\tilde{f}$ as a low-rank Tensor-Train. This is achieved using TT-cross interpolation, which constructs the TT representation by querying the function at a small number of adaptive grid points. The resulting 3$d$-dimensional TT consists of triplets of cores for each dimension, capturing correlations within and between dimensions.

2. Tensor Contraction for Inversion:
   Once the TT representation of $\tilde{f}$ is constructed, the inversion is performed by contracting the quadrature indices ($p_k$ and $j_k$) with pre-computed Euler summation weights. This operation is separable across dimensions; the Euler weights act independently on the cores of each triplet. Consequently, the contraction reduces the 3$d$-dimensional TT to a $d$-dimensional TT representing the recovered density $\bar{f}(t)$, without inflating the inter-dimensional bond dimensions.

Key Contributions
The paper makes three primary contributions:

• Operator Formulation: It derives a tensor-network formulation of the multidimensional inverse Laplace transform, expressing the inversion as a sequence of local tensor contractions acting on a low-rank approximation of the transformed function.
• Numerical Algorithm: It proposes a practical inversion algorithm combining the operator structure with TT-cross interpolation. This allows the method to exploit the low-rank structure of $\tilde{f}$ on the complex grid.
• Empirical Validation: The method is demonstrated on three classes of multivariate distributions: Multivariate Normal-Inverse Gaussian (MNIG), Wishart distributions, and Correlated Gamma-type models. The performance is benchmarked against Monte Carlo estimates (using Kernel Density Estimation) and exact analytical references where available.

Results
Numerical experiments demonstrate that the TT inversion method successfully reconstructs multivariate probability densities in dimensions where direct inversion is infeasible.

• Accuracy vs. Cost: For the MNIG and Wishart examples, the TT method achieves relative errors significantly lower than Monte Carlo estimates with comparable or fewer function evaluations. For instance, in a 4-dimensional MNIG case, the TT method reached the discretization error floor ($\approx 10^{-6}$) with roughly $10^8$ evaluations, whereas Monte Carlo required $N=10^8$ samples to reach similar accuracy, with convergence rates scaling as $O(N^{-2/(d+4)})$.
• Rank Behavior: The computational cost depends heavily on the maximum TT bond dimension $\chi$. The results show that $\chi$ remains moderate for dimensions up to $d=8$ provided correlations are not extreme. However, as the correlation parameter $\rho$ increases, the bond dimension required to maintain a fixed accuracy grows, reflecting the decreasing compressibility of the transformed function.
• Distributional Queries: The recovered TT density serves as a reusable representation. The authors demonstrate that marginal densities, conditional cumulative distribution functions (CDFs), and mutual information can be computed via deterministic tensor contractions without additional sampling or constructing the full dense tensor. This is particularly effective for rare-event probabilities (e.g., conditional CDFs for tail events) where Monte Carlo estimation suffers from high variance.

Significance and Claims
The authors claim that the central observation of this work is that multidimensional Laplace inversion possesses a natural tensor-network structure due to the separability of the quadrature nodes in the inversion formula. By approximating the transformed function (which often has a simple analytic form) rather than the density itself (which may involve complex special functions like Bessel functions), the method shifts the computational burden to evaluations on a contour grid.

The significance lies in reducing the computational complexity from exponential to polynomial in the dimension $d$, provided the relevant bond dimensions remain bounded. The method offers a deterministic alternative to sampling-based approaches, providing error estimations and the ability to compute various statistical quantities directly from the compressed representation. The authors note limitations, specifically that the method's efficiency relies on the low-rank compressibility of the transformed function; highly correlated systems or transforms with singularities near the inversion contour may require larger bond dimensions or finer grids, potentially increasing costs.