Local Interpolation via Low-Rank Tensor Trains

TL;DR

This work tackles the challenge of high-dimensional interpolation on grid data by introducing Tensor Train Interpolation (TTI), which refines a coarse TT representation into a finer one while preserving low rank. By formulating a TT-O operator built from shift matrices and interpolation kernels, TTI achieves $oldsymbol{2}$-norm error bounds that do not grow with resolution and provides exponential compression with logarithmic complexity in the number of grid points. The method unifies QTT-I and QTT-Tucker encodings to maintain low tail ranks and enable efficient multi-dimensional interpolation without forming full tensors. Empirically, TTI delivers constant-time refinement and strong compression for 1D–3D soft masks and synthetic noise/turbulence fields, enabling scalable TT-native solvers for complex geometries and multiscale generative models with broad implications for scientific simulation, imaging, and real-time graphics.

Abstract

Tensor Train (TT) decompositions provide a powerful framework to compress grid-structured data, such as sampled function values, on regular Cartesian grids. Such high compression, in turn, enables efficient high-dimensional computations. Exact TT representations are only available for simple analytic functions. Furthermore, global polynomial or Fourier expansions typically yield TT-ranks that grow proportionally with the number of basis terms. State-of-the-art methods are often prohibitively expensive or fail to recover the underlying low-rank structure. We propose a low-rank TT interpolation framework that, given a TT describing a discrete (scalar-, vector-, or tensor-valued) function on a coarse regular grid with $n$ cores, constructs a finer-scale version of the same function represented by a TT with $n+m$ cores, where the last $m$ cores maintain constant rank. Our method guarantees a $\ell^{2}$-norm error bound independent of the total number of cores, achieves exponential compression at fixed accuracy, and admits logarithmic complexity with respect of the number of grid points. We validate its performance through numerical experiments, including 1D, 2D, and 3D applications such as: 2D and 3D airfoil mask embeddings, image super-resolution, and synthetic noise fields such as 3D synthetic turbulence. In particular, we generate fractal noise fields directly in TT format with logarithmic complexity and memory. This work opens a path to scalable TT-native solvers with complex geometries and multiscale generative models, with implications from scientific simulation to imaging and real-time graphics.

Figures (15)

• Figure 1: Tensor-Train Interpolation (TTI) framework. Schematic overview of tensor-train interpolation (TTI) in one and multiple dimensions. (a) Diagrammatic view of the TTI algorithm: tensor-network representations of the shift matrices $S^{(k)}_{\boldsymbol{a}\boldsymbol{a}'}$, the polynomial weights $P^{(k)}_{\boldsymbol{b}}$, and the MPO--MPS TTI operator ("TTI-O", see Eq. \ref{['eq: tto']}), together with representative interpolation kernels. (b) One-dimensional TTI applied to a discrete function $f_{\boldsymbol{a}}$. The TTI-O is represented as an MPO with $n$ input (coarse-grid) indices and $m$ output (fine-grid) indices; acting on $f_{\boldsymbol{a}}$ produces values on a refined grid. The MPO ranks of $\mathrm{TTI\text{-}O}$ are bounded by $q+1$, where $q$ is the number of integer grid points in the support of the kernel $\phi$, while the MPS ranks are bounded by $p+1$, where $p$ is the polynomial degree of the interpolation scheme. In (c) and (d), colors indicate different spatial dimensions. (c) Multidimensional QTT-interleaved encoding and interpolation: TT cores are labeled by $G_{m,k}$ in dimension--scale ordering, together with the corresponding multidimensional $\mathrm{TTI\text{-}O}$ adapted to this encoding, whose cores are defined in Eq. \ref{['eq:multicore']}. (d) Multidimensional QTT-Tucker encoding and interpolation: the core (gray) TT is denoted by $G_m$ and the associated Tucker unitaries (see App. \ref{['sec:tucker']}) are labeled by $U_{m,k}$. We also illustrate the parallel application of the one-dimensional $\mathrm{TTI\text{-}O}$ along each dimension.
• Figure 2: 1D function encoding. Demonstration of tensor-train interpolation (TTI) in one dimension. A $C^{2}$ function is encoded in QTT format using TTI (labeled QTT-I). We begin from a coarse QTT with $18$ cores (grid spacing $h = 2^{-18} \approx 10^{-6}$) and extend its definition to finer scales up to $28$ cores. For comparison, the same function is encoded using TT-Cross (labeled QTT-C). (a) Root-mean-square error (RMSE) of the function and its first two derivatives, computed via TT sampling Ferris2012-mg. Derivatives for QTT-I are obtained analytically from the interpolation, whereas derivatives for QTT-C are computed using a finite MPO differentiation operator. (b) Runtime of TTI and TT-Cross for a fixed target precision together with an inset of the interpolated $C^2$ function. (c) Maximum QTT rank for both methods, with an inset showing the corresponding compression ratios. For highly refined grids (large core counts), TT-Cross tends to overestimate the ranks relative to the interpolation-based approach.
• Figure 3: 3D soft-mask encoding. Three-dimensional soft indicator function of a tampered airfoil embedded in a computational domain that is four times larger than the object. The results compare three tensor-network approaches: TT-Cross (QTT-C), tensor-train interpolation in the QTT-interleaved architecture (QTT-SVD), and tensor-train interpolation in the QTT–Tucker architecture (QTT-T). (a) Runtime of the three methods (solid lines), together with the root-mean-square error (RMSE) computed via TT sampling (dashed lines). (b) Data compression ratio (solid lines) and maximum TT rank (dashed lines) for each tensor-network architecture. (c) Visualization of the softened mask of the airfoil in the full 3D domain. A hard mask is shown in purple together with a softened cloud-point for visualization. Panels (a) and (b) demonstrate that our interpolation-based method yields approximately constant runtime, maximal rank, and error across increasing core counts. The interpolation kernel used was a $C^1$ cubic kernel.
• Figure 4: TN synthetic turbulence. 3D turbulence metrics for two tensor-network variants: QTT-Interleaved (QTT-I) and QTT-Tucker (QTT-T). (a) Energy spectrum $E(k)$ vs. wavenumber $k=|\mathbf{k}|$ (log–log); dashed red and purple reference lines (QTT-I and QTT-T, respectively) indicate $\propto k^{-5/3}$, confirming the expected Kolmogorov inertial-range scaling; inset: representative 3D snapshot of the velocity magnitude. (b) Flatness (kurtosis) of velocity increments as a function of separation $r$ (log–log); systematic departures from a constant baseline highlight intermittency (non-Gaussian fluctuations) across scales. (c) Maximum bond dimension $r_{\text{max}}$ versus the number of scales $M$ (circles, dashed lines; one third of the TN cores), showing approximately linear growth. Compression ratios (squares, solid lines; TN parameters divided by grid points) are plotted on the secondary axis to quantify storage efficiency. Results are averaged over 20 random seeds; shaded regions indicate $\pm 1$ standard deviation.
• Figure C1: Interpolation Schemes. This figure shows different interpolation schemes using 4 data points on a regular grid, $F(x) = \sum_{i=-1}^2 \phi_{3}\!\Bigl(\tfrac{x - x_{i}}{h}\Bigr)\,c_{i} \quad x \in [x_i,x_{i+1})$. Figure (a) shows a $C^1$ cubic interpolant (Keys' kernel Keys1981 , see Appendix \ref{['app:interpolation_methods']}), while figure (b) shows a $C^2$ B-spline quasi interpolant Schoenberg1988. The first method has a convergence error rate of $\mathcal{O}(h^3)$, whereas the second has a rate of $\mathcal{O}(h^2)$, where $h$ is the grid size.