High-Resolution Tensor-Network Fourier Methods for Exponentially Compressed Non-Gaussian Aggregate Distributions

Juan José Rodríguez-Aldavero; Juan José García-Ripoll

stat.MLMar 24, 2026

High-Resolution Tensor-Network Fourier Methods for Exponentially Compressed Non-Gaussian Aggregate Distributions

Authors: Juan José Rodríguez-Aldavero, Juan José García-Ripoll

Organizations: Institute of Fundamental Physics IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain

Abstract

Characteristic functions of weighted sums of independent random variables exhibit low-rank structure in the quantized tensor train (QTT) representation, also known as matrix product states (MPS), enabling up to exponential compression of their fully non-Gaussian probability distributions. Under variable independence, the global characteristic function factorizes into local terms. Its low-rank QTT structure arises from intrinsic spectral smoothness in continuous models, or from spectral energy concentration as the number of components $D$ grows in discrete models. We demonstrate this on weighted sums of Bernoulli and lognormal random variables. In the former, despite an adversarial, incompressible small- $D$ regime, the characteristic function undergoes a sharp bond-dimension collapse for $D \gtrsim 300$ components, enabling polylogarithmic time and memory scaling. In the latter, the approach reaches high-resolution discretizations of $N = 2^{30}$ frequency modes on standard hardware, far beyond the $N = 2^{24}$ ceiling of dense implementations. These compressed representations enable efficient computation of Value at Risk (VaR) and Expected Shortfall (ES), supporting applications in quantitative finance and beyond.

Figures & tables

Figure 1: QTT diagram with $n$ rank-3 cores, carrying free indices $s_{k}$ of size $\ell_{k}$ and mutually contracted along virtual bonds $\alpha_{k}$ of dimension $\chi_{k}$ .

Figure 2: Experimental setting used for the discrete experiments. (a) Beta

\mathcal{B}(2,10)

distribution used to sample the weights. (b) Spectral filters considered, including the raised cosine filter

\sigma_{\mathrm{rc}}

(blue,

q=2

), sharpened raised cosine filter

\sigma_{\mathrm{src}}

(orange,

q=8

), and exponential (Gaussian) filter

\sigma_{\mathrm{exp}}

(green).

Figure 3: Spectral reconstruction of the binomial distribution with

D=3

Bernoulli variables and 256 frequency modes, using the filters shown in Fig. 2 b. (a) Exact PMF and spectral reconstructions. (b) Reconstructed CDF. (c) Pointwise error of the CDF, showing Gibbs oscillations surrounding the jump discontinuities.

Figure 4: Error analysis for a binomial model with

D=3

components. (a) Decay of the median error (50% quantile of pointwise errors), reflecting the filter-dependent spectral convergence rates in smooth regions. (b) Decay of the global

\mathrm{L}^{1}

error and the widths of the Gibbs bands.

Figure 5: Error analysis for WPB models of increasing size

D

. (a) Error in the global

\mathrm{L}^{1}

norm, largely independent of

D

. (b) Error in the pointwise

\mathrm{L}^{\infty}

norm, showing an exponential decay of Gibbs oscillation amplitudes due to diminishing jump heights. (c) Median error, deteriorating with increasing

D

as the measure of the smooth region shrinks.

Figure 6: Performance comparison between spectral and Monte Carlo methods for the reconstruction of the WPB CDF (in

\mathrm{L}^{1}

norm) and computation of

99\%

VaR and ES. Exact reference solutions are obtained via recursive convolution (see Appendix A.2 ). A convergence triangle is shown, illustrating the expected quadratic advantage of spectral methods.

Figure 7: Compressibility of the QTT characteristic function for discrete models. (a) Binomial model with

n=20

cores, determined by the 2-adic valuation

v_{2}(D)

. (b) WPB model averaged over 10 random instances, showing near-maximal rank growth at small

D

. (c) WPB model, showing a sharp bond-dimension collapse with increasing

D

and entering a compressible low-rank regime for

D\gtrsim 300

Figure 8: Compressibility of the WPB characteristic function ( 38 ) with

n=18

and

D=300

. (a) Heat map showing the maximum bond dimension at each virtual index

s_{k}

after incorporating each variable

X_{d}

. (b) Slice of the previous heat map at

s_{7}

averaged over 10 independent random instances, showing an initial filter-dependent exponential barrier followed by strong compression.

Figure 9: Performance comparison between QTT and dense-vector spectral methods for the computation of the filtered characteristic function. Top row (fixed

D=3000

, varying

n

): (a) peak and truncated bond dimensions, well below the maximal

2^{n/2}

exponential value; (b) memory usage in bytes; (c) wall-clock time. Bottom row [(d)-(f)] show the same metrics at fixed

n=16

while varying

D

Figure 10: Performance comparison between QTT and dense-vector spectral methods for the computation of the CDF, VaR, and ES. (a) Wall-clock time required to reach a prescribed

\mathrm{L}^{1}

tolerance for Monte Carlo, dense (black) and QTT (green) spectral methods. (b) Wall-clock time required to compute VaR and ES given a CDF in dense-vector or QTT format. (c) Maximum pointwise deviation (

\mathrm{L}^{\infty}

) between the dense-vector and QTT representations of the model’s CDF.

Figure 11: Spectral reconstruction of a single lognormal density in Eq. ( 45 ) with

\mu=0

\sigma=1

, and tail tolerance

\delta=10^{-8}

(

b\approx 273

). (a) Density function, reconstructed using dense-vector and QTT spectral methods. (b) Reconstruction error for both representations.

Figure 12: Performance comparison between QTT and dense-vector spectral methods for the computation of a single lognormal density function. (a)

\mathrm{L}^{1}

reconstruction error for several distinct variances shown in the legend. (b) Wall-clock time for increasing resolutions

n=\log_{2}N

. (c) Memory usage (in bytes).

Figure 13: Performance comparison between QTT and dense-vectors for the evaluation of a weighted sum of

D=100

lognormal variables. The results are averaged across

10

random instances. (a) Self-error for the density

f^{n}

(absolute

\mathrm{L}^{2}

norm) and CDF

F^{n}

(relative) for increasing resolutions. Dense and QTT results match exactly, so the dense curves are omitted for clarity. The vertical dashed line indicates the largest dense resolution achieved (

n=24

). (b) Wall-clock time versus CDF self-error, comparing dense (black, dashed) and QTT (blue, solid) implementations. (c) Maximum bond dimension of the QTT CDF, saturating at

n\approx 20

and indicating strong compressibility.

Low-Rank Compression

Low-rank compression reduces model size, memory use, and computation by representing weights, activations, or caches with compact factors.

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Tensor Decomposition TT Decomposition Low-Rank Compression

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We introduce the tubal tensor train (TTT) decomposition, a tensor-network model that combines the t-product algebra of the tensor singular value decomposition (T-SVD) with the low-order core structure of the tensor train (TT) format. For an order-

(N+1)

tensor with a distinguished tube mode, the proposed representation consists of two third-order boundary cores and

N-2

fourth-order interior cores linked through the t-product. As a result, for bounded tubal ranks, the storage scales linearly with the number of modes, in contrast to direct high-order extensions of T-SVD. We present two computational strategies: a sequential fixed-rank construction, called TTT-SVD, and a Fourier-slice alternating scheme based on the alternating two-cores update (ATCU). We also state a TT-SVD-type error bound for TTT-SVD and illustrate the practical performance of the proposed model on image compression, video compression, tensor completion, and hyperspectral imaging.

Abstract

Low-Rank Compression

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