Fourier-cosine Tensor Trains for Density Recovery and Expectation Calculation

1 Introduction

• COS-TT compresses the COS coefficient tensor to make online synthesis efficient, but its offline construction still requires exponentially many characteristic-function evaluations.It is also simpler to implement, useful in moderate dimensions, and inherits standard COS error control.Source: §1 · p. 2
• COS-TT-CHF compresses characteristic-function samples and transforms them into Fourier–cosine coefficient cores, with linear dimension scaling apart from the cost of each characteristic-function evaluation.Its frequency-truncation and discretization errors are controlled by parameter-selection rules, and density recovery does not amplify coefficient-level approximation error.Source: §1 · p. 2
• For nonseparable functions of additive scalar aggregates, the method computes the aggregate’s characteristic function and reduces the expectation to a one-dimensional COS calculation.For weighted sums, closed-form contraction integrals yield a semi-analytical formula.Source: §1 · p. 2
• COS-TT-CHF remains accurate through 150 dimensions for GBM and 100 for VG within a 30-minute offline computation budget, with error growing algebraically.The experiments use the proposed parameter-selection rules.Source: §1 · p. 2
• The derivations also provide cosine-basis functional tensor trains under weaker assumptions than existing spectral derivations.Source: §1 · p. 3

2 Related work

• Existing low-rank tensor approaches are distinguished by their assumptions and methodology: this paper assumes the characteristic function is known while the density need not be, and compresses the density rather than the target.Source: §2 · p. 3
• TT-cross integration requires pointwise integrand evaluations, including density access for direct expectation integration, and does not itself provide the characteristic-function-to-density construction studied here.It can also be used with other evaluable integrand representations, including suitable Fourier formulations.Source: §2 · p. 3
• Chebyshev–TT pricing compresses price dependence on model and payoff parameters, whereas tensorized COS compresses the density expansion over risk factors; the constructions may be complementary.Source: §2 · p. 3
• Fourier–TT pricing from characteristic-function samples requires evaluable payoff transforms and suitable complex contour shifts; reported studies reach 15 assets for Kastoryano and Pancotti [2022] and 11 for Sakurai et al. [2025].Kastoryano and Pancotti [2022] report TT-cross convergence difficulties beyond 15 assets; Sakurai et al. [2025] target repeated pricing as model parameters vary.Source: §2 · p. 3
• For nonseparable targets of an additive scalar aggregate, the approach obtains the aggregate characteristic function through TT contractions, then applies one-dimensional COS; weighted sums admit closed-form contraction integrals.The reduction applies to the stated class of additive aggregates, not arbitrary multivariate targets.Source: §2 · pp. 3–4
• The paper’s cosine-basis FTT construction requires only the characteristic function and converges under square integrability, unlike spectral FTT derivations that assume Hölder continuity and Sobolev regularity.Source: §2 · p. 4
• The companion CP realization becomes an offline bottleneck beyond 20 dimensions, motivating this paper’s TT realization and COS-TT-CHF construction.Source: §2 · p. 4
• The present paper develops methods first documented in an MSc thesis into a framework with systematic derivations, error analysis, proofs, parameter rules, and higher-dimensional tests; an independent reimplementation appeared during preparation.The independent implementation followed the thesis construction without methodological changes.Source: §2 · p. 4

3.1 The COS method

• In one dimension, the COS series converges uniformly under the stated piecewise-C1 assumption; the multidimensional expansion is interpreted in L2, with stronger assumptions needed for Fourier inversion and convergence rates.The multidimensional density is assumed to lie in L2 on the truncated domain; convergence rates require polynomial characteristic-function decay of order p > d/2 and COS-admissibility.Source: §3.1 · pp. 4–5
• The multidimensional COS formula recovers density by replacing Fourier–cosine coefficients with characteristic-function-based coefficients summed over 2^(d−1) sign patterns.The sign vectors fix their first entry to +1.Source: §3.1 · p. 6
• Direct COS construction and synthesis are exponential in dimension: the coefficient tensor has product-sized entries, and each entry requires exponentially many characteristic-function evaluations.For isotropic term count K, the stated costs are O(K^d) for synthesis and O(2^(d−1)K^d) for coefficient-tensor construction.Source: §3.1 · p. 6

3.2 TT decomposition

• For bounded ranks, TT storage and separable contraction scale linearly with dimension, and bounded-rank TT sets are closed so best fixed-rank approximations exist.TT-SVD is ruled out here because it requires the full tensor; practical cross methods select entries but lack the maximum-volume quasi-optimality guarantee.Source: §3.2 · pp. 6–7
• TT-DMRG-greedy builds cores from tensor entries without forming the full tensor; for bounded coordinate size and rank, its TT construction is linear in dimension.The tensor algorithm requires O(dnr^2) entry evaluations and O(dnr^3) additional algebraic operations; entry-evaluation costs must be counted separately.Source: §3.2 · p. 7

3.3 Functional tensor trains (FTTs)

• A functional TT represents functions in tensor-product Hilbert spaces, and contraction against a separable function factorizes into an ordered matrix product of one-dimensional integrals.The separable-contraction result holds on products of possibly unbounded intervals when the integrals exist.Source: §3.3 · pp. 7–8
• The separable-contraction theorem does not require bounded domains, supporting its later use for frequency variables on R^d.Source: §3.3 · p. 8

4.1 COS-TT: via TT decomposition of the COS coefficient tensor

• COS-TT rewrites the COS approximation using a TT approximation of its coefficient tensor, but offline coefficient queries still require exponentially many characteristic-function evaluations.After the cores are available, density recovery costs O(dK_maxr^2).Source: §4.1 · p. 9

4.2 COS-TT-CHF: via TT decomposition of the ch.f. sample tensor

• COS-TT-CHF bypasses the COS coefficient tensor by using the exact relation between the characteristic function and the original Fourier–cosine coefficients.It compresses sampled characteristic-function values and transforms them into coefficient cores.Source: §4.2 · p. 9

4.2.1 The alternative COS representation

• The alternative COS representation expresses multidimensional Fourier–cosine coefficients through a separable kernel linking the density and characteristic function.Each one-dimensional kernel factor is available analytically and can be tabulated independently of the probabilistic model.Source: §4.2.1 · p. 10

4.2.2 Core-wise relation between the ch.f. and the Fourier–cosine coefficient tensor

• The core-wise characteristic-function-to-coefficient transformation acts independently by dimension, preserves TT ranks, and is exact apart from the characteristic-function FTT approximation.Each output coefficient core depends only on the corresponding characteristic-function core.Source: §4.2.2 · p. 10

4.2.3 COS-TT-CHF: the same derivation in discretized form

• The discretized route truncates the frequency domain, samples the characteristic function, and applies quadrature to transform its discrete TT cores into coefficient cores.TT-cross obtains sampled characteristic-function cores using one evaluation per entry, rather than the exponentially many evaluations required per COS coefficient.Source: §4.2.2 · p. 11, §4.2.3 · pp. 11–12

4.3 Relation to functional tensor trains

• For finite cosine expansions, coefficient tensors have TT ranks bounded by the functional TT rank, and cosine synthesis gives orthogonal projections converging in L2.The representation-level convergence requires only f ∈ L2(D), weaker than the cited spectral FTT assumptions; it guarantees convergence but not a rate.Source: §4.3 · pp. 12–13

5.1 Frequency-domain truncation error εΩ

• Frequency truncation must cover retained cosine frequencies and account for omitted characteristic-function tails; tail bounds enable tolerance-based range selection.When tail constants are unavailable, the paper suggests starting with Ω_j ≥ κ_j and increasing Ω_j until computed quantities stabilize.Source: §5.1 · pp. 13–14

5.2 Quadrature error εquad

• Quadrature resolution depends on oscillation as well as smoothness; the general phase-resolution scaling is heuristic, and Ω and M should be selected jointly.Increasing Ω_j reduces frequency-truncation error but increases required quadrature resolution.Source: §5.2 · pp. 15–16

5.3 Propagated TT-decomposition error: εφTT

• Once quadrature resolves the retained kernel terms, the characteristic-function-to-coefficient transformation introduces no explicit adverse dependence on K_j.The decomposition error itself is not analyzed; the analysis assumes a prescribed TT-decomposition tolerance.Source: §5.3 · pp. 16, 18

6.1 Expectations of separable functions

• For separable targets, contracting their cosine coefficients with the density train computes expectations without adding approximation beyond the errors already present in the density train.Univariate target coefficients depend only on the target and basis, so they can be precomputed analytically or by one-dimensional quadrature.Source: §6.1 · pp. 18–19
• For separable-target expectations, the density approximation error is amplified by at most the target’s L2 norm on D, apart from physical-domain truncation.The bound applies to either tensorized COS method.Source: §6.1 · p. 19

6.2 Expectations of nonseparable functions

• For nonseparable G(H(X)), where H is an additive scalar aggregate, contracting the density train with the separable exponential of H approximates the aggregate’s characteristic function for one-dimensional COS.The multidimensional step depending on H is the TT contraction; the remaining calculations are standard one-dimensional COS computations.Source: §6.2 · p. 19
• Changing G affects only the final one-dimensional integral, so the same density train and aggregate density can serve multiple target functions.The method does not require a tensor representation of G; for fixed component functions and domain, the inner integrals can be precomputed.Source: §6.2 · p. 20

6.3 Application: semi-analytical formula for basket option pricing

• Closed-form contraction integrals avoid repeated numerical quadrature for weighted-sum basket payoffs, whose oscillatory cosine factors require resolution that grows with the expansion frequency.The resulting aggregate characteristic function feeds the one-dimensional COS calculation; the basket payoff is then handled by the final one-dimensional integration.Source: §6.3 · pp. 20–21

7.1 Setup

• Tests cover GBM, with a Gaussian characteristic function, and VG, with rapid and algebraic characteristic-function decay, using exact Normal density references for GBM and refined COS references for VG.The VG reference uses K=512 and λ1=16 and is treated as exact to a resolution of 10^-13.Source: §7.1 · p. 22

7.2 Frequency-domain truncation and discretization

• After rescaling by configuration-specific κ, 15 truncation-error curves approach their plateaus near Ω=κ; κ locates the transition, while K and λ1 set its level.Three configurations with λ1=6 were already at their plateau at Ω=0.8κ because physical-domain truncation error dominated.Source: §7.2 · pp. 23–24
• Flatness occurs at Mj/χj=0.86–1.16 for six GBM configurations and 0.71–1.17 for six VG configurations; the authors recommend Mj≥1.3χj.Below Mj=0.6χj, some curves rise again as aliasing replaces ordinary quadrature error.Source: §7.2 · p. 25

7.3 The physical-domain truncation range, the expansion-term count and the rank

• Wider physical windows require more K, whereas narrow windows truncate too much density mass; COS-TT-CHF matches COS-TT accuracy when frequency settings satisfy the guide.The experiments show that the remaining error control reduces to the familiar COS choices of K and λ1.Source: §7.3 · p. 26
• Required rank is more model-dependent than dimension: GBM converges at ranks 37–38 for d=10, 15, and 20, whereas VG requires ranks 153–178 at d=4 and 6.Neither method reached the rank cap in these comparisons.Source: §7.3 · p. 26

7.4 Cost and dimensional scaling

• At d=150 for GBM and d=100 for VG, COS-TT-CHF took 745 s and 1361 s; GBM density error stayed at or below 10^-4, and price errors below 1.5 Sobol half-widths.These runs used an offline budget of about 30 minutes per run.Source: §7.4 · pp. 28–29
• COS-TT-CHF offline wall time grows algebraically, with fitted exponents 2.34 for GBM and 2.24 for VG, versus 2.4× and 2.5× per dimension for COS-TT.For GBM dimensions 4–50, density-error growth is also algebraic, with fitted exponents 1.6, 2.5, and 1.2 for K=48, 64, and 96.Source: §7.4 · pp. 28–29
• Online time grows approximately as d^1.04 for GBM and d^1.19 for VG; COS-TT-CHF reuses one characteristic-function compression across (K, λ1) choices, unlike repeated COS-TT compressions.The authors recommend choosing λ1 from price or mass defect, K from coefficient decay, and setting Ωj=1.2κj (or 1.5κj conservatively) and Mj≥1.3χj.Source: §7.4 · pp. 29–30

8 Conclusions and future work

• For fixed cosine-term counts, bounded TT ranks, and fixed cost per characteristic-function evaluation, COS-TT-CHF construction and online computation scale linearly with dimension while density error grows algebraically.Experiments report good accuracy at d = 150 for GBM and d = 100 for VG within a 30-minute offline budget.Source: §8 · p. 30
• COS-TT also has linear online cost, but its offline construction remains exponential because each COS coefficient requires 2^d−1 characteristic-function evaluations.Its simpler implementation and standard COS error control make it useful in moderate dimensions, with computations reported up to dimension 26.Source: §8 · p. 30
• For functions of scalar aggregates, contracting the compressed density reduces a d-dimensional expectation to a one-dimensional COS calculation; weighted-sum contractions have closed forms.Because the payoff is not tensorized, one compressed density can be reused across many payoffs and strikes.Source: §8 · p. 31
• Potential applications include counterparty credit risk exposure profiles and multi-asset options with contingent or exotic features.These are identified as candidates for future work.Source: §8 · p. 31
• The method offers a route to revisiting Fourier-based methods for high-dimensional problems that were previously out of reach.Source: §8 · p. 31
• Sharper error control requires model-specific tail constants and oscillation scales to turn truncation and quadrature conditions into explicit accuracy bounds and jointly optimize Ω and M.Oscillation-aware quadrature may reduce M while preserving the core-wise TT construction.Source: §8 · p. 31

A Error analysis of COS-TT

• The approximation chain in (20) yields three additive error components.Source: A Error analysis of COS-TT · p. 31
• The first two error components belong to the underlying COS method.Source: A Error analysis of COS-TT · p. 31
• Junike and Stier [2025, Thm. 3.7 and Prop. A.1] give multidimensional bounds for the first two components and the trade-off between physical-domain truncation range and expansion-term count.The paper does not repeat those bounds.Source: A Error analysis of COS-TT · p. 31

A.1 TT decomposition error εCTT

• The reconstructed-density error εCTT includes both rank truncation and numerical error from the decomposition algorithm.Source: §A.1 · p. 31
• The analysis assumes the decomposition returns cores whose reconstruction satisfies a prescribed tolerance, then converts that coefficient accuracy into density error.Source: §A.1 · pp. 31–32
• Theorem A.1 states that the density error from the decomposition equals the Frobenius-norm error of the approximated coefficient tensor.The equality follows from orthonormality of the product cosine basis and also holds for any pair of coefficient tensors expanded in that basis.Source: §A.1 · p. 32
• The decomposition tolerance δTT is not expressed in terms of K, D, or d, so resulting accuracy is conditional on the decomposition stage but explicit in the COS parameters.Source: §A.1 · p. 32
• Assumption A.2 is not directly verified in the reported high-dimensional experiments because the full tensors cannot be formed; its practical effect is assessed through density and expectation errors.When the full tensor is available, its Frobenius error can be evaluated directly.Source: §A.1 · p. 32
• For cross interpolation, the error is bounded by the best rank-r error times a polynomial-in-dimension amplification factor, but constants depend on interpolation-set quality [Savostyanov, 2014, Osinsky, 2019, Qin et al., 2022].The paper does not specialize these results to the TT-DMRG-greedy variant.Source: §A.1 · p. 32

Competing Interests

• The authors report no other related financial or non-financial interests; FF Quant Advisory B.V. supported earlier MSc proof-of-concept studies and had no role in this work’s design, analysis, or publication decision.Source: Competing Interests · p. 32

• No organization supported the submitted work.Source: Funding · p. 32

Data and Code Availability

• Experiments used synthetic model parameters; no datasets were generated or analyzed, and code is available from the corresponding author on reasonable request, with release planned on publication.Source: Data and Code Availability · p. 33

Declaration of AI Use

• The authors used Claude Code and ChatGPT-5 for debugging, tests, derivation cross-checks, reference checks, and copy editing, while reporting that they derived and verified results themselves.Source: Declaration of AI Use · p. 33