STN-GPR: A Singularity Tensor Network Framework for Efficient Option Pricing

We develop a tensor-network surrogate for option pricing, targeting large-scale portfolio revaluation problems arising in market risk management (e.g., VaR and Expected Shortfall computations). The method involves representing high-dimensional price surfaces in tensor-train (TT) form using TT-cross approximation, constructing the surrogate directly from black-box price evaluations without materializing the full training tensor. For inference, we use a Laplacian kernel and derive TT representations of the kernel matrix and its closed-form inverse in the noise-free setting, enabling TT-based Gaussian process regression without dense matrix factorization or iterative linear solves. We found that hyperparameter optimization consistently favors a large kernel length-scale and show that in this regime the GPR predictor reduces to multilinear interpolation for off-grid inputs; we also derive a low-rank TT representation for this limit. We evaluate the approach on five-asset basket options over an eight dimensional parameter space (asset spot levels, strike, interest rate, and time to maturity). For European geometric basket puts, the tensor surrogate achieves lower test error at shorter training times than standard GPR by scaling to substantially larger effective training sets. For American arithmetic basket puts trained on LSMC data, the surrogate exhibits more favorable scaling with training-set size while providing millisecond-level evaluation per query, with overall runtime dominated by data generation.

Approximation with Tensor Networks. Part III: Multivariate Approximation

We study the approximation of multivariate functions with tensor networks (TNs). The main conclusion of this work is an answer to the following two questions: "What are the approximation capabilities of TNs?" and "What is an appropriate model class of functions that can be approximated with TNs?" To answer the former: we show that TNs can (near to) optimally replicate $h$-uniform and $h$-adaptive approximation, for any smoothness order of the target function. Tensor networks thus exhibit universal expressivity w.r.t. isotropic, anisotropic and mixed smoothness spaces that is comparable with more general neural networks families such as deep rectified linear unit (ReLU) networks. Put differently, TNs have the capacity to (near to) optimally approximate many function classes -- without being adapted to the particular class in question. To answer the latter: as a candidate model class we consider approximation classes of TNs and show that these are (quasi-)Banach spaces, that many types of classical smoothness spaces are continuously embedded into said approximation classes and that TN approximation classes are themselves not embedded in any classical smoothness space.

Approximation of Smoothness Classes by Deep ReLU Networks

We consider approximation rates of sparsely connected deep rectified linear unit (ReLU) and rectified power unit (RePU) neural networks for functions in Besov spaces $B^\alpha_{q}(L^p)$ in arbitrary dimension $d$, on bounded or unbounded domains. We show that RePU networks with a fixed activation function attain optimal approximation rates for functions in the Besov space $B^\alpha_{\tau}(L^\tau)$ on the critical embedding line $1/\tau=\alpha/d+1/p$ for arbitrary smoothness order $\alpha>0$. Moreover, we show that ReLU networks attain near to optimal rates for any Besov space strictly above the critical line. Using interpolation theory, this implies that the entire range of smoothness classes at or above the critical line is (near to) optimally approximated by deep ReLU/RePU networks.

Approximation with Tensor Networks. Part II: Approximation Rates for Smoothness Classes

We study the approximation by tensor networks (TNs) of functions from classical smoothness classes. The considered approximation tool combines a tensorization of functions in $L^p([0,1))$, which allows to identify a univariate function with a multivariate function (or tensor), and the use of tree tensor networks (the tensor train format) for exploiting low-rank structures of multivariate functions. The resulting tool can be interpreted as a feed-forward neural network, with first layers implementing the tensorization, interpreted as a particular featuring step, followed by a sum-product network with sparse architecture. In part I of this work, we presented several approximation classes associated with different measures of complexity of tensor networks and studied their properties. In this work (part II), we show how classical approximation tools, such as polynomials or splines (with fixed or free knots), can be encoded as a tensor network with controlled complexity. We use this to derive direct (Jackson) inequalities for the approximation spaces of tensor networks. This is then utilized to show that Besov spaces are continuously embedded into these approximation spaces. In other words, we show that arbitrary Besov functions can be approximated with optimal or near to optimal rate. We also show that an arbitrary function in the approximation class possesses no Besov smoothness, unless one limits the depth of the tensor network.

Approximation with Tensor Networks. Part I: Approximation Spaces

We study the approximation of functions by tensor networks (TNs). We show that Lebesgue $L^p$-spaces in one dimension can be identified with tensor product spaces of arbitrary order through tensorization. We use this tensor product structure to define subsets of $L^p$ of rank-structured functions of finite representation complexity. These subsets are then used to define different approximation classes of tensor networks, associated with different measures of complexity. These approximation classes are shown to be quasi-normed linear spaces. We study some elementary properties and relationships of said spaces. In part II of this work, we will show that classical smoothness (Besov) spaces are continuously embedded into these approximation classes. We will also show that functions in these approximation classes do not possess any Besov smoothness, unless one restricts the depth of the tensor networks. The results of this work are both an analysis of the approximation spaces of TNs and a study of the expressivity of a particular type of neural networks (NN) -- namely feed-forward sum-product networks with sparse architecture. The input variables of this network result from the tensorization step, interpreted as a particular featuring step which can also be implemented with a neural network with a specific architecture. We point out interesting parallels to recent results on the expressivity of rectified linear unit (ReLU) networks -- currently one of the most popular type of NNs.