用张量网络压缩非高斯分布，实现高效风险计算。
High-Resolution Tensor-Network Fourier Methods for Exponentially Compressed Non-Gaussian Aggregate Distributions
- 利用量子张量列车表示特征函数的低秩结构
- 在D≥300时实现多项式对数级计算开销
- 适合金融风险分析等需要高精度分布计算的场景
独立随机变量加权和的特征函数在量化张量列车（QTT）表示中具有低秩结构，即矩阵乘积态（MPS），可实现其完全非高斯概率分布的指数级压缩。在变量独立条件下，全局特征函数可分解为局部项。其低秩特性源于连续模型中的内在谱光滑性，或离散模型中随分量数D增大而产生的谱能量集中。我们在伯努利与对数正态随机变量的加权和上进行了验证。前者在小D时存在难以压缩的对抗性情形，但当D≳300时，特征函数发生急剧的张量阶数坍缩，实现多项式对数时间与内存复杂度。后者可在标准硬件上达到N=2^30个频率模式的高分辨率离散化，远超密集实现的N=2^24上限。这些压缩表示支持高效计算风险价值（VaR）与预期短缺（ES），适用于量化金融等领域。
原文摘要 · Abstract (English)
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.
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