We introduce a novel approach for prompt mixing, aiming to generate images at
the intersection of multiple text prompts using pre-trained text-to-image
diffusion models. At each time step during diffusion denoising, our algorithm
forecasts predictions w.r.t. the generated image and makes informed text
conditioning decisions. To do so, we leverage the connection between diffusion
models (rooted in non-equilibrium thermodynamics) and the Black-Scholes model
for pricing options in Finance, and draw analogies between the variables in
both contexts to derive an appropriate algorithm for prompt mixing using the
Black Scholes model. Specifically, the parallels between diffusion models and
the Black-Scholes model enable us to leverage properties related to the
dynamics of the Markovian model derived in the Black-Scholes algorithm. Our
prompt-mixing algorithm is data-efficient, meaning it does not need additional
training. Furthermore, it operates without human intervention or hyperparameter
tuning. We highlight the benefits of our approach by comparing it qualitatively
and quantitatively to other prompt mixing techniques, including linear
interpolation, alternating prompts, step-wise prompt switching, and CLIP-guided
prompt selection across various scenarios such as single object per text
prompt, multiple objects per text prompt and objects against backgrounds. Code
is available at this https URL

我们介绍了一种新颖的方法来进行提示混合，通过使用预训练的文本到图像扩散模型，在扩散去噪的每个时间步中，我们的算法根据生成的图像预测预测，并做出明智的文本调节决策。通过借鉴非平衡热力学中与金融中期权定价的 Black-Scholes 模型的联系，并在两个情境中的变量之间进行类比，我们推导出了使用 Black-Scholes 模型进行提示混合的适当算法。我们的提示混合算法是数据高效的，不需要额外的训练，操作不需要人工干预或超参数调整。我们通过与其他提示混合技术（包括线性插值、交替提示、逐步提示切换和 CLIP 引导提示选择）在单个文本提示、多个物体提示以及物体与背景之间的各种情景下进行定性和定量比较来突出我们方法的优势。

扩散模型中的混合使用 Black Scholes 算法

Prompt Mixing in Diffusion Models using the Black Scholes Algorithm

We consider the supervised learning problem of learning the price of an
option or the implied volatility given appropriate input data (model
parameters) and corresponding output data (option prices or implied
volatilities). The majority of articles in this literature considers a (plain)
feed forward neural network architecture in order to connect the neurons used
for learning the function mapping inputs to outputs. In this article, motivated
by methods in image classification and recent advances in machine learning
methods for PDEs, we investigate empirically whether and how the choice of
network architecture affects the accuracy and training time of a machine
learning algorithm. We find that for option pricing problems, where we focus on
the Black--Scholes and the Heston model, the generalized highway network
architecture outperforms all other variants, when considering the mean squared
error and the training time as criteria. Moreover, for the computation of the
implied volatility, after a necessary transformation, a variant of the DGM
architecture outperforms all other variants, when considering again the mean
squared error and the training time as criteria.

本文考虑了通过监督学习来学习期权价格或隐含波动率的问题，并发现在所选择的网络体系结构方面使用广义高速公路网络的精度比其它变体高，对于计算隐含波动率，采用变换后的 DGM 架构是最优的。

期权定价的机器学习：对网络架构的实证研究

Machine learning for option pricing: an empirical investigation of  network architectures

We analyze approximation rates by deep ReLU networks of a class of
multi-variate solutions of Kolmogorov equations which arise in option pricing.
Key technical devices are deep ReLU architectures capable of efficiently
approximating tensor products. Combining this with results concerning the
approximation of well behaved (i.e. fulfilling some smoothness properties)
univariate functions, this provides insights into rates of deep ReLU
approximation of multi-variate functions with tensor structures. We apply this
in particular to the model problem given by the price of a European maximum
option on a basket of $d$ assets within the Black-Scholes model for European
maximum option pricing. We prove that the solution to the $d$-variate option
pricing problem can be approximated up to an $\varepsilon$-error by a deep ReLU
network with depth $\mathcal{O}\big(\ln(d)\ln(\varepsilon^{-1})+\ln(d)^2\big)$
and $\mathcal{O}\big(d^{2+\frac{1}{n}}\varepsilon^{-\frac{1}{n}}\big)$ non-zero
weights, where $n\in \mathbb{N}$ is arbitrary (with the constant implied in
$\mathcal{O}(\cdot)$ depending on $n$). The techniques developed in the
constructive proof are of independent interest in the analysis of the
expressive power of deep neural networks for solution manifolds of PDEs in high
dimension.

该研究使用深度 ReLU 网络对 Kolmogorov 方程的一类多元解的逼近速度进行了分析，并应用于对欧式最大期权价格模型问题的求解，证明了高维解空间中的 PDE 解可以通过深度 ReLU 网络进行逼近，并提供了深层神经网络表达能力分析的技术。