Convolution-Based Converter : A Weak-Prior Approach For Modeling Stochastic Processes Based On Conditional Density Estimation

TL;DR

The paper addresses the limitations of traditional stochastic-process models that rely on strong priors (e.g., SDEs, Markov models, Gaussian processes) and may fail when those priors are misaligned with the data. It introduces the Convolution-Based Converter (CBC), a weak-prior approach that implicitly learns the conditional distribution $P(X(T)\mid X(S)=O)$ by transforming an initial stochastic trajectory into an observation-consistent target trajectory through a convolutional neural architecture. Key contributions include a CBC framework composed of an initial stochastic process, a dependency constructor, and a convolution-based transformer, along with a differentiable loss $\mathcal{L}(\theta)$ that enforces observations; the method demonstrates robustness in limited-data scenarios and competitive performance across 1-D and 2-D tasks, including image completion on MNIST and CIFAR. The results suggest CBC's flexible modeling of complex dependencies without explicit prior assumptions, offering practical utility for stochastic modeling in uncertain or data-scarce environments.

Abstract

In this paper, a Convolution-Based Converter (CBC) is proposed to develop a methodology for removing the strong or fixed priors in estimating the probability distribution of targets based on observations in the stochastic process. Traditional approaches, e.g., Markov-based and Gaussian process-based methods, typically leverage observations to estimate targets based on strong or fixed priors (such as Markov properties or Gaussian prior). However, the effectiveness of these methods depends on how well their prior assumptions align with the characteristics of the problem. When the assumed priors are not satisfied, these approaches may perform poorly or even become unusable. To overcome the above limitation, we introduce the Convolution-Based converter (CBC), which implicitly estimates the conditional probability distribution of targets without strong or fixed priors, and directly outputs the expected trajectory of the stochastic process that satisfies the constraints from observations. This approach reduces the dependence on priors, enhancing flexibility and adaptability in modeling stochastic processes when addressing different problems. Experimental results demonstrate that our method outperforms existing baselines across multiple metrics.

Figures (4)

• Figure 1: The framework of Convolution-Based Converter (CBC) consists of four successive parts:(a) a input initial stochastic process ,(b) a constructor that establishes preliminary dependency (c) A Convolution-Based Converter that constructs the dependencies among random variables in the stochastic process (d) The output expected stochastic process. The modeling effect is shown in (d2), where (d1) represents restricting the output space to the [0,1] range.
• Figure 2: Smooth Convolution-Converter:the framework is as the same as the Fig\ref{['fig:model']},the difference is replacing the base Convolution-Converter with a Smooth Convolution-Converter to accommodate the modeling of smooth stochastic processes,and d(3) is the expected stochastic process
• Figure 3: 1-D Stochastic Process. We present the modeling results on 1-D data generated from the Gaussian Process, Uniform Process, and Markov Process (arranged from top to bottom). For each dataset, observations are set at positions [20, 30, 50]. The red markers indicate the observations,and the blue regions represent the estimating results of the methods. Left: the estimating results of CBC Right: the estimating results of GP.
• Figure 4: 2-D stochastic process on the MNIST dataset. Left: Original Image. Right: From bottom to top, there are images with [100, 200, 500, and 784] observations(blue pixels denote the pixels of targets that have been masked ), respectively, the middle row displays sampled images estimated by CBC, and the top row shows the conditional probability distribution estimated by CBC. As the number of observations increases, the conditional probability estimation becomes more precise.