Peiman Mohseni

Peiman Mohseni, Nick Duffield, Raymond K. W. Wong

Modeling unknown latent functions from finite, irregularly sampled measurements is a recurring challenge across science and engineering. Neural processes (NPs), a family of probabilistic functional models, are promising solutions -- especially when endowed with domain-specific symmetries like translation equivariance, which improve sample efficiency and generalization. Yet existing translation-equivariant NPs face two limitations: (i) they stack generic components with non-linearities, obscuring the induced function class and limiting interpretability; and (ii) convolutional designs rely on kernels with local receptive fields and require dense uniform input grids, while attention-based methods avoid these issues but scale quadratically with the number of observations. We address both with two contributions. First, using the Volterra expansion, we characterize continuous translation-equivariant operators as sums of higher-order convolutions, yielding analytical transparency while admitting efficient approximation by first-order convolutions. Second, we introduce set Fourier convolutions (SFConvs), a frequency-domain parameterization that operates directly on irregularly sampled points, achieves approximately global receptive fields, and scales linearly in the number of observations. Building on these ideas, we propose two conditional NPs (CNPs): SFConvCNPs, which stack SFConv blocks with non-linearities, and SFVConvCNPs, which integrate the Volterra formulation. Experiments on synthetic and real-world datasets demonstrate our methods' efficacy against state-of-the-art baselines.

Peiman Mohseni, Nick Duffield

Conditional Neural Processes (CNPs) constitute a family of probabilistic models that harness the flexibility of neural networks to parameterize stochastic processes. Their capability to furnish well-calibrated predictions, combined with simple maximum-likelihood training, has established them as appealing solutions for addressing various learning problems, with a particular emphasis on meta-learning. A prominent member of this family, Convolutional Conditional Neural Processes (ConvCNPs), utilizes convolution to explicitly introduce translation equivariance as an inductive bias. However, ConvCNP's reliance on local discrete kernels in its convolution layers can pose challenges in capturing long-range dependencies and complex patterns within the data, especially when dealing with limited and irregularly sampled observations from a new task. Building on the successes of Fourier neural operators (FNOs) for approximating the solution operators of parametric partial differential equations (PDEs), we propose Spectral Convolutional Conditional Neural Processes (SConvCNPs), a new addition to the NPs family that allows for more efficient representation of functions in the frequency domain.

Peiman Mohseni, Nick Duffield, Bani Mallick et al.

Neural processes are a family of probabilistic models that inherit the flexibility of neural networks to parameterize stochastic processes. Despite providing well-calibrated predictions, especially in regression problems, and quick adaptation to new tasks, the Gaussian assumption that is commonly used to represent the predictive likelihood fails to capture more complicated distributions such as multimodal ones. To overcome this limitation, we propose Conditional Quantile Neural Processes (CQNPs), a new member of the neural processes family, which exploits the attractive properties of quantile regression in modeling the distributions irrespective of their form. By introducing an extension of quantile regression where the model learns to focus on estimating informative quantiles, we show that the sampling efficiency and prediction accuracy can be further enhanced. Our experiments with real and synthetic datasets demonstrate substantial improvements in predictive performance compared to the baselines, and better modeling of heterogeneous distributions' characteristics such as multimodality.