We propose refined GRFs (GRFs++), a new class of Graph Random Features (GRFs) for efficient and accurate computations involving kernels defined on the nodes of a graph. GRFs++ resolve some of the long-standing limitations of regular GRFs, including difficulty modeling relationships between more distant nodes. They reduce dependence on sampling long graph random walks via a novel walk-stitching technique, concatenating several shorter walks without breaking unbiasedness. By applying these techniques, GRFs++ inherit the approximation quality provided by longer walks but with greater efficiency, trading sequential, inefficient sampling of a long walk for parallel computation of short walks and matrix-matrix multiplication. Furthermore, GRFs++ extend the simplistic GRFs walk termination mechanism (Bernoulli schemes with fixed halting probabilities) to a broader class of strategies, applying general distributions on the walks' lengths. This improves the approximation accuracy of graph kernels, without incurring extra computational cost. We provide empirical evaluations to showcase all our claims and complement our results with theoretical analysis.

Applying BO with GPs on combinatorial spaces is, therefore, not straightforward.

We build upon a recently introduced class of quasi-graph random features (q-GRFs), which have demonstrated the ability to yield lower variance estimators of the 2-regularized Laplacian kernel (Choromanski 2023). Our research investigates whether similar results can be achieved with alternative kernel functions, specifically the Diffusion (or Heat), Mat\'ern, and Inverse Cosine kernels. We find that the Diffusion kernel performs most similarly to the 2-regularized Laplacian, and we further explore graph types that benefit from the previously established antithetic termination procedure. Specifically, we explore Erd\H{o}s-R\'enyi and Barab\'asi-Albert random graph models, Binary Trees, and Ladder graphs, with the goal of identifying combinations of specific kernel and graph type that benefit from antithetic termination. We assert that q-GRFs achieve lower variance estimators of the Diffusion (or Heat) kernel on Ladder graphs. However, the number of rungs on the Ladder graphs impacts the algorithm's performance; further theoretical results supporting our experimentation are forthcoming. This work builds upon some of the earliest Quasi-Monte Carlo methods for kernels defined on combinatorial objects, paving the way for kernel-based learning algorithms and future real-world applications in various domains.

Diffusion models have emerged as a robust framework for various generative tasks, such as image and audio synthesis, and have also demonstrated a remarkable ability to generate mixed-type tabular data comprising both continuous and discrete variables. However, current approaches to training diffusion models on mixed-type tabular data tend to inherit the imbalanced distributions of features present in the training dataset, which can result in biased sampling. In this research, we introduce a fair diffusion model designed to generate balanced data on sensitive attributes. We present empirical evidence demonstrating that our method effectively mitigates the class imbalance in training data while maintaining the quality of the generated samples. Furthermore, we provide evidence that our approach outperforms existing methods for synthesizing tabular data in terms of performance and fairness.

Diffusion models have risen as a powerful tool in robotics due to their flexibility and multi-modality. While some of these methods effectively address complex problems, they often depend heavily on inference-time obstacle detection and require additional equipment. Addressing these challenges, we present a method that, during inference time, simultaneously generates only reachable goals and plans motions that avoid obstacles, all from a single visual input. Central to our approach is the novel use of a collision-avoiding diffusion kernel for training. Through evaluations against behavior-cloning and classical diffusion models, our framework has proven its robustness. It is particularly effective in multi-modal environments, navigating toward goals and avoiding unreachable ones blocked by obstacles, while ensuring collision avoidance.

Diffusion models have achieved state-of-the-art performance in generating many different kinds of data, including images, text, and videos. Despite their success, there has been limited research on how the underlying diffusion process and the final convergent prior can affect generative performance; this research has also been limited to continuous data types and a score-based diffusion framework. To fill this gap, we explore how different discrete diffusion kernels (which converge to different prior distributions) affect the performance of diffusion models for graphs. To this end, we developed a novel formulation of a family of discrete diffusion kernels which are easily adjustable to converge to different Bernoulli priors, and we study the effect of these different kernels on generative performance. We show that the quality of generated graphs is sensitive to the prior used, and that the optimal choice cannot be explained by obvious statistics or metrics, which challenges the intuitions which previous works have suggested.

Diffusion maps (DM) [4] are used in machine-learning to achieve dimension reduction for data that are assumed to be sampled from a lower-dimensional manifold within a higherdimensional setting; they are related to other kernel eigenmap methods such as Laplacian eigenmaps [1], local linear embedding [10], Hessian eigenmaps [6] and local tangent space alignment [12]. The basic idea is simple: diffusion on a manifold is governed by the semigroup generated by the manifold's Laplace-Beltrami operator; the spectral analysis of the diffusion operator thus provides information about the manifold that can be used to provide a lower-dimensional parametrization for the data that also removes "noise" from the data inconsistent with the manifold hypothesis. One can (approximately) simulate a random walk or diffusion process on the (unknown) manifold by taking small steps within the data set according to probabilities estimated from the distances between data points.

We consider the problem of optimizing hybrid structures (mixture of discrete and continuous input variables) via expensive black-box function evaluations. This problem arises in many real-world applications. For example, in materials design optimization via lab experiments, discrete and continuous variables correspond to the presence/absence of primitive elements and their relative concentrations respectively. The key challenge is to accurately model the complex interactions between discrete and continuous variables. In this paper, we propose a novel approach referred as Hybrid Bayesian Optimization (HyBO) by utilizing diffusion kernels, which are naturally defined over continuous and discrete variables. We develop a principled approach for constructing diffusion kernels over hybrid spaces by utilizing the additive kernel formulation, which allows additive interactions of all orders in a tractable manner. We theoretically analyze the modeling strength of additive hybrid kernels and prove that it has the universal approximation property. Our experiments on synthetic and six diverse real-world benchmarks show that HyBO significantly outperforms the state-of-the-art methods.

Bayesian optimization (BO) is an efficient framework for solving black-box optimization problems with expensive function evaluations. This paper addresses the BO problem setting for combinatorial spaces (e.g., sequences and graphs) that occurs naturally in science and engineering applications. A prototypical example is molecular optimization guided by expensive experiments. The key challenge is to balance the complexity of statistical models and tractability of search to select combinatorial structures for evaluation. In this paper, we propose an efficient approach referred as Mercer Features for Combinatorial Bayesian Optimization (MerCBO). The key idea behind MerCBO is to provide explicit feature maps for diffusion kernels over discrete objects by exploiting the structure of their combinatorial graph representation. These Mercer features combined with Thompson sampling as the acquisition function allows us to employ tractable solvers to find next structures for evaluation. Experiments on diverse real-world benchmarks demonstrate that MerCBO performs similarly or better than prior methods. The source code is available at https://github.com/aryandeshwal/MerCBO .